tdehaeze/digital-brain
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Update Content - 2022-03-15

Browse Source
This commit is contained in:
Thomas Dehaeze
2022-03-15 16:40:48 +01:00
parent e6390908c4
commit 22fb3361a5
148 changed files with 3981 additions and 3197 deletions
Download Patch File Download Diff File Expand all files Collapse all files

31
content/book/du10_model_contr_vibrat_mechan_system.md
View File

@@ -1,17 +1,17 @@
+++
title = "Modeling and control of vibration in mechanical systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
draft = true
+++
Tags
: [Stewart Platforms]({{< relref "stewart_platforms" >}}), [Vibration Isolation]({{< relref "vibration_isolation" >}})
: [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [Vibration Isolation]({{< relref "vibration_isolation.md" >}})
Reference
: ([Du and Xie 2010](#orga475b60))
: (<a href="#citeproc_bib_item_1">Du and Xie 2010</a>)
Author(s)
: Du, C., & Xie, L.
: Du, C., &amp; Xie, L.
Year
: 2010
@@ -110,7 +110,7 @@ Year
### 2.5 Conclusion {#2-dot-5-conclusion}
## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
## 3. Modeling of [Stewart Platforms]({{< relref "stewart_platforms.md" >}}) {#3-dot-modeling-of-stewart-platforms--stewart-platforms-dot-md}
### 3.1 Introduction {#3-dot-1-introduction}
@@ -152,7 +152,7 @@ Year
#### 4.2.4 Suspension {#4-dot-2-dot-4-suspension}
#### 4.2.5 An application example &#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
#### 4.2.5 An application example &amp;#8211; Disk vibration reduction via stacked disks {#4-dot-2-dot-5-an-application-example-and-8211-disk-vibration-reduction-via-stacked-disks}
### 4.3 Self-adapting systems {#4-dot-3-self-adapting-systems}
@@ -179,13 +179,13 @@ Year
### 5.1 Introduction {#5-dot-1-introduction}
### 5.2 H2 and H&#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
### 5.2 H2 and H&amp;#8734; norms {#5-dot-2-h2-and-h-and-8734-norms}
#### 5.2.1 H2 norm {#5-dot-2-dot-1-h2-norm}
#### 5.2.2 H&#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
#### 5.2.2 H&amp;#8734; norm {#5-dot-2-dot-2-h-and-8734-norm}
### 5.3 H2 optimal control {#5-dot-3-h2-optimal-control}
@@ -197,7 +197,7 @@ Year
#### 5.3.2 Discrete-time case {#5-dot-3-dot-2-discrete-time-case}
### 5.4 H&#8734; control {#5-dot-4-h-and-8734-control}
### 5.4 H&amp;#8734; control {#5-dot-4-h-and-8734-control}
#### 5.4.1 Continuous-time case {#5-dot-4-dot-1-continuous-time-case}
@@ -227,13 +227,13 @@ Year
### 5.8 Conclusion {#5-dot-8-conclusion}
## 6. Mixed H2/H&#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
## 6. Mixed H2/H&amp;#8734; Control Design for Vibration Rejection {#6-dot-mixed-h2-h-and-8734-control-design-for-vibration-rejection}
### 6.1 Introduction {#6-dot-1-introduction}
### 6.2 Mixed H2/H&#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
### 6.2 Mixed H2/H&amp;#8734; control problem {#6-dot-2-mixed-h2-h-and-8734-control-problem}
### 6.3 Method 1: slack variable approach {#6-dot-3-method-1-slack-variable-approach}
@@ -266,7 +266,7 @@ Year
### 7.3 Design in continuous-time domain {#7-dot-3-design-in-continuous-time-domain}
#### 7.3.1 H&#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
#### 7.3.1 H&amp;#8734; loop shaping for low-hump sensitivity functions {#7-dot-3-dot-1-h-and-8734-loop-shaping-for-low-hump-sensitivity-functions}
#### 7.3.2 Application examples {#7-dot-3-dot-2-application-examples}
@@ -395,7 +395,7 @@ Year
### 10.5 Conclusion {#10-dot-5-conclusion}
## 11. H&#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
## 11. H&amp;#8734;-Based Design for Disturbance Observer {#11-dot-h-and-8734-based-design-for-disturbance-observer}
### 11.1 Introduction {#11-dot-1-introduction}
@@ -533,7 +533,8 @@ Year
### 15.6 Conclusion {#15-dot-6-conclusion}
## Bibliography {#bibliography}
<a id="orga475b60"></a>Du, Chunling, and Lihua Xie. 2010. _Modeling and Control of Vibration in Mechanical Systems_. Automation and Control Engineering. CRC Press. <https://doi.org/10.1201/9781439817995>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Lihua Xie. 2010. <i>Modeling and Control of Vibration in Mechanical Systems</i>. Automation and Control Engineering. CRC Press. doi:<a href="https://doi.org/10.1201/9781439817995">10.1201/9781439817995</a>.</div>
</div>

113
content/book/du19_multi_actuat_system_contr.md
View File

@@ -1,6 +1,6 @@
+++
title = "Multi-stage actuation systems and control"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Proposes a way to combine multiple actuators (short stroke and long stroke) for control."
keywords = ["Control", "Mechatronics"]
draft = false
@@ -11,10 +11,10 @@ Tags
Reference
: ([Du and Pang 2019](#org2403f17))
: (<a href="#citeproc_bib_item_1">Du and Pang 2019</a>)
Author(s)
: Du, C., & Pang, C. K.
: Du, C., &amp; Pang, C. K.
Year
: 2019
@@ -43,11 +43,11 @@ When high bandwidth, high position accuracy and long stroke are required simulta
Popular choices for coarse actuator are:
- DC motor
- [Voice Coil Motors]({{< relref "voice_coil_actuators" >}}) (VCM)
- [Voice Coil Motors]({{< relref "voice_coil_actuators.md" >}}) (VCM)
- Permanent magnet stepper motor
- Permanent magnet linear synchronous motor
As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators" >}}) are used.
As fine actuators, most of the time [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}) are used.
In order to overcome fine actuator stringent stroke limitation and increase control bandwidth, three-stage actuation systems are necessary in practical applications.
@@ -75,19 +75,19 @@ which includes the resonance model
and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
\begin{align\*}
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\\\
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\end{align\*}
#### Secondary Actuators {#secondary-actuators}
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#org4cc1c22)) and the microactuator.
We here consider two types of secondary actuators: the PZT milliactuator (figure [1](#figure--fig:pzt-actuator)) and the microactuator.
<a id="org4cc1c22"></a>
<a id="figure--fig:pzt-actuator"></a>
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="Figure 1: A PZT-actuator suspension" >}}
{{< figure src="/ox-hugo/du19_pzt_actuator.png" caption="<span class=\"figure-number\">Figure 1: </span>A PZT-actuator suspension" >}}
There are three popular types of micro-actuators: electrostatic moving-slider microactuator, PZT slider-driven microactuator and thermal microactuator.
There characteristics are shown on table [1](#table--tab:microactuator).
@@ -107,11 +107,11 @@ There characteristics are shown on table [1](#table--tab:microactuator).
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
A typical closed-loop control system is shown on figure [2](#org3b2af5e), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
A typical closed-loop control system is shown on figure [2](#figure--fig:single-stage-control), where \\(P\_v(s)\\) and \\(C(z)\\) represent the actuator system and its controller.
<a id="org3b2af5e"></a>
<a id="figure--fig:single-stage-control"></a>
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="Figure 2: Block diagram of a single-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_single_stage_control.png" caption="<span class=\"figure-number\">Figure 2: </span>Block diagram of a single-stage actuation system" >}}
### Dual-Stage Actuation Systems {#dual-stage-actuation-systems}
@@ -119,9 +119,9 @@ A typical closed-loop control system is shown on figure [2](#org3b2af5e), where
Dual-stage actuation mechanism for the hard disk drives consists of a VCM actuator and a secondary actuator placed between the VCM and the sensor head.
The VCM is used as the primary stage to provide long track seeking but with poor accuracy and slow response time, while the secondary stage actuator is used to provide higher positioning accuracy and faster response but with a stroke limit.
<a id="org9af6d44"></a>
<a id="figure--fig:dual-stage-control"></a>
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="Figure 3: Block diagram of dual-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_dual_stage_control.png" caption="<span class=\"figure-number\">Figure 3: </span>Block diagram of dual-stage actuation system" >}}
### Three-Stage Actuation Systems {#three-stage-actuation-systems}
@@ -145,7 +145,7 @@ In view of this, the controller design for dual-stage actuation systems adopts a
### Control Schemes {#control-schemes}
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#org0221f39).
A popular control scheme for dual-stage actuation system is the **decoupled structure** as shown in figure [4](#figure--fig:decoupled-control).
- \\(C\_v(z)\\) and \\(C\_p(z)\\) are the controllers respectively, for the primary VCM actuator \\(P\_v(s)\\) and the secondary actuator \\(P\_p(s)\\).
- \\(\hat{P}\_p(z)\\) is an approximation of \\(P\_p\\) to estimate \\(y\_p\\).
@@ -153,9 +153,9 @@ A popular control scheme for dual-stage actuation system is the **decoupled stru
- \\(n\\) is the measurement noise
- \\(d\_u\\) stands for external vibration
<a id="org0221f39"></a>
<a id="figure--fig:decoupled-control"></a>
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="Figure 4: Decoupled control structure for the dual-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_decoupled_control.png" caption="<span class=\"figure-number\">Figure 4: </span>Decoupled control structure for the dual-stage actuation system" >}}
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) + P\_v(z) C\_v(z) \hat{P}\_p(z) C\_p(z) \\]
@@ -175,16 +175,16 @@ The sensitivity functions of the VCM loop and the secondary actuator loop are
And we obtain that the dual-stage sensitivity function \\(S(z)\\) is the product of \\(S\_v(z)\\) and \\(S\_p(z)\\).
Thus, the dual-stage system control design can be decoupled into two independent controller designs.
Another type of control scheme is the **parallel structure** as shown in figure [5](#org9edcb9b).
Another type of control scheme is the **parallel structure** as shown in figure [5](#figure--fig:parallel-control-structure).
The open-loop transfer function from \\(pes\\) to \\(y\\) is
\\[ G(z) = P\_p(z) C\_p(z) + P\_v(z) C\_v(z) \\]
The overall sensitivity function of the closed-loop system from \\(r\\) to \\(pes\\) is
\\[ S(z) = \frac{1}{1 + G(z)} = \frac{1}{1 + P\_p(z) C\_p(z) + P\_v(z) C\_v(z)} \\]
<a id="org9edcb9b"></a>
<a id="figure--fig:parallel-control-structure"></a>
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="Figure 5: Parallel control structure for the dual-stage actuator system" >}}
{{< figure src="/ox-hugo/du19_parallel_control_structure.png" caption="<span class=\"figure-number\">Figure 5: </span>Parallel control structure for the dual-stage actuator system" >}}
Because of the limited displacement range of the secondary actuator, the control efforts for the two actuators should be distributed properly when designing respective controllers to meet the required performance, make the actuators not conflict with each other, as well as prevent the saturation of the secondary actuator.
@@ -192,7 +192,7 @@ Because of the limited displacement range of the secondary actuator, the control
### Controller Design Method in the Continuous-Time Domain {#controller-design-method-in-the-continuous-time-domain}
\\(\mathcal{H}\_\infty\\) loop shaping method is used to design the controllers for the primary and secondary actuators.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#org24873cb) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
The structure of the \\(\mathcal{H}\_\infty\\) loop shaping method is plotted in figure [6](#figure--fig:h-inf-diagram) where \\(W(s)\\) is a weighting function relevant to the designed control system performance such as the sensitivity function.
For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such that the closed-loop system is stable and
@@ -202,11 +202,11 @@ For a plant model \\(P(s)\\), a controller \\(C(s)\\) is to be designed such tha
is satisfied, where \\(T\_{zw}\\) is the transfer function from \\(w\\) to \\(z\\): \\(T\_{zw} = S(s) W(s)\\).
<a id="org24873cb"></a>
<a id="figure--fig:h-inf-diagram"></a>
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="Figure 6: Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
{{< figure src="/ox-hugo/du19_h_inf_diagram.png" caption="<span class=\"figure-number\">Figure 6: </span>Block diagram for \\(\mathcal{H}\_\infty\\) loop shaping method to design the controller \\(C(s)\\) with the weighting function \\(W(s)\\)" >}}
Equation [1](#orga734f85) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
Equation [1](#org563f2ec) means that \\(S(s)\\) can be shaped similarly to the inverse of the chosen weighting function \\(W(s)\\).
One form of \\(W(s)\\) is taken as
\begin{equation}
@@ -219,18 +219,18 @@ The controller can then be synthesis using the linear matrix inequality (LMI) ap
The primary and secondary actuator control loops are designed separately for the dual-stage control systems.
But when designing their respective controllers, certain performances are required for the two actuators, so that control efforts for the two actuators are distributed properly and the actuators don't conflict with each other's control authority.
As seen in figure [7](#orgb5c1410), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
As seen in figure [7](#figure--fig:dual-stage-loop-gain), the VCM primary actuator open loop has a higher gain at low frequencies, and the secondary actuator open loop has a higher gain in the high-frequency range.
<a id="orgb5c1410"></a>
<a id="figure--fig:dual-stage-loop-gain"></a>
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="Figure 7: Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
{{< figure src="/ox-hugo/du19_dual_stage_loop_gain.png" caption="<span class=\"figure-number\">Figure 7: </span>Frequency responses of \\(G\_v(s) = C\_v(s)P\_v(s)\\) (solid line) and \\(G\_p(s) = C\_p(s) P\_p(s)\\) (dotted line)" >}}
The sensitivity functions are shown in figure [8](#orgd91ec4c), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
The sensitivity functions are shown in figure [8](#figure--fig:dual-stage-sensitivity), where the hump of \\(S\_v\\) is arranged within the bandwidth of \\(S\_p\\) and the hump of \\(S\_p\\) is lowered as much as possible.
This needs to decrease the bandwidth of the primary actuator loop and increase the bandwidth of the secondary actuator loop.
<a id="orgd91ec4c"></a>
<a id="figure--fig:dual-stage-sensitivity"></a>
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="Figure 8: Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
{{< figure src="/ox-hugo/du19_dual_stage_sensitivity.png" caption="<span class=\"figure-number\">Figure 8: </span>Frequency response of \\(S\_v(s)\\) and \\(S\_p(s)\\)" >}}
A basic requirement of the dual-stage actuation control system is to make the individual primary and secondary loops stable.
It also required that the primary actuator path has a higher gain than the secondary actuator path at low frequency range and the secondary actuator path has a higher gain than the primary actuator path in high-frequency range.
@@ -261,15 +261,15 @@ A VCM actuator is used as the first-stage actuator denoted by \\(P\_v(s)\\), a P
### Control Strategy and Controller Design {#control-strategy-and-controller-design}
Figure [9](#org4bda714) shows the control structure for the three-stage actuation system.
Figure [9](#figure--fig:three-stage-control) shows the control structure for the three-stage actuation system.
The control scheme is based on the decoupled master-slave dual-stage control and the third stage microactuator is added in parallel with the dual-stage control system.
The parallel format is advantageous to the overall control bandwidth enhancement, especially for the microactuator having limited stroke which restricts the bandwidth of its own loop.
The reason why the decoupled control structure is adopted here is that its overall sensitivity function is the product of those of the two individual loops, and the VCM and the PTZ controllers can be designed separately.
<a id="org4bda714"></a>
<a id="figure--fig:three-stage-control"></a>
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="Figure 9: Control system for the three-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_three_stage_control.png" caption="<span class=\"figure-number\">Figure 9: </span>Control system for the three-stage actuation system" >}}
The open-loop transfer function of the three-stage actuation system is derived as
@@ -280,8 +280,8 @@ The open-loop transfer function of the three-stage actuation system is derived a
with
\begin{align\*}
G\_v(z) &= P\_v(z) C\_v(z) \\\\\\
G\_p(z) &= P\_p(z) C\_p(z) \\\\\\
G\_v(z) &= P\_v(z) C\_v(z) \\\\
G\_p(z) &= P\_p(z) C\_p(z) \\\\
G\_m(z) &= P\_m(z) C\_m(z)
\end{align\*}
@@ -296,17 +296,17 @@ The PZT actuated milliactuator \\(P\_p(s)\\) works under a reasonably high bandw
The third-stage actuator \\(P\_m(s)\\) is used to further push the bandwidth as high as possible.
The control performances of both the VCM and the PZT actuators are limited by their dominant resonance modes.
The open-loop frequency responses of the three stages are shown on figure [10](#orgded6e76).
The open-loop frequency responses of the three stages are shown on figure [10](#figure--fig:open-loop-three-stage).
<a id="orgded6e76"></a>
<a id="figure--fig:open-loop-three-stage"></a>
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="Figure 10: Frequency response of the open-loop transfer function" >}}
{{< figure src="/ox-hugo/du19_open_loop_three_stage.png" caption="<span class=\"figure-number\">Figure 10: </span>Frequency response of the open-loop transfer function" >}}
The obtained sensitivity function is shown on figure [11](#orgde9819c).
The obtained sensitivity function is shown on figure [11](#figure--fig:sensitivity-three-stage).
<a id="orgde9819c"></a>
<a id="figure--fig:sensitivity-three-stage"></a>
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="Figure 11: Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
{{< figure src="/ox-hugo/du19_sensitivity_three_stage.png" caption="<span class=\"figure-number\">Figure 11: </span>Sensitivity function of the VCM single stage, the dual-stage and the three-stage loops" >}}
### Performance Evaluation {#performance-evaluation}
@@ -319,13 +319,13 @@ Otherwise, saturation will occur in the control loop and the control system perf
Therefore, the stroke specification of the actuators, especially milliactuator and microactuators, is very important for achievable control performance.
Higher stroke actuators have stronger abilities to make sure that the control performances are not degraded in the presence of external vibrations.
For the three-stage control architecture as shown on figure [9](#org4bda714), the position error is
For the three-stage control architecture as shown on figure [9](#figure--fig:three-stage-control), the position error is
\\[ e = -S(P\_v d\_1 + d\_2 + d\_e) + S n \\]
The control signals and positions of the actuators are given by
\begin{align\*}
u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\\\
u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\\\
u\_p &= C\_p e,\ y\_p = P\_p C\_p e \\\\
u\_m &= C\_m e,\ y\_m = P\_m C\_m e \\\\
u\_v &= C\_v ( 1 + \hat{P}\_pC\_p ) e,\ y\_v = P\_v ( u\_v + d\_1 )
\end{align\*}
@@ -335,11 +335,11 @@ Higher bandwidth/higher level of disturbance generally means high stroke needed.
### Different Configurations of the Control System {#different-configurations-of-the-control-system}
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#orga3b472d)).
A decoupled control structure can be used for the three-stage actuation system (see figure [12](#figure--fig:three-stage-decoupled)).
The overall sensitivity function is
\\[ S(z) = \approx S\_v(z) S\_p(z) S\_m(z) \\]
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org442b5f7) and
with \\(S\_v(z)\\) and \\(S\_p(z)\\) are defined in equation [1](#org9bf2b8d) and
\\[ S\_m(z) = \frac{1}{1 + P\_m(z) C\_m(z)} \\]
Denote the dual-stage open-loop transfer function as \\(G\_d\\)
@@ -348,23 +348,23 @@ Denote the dual-stage open-loop transfer function as \\(G\_d\\)
The open-loop transfer function of the overall system is
\\[ G(z) = G\_d(z) + G\_m(z) + G\_d(z) G\_m(z) \\]
<a id="orga3b472d"></a>
<a id="figure--fig:three-stage-decoupled"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="Figure 12: Decoupled control structure for the three-stage actuation system" >}}
{{< figure src="/ox-hugo/du19_three_stage_decoupled.png" caption="<span class=\"figure-number\">Figure 12: </span>Decoupled control structure for the three-stage actuation system" >}}
The control signals and the positions of the three actuators are
\begin{align\*}
u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\\\
u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\\\
u\_p &= C\_p(1 + \hat{P}\_m C\_m) e, \ y\_p = P\_p u\_p \\\\
u\_m &= C\_m e, \ y\_m = P\_m M\_m e \\\\
u\_v &= C\_v(1 + \hat{P}\_p C\_p) (1 + \hat{P}\_m C\_m) e, \ y\_v = P\_v u\_v
\end{align\*}
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#org5311716)).
The decoupled configuration makes the low frequency gain much higher, and consequently there is much better rejection capability at low frequency compared to the parallel architecture (see figure [13](#figure--fig:three-stage-decoupled-loop-gain)).
<a id="org5311716"></a>
<a id="figure--fig:three-stage-decoupled-loop-gain"></a>
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="Figure 13: Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
{{< figure src="/ox-hugo/du19_three_stage_decoupled_loop_gain.png" caption="<span class=\"figure-number\">Figure 13: </span>Frequency responses of the open-loop transfer functions for the three-stages parallel and decoupled structure" >}}
### Conclusion {#conclusion}
@@ -671,7 +671,8 @@ Using PZT elements as a sensor to deal with high-frequency vibration beyond the
As a more advanced concept, PZT elements being used as actuator and sensor simultaneously has also been addressed in this book with detailed scheme and controller design methodology for effective utilization.
## Bibliography {#bibliography}
<a id="org2403f17"></a>Du, Chunling, and Chee Khiang Pang. 2019. _Multi-Stage Actuation Systems and Control_. Boca Raton, FL: CRC Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Du, Chunling, and Chee Khiang Pang. 2019. <i>Multi-Stage Actuation Systems and Control</i>. Boca Raton, FL: CRC Press.</div>
</div>

827
content/book/ewins00_modal.md
View File

File diff suppressed because it is too large Load Diff

21
content/book/fleming14_desig_model_contr_nanop_system.md
View File

@@ -1,19 +1,19 @@
+++
title = "Design, modeling and control of nanopositioning systems"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Talks about various topics related to nano-positioning systems."
keywords = ["Control", "Metrology", "Flexible Joints"]
draft = false
+++
Tags
: [Piezoelectric Actuators]({{<relref "piezoelectric_actuators.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}})
: [Piezoelectric Actuators]({{< relref "piezoelectric_actuators.md" >}}), [Flexible Joints]({{< relref "flexible_joints.md" >}})
Reference
: ([Fleming and Leang 2014](#orgd16fb21))
: (<a href="#citeproc_bib_item_1">Fleming and Leang 2014</a>)
Author(s)
: Fleming, A. J., & Leang, K. K.
: Fleming, A. J., &amp; Leang, K. K.
Year
: 2014
@@ -728,16 +728,15 @@ Year
### Amplifier and Piezo electrical models {#amplifier-and-piezo-electrical-models}
<a id="orgb084203"></a>
<a id="figure--fig:fleming14-amplifier-model"></a>
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="Figure 1: A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
{{< figure src="/ox-hugo/fleming14_amplifier_model.png" caption="<span class=\"figure-number\">Figure 1: </span>A voltage source \\(V\_s\\) driving a piezoelectric load. The actuator is modeled by a capacitance \\(C\_p\\) and strain-dependent voltage source \\(V\_p\\). The resistance \\(R\_s\\) is the output impedance and \\(L\\) the cable inductance." >}}
Consider the electrical circuit shown in Figure [1](#orgb084203) where a voltage source is connected to a piezoelectric actuator.
Consider the electrical circuit shown in Figure [1](#figure--fig:fleming14-amplifier-model) where a voltage source is connected to a piezoelectric actuator.
The actuator is modeled as a capacitance \\(C\_p\\) in series with a strain-dependent voltage source \\(V\_p\\).
The resistance \\(R\_s\\) and inductance \\(L\\) are the source impedance and the cable inductance respectively.
<div class="exampl">
<div></div>
Typical inductance of standard RG-58 coaxial cable is \\(250 nH/m\\).
Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
@@ -810,7 +809,6 @@ For sinusoidal signals, the amplifiers slew rate must exceed:
where \\(V\_{p-p}\\) is the peak to peak voltage and \\(f\\) is the frequency.
<div class="exampl">
<div></div>
If a 300kHz sine wave is to be reproduced with an amplitude of 10V, the required slew rate is \\(\approx 20 V/\mu s\\).
@@ -853,7 +851,8 @@ The bandwidth limitations of standard piezoelectric drives were identified as:
### References {#references}
## Bibliography {#bibliography}
<a id="orgd16fb21"></a>Fleming, Andrew J., and Kam K. Leang. 2014. _Design, Modeling and Control of Nanopositioning Systems_. Advances in Industrial Control. Springer International Publishing. <https://doi.org/10.1007/978-3-319-06617-2>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Fleming, Andrew J., and Kam K. Leang. 2014. <i>Design, Modeling and Control of Nanopositioning Systems</i>. Advances in Industrial Control. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-06617-2">10.1007/978-3-319-06617-2</a>.</div>
</div>

316
content/book/hatch00_vibrat_matlab_ansys.md
View File

@@ -1,16 +1,16 @@
+++
title = "Vibration Simulation using Matlab and ANSYS"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Nice techniques to analyze resonant systems with Ansys and Matlab."
keywords = ["Modal Analysis", "FEM"]
draft = false
+++
Tags
: [Finite Element Model]({{< relref "finite_element_model" >}})
: [Finite Element Model]({{< relref "finite_element_model.md" >}})
Reference
: ([Hatch 2000](#org4036e02))
: (<a href="#citeproc_bib_item_1">Hatch 2000</a>)
Author(s)
: Hatch, M. R.
@@ -23,17 +23,16 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<a id="org96f8e54"></a>
<span class="org-target" id="org-target--sec:introduction"></span>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#org97c03ca) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
<div></div>
The steps are:
@@ -48,9 +47,9 @@ The steps are:
</div>
<a id="org97c03ca"></a>
<a id="figure--fig:hatch00-modal-analysis-flowchart"></a>
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="<span class=\"figure-number\">Figure 1: </span>Modal analysis method flowchart" >}}
### Model Size Reduction {#model-size-reduction}
@@ -58,9 +57,8 @@ The steps are:
Because finite element models usually have a very large number of states, an important step is the reduction of the number of states while still providing correct responses for the forcing function input and desired output points.
<div class="important">
<div></div>
Figure [2](#orgdbb9ffa) shows such process, the steps are:
Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -73,14 +71,14 @@ Figure [2](#orgdbb9ffa) shows such process, the steps are:
</div>
<a id="orgdbb9ffa"></a>
<a id="figure--fig:hatch00-model-reduction-chart"></a>
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="<span class=\"figure-number\">Figure 2: </span>Model size reduction flowchart" >}}
### Notations {#notations}
Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
@@ -129,22 +127,21 @@ Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<a id="orgca1a04d"></a>
<span class="org-target" id="org-target--sec:zeros_siso_systems"></span>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#org4819d7f) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
Figure [3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="org4819d7f"></a>
<a id="figure--fig:hatch00-n-dof-zeros"></a>
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="<span class=\"figure-number\">Figure 3: </span>n dof system showing various SISO input/output configurations" >}}
<div class="important">
<div></div>
([Miu 1993](#orgcda3e53)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
(<a href="#citeproc_bib_item_2">Miu 1993</a>) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
The resonances of the "overhanging appendages" of the constrained system create the zeros.
@@ -153,17 +150,16 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<a id="orgc4e6e06"></a>
<span class="org-target" id="org-target--sec:state_space_analysis"></span>
## Modal Analysis {#modal-analysis}
<a id="orge1af07f"></a>
<span class="org-target" id="org-target--sec:modal_analysis"></span>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
<div class="important">
<div></div>
The modal method allows one to replace the n-coupled differential equations with n-uncoupled equations, where each uncoupled equation represents the motion of the system for that mode of vibration.
@@ -175,7 +171,6 @@ Heavily damped structures or structures which explicit damping elements, such as
Thus, the present methods only works for lightly damped structures.
<div class="important">
<div></div>
Summarizing the modal analysis method of analyzing linear mechanical systems and the benefits derived:
@@ -198,34 +193,34 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
#### Equation of Motion {#equation-of-motion}
Let's consider the model shown in Figure [4](#orgde2ed42) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in Figure [4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="orgde2ed42"></a>
<a id="figure--fig:hatch00-undamped-tdof-model"></a>
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="<span class=\"figure-number\">Figure 4: </span>Undamped tdof model" >}}
The equations of motions are:
\begin{equation}
\begin{bmatrix}
m & 0 & 0 \\\\\\
0 & m & 0 \\\\\\
m & 0 & 0 \\\\
0 & m & 0 \\\\
0 & 0 & m
\end{bmatrix} \begin{bmatrix}
\ddot{z}\_1 \\\\\\
\ddot{z}\_2 \\\\\\
\ddot{z}\_1 \\\\
\ddot{z}\_2 \\\\
\ddot{z}\_3
\end{bmatrix} + \begin{bmatrix}
k & -k & 0 \\\\\\
-k & 2k & -k \\\\\\
k & -k & 0 \\\\
-k & 2k & -k \\\\
0 & -k & k
\end{bmatrix} \begin{bmatrix}
z\_1 \\\\\\
z\_2 \\\\\\
z\_1 \\\\
z\_2 \\\\
z\_3
\end{bmatrix} = \begin{bmatrix}
0 \\\\\\
0 \\\\\\
0 \\\\
0 \\\\
0
\end{bmatrix} \label{eq:tdof\_eom}
\end{equation}
@@ -236,7 +231,6 @@ The equations of motions are:
Since the system is conservative (it has no damping), normal modes of vibration will exist.
<div class="important">
<div></div>
Having normal modes means that at certain frequencies all points in the system will vibrate at the same frequency and in phase, i.e., **all points in the system will reach their minimum and maximum displacements at the same point in time**.
@@ -258,7 +252,7 @@ where:
#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
Re-injecting normal modes \eqref{eq:principal_mode} into the equation of motion \eqref{eq:tdof_eom} gives the eigenvalue problem:
Re-injecting normal modes <eq:principal_mode> into the equation of motion <eq:tdof_eom> gives the eigenvalue problem:
\begin{equation}
(\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
@@ -285,45 +279,45 @@ One then find:
\begin{equation}
\bm{z}\_1 = \begin{bmatrix}
1 \\\\\\
1 \\\\\\
1 \\\\
1 \\\\
1
\end{bmatrix}, \quad \bm{z}\_2 = \begin{bmatrix}
1 \\\\\\
0 \\\\\\
1 \\\\
0 \\\\
-1
\end{bmatrix}, \quad \bm{z}\_3 = \begin{bmatrix}
1 \\\\\\
-2 \\\\\\
1 \\\\
-2 \\\\
1
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc0f09b0), [6](#org88e7153) and [7](#org8225e3c).
Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig:hatch00-tdof-mode-1), [6](#figure--fig:hatch00-tdof-mode-2) and [7](#figure--fig:hatch00-tdof-mode-3).
<a id="orgc0f09b0"></a>
<a id="figure--fig:hatch00-tdof-mode-1"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="<span class=\"figure-number\">Figure 5: </span>Rigid-Body Mode, 0rad/s" >}}
<a id="org88e7153"></a>
<a id="figure--fig:hatch00-tdof-mode-2"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="<span class=\"figure-number\">Figure 6: </span>Second Model, Middle Mass Stationary, 1rad/s" >}}
<a id="org8225e3c"></a>
<a id="figure--fig:hatch00-tdof-mode-3"></a>
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="<span class=\"figure-number\">Figure 7: </span>Third Mode, 1.7rad/s" >}}
#### Modal Matrix {#modal-matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal_matrix}
The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. <eq:modal_matrix>
\begin{equation}
\bm{z}\_m = \begin{bmatrix}
\bm{z}\_1 & \bm{z}\_2 & \bm{z}\_3
\end{bmatrix} = \begin{bmatrix}
z\_{m11} & z\_{m12} & z\_{m13} \\\\\\
z\_{m21} & z\_{m22} & z\_{m23} \\\\\\
z\_{m11} & z\_{m12} & z\_{m13} \\\\
z\_{m21} & z\_{m22} & z\_{m23} \\\\
z\_{m31} & z\_{m32} & z\_{m33}
\end{bmatrix} \label{eq:modal\_matrix}
\end{equation}
@@ -339,7 +333,6 @@ It is thus useful to **transform the n-coupled second order differential equatio
In linear algebra terms, the transformation from physical to principal coordinates is known as a **change of basis**.
<div class="important">
<div></div>
There are many options for change of basis, but we will show that **when eigenvectors are used for the transformation, the principal coordinate system has a physical meaning: each of the uncoupled sdof systems represents the motion of a specific mode of vibration**.
@@ -348,11 +341,11 @@ There are many options for change of basis, but we will show that **when eigenve
The n-uncoupled equations in the principal coordinate system can then be solved for the responses in the principal coordinate system using the well known solutions for the single dof systems.
The n-responses in the principal coordinate system can then be **transformed back** to the physical coordinate system to provide the actual response in physical coordinate.
This procedure is schematically shown in Figure [8](#org0f0be39).
This procedure is schematically shown in Figure [8](#figure--fig:hatch00-schematic-modal-solution).
<a id="org0f0be39"></a>
<a id="figure--fig:hatch00-schematic-modal-solution"></a>
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="<span class=\"figure-number\">Figure 8: </span>Roadmap for Modal Solution" >}}
The condition to guarantee diagonalization is the existence of n-linearly independent eigenvectors, which is always the case if either:
@@ -407,12 +400,12 @@ One method is to normalize with respect to unity, making the **largest** element
\begin{equation}
\bm{z}\_m = \begin{bmatrix}
1 & 1 & 1 \\\\\\
1 & 0 & -2 \\\\\\
1 & 1 & 1 \\\\
1 & 0 & -2 \\\\
1 & -1 & 1
\end{bmatrix} \Longrightarrow \bm{z}\_n \begin{bmatrix}
1 & 1 & -0.5 \\\\\\
1 & 0 & 1 \\\\\\
1 & 1 & -0.5 \\\\
1 & 0 & 1 \\\\
1 & -1 & -0.5
\end{bmatrix}
\end{equation}
@@ -423,12 +416,12 @@ Transforming the mass and stiffness matrices give:
\begin{equation}
\bm{m}\_n = \bm{z}\_n^T \bm{m} \bm{z}\_n = \begin{bmatrix}
3m & 0 & 0 \\\\\\
0 & 2m & 0 \\\\\\
3m & 0 & 0 \\\\
0 & 2m & 0 \\\\
0 & 0 & 1.5m
\end{bmatrix}; \quad \bm{k}\_n = \bm{z}\_n^T \bm{k} \bm{z}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 2k & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 2k & 0 \\\\
0 & 0 & 4.5k
\end{bmatrix}
\end{equation}
@@ -455,12 +448,12 @@ And the normalized mass and stiffness matrices are:
\begin{equation}
\bm{m}\_n = \begin{bmatrix}
1 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
1 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 1
\end{bmatrix}; \quad \bm{k}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 3
\end{bmatrix} \frac{k}{m}
\end{equation}
@@ -471,7 +464,6 @@ The normalized stiffness matrix is known as the **spectral matrix**.
Normalizing with respect to mass results in an identify principal mass matrix and squares of the eigenvalues on the diagonal in the principal stiffness matrix, this normalization technique is thus very useful for the following reason.
<div class="important">
<div></div>
Since we know the form of the principal matrices when normalizing with respect to mass, no multiplying of modal matrices is actually required: **the homogeneous principal equations of motion can be written by inspection knowing only the eigenvalues**.
@@ -498,7 +490,6 @@ Pre-multiplying by \\(\bm{z}\_n^T\\) and inserting \\(I = \bm{z}\_n \bm{z}\_n^{-
Which is re-written in the following form:
<div class="important">
<div></div>
\begin{equation}
\bm{m}\_p \ddot{\bm{z}}\_p + \bm{k}\_p \bm{z}\_p = \bm{F}\_p
@@ -517,7 +508,7 @@ where:
The vectors of initial displacements \\(\bm{z}\_{op}\\) and velocities \\(\dot{\bm{z}}\_{op}\\) in the principal coordinate system can be expressed as:
\begin{align}
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\\\
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\
\dot{\bm{z}}\_{op} &= \bm{z}\_n^{-1} \dot{\bm{z}}\_0
\end{align}
@@ -529,7 +520,6 @@ where \\(\bm{z}\_0\\) and \\(\dot{\bm{z}}\_0\\) are the vectors of initial displ
We have now everything required to solve the equations in the principal coordinate system.
<div class="important">
<div></div>
The variables in physical coordinates are the positions and velocities of the masses.
The variables in principal coordinates are the displacements and velocities of each mode of vibration.
@@ -568,12 +558,12 @@ Let's first examine the force transformation from physical to principal coordina
\begin{equation}
\bm{F}\_p = \bm{z}\_n^T \bm{F} = \begin{bmatrix}
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
z\_{n11} & z\_{n12} & z\_{n13} \\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\
z\_{n31} & z\_{n32} & z\_{n33}
\end{bmatrix}^T \begin{bmatrix}
F\_1 \\\\\\
F\_2 \\\\\\
F\_1 \\\\
F\_2 \\\\
F\_3
\end{bmatrix}
\end{equation}
@@ -584,12 +574,12 @@ Let's now examine the displacement transformation from principal to physical coo
\begin{equation}
\bm{z} = \bm{z}\_n \bm{z}\_p = \begin{bmatrix}
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
z\_{n11} & z\_{n12} & z\_{n13} \\\\
z\_{n21} & z\_{n22} & z\_{n23} \\\\
z\_{n31} & z\_{n32} & z\_{n33}
\end{bmatrix} \begin{bmatrix}
z\_{p1} \\\\\\
z\_{p2} \\\\\\
z\_{p1} \\\\
z\_{p2} \\\\
z\_{p3}
\end{bmatrix}
\end{equation}
@@ -597,7 +587,6 @@ Let's now examine the displacement transformation from principal to physical coo
And thus, if we are only interested in the physical displacement of the mass 2 (\\(z\_2 = z\_{n21} z\_{p1} + z\_{n22} z\_{p2} + z\_{n23} z\_{p3}\\)), only the second row of the modal matrix is required to transform the three displacements \\(z\_{p1}\\), \\(z\_{p2}\\), \\(z\_{p3}\\) in principal coordinates to \\(z\_2\\).
<div class="important">
<div></div>
**Only the rows of the modal matrix that correspond to degrees of freedom to which forces are applied and/or for which displacements are desired are required to complete the model.**
@@ -698,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<a id="org027da35"></a>
<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
The procedure to obtain the frequency response from a modal form is as follow:
@@ -706,11 +695,11 @@ The procedure to obtain the frequency response from a modal form is as follow:
- use Laplace transform to obtain the transfer functions in principal coordinates
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
This will be applied to the model shown in Figure [9](#orgafc54fa).
This will be applied to the model shown in Figure [9](#figure--fig:hatch00-tdof-model).
<a id="orgafc54fa"></a>
<a id="figure--fig:hatch00-tdof-model"></a>
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="<span class=\"figure-number\">Figure 9: </span>tdof undamped model for modal analysis" >}}
### Review from Previous Results {#review-from-previous-results}
@@ -725,8 +714,8 @@ From previous analysis, we know the eigenvalues and eigenvectors normalized with
\begin{equation}
\bm{z}\_n = \frac{1}{\sqrt{m}} \begin{bmatrix}
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\\\
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\\\
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\
\frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{6}}
\end{bmatrix}
\end{equation}
@@ -735,13 +724,13 @@ Knowing that in principal coordinates the mass matrix is the identify matrix and
\begin{equation}
\bm{m}\_n = \begin{bmatrix}
1 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
1 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 1
\end{bmatrix}, \quad
\bm{k}\_n = \begin{bmatrix}
0 & 0 & 0 \\\\\\
0 & 1 & 0 \\\\\\
0 & 0 & 0 \\\\
0 & 1 & 0 \\\\
0 & 0 & 3
\end{bmatrix} \frac{k}{m}
\end{equation}
@@ -761,8 +750,8 @@ The equations of motion in principal coordinates are then:
which give:
\begin{align}
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\\\
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\\\
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\
\ddot{z}\_{p3} + \frac{3k}{m} z\_{p3} &= (F\_1 - 2 F\_2 + F\_3) \frac{1}{\sqrt{6m}}
\end{align}
@@ -773,48 +762,48 @@ Taking the Laplace transform of each equation gives:
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{1}} \\\\\\
\frac{z\_{p2}}{F\_{1}} \\\\\\
\frac{z\_{p1}}{F\_{1}} \\\\
\frac{z\_{p2}}{F\_{1}} \\\\
\frac{z\_{p3}}{F\_{1}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p11} \\\\\\
z\_{p21} \\\\\\
z\_{p11} \\\\
z\_{p21} \\\\
z\_{p31}
\end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{2}} \\\\\\
\frac{z\_{p2}}{F\_{2}} \\\\\\
\frac{z\_{p1}}{F\_{2}} \\\\
\frac{z\_{p2}}{F\_{2}} \\\\
\frac{z\_{p3}}{F\_{2}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
0 \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
0 \\\\
\frac{-2}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p12} \\\\\\
z\_{p22} \\\\\\
z\_{p12} \\\\
z\_{p22} \\\\
z\_{p32}
\end{bmatrix}
\end{equation}
\begin{equation}
\begin{bmatrix}
\frac{z\_{p1}}{F\_{3}} \\\\\\
\frac{z\_{p2}}{F\_{3}} \\\\\\
\frac{z\_{p1}}{F\_{3}} \\\\
\frac{z\_{p2}}{F\_{3}} \\\\
\frac{z\_{p3}}{F\_{3}}
\end{bmatrix} = \begin{bmatrix}
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
\frac{1}{s^{2}\sqrt{3m}} \\\\
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
\end{bmatrix} = \begin{bmatrix}
z\_{p13} \\\\\\
z\_{p23} \\\\\\
z\_{p13} \\\\
z\_{p23} \\\\
z\_{p33}
\end{bmatrix}
\end{equation}
@@ -839,7 +828,7 @@ And the transfer functions \\(\frac{z\_i}{F\_j}\\) can be computed.
For instance, the contributions to the transfer function \\(\frac{z\_1}{F\_1}\\) are:
\begin{align}
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\\\
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\
& = \frac{\frac{1}{3m}}{s^2} + \frac{\frac{1}{2m}}{s^2 + \omega\_2^2} + \frac{\frac{1}{6m}}{s^2 + \omega\_3^2}
\end{align}
@@ -858,7 +847,6 @@ The forces transform in the principal coordinates using:
\end{equation}
<div class="important">
<div></div>
Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized eigenvector), then \\(\bm{F}\_p\\) will be null except for its i'th term and only the i'th mode will be excited.
@@ -870,7 +858,6 @@ Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized ei
Any transfer function derived from the modal analysis is an additive combination of sdof systems.
<div class="important">
<div></div>
Each single degree of freedom system has a gain determined by the appropriate eigenvector entries and a resonant frequency given by the appropriate eigenvalue.
@@ -886,33 +873,33 @@ If modes have some damping:
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
\end{equation}
Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
</div>
Figure [10](#orgf64b6e5) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="orgf64b6e5"></a>
<a id="figure--fig:hatch00-z11-tf-example"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 10: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
The zeros for SISO transfer functions are the roots of the numerator, however, from modal analysis we can see that the zeros arise when modes combine with appropriate phase such that the resulting motion is null.
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
<a id="org39bd7f2"></a>
<span class="org-target" id="org-target--sec:siso_state_space"></span>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgc285575).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
<a id="orgc285575"></a>
<a id="figure--fig:hatch00-cantilever-beam"></a>
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="<span class=\"figure-number\">Figure 11: </span>Cantilever beam with forcing function at midpoint" >}}
The steps to define the smallest model are:
@@ -952,7 +939,7 @@ We will discuss in this section two methods of sorting, one which is applicable
The general equation for the overall transfer function of undamped and damped systems are:
\begin{align}
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\\\
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\end{align}
@@ -989,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<a id="org658f39a"></a>
<span class="org-target" id="org-target--sec:ground_acceleration"></span>
### Model Description {#model-description}
@@ -1003,25 +990,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<a id="orgcd094f5"></a>
<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org97a4ded)).
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-model)).
### Actuator Description {#actuator-description}
<a id="org97a4ded"></a>
<a id="figure--fig:hatch00-disk-drive-siso-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orga92b66d)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
<a id="orga92b66d"></a>
<a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#orga92b66d) shows the nodes used for the reduced Matlab model.
Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@@ -1045,10 +1032,8 @@ A recommended sequence for analyzing dynamic finite element models is:
A small section of the exported `.eig` file from ANSYS is shown bellow..
<div class="exampl">
<div></div>
<div class="monoblock">
<div></div>
LOAD STEP= 1 SUBSTEP= 1
FREQ= 8.1532 LOAD CASE= 0
@@ -1089,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<a id="org58a3a47"></a>
<span class="org-target" id="org-target--sec:balanced_reduction"></span>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@@ -1117,7 +1102,7 @@ A mode which cannot be excited by the applied force is said to be **uncontrollab
For a state space system described by:
\begin{align\*}
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\\\
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\
\bm{y} &= \bm{C} \bm{x}
\end{align\*}
@@ -1159,7 +1144,7 @@ A similar set of definitions can be made for observability:
\begin{equation}
\bm{\mathcal{O}} = \begin{bmatrix}
\bm{C} \\ \bm{C} \bm{A} \\ \bm{C} \bm{A}^{2} \\ \vdots \\ \bm{C} \bm{A}^{n-1}
\bm{C} \\\ \bm{C} \bm{A} \\\ \bm{C} \bm{A}^{2} \\\ \vdots \\\ \bm{C} \bm{A}^{n-1}
\end{bmatrix}
\end{equation}
@@ -1204,16 +1189,16 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<a id="orgf33e1dd"></a>
<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org59e7525)).
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
### Actuator Description {#actuator-description}
<a id="org59e7525"></a>
<a id="figure--fig:hatch00-disk-drive-mimo-schematic"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="<span class=\"figure-number\">Figure 14: </span>Drawing of actuator/suspension system" >}}
A piezo-actuator is now bounded into one side of each of the arms.
The piezo actuator consists of a ceramic element that changes size when a voltage is applied.
@@ -1221,7 +1206,6 @@ The piezo actuator consists of a ceramic element that changes size when a voltag
Then the fine positioning motion of the piezo is used in conjunction with VCM's coarse positioning motion, higher servo bandwidth is possible.
<div class="important">
<div></div>
Instead of applying voltage as the input into the piezo elements, we will assume that we have calculated an equivalent set of forces which can be applied at the ends of the element that will replicate the voltage force function.
In this model, we will be applying forces to multiple nodes at the ends of both piezo elements.
@@ -1233,11 +1217,11 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#org5f31090) are shown the principal nodes used for the model.
In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
<a id="org5f31090"></a>
<a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="<span class=\"figure-number\">Figure 15: </span>Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
### Matlab Model {#matlab-model}
@@ -1354,13 +1338,13 @@ And we note:
G = zn * Gp;
```
<a id="orgbe6df95"></a>
<a id="figure--fig:hatch00-z13-tf"></a>
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="<span class=\"figure-number\">Figure 16: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
<a id="orgcec939e"></a>
<a id="figure--fig:hatch00-z11-tf"></a>
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 17: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
## Matlab with ANSYS {#matlab-with-ansys}
@@ -1456,15 +1440,15 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#org1183b44)).
Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
<a id="org1183b44"></a>
<a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="<span class=\"figure-number\">Figure 18: </span>Frequency of the modes" >}}
<a id="org350c1cb"></a>
<a id="figure--fig:hatch00-cant-beam-unsorted-dc-gains"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 19: </span>Unsorted DC Gains" >}}
Let's keep only the first 10 modes.
@@ -1531,9 +1515,9 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
<a id="orgd64190f"></a>
<a id="figure--fig:hatch00-cant-beam-sorted-dc-gains"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 20: </span>Sorted DC Gains" >}}
Let's keep only the first 10 **sorted** modes.
@@ -1875,9 +1859,9 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms
<a id="orgbdc6b3b"></a>
<a id="figure--fig:hatch00-gramians"></a>
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="<span class=\"figure-number\">Figure 21: </span>Observability and Controllability Gramians" >}}
We use `balreal` to rank oscillatory states.
@@ -1893,9 +1877,9 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
<a id="org2787898"></a>
<a id="figure--fig:hatch00-cant-beam-gramian-balanced"></a>
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="<span class=\"figure-number\">Figure 22: </span>Sorted values of the Gramian of the balanced realization" >}}
Now we can choose the number of states to keep.
@@ -2136,9 +2120,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
```
## Bibliography {#bibliography}
<a id="org4036e02"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
<a id="orgcda3e53"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
</div>

11
content/book/horowitz15_art_of_elect_third_edition.md
View File

@@ -1,16 +1,16 @@
+++
title = "The Art of Electronics - Third Edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "One of the best book in electronics. Cover most topics (both analog and digital)."
keywords = ["electronics"]
draft = false
+++
Tags
: [Reference Books]({{< relref "reference_books" >}}), [Electronics]({{< relref "electronics" >}})
: [Reference Books]({{< relref "reference_books.md" >}}), [Electronics]({{< relref "electronics.md" >}})
Reference
: ([Horowitz 2015](#org8eab88c))
: (<a href="#citeproc_bib_item_1">Horowitz 2015</a>)
Author(s)
: Horowitz, P.
@@ -19,7 +19,8 @@ Year
: 2015
## Bibliography {#bibliography}
<a id="org8eab88c"></a>Horowitz, Paul. 2015. _The Art of Electronics - Third Edition_. New York, NY, USA: Cambridge University Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Horowitz, Paul. 2015. <i>The Art of Electronics - Third Edition</i>. New York, NY, USA: Cambridge University Press.</div>
</div>

15
content/book/leach14_fundam_princ_engin_nanom.md
View File

@@ -1,15 +1,15 @@
+++
title = "Fundamental principles of engineering nanometrology"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
keywords = ["Metrology"]
draft = false
+++
Tags
: [Metrology]({{<relref "metrology.md#" >}})
: [Metrology]({{< relref "metrology.md" >}})
Reference
: ([Leach 2014](#org27b4df3))
: (<a href="#citeproc_bib_item_1">Leach 2014</a>)
Author(s)
: Leach, R.
@@ -64,8 +64,8 @@ The second order nature means that cosine error quickly diminish as the alignmen
## Latest advances in displacement interferometry {#latest-advances-in-displacement-interferometry}
Commercial interferometers
=> fused silica optics housed in Invar mounts
=> all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
=&gt; fused silica optics housed in Invar mounts
=&gt; all the optical components are mounted to one central optic to reduce the susceptibility to thermal variations
One advantage that homodyme systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer.
@@ -88,7 +88,8 @@ The measurement of angles is then relative.
This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
## Bibliography {#bibliography}
<a id="org27b4df3"></a>Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. <https://doi.org/10.1016/c2012-0-06010-3>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard. 2014. <i>Fundamental Principles of Engineering Nanometrology</i>. Elsevier. doi:<a href="https://doi.org/10.1016/c2012-0-06010-3">10.1016/c2012-0-06010-3</a>.</div>
</div>

13
content/book/leach18_basic_precis_engin_edition.md
View File

@@ -1,24 +1,25 @@
+++
title = "Basics of precision engineering - 1st edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
keywords = ["Metrology", "Mechatronics"]
draft = true
+++
Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
: [Precision Engineering]({{< relref "precision_engineering.md" >}})
Reference
: ([Leach and Smith 2018](#org02e139c))
: (<a href="#citeproc_bib_item_1">Leach and Smith 2018</a>)
Author(s)
: Leach, R., & Smith, S. T.
: Leach, R., &amp; Smith, S. T.
Year
: 2018
## Bibliography {#bibliography}
<a id="org02e139c"></a>Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Leach, Richard, and Stuart T. Smith. 2018. <i>Basics of Precision Engineering - 1st Edition</i>. CRC Press.</div>
</div>

139
content/book/morrison16_groun_shiel.md
View File

@@ -1,16 +1,16 @@
+++
title = "Grounding and Shielding: Circuits and Interference"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Explains in a clear manner what is grounding and shielding and what are the fundamental physics behind these terms."
keywords = ["Electronics"]
draft = false
+++
Tags
: [Electronics]({{< relref "electronics" >}})
: [Electronics]({{< relref "electronics.md" >}})
Reference
: ([Morrison 2016](#org7a49345))
: (<a href="#citeproc_bib_item_1">Morrison 2016</a>)
Author(s)
: Morrison, R.
@@ -22,7 +22,6 @@ Year
## Voltage and Capacitors {#voltage-and-capacitors}
<div class="sum">
<div></div>
This first chapter described the electric field that is basic to all electrical activity.
The electric or \\(E\\) field represents forces between charges.
@@ -53,9 +52,9 @@ This displacement current flows when charges are added or removed from the plate
### Field representation {#field-representation}
<a id="orga3615d0"></a>
<a id="figure--fig:morrison16-E-field-charge"></a>
{{< figure src="/ox-hugo/morrison16_E_field_charge.svg" caption="Figure 1: The force field lines around a positively chaged conducting sphere" >}}
{{< figure src="/ox-hugo/morrison16_E_field_charge.svg" caption="<span class=\"figure-number\">Figure 1: </span>The force field lines around a positively chaged conducting sphere" >}}
### The definition of voltage {#the-definition-of-voltage}
@@ -64,22 +63,22 @@ This displacement current flows when charges are added or removed from the plate
### Equipotential surfaces {#equipotential-surfaces}
### The force field or \\(E\\) field between two conducting plates {#the-force-field-or--e--field-between-two-conducting-plates}
### The force field or \\(E\\) field between two conducting plates {#the-force-field-or-e-field-between-two-conducting-plates}
<a id="org82b88ec"></a>
<a id="figure--fig:morrison16-force-field-plates"></a>
{{< figure src="/ox-hugo/morrison16_force_field_plates.svg" caption="Figure 2: The force field between two conducting plates with equal and opposite charges and spacing distance \\(h\\)" >}}
{{< figure src="/ox-hugo/morrison16_force_field_plates.svg" caption="<span class=\"figure-number\">Figure 2: </span>The force field between two conducting plates with equal and opposite charges and spacing distance \\(h\\)" >}}
### Electric field patterns {#electric-field-patterns}
<a id="org16f20a9"></a>
<a id="figure--fig:morrison16-electric-field-ground-plane"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_ground_plane.svg" caption="Figure 3: The electric field pattern of one circuit trace and two circuit traces over a ground plane" >}}
{{< figure src="/ox-hugo/morrison16_electric_field_ground_plane.svg" caption="<span class=\"figure-number\">Figure 3: </span>The electric field pattern of one circuit trace and two circuit traces over a ground plane" >}}
<a id="org38210cb"></a>
<a id="figure--fig:morrison16-electric-field-shielded-conductor"></a>
{{< figure src="/ox-hugo/morrison16_electric_field_shielded_conductor.svg" caption="Figure 4: Field configuration around a shielded conductor" >}}
{{< figure src="/ox-hugo/morrison16_electric_field_shielded_conductor.svg" caption="<span class=\"figure-number\">Figure 4: </span>Field configuration around a shielded conductor" >}}
### The energy stored in an electric field {#the-energy-stored-in-an-electric-field}
@@ -88,11 +87,11 @@ This displacement current flows when charges are added or removed from the plate
### Dielectrics {#dielectrics}
### The \\(D\\) field {#the--d--field}
### The \\(D\\) field {#the-d-field}
<a id="org5a4329e"></a>
<a id="figure--fig:morrison16-E-D-fields"></a>
{{< figure src="/ox-hugo/morrison16_E_D_fields.svg" caption="Figure 5: The electric field pattern in the presence of a dielectric" >}}
{{< figure src="/ox-hugo/morrison16_E_D_fields.svg" caption="<span class=\"figure-number\">Figure 5: </span>The electric field pattern in the presence of a dielectric" >}}
### Capacitance {#capacitance}
@@ -122,7 +121,6 @@ This displacement current flows when charges are added or removed from the plate
## Magnetics {#magnetics}
<div class="sum">
<div></div>
This chapter discusses magnetic fields.
As in the electric field, there are two measures of the same magnetic field.
@@ -150,11 +148,11 @@ In a few elements, the atomic structure is such that atoms align to generate a n
The flow of electrons is another way to generate a magnetic field.
The letter \\(H\\) is reserved for the magnetic field generated by a current.
Figure [6](#org9b0e888) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
Figure [6](#figure--fig:morrison16-H-field) shows the shape of the \\(H\\) field around a long, straight conductor carrying a direct current \\(I\\).
<a id="org9b0e888"></a>
<a id="figure--fig:morrison16-H-field"></a>
{{< figure src="/ox-hugo/morrison16_H_field.svg" caption="Figure 6: The \\(H\\) field around a current-carrying conductor" >}}
{{< figure src="/ox-hugo/morrison16_H_field.svg" caption="<span class=\"figure-number\">Figure 6: </span>The \\(H\\) field around a current-carrying conductor" >}}
The magnetic field is a force field.
This force can only be exerted on another magnetic field.
@@ -169,7 +167,7 @@ Ampere's law states that the integral of the \\(H\\) field intensity in a closed
\boxed{\oint H dl = I}
\end{equation}
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#org9b0e888), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
The simplest path to use for this integration is the one of the concentric circles in Figure [6](#figure--fig:morrison16-H-field), where \\(H\\) is constant and \\(r\\) is the distance from the conductor.
Solving for \\(H\\), we obtain
\begin{equation}
@@ -181,29 +179,29 @@ And we see that \\(H\\) has units of amperes per meter.
### The solenoid {#the-solenoid}
The magnetic field of a solenoid is shown in Figure [7](#orgd3a9cf9).
The magnetic field of a solenoid is shown in Figure [7](#figure--fig:morrison16-solenoid).
The field intensity inside the solenoid is nearly constant, while outside its intensity falls of rapidly.
Using Ampere's law \eqref{eq:ampere_law}:
Using Ampere's law <eq:ampere_law>:
\begin{equation}
\oint H dl \approx n I l
\end{equation}
<a id="orgd3a9cf9"></a>
<a id="figure--fig:morrison16-solenoid"></a>
{{< figure src="/ox-hugo/morrison16_solenoid.svg" caption="Figure 7: The \\(H\\) field around a solenoid" >}}
{{< figure src="/ox-hugo/morrison16_solenoid.svg" caption="<span class=\"figure-number\">Figure 7: </span>The \\(H\\) field around a solenoid" >}}
### Faraday's law and the induction field {#faraday-s-law-and-the-induction-field}
When a conducting coil is moved through a magnetic field, a voltage appears at the open ends of the coil.
This is illustrated in Figure [8](#org4b2f5c1).
This is illustrated in Figure [8](#figure--fig:morrison16-voltage-moving-coil).
The voltage depends on the number of turns in the coil and the rate at which the flux is changing.
<a id="org4b2f5c1"></a>
<a id="figure--fig:morrison16-voltage-moving-coil"></a>
{{< figure src="/ox-hugo/morrison16_voltage_moving_coil.svg" caption="Figure 8: A voltage induced into a moving coil" >}}
{{< figure src="/ox-hugo/morrison16_voltage_moving_coil.svg" caption="<span class=\"figure-number\">Figure 8: </span>A voltage induced into a moving coil" >}}
The magnetic field has two measured.
The \\(H\\) or magnetic field that is proportional to current flow.
@@ -232,14 +230,13 @@ The inverse is also true.
### The definition of inductance {#the-definition-of-inductance}
<div class="definition">
<div></div>
Inductance is defined as the ratio of magnetic flux generated per unit current.
The unit of inductance if the henry.
</div>
For the coil in Figure [7](#orgd3a9cf9):
For the coil in Figure [7](#figure--fig:morrison16-solenoid):
\begin{equation} \label{eq:inductance\_coil}
V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
@@ -247,12 +244,12 @@ V = n^2 A k \mu\_0 \frac{dI}{dt} = L \frac{dI}{dt}
where \\(k\\) relates to the geometry of the coil.
Equation \eqref{eq:inductance_coil} states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
Equation <eq:inductance_coil> states that if \\(V\\) is one volt, then for an inductance of one henry, the current will rise at the rate of one ampere per second.
### The energy stored in an inductance {#the-energy-stored-in-an-inductance}
One way to calculate the work stored in a magnetic field is to use Eq. \eqref{eq:inductance_coil}.
One way to calculate the work stored in a magnetic field is to use Eq. <eq:inductance_coil>.
The voltage \\(V\\) applied to a coil results in a linearly increasing current.
At any time \\(t\\), the power \\(P\\) supplied is equal to \\(VI\\).
Power is the rate of change of energy or \\(P = d\bm{E}/dt\\) where \\(\bm{E}\\) is the stored energy in the inductance.
@@ -263,7 +260,6 @@ We then have the stored energy in an inductance:
\end{equation}
<div class="important">
<div></div>
An inductor stores field energy.
It does not dissipate energy.
@@ -275,7 +271,6 @@ The movement of energy into the inductor thus requires both an electric and a ma
This is due to the Faraday's law that requires a voltage when changing magnetic flux couples to a coil.
<div class="exampl">
<div></div>
Consider a 1mH inductor carrying a current of 0.1A.
The stored energy is \\(5 \times 10^{-4} J\\).
@@ -309,7 +304,6 @@ In a typical circuit, conductor carrying current, the average electron velocity
## Digital Electronics {#digital-electronics}
<div class="sum">
<div></div>
This chapter shows that both electric and magnetic field are needed to move energy over pairs of conductors.
The idea of transporting electrical energy in field is extended to traces and conducting planes on printed circuit boards.
@@ -415,7 +409,6 @@ Radiation occurs at the leading edge of a wave as it moves down the transmission
## Analog Circuits {#analog-circuits}
<div class="sum">
<div></div>
This chapter treats the general problem of analog instrumentation.
The signals of interest are often generated while testing functioning hardware.
@@ -451,7 +444,6 @@ There are many transducers that can measure temperature, strain, stress, positio
The signals generated are usually in the milli-volt range and must be amplified, conditioned, and then recorded for later analysis.
<div class="important">
<div></div>
It can be very difficult to verify that the measurement is valid.
For example, signals that overload an input stage can produce noise that may look like signal.
@@ -459,7 +451,6 @@ For example, signals that overload an input stage can produce noise that may loo
</div>
<div class="definition">
<div></div>
1. **Reference Conductor**.
Any conductor used as the zero of voltage.
@@ -485,39 +476,39 @@ For example, signals that overload an input stage can produce noise that may loo
### The basic shield enclosure {#the-basic-shield-enclosure}
Consider the simple amplifier circuit shown in Figure [9](#org3286d62) with:
Consider the simple amplifier circuit shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) with:
- \\(V\_1\\) the input lead
- \\(V\_2\\) the output lead
- \\(V\_3\\) the conducting enclosure which is floating and taken as the reference conductor
- \\(V\_4\\) a signal common or reference conductor
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#org3286d62) (b).
The equivalent circuit is shown in Figure [9](#org3286d62) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
Every conductor pair has a mutual capacitance, which are shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (b).
The equivalent circuit is shown in Figure [9](#figure--fig:morrison16-parasitic-capacitance-amp) (c) and it is apparent that there is some feedback from the output to the input or the amplifier.
<a id="org3286d62"></a>
<a id="figure--fig:morrison16-parasitic-capacitance-amp"></a>
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="Figure 9: Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
{{< figure src="/ox-hugo/morrison16_parasitic_capacitance_amp.svg" caption="<span class=\"figure-number\">Figure 9: </span>Parasitic capacitances in a simple circuit. (a) Field lines in a circuit. (b) Mutual capacitance diagram. (b) Circuit representation" >}}
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#org9f3c9db)).
It is common practice in analog design to connect the enclosure to circuit common (Figure [10](#figure--fig:morrison16-grounding-shield-amp)).
When this connection is made, the feedback is removed and the enclosure no longer couples signals into the feedback structure.
The conductive enclosure is called a **shield**.
Connecting the signal common to the conductive enclosure is called "**grounding the shield**".
This "grounding" usually removed "hum" from the circuit.
<a id="org9f3c9db"></a>
<a id="figure--fig:morrison16-grounding-shield-amp"></a>
{{< figure src="/ox-hugo/morrison16_grounding_shield_amp.svg" caption="Figure 10: Grounding the shield to limit feedback" >}}
{{< figure src="/ox-hugo/morrison16_grounding_shield_amp.svg" caption="<span class=\"figure-number\">Figure 10: </span>Grounding the shield to limit feedback" >}}
Most practical circuits provide connections to external points.
To see the effect of making a _single_ external connection, open the conductive enclosure and connect the input circuit common to an external ground.
Figure [11](#orgc4242ae) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a) shows this grounded connection surrounded by an extension of the enclosure called the _cable shield_.
A problem can be caused by an incorrect location of the connection between the cable shield and the enclosure.
In Figure [11](#orgc4242ae) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\,\Omega\\), this current induced voltage that it added to the transmitted signal.
In Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (a), the electromagnetic field in the area induces a voltage in the loop and a resulting current to flow in conductor (1)-(2).
This conductor being the common ground that might have a resistance \\(R\\) or \\(1\\,\Omega\\), this current induced voltage that it added to the transmitted signal.
Our goal in this chapter is to find ways of keeping interference currents from flowing in any input signal conductor.
To remove this coupling, the shield connection to circuit common must be made at the point, where the circuit common connects to the external ground.
This connection is shown in Figure [11](#orgc4242ae) (b).
This connection is shown in Figure [11](#figure--fig:morrison16-enclosure-shield-1-2-leads) (b).
This connection keeps the circulation of interference current on the outside of the shield.
There is only one point of zero signal potential external to the enclosure and that is where the signal common connects to an external hardware ground.
@@ -527,7 +518,6 @@ If there is an external electromagnetic field, there will be current flow in the
A voltage gradient will couple interference capacitively to the signal conductors.
<div class="important">
<div></div>
An input circuit shield should connect to the circuit common, where the signal common makes its connection to the source of signal.
Any other shield connection will introduce interference.
@@ -535,16 +525,15 @@ Any other shield connection will introduce interference.
</div>
<div class="important">
<div></div>
Shielding is not an issue of finding a "really good ground".
It is an issue of using the _right_ ground.
</div>
<a id="orgc4242ae"></a>
<a id="figure--fig:morrison16-enclosure-shield-1-2-leads"></a>
{{< figure src="/ox-hugo/morrison16_enclosure_shield_1_2_leads.png" caption="Figure 11: (a) The problem of bringing one lead out of a shielded region. Unwanted current circulates in the signal lead 2. (b) The \\(E\\) field circulate current in the shield, not in the signal conductor." >}}
{{< figure src="/ox-hugo/morrison16_enclosure_shield_1_2_leads.png" caption="<span class=\"figure-number\">Figure 11: </span>(a) The problem of bringing one lead out of a shielded region. Unwanted current circulates in the signal lead 2. (b) The \\(E\\) field circulate current in the shield, not in the signal conductor." >}}
### The enclosure and utility power {#the-enclosure-and-utility-power}
@@ -554,9 +543,9 @@ The power transformer couples fields from the external environment into the encl
The obvious coupling results from capacitance between the primary coil and the secondary coil.
Note that the secondary coil is connected to the circuit common conductor.
<a id="org5995e31"></a>
<a id="figure--fig:morrison16-power-transformer-enclosure"></a>
{{< figure src="/ox-hugo/morrison16_power_transformer_enclosure.png" caption="Figure 12: A power transformer added to the circuit enclosure" >}}
{{< figure src="/ox-hugo/morrison16_power_transformer_enclosure.png" caption="<span class=\"figure-number\">Figure 12: </span>A power transformer added to the circuit enclosure" >}}
### The two-ground problem {#the-two-ground-problem}
@@ -566,9 +555,9 @@ Note that the secondary coil is connected to the circuit common conductor.
The basic analog problem is to condition a signal associated with one ground reference potential and transport this signal to a second ground reference potential without adding interference.
<a id="org3228c82"></a>
<a id="figure--fig:morrison16-two-ground-problem"></a>
{{< figure src="/ox-hugo/morrison16_two_ground_problem.svg" caption="Figure 13: The two-circuit enclosures used to transport signals between grounds" >}}
{{< figure src="/ox-hugo/morrison16_two_ground_problem.svg" caption="<span class=\"figure-number\">Figure 13: </span>The two-circuit enclosures used to transport signals between grounds" >}}
### Strain-gauge instrumentation {#strain-gauge-instrumentation}
@@ -582,9 +571,9 @@ The basic analog problem is to condition a signal associated with one ground ref
### The basic low-gain differential amplifier (forward referencing amplifier) {#the-basic-low-gain-differential-amplifier--forward-referencing-amplifier}
<a id="org4f33add"></a>
<a id="figure--fig:morrison16-low-gain-diff-amp"></a>
{{< figure src="/ox-hugo/morrison16_low_gain_diff_amp.svg" caption="Figure 14: The low-gain differential amplifier applied to the two-ground problem" >}}
{{< figure src="/ox-hugo/morrison16_low_gain_diff_amp.svg" caption="<span class=\"figure-number\">Figure 14: </span>The low-gain differential amplifier applied to the two-ground problem" >}}
### Shielding in power transformers {#shielding-in-power-transformers}
@@ -599,7 +588,6 @@ The basic analog problem is to condition a signal associated with one ground ref
### Signal flow paths in analog circuits {#signal-flow-paths-in-analog-circuits}
<div class="important">
<div></div>
Here are a few rule that will help in analog board layout:
@@ -625,13 +613,13 @@ Here are a few rule that will help in analog board layout:
### Feedback theory {#feedback-theory}
<a id="org4a09d89"></a>
<a id="figure--fig:morrison16-basic-feedback-circuit"></a>
{{< figure src="/ox-hugo/morrison16_basic_feedback_circuit.svg" caption="Figure 15: The basic feedback circuit" >}}
{{< figure src="/ox-hugo/morrison16_basic_feedback_circuit.svg" caption="<span class=\"figure-number\">Figure 15: </span>The basic feedback circuit" >}}
<a id="orgf414d06"></a>
<a id="figure--fig:morrison16-LR-stabilizing-network"></a>
{{< figure src="/ox-hugo/morrison16_LR_stabilizing_network.svg" caption="Figure 16: An LR-stabilizing network" >}}
{{< figure src="/ox-hugo/morrison16_LR_stabilizing_network.svg" caption="<span class=\"figure-number\">Figure 16: </span>An LR-stabilizing network" >}}
### Output loads and circuit stability {#output-loads-and-circuit-stability}
@@ -667,27 +655,26 @@ If the resistors are replaced by capacitors, the gain is the ratio of reactances
This feedback circuit is called a **charge converter**.
The charge on the input capacitor is transferred to the feedback capacitor.
If the feedback capacitor is smaller than the transducer capacitance by a factor of 100, then the voltage across the feedback capacitor will be 100 times greater than the open-circuit transducer voltage.
This feedback arrangement is shown in Figure [17](#org74f6090).
This feedback arrangement is shown in Figure [17](#figure--fig:morrison16-charge-amplifier).
The open-circuit input signal voltage is \\(Q/C\_T\\).
The output voltage is \\(Q/C\_{FB}\\).
The voltage gain is therefore \\(C\_T/C\_{FB}\\).
Note that there is essentially no voltage at the summing node \\(s\_p\\).
<div class="important">
<div></div>
A charge converter does not amplifier charge.
It converts a charge signal to a voltage.
</div>
<a id="org74f6090"></a>
<a id="figure--fig:morrison16-charge-amplifier"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier.svg" caption="Figure 17: A basic charge amplifier" >}}
{{< figure src="/ox-hugo/morrison16_charge_amplifier.svg" caption="<span class=\"figure-number\">Figure 17: </span>A basic charge amplifier" >}}
<a id="orgb9f996c"></a>
<a id="figure--fig:morrison16-charge-amplifier-feedback-resistor"></a>
{{< figure src="/ox-hugo/morrison16_charge_amplifier_feedback_resistor.svg" caption="Figure 18: The resistor feedback arrangement to control the low-frequency response" >}}
{{< figure src="/ox-hugo/morrison16_charge_amplifier_feedback_resistor.svg" caption="<span class=\"figure-number\">Figure 18: </span>The resistor feedback arrangement to control the low-frequency response" >}}
### DC power supplies {#dc-power-supplies}
@@ -705,7 +692,6 @@ It converts a charge signal to a voltage.
## Utility Power and Facility Grounding {#utility-power-and-facility-grounding}
<div class="sum">
<div></div>
This chapter discusses the relationship between utility power and the performance of electrical circuits.
Utility installations in facilities are controller by the NEC (National Electrical Code).
@@ -798,7 +784,7 @@ Listed equipment
### Neutral conductors {#neutral-conductors}
### \\(k\\) factor in transformers {#k--factor-in-transformers}
### \\(k\\) factor in transformers {#k-factor-in-transformers}
### Power factor correction {#power-factor-correction}
@@ -858,7 +844,6 @@ Listed equipment
## Radiation {#radiation}
<div class="sum">
<div></div>
This chapter discusses radiation from circuit boards, transmission lines, conductor loops, and antennas.
The frequency spectrum of square waves and pulses is presented.
@@ -917,7 +902,6 @@ Simple tools for locating sources of radiation are suggested.
## Shielding from Radiation {#shielding-from-radiation}
<div class="sum">
<div></div>
Cable shields are often made of aluminum foil or tinned copper braid.
Drain wires make it practical to connect to the foil.
@@ -1033,7 +1017,8 @@ To transport RF power without reflections, the source impedance and the terminat
### Shielded and screen rooms {#shielded-and-screen-rooms}
## Bibliography {#bibliography}
<a id="org7a49345"></a>Morrison, Ralph. 2016. _Grounding and Shielding: Circuits and Interference_. John Wiley & Sons.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Morrison, Ralph. 2016. <i>Grounding and Shielding: Circuits and Interference</i>. John Wiley &#38; Sons.</div>
</div>

215
content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
View File

@@ -1,16 +1,16 @@
+++
title = "Vibration Control of Active Structures - Fourth Edition"
author = ["Thomas Dehaeze"]
author = ["Dehaeze Thomas"]
description = "Gives a broad overview of vibration control."
keywords = ["Control", "Vibration"]
draft = false
+++
Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
: [Vibration Isolation]({{< relref "vibration_isolation.md" >}}), [Reference Books]({{< relref "reference_books.md" >}}), [Stewart Platforms]({{< relref "stewart_platforms.md" >}}), [HAC-HAC]({{< relref "hac_hac.md" >}})
Reference
: ([Preumont 2018](#orgf75c814))
: (<a href="#citeproc_bib_item_1">Preumont 2018</a>)
Author(s)
: Preumont, A.
@@ -63,11 +63,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
<a id="org30e8b62"></a>
<a id="figure--fig:classical-feedback-small"></a>
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="<span class=\"figure-number\">Figure 1: </span>Principle of feedback control" >}}
The principle of feedback is represented on figure [1](#org30e8b62). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The principle of feedback is represented on figure [1](#figure--fig:classical-feedback-small). The output \\(y\\) of the system is compared to the reference signal \\(r\\), and the error signal \\(\epsilon = r-y\\) is passed into a compensator \\(K(s)\\) and applied to the system \\(G(s)\\), \\(d\\) is the disturbance.
The design problem consists of finding the appropriate compensator \\(K(s)\\) such that the closed-loop system is stable and behaves in the appropriate manner.
In the control of lightly damped structures, feedback control is used for two distinct and complementary purposes: **active damping** and **model-based feedback**.
@@ -89,12 +89,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
<a id="org0cb2cac"></a>
<a id="figure--fig:feedforward-adaptative"></a>
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: Principle of feedforward control" >}}
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="<span class=\"figure-number\">Figure 2: </span>Principle of feedforward control" >}}
The method relies on the availability of a **reference signal correlated to the primary disturbance**.
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#org0cb2cac).
The idea is to produce a second disturbance such that is cancels the effect of the primary disturbance at the location of the sensor error. Its principle is explained in figure [2](#figure--fig:feedforward-adaptative).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@@ -125,11 +125,11 @@ The table [1](#table--tab:adv-dis-type-control) summarizes the main features of
### The Various Steps of the Design {#the-various-steps-of-the-design}
<a id="org5fed023"></a>
<a id="figure--fig:design-steps"></a>
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="<span class=\"figure-number\">Figure 3: </span>The various steps of the design" >}}
The various steps of the design of a controlled structure are shown in figure [3](#org5fed023).
The various steps of the design of a controlled structure are shown in figure [3](#figure--fig:design-steps).
The **starting point** is:
@@ -156,21 +156,20 @@ If the dynamics of the sensors and actuators may significantly affect the behavi
### Plant Description, Error and Control Budget {#plant-description-error-and-control-budget}
From the block diagram of the control system (figure [4](#orgc558cd1)):
From the block diagram of the control system (figure [4](#figure--fig:general-plant)):
\begin{align\*}
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\\\
y &= (I - G\_{yu}H)^{-1} G\_{yw} w\\\\
z &= T\_{zw} w = [G\_{zw} + G\_{zu}H(I - G\_{yu}H)^{-1} G\_{yw}] w
\end{align\*}
<a id="orgc558cd1"></a>
<a id="figure--fig:general-plant"></a>
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="<span class=\"figure-number\">Figure 4: </span>Block diagram of the control System" >}}
The frequency content of the disturbance \\(w\\) is usually described by its **power spectral density** \\(\Phi\_w (\omega)\\) which describes the frequency distribution of the meas-square value.
<div class="cbox">
<div></div>
\\[\sigma\_w = \sqrt{\int\_0^\infty \Phi\_w(\omega) d\omega}\\]
@@ -179,7 +178,6 @@ The frequency content of the disturbance \\(w\\) is usually described by its **p
Even more interesting for the design is the **Cumulative Mean Square** response defined by the integral of the PSD in the frequency range \\([\omega, \infty[\\).
<div class="cbox">
<div></div>
\\[\sigma\_z^2(\omega) = \int\_\omega^\infty \Phi\_z(\nu) d\nu = \int\_\omega^\infty |T\_{zw}|^2 \Phi\_w(\nu) d\nu \\]
@@ -188,14 +186,14 @@ Even more interesting for the design is the **Cumulative Mean Square** response
It is a monotonously decreasing function of frequency and describes the contribution of all frequencies above \\(\omega\\) to the mean-square value of \\(z\\).
\\(\sigma\_z(0)\\) is then the global RMS response.
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orgd0ed9cf).
A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#figure--fig:cas-plot).
It is useful to **identify the critical modes** in a design, at which the effort should be targeted.
The diagram can also be used to **assess the control laws** and compare different actuator and sensor configuration.
<a id="orgd0ed9cf"></a>
<a id="figure--fig:cas-plot"></a>
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="<span class=\"figure-number\">Figure 5: </span>Error budget distribution in OL and CL for increasing gains" >}}
### Pseudo-inverse {#pseudo-inverse}
@@ -254,7 +252,6 @@ This will have usually little impact of the fitting error while reducing conside
The general form of the equation of motion governing the dynamic equilibrium between the external, elastic, inertia and damping forces acting on a discrete, flexible structure with a finite number \\(n\\) of degrees of freedom is
<div class="cbox">
<div></div>
\begin{equation}
M \ddot{x} + C \dot{x} + K x = f
@@ -271,7 +268,6 @@ With:
The damping matrix \\(C\\) represents the various dissipation mechanisms in the structure, which are usually poorly known. One of the popular hypotheses is the Rayleigh damping.
<div class="cbox">
<div></div>
\begin{equation}
C = \alpha M + \beta K
@@ -299,14 +295,14 @@ The number of mode shapes is equal to the number of degrees of freedom \\(n\\).
The mode shapes are orthogonal with respect to the stiffness and mass matrices:
\begin{align}
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\\\
\phi\_i^T M \phi\_j &= \mu\_i \delta\_{ij} \\\\
\phi\_i^T K \phi\_j &= \mu\_i \omega\_i^2 \delta\_{ij}
\end{align}
With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode \\(i\\).
### [Modal Decomposition]({{< relref "modal_decomposition" >}}) {#modal-decomposition--modal-decomposition-dot-md}
### [Modal Decomposition]({{< relref "modal_decomposition.md" >}}) {#modal-decomposition--modal-decomposition-dot-md}
#### Structure Without Rigid Body Modes {#structure-without-rigid-body-modes}
@@ -314,7 +310,6 @@ With \\(\mu\_i\\) the **modal mass** (also called the generalized mass) of mode
Let perform a change of variable from physical coordinates \\(x\\) to modal coordinates \\(z\\).
<div class="cbox">
<div></div>
\begin{equation}
x = \Phi z
@@ -336,12 +331,11 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
One can verify that the Rayleigh damping \eqref{eq:rayleigh_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
One can verify that the Rayleigh damping <eq:rayleigh_damping> complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
And we obtain decoupled modal equations \eqref{eq:modal_eom}.
And we obtain decoupled modal equations <eq:modal_eom>.
<div class="cbox">
<div></div>
\begin{equation}
\ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
@@ -355,7 +349,7 @@ with:
</div>
Typical values of the modal damping ratio are summarized on table [tab:damping_ratio](#tab:damping_ratio).
Typical values of the modal damping ratio are summarized on table <tab:damping_ratio>.
<a id="table--tab:damping-ratio"></a>
<div class="table-caption">
@@ -372,15 +366,15 @@ Typical values of the modal damping ratio are summarized on table [tab:damping_r
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general_eom} and \eqref{eq:modal_eom} lies in the change of coordinates.
If one accepts the assumption of classical damping, the only difference between equation <eq:general_eom> and <eq:modal_eom> lies in the change of coordinates.
However, in physical coordinates, the number of degrees of freedom is usually very large.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal_eom} can often be restricted to theses modes.
If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation <eq:modal_eom> can often be restricted to theses modes.
Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
#### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
If we consider the steady-state response of equation \eqref{eq:general_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
If we consider the steady-state response of equation <eq:general_eom> to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
\\[ X = G(\omega) F \\]
Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@@ -400,11 +394,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
<a id="orgeec9f86"></a>
<a id="figure--fig:neglected-modes"></a>
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="Figure 6: Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
{{< figure src="/ox-hugo/preumont18_neglected_modes.png" caption="<span class=\"figure-number\">Figure 6: </span>Fourier spectrum of the excitation \\(F\\) and dynamic amplitification \\(D\_i\\) of mode \\(i\\) and \\(k\\) such that \\(\omega\_i < \omega\_b\\) and \\(\omega\_k \gg \omega\_b\\)" >}}
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#orgeec9f86)).
If the excitation has a limited bandwidth \\(\omega\_b\\), the contribution of the high frequency modes \\(\omega\_k \gg \omega\_b\\) can be evaluated by assuming \\(D\_k(\omega) \approx 1\\) (as shown on figure [6](#figure--fig:neglected-modes)).
And \\(G(\omega)\\) can be rewritten on terms of the **low frequency modes only**:
\\[ G(\omega) \approx \sum\_{i=1}^m \frac{\phi\_i \phi\_i^T}{\mu\_i \omega\_i^2} D\_i(\omega) + R \\]
@@ -418,7 +412,6 @@ The quasi-static correction of the high frequency modes \\(R\\) is called the **
### Collocated Control System {#collocated-control-system}
<div class="cbox">
<div></div>
A **collocated control system** is a control system where:
@@ -443,30 +436,28 @@ The open-loop FRF of a collocated system corresponds to a diagonal component of
If we assumes that the collocated system is undamped and is attached to the DoF \\(k\\), the open-loop FRF is purely real:
\\[ G\_{kk}(\omega) = \sum\_{i=1}^m \frac{\phi\_i^2(k)}{\mu\_i (\omega\_i^2 - \omega^2)} + R\_{kk} \\]
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org2389144)).
\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#figure--fig:collocated-control-frf)).
<a id="org2389144"></a>
<a id="figure--fig:collocated-control-frf"></a>
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="<span class=\"figure-number\">Figure 7: </span>Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
The amplitude of the FRF goes from \\(-\infty\\) at the resonance frequencies \\(\omega\_i\\) to \\(+\infty\\) at the next resonance frequency \\(\omega\_{i+1}\\). Therefore, in every interval, there is a frequency \\(z\_i\\) such that \\(\omega\_i < z\_i < \omega\_{i+1}\\) where the amplitude of the FRF vanishes. The frequencies \\(z\_i\\) are called **anti-resonances**.
<div class="cbox">
<div></div>
Undamped **collocated control systems** have **alternating poles and zeros** on the imaginary axis.
For lightly damped structure, the poles and zeros are just moved a little bit in the left-half plane, but they are still interlacing.
</div>
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#org9a738f7).
If the undamped structure is excited harmonically by the actuator at the frequency of the transmission zero \\(z\_i\\), the amplitude of the response of the collocated sensor vanishes. That means that the structure oscillates at the frequency \\(z\_i\\) according to the mode shape shown in dotted line figure [8](#figure--fig:collocated-zero).
<a id="org9a738f7"></a>
<a id="figure--fig:collocated-zero"></a>
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="Figure 8: Structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_collocated_zero.png" caption="<span class=\"figure-number\">Figure 8: </span>Structure with collocated actuator and sensor" >}}
<div class="cbox">
<div></div>
The frequency of the transmission zero \\(z\_i\\) and the mode shape associated are the **natural frequency** and the **mode shape** of the system obtained by **constraining the d.o.f. on which the control systems acts**.
@@ -476,11 +467,11 @@ The open-loop poles are independant of the actuator and sensor configuration whi
</div>
By looking at figure [7](#org2389144), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
By looking at figure [7](#figure--fig:collocated-control-frf), we see that neglecting the residual mode in the modelling amounts to translating the FRF diagram vertically. That produces a shift in the location of the transmission zeros to the right.
<a id="org52c26c5"></a>
<a id="figure--fig:alternating-p-z"></a>
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="Figure 9: Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
{{< figure src="/ox-hugo/preumont18_alternating_p_z.png" caption="<span class=\"figure-number\">Figure 9: </span>Bode plot of a lighly damped structure with collocated actuator and sensor" >}}
The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
@@ -488,7 +479,7 @@ The open-loop transfer function of a lighly damped structure with a collocated a
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in figure [9](#org52c26c5). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
The corresponding Bode plot is represented in figure [9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
In this way, the phase diagram is always contained between \\(\SI{0}{\degree}\\) and \\(\SI{-180}{\degree}\\) as a consequence of the interlacing property.
@@ -510,14 +501,14 @@ Two broad categories of actuators can be distinguish:
A voice coil transducer is an energy transformer which converts electrical power into mechanical power and vice versa.
The system consists of (see figure [10](#orga1a9b67)):
The system consists of (see figure [10](#figure--fig:voice-coil-schematic)):
- A permanent magnet which produces a uniform flux density \\(B\\) normal to the gap
- A coil which is free to move axially
<a id="orga1a9b67"></a>
<a id="figure--fig:voice-coil-schematic"></a>
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="Figure 10: Physical principle of a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_voice_coil_schematic.png" caption="<span class=\"figure-number\">Figure 10: </span>Physical principle of a voice coil transducer" >}}
We note:
@@ -527,7 +518,6 @@ We note:
- \\(i\\) the current into the coil
<div class="cbox">
<div></div>
**Faraday's law**:
@@ -553,11 +543,11 @@ Thus, at any time, there is an equilibrium between the electrical power absorbed
#### Proof-Mass Actuator {#proof-mass-actuator}
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#orgc439137)).
A reaction mass \\(m\\) is conected to the support structure by a spring \\(k\\) , and damper \\(c\\) and a force actuator \\(f = T i\\) (figure [11](#figure--fig:proof-mass-actuator)).
<a id="orgc439137"></a>
<a id="figure--fig:proof-mass-actuator"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="Figure 11: Proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_actuator.png" caption="<span class=\"figure-number\">Figure 11: </span>Proof-mass actuator" >}}
If we apply the second law of Newton on the mass:
\\[ m\ddot{x} + c\dot{x} + kx = f = Ti \\]
@@ -571,7 +561,6 @@ The total force applied on the support is:
The transfer function between the total force and the current \\(i\\) applied to the coil is :
<div class="cbox">
<div></div>
\begin{equation}
\frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi\_p \omega\_p s + \omega\_p^2}
@@ -585,11 +574,11 @@ with:
</div>
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#org3b93a8e)).
Above some critical frequency \\(\omega\_c \approx 2\omega\_p\\), **the proof-mass actuator can be regarded as an ideal force generator** (figure [12](#figure--fig:proof-mass-tf)).
<a id="org3b93a8e"></a>
<a id="figure--fig:proof-mass-tf"></a>
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="Figure 12: Bode plot \\(F/i\\) of the proof-mass actuator" >}}
{{< figure src="/ox-hugo/preumont18_proof_mass_tf.png" caption="<span class=\"figure-number\">Figure 12: </span>Bode plot \\(F/i\\) of the proof-mass actuator" >}}
#### Geophone {#geophone}
@@ -600,7 +589,7 @@ The voltage \\(e\\) of the coil is used as the sensor output.
If \\(x\_0\\) is the displacement of the support and if the voice coil is open (\\(i=0\\)), the governing equations are:
\begin{align\*}
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\\\
m\ddot{x} + c(\dot{x}-\dot{x\_0}) + k(x-x\_0) &= 0\\\\
T(\dot{x}-\dot{x\_0}) &= e
\end{align\*}
@@ -612,25 +601,25 @@ By using the two equations, we obtain:
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
<a id="org7ded49f"></a>
<a id="figure--fig:geophone"></a>
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="Figure 13: Model of a geophone based on a voice coil transducer" >}}
{{< figure src="/ox-hugo/preumont18_geophone.png" caption="<span class=\"figure-number\">Figure 13: </span>Model of a geophone based on a voice coil transducer" >}}
Designing geophones with very low corner frequency is in general difficult. Active geophones where the frequency is lowered electronically may constitute a good alternative option.
### General Electromechanical Transducer {#general-electromechanical-transducer}
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#org82c090c).
The consitutive behavior of a wide class of electromechanical transducers can be modelled as in figure [14](#figure--fig:electro-mechanical-transducer).
<a id="org82c090c"></a>
<a id="figure--fig:electro-mechanical-transducer"></a>
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="Figure 14: Electrical analog representation of an electromechanical transducer" >}}
{{< figure src="/ox-hugo/preumont18_electro_mechanical_transducer.png" caption="<span class=\"figure-number\">Figure 14: </span>Electrical analog representation of an electromechanical transducer" >}}
In Laplace form the constitutive equations read:
\begin{align}
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\\\
e & = Z\_e i + T\_{em} v \label{eq:gen\_trans\_e} \\\\
f & = T\_{em} i + Z\_m v \label{eq:gen\_trans\_f}
\end{align}
@@ -645,10 +634,10 @@ With:
- \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(\si{\newton\per\ampere}\\))
- \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
Equation \eqref{eq:gen_trans_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Equation <eq:gen_trans_e> shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
To do so, the bridge circuit as shown on figure [15](#org8e1c5fb) can be used.
To do so, the bridge circuit as shown on figure [15](#figure--fig:bridge-circuit) can be used.
We can show that
@@ -658,19 +647,19 @@ We can show that
which is indeed a linear function of the velocity \\(v\\) at the mechanical terminals.
<a id="org8e1c5fb"></a>
<a id="figure--fig:bridge-circuit"></a>
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="Figure 15: Bridge circuit for self-sensing actuation" >}}
{{< figure src="/ox-hugo/preumont18_bridge_circuit.png" caption="<span class=\"figure-number\">Figure 15: </span>Bridge circuit for self-sensing actuation" >}}
### Smart Materials {#smart-materials}
Smart materials have the ability to respond significantly to stimuli of different physical nature.
Figure [16](#org29efe87) lists various effects that are observed in materials in response to various inputs.
Figure [16](#figure--fig:smart-materials) lists various effects that are observed in materials in response to various inputs.
<a id="org29efe87"></a>
<a id="figure--fig:smart-materials"></a>
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="Figure 16: Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
{{< figure src="/ox-hugo/preumont18_smart_materials.png" caption="<span class=\"figure-number\">Figure 16: </span>Stimulus response relations indicating various effects in materials. The smart materials corresponds to the non-diagonal cells" >}}
### Piezoelectric Transducer {#piezoelectric-transducer}
@@ -678,14 +667,12 @@ Figure [16](#org29efe87) lists various effects that are observed in materials in
Piezoelectric materials exhibits two effects described below.
<div class="cbox">
<div></div>
Ability to generate an electrical charge in proportion to an external applied force.
</div>
<div class="cbox">
<div></div>
An electric filed parallel to the direction of polarization induces an expansion of the material.
@@ -696,11 +683,10 @@ The most popular piezoelectric materials are Lead-Zirconate-Titanate (PZT) which
We here consider a transducer made of one-dimensional piezoelectric material.
<div class="cbox">
<div></div>
\begin{subequations}
\begin{align}
D & = \epsilon^T E + d\_{33} T\\\\\\
D & = \epsilon^T E + d\_{33} T\\\\
S & = d\_{33} E + s^E T
\end{align}
\end{subequations}
@@ -720,16 +706,16 @@ With:
#### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
The set of equations \eqref{eq:piezo_eq} can be written in a matrix form:
The set of equations <eq:piezo_eq> can be written in a matrix form:
\begin{equation}
\begin{bmatrix}D\\S\end{bmatrix}
\begin{bmatrix}D\\\S\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T & d\_{33}\\\\\\
\epsilon^T & d\_{33}\\\\
d\_{33} & s^E
\end{bmatrix}
\begin{bmatrix}E\\T\end{bmatrix}
\begin{bmatrix}E\\\T\end{bmatrix}
\end{equation}
Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the dependent variable.
@@ -737,13 +723,13 @@ Where \\((E, T)\\) are the independent variables and \\((D, S)\\) are the depend
If \\((E, S)\\) are taken as independant variables:
\begin{equation}
\begin{bmatrix}D\\T\end{bmatrix}
\begin{bmatrix}D\\\T\end{bmatrix}
=
\begin{bmatrix}
\epsilon^T(1-k^2) & e\_{33}\\\\\\
\epsilon^T(1-k^2) & e\_{33}\\\\
-e\_{33} & c^E
\end{bmatrix}
\begin{bmatrix}E\\S\end{bmatrix}
\begin{bmatrix}E\\\S\end{bmatrix}
\end{equation}
With:
@@ -752,7 +738,6 @@ With:
- \\(e\_{33} = \frac{d\_{33}}{s^E}\\) is the constant relating the electric displacement to the strain for short-circuited electrodes \\([C/m^2]\\)
<div class="cbox">
<div></div>
\begin{equation}
k^2 = \frac{{d\_{33}}^2}{s^E \epsilon^T} = \frac{{e\_{33}}^2}{c^E \epsilon^T}
@@ -763,16 +748,16 @@ It measures the efficiency of the conversion of the mechanical energy into elect
</div>
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#org226015b)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo_eq_matrix_bis} over the volume of the transducer:
If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see figure [17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating <eq:piezo_eq_matrix_bis> over the volume of the transducer:
\begin{equation}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
=
\begin{bmatrix}
C & nd\_{33}\\\\\\
C & nd\_{33}\\\\
nd\_{33} & 1/K\_a
\end{bmatrix}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
\end{equation}
where
@@ -784,27 +769,27 @@ where
- \\(C = \epsilon^T A n^2/l\\) is the capacitance of the transducer with no external load (\\(f = 0\\))
- \\(K\_a = A/s^El\\) is the stiffness with short-circuited electrodes (\\(V = 0\\))
<a id="org226015b"></a>
<a id="figure--fig:piezo-stack"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="Figure 17: Piezoelectric linear transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class=\"figure-number\">Figure 17: </span>Piezoelectric linear transducer" >}}
Equation \eqref{eq:piezo_stack_eq} can be inverted to obtain
Equation <eq:piezo_stack_eq> can be inverted to obtain
\begin{equation}
\begin{bmatrix}V\\f\end{bmatrix}
\begin{bmatrix}V\\\f\end{bmatrix}
=
\frac{K\_a}{C(1-k^2)}
\begin{bmatrix}
1/K\_a & -nd\_{33}\\\\\\
1/K\_a & -nd\_{33}\\\\
-nd\_{33} & C
\end{bmatrix}
\begin{bmatrix}Q\\\Delta\end{bmatrix}
\begin{bmatrix}Q\\\\Delta\end{bmatrix}
\end{equation}
#### Energy Stored in the Piezoelectric Transducer {#energy-stored-in-the-piezoelectric-transducer}
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#org4316115).
Let us write the total stored electromechanical energy of a discrete piezoelectric transducer as shown on figure [18](#figure--fig:piezo-discrete).
The total power delivered to the transducer is the sum of electric power \\(V i\\) and the mechanical power \\(f \dot{\Delta}\\). The net work of the transducer is
@@ -812,11 +797,11 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
dW = V i dt + f \dot{\Delta} dt = V dQ + f d\Delta
\end{equation}
<a id="org4316115"></a>
<a id="figure--fig:piezo-discrete"></a>
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="Figure 18: Discrete Piezoelectric Transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class=\"figure-number\">Figure 18: </span>Discrete Piezoelectric Transducer" >}}
By integrating equation \eqref{eq:piezo_work} and using the constitutive equations \eqref{eq:piezo_stack_eq_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
By integrating equation <eq:piezo_work> and using the constitutive equations <eq:piezo_stack_eq_inv>, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
\begin{equation}
W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@@ -830,7 +815,7 @@ The constitutive equations can be recovered by differentiate the stored energy:
\\[ f = \frac{\partial W\_e}{\partial \Delta}, \quad V = \frac{\partial W\_e}{\partial Q} \\]
#### Interpretation of \\(k^2\\) {#interpretation-of--k-2}
#### Interpretation of \\(k^2\\) {#interpretation-of-k-2}
Consider a piezoelectric transducer subjected to the following mechanical cycle: first, it is loaded with a force \\(F\\) with short-circuited electrodes; the resulting extension is \\(\Delta\_1 = F/K\_a\\) where \\(K\_a = A/(s^El)\\) is the stiffness with short-circuited electrodes.
The energy stored in the system is:
@@ -846,12 +831,12 @@ The ratio between the remaining stored energy and the initial stored energy is
#### Admittance of the Piezoelectric Transducer {#admittance-of-the-piezoelectric-transducer}
Consider the system of figure [19](#orgcdbb831), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
Consider the system of figure [19](#figure--fig:piezo-stack-admittance), where the piezoelectric transducer is assumed massless and is connected to a mass \\(M\\).
The force acting on the mass is negative of that acting on the transducer, \\(f = -M \ddot{x}\\).
<a id="orgcdbb831"></a>
<a id="figure--fig:piezo-stack-admittance"></a>
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="Figure 19: Elementary dynamical model of the piezoelectric transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_stack_admittance.png" caption="<span class=\"figure-number\">Figure 19: </span>Elementary dynamical model of the piezoelectric transducer" >}}
From the constitutive equations, one finds
@@ -868,11 +853,11 @@ And one can see that
\frac{z^2 - p^2}{z^2} = k^2
\end{equation}
Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#org15dd7b6)).
Equation <eq:distance_p_z> constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement (figure [20](#figure--fig:piezo-admittance-curve)).
<a id="org15dd7b6"></a>
<a id="figure--fig:piezo-admittance-curve"></a>
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="Figure 20: Typical admittance FRF of the transducer" >}}
{{< figure src="/ox-hugo/preumont18_piezo_admittance_curve.png" caption="<span class=\"figure-number\">Figure 20: </span>Typical admittance FRF of the transducer" >}}
## Piezoelectric Beam, Plate and Truss {#piezoelectric-beam-plate-and-truss}
@@ -1004,13 +989,12 @@ Equation \eqref{eq:distance_p_z} constitutes a practical way to determine the el
#### Equivalent Damping Ratio {#equivalent-damping-ratio}
## Collocated Versus Non-collocated Control {#collocated-versus-non-collocated-control}
## BKMK Collocated Versus Non-collocated Control {#bkmk-collocated-versus-non-collocated-control}
### Pole-Zero Flipping {#pole-zero-flipping}
<div class="cbox">
<div></div>
The Root Locus shows, in a graphical form, the evolution of the poles of the closed-loop system as a function of the scalar gain \\(g\\) applied to the compensator.
The Root Locus is the locus of the solution \\(s\\) of the closed loop characteristic equation \\(1 + gG(s)H(s) = 0\\) when \\(g\\) goes from zero to infinity.
@@ -1380,7 +1364,7 @@ Weakness of LQG:
- use frequency independant cost function
- use noise statistics with uniform distribution
To overcome the weakness => frequency shaping either by:
To overcome the weakness =&gt; frequency shaping either by:
- considering a frequency dependant cost function
- using colored noise statistics
@@ -1568,7 +1552,7 @@ Their design requires a model of the structure, and there is usually a trade-off
When collocated actuator/sensor pairs can be used, stability can be achieved using positivity concepts, but in many situations, collocated pairs are not feasible for HAC.
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#org0c9fed0).
The HAC/LAC approach consist of combining the two approached in a dual-loop control as shown in Figure [21](#figure--fig:hac-lac-control).
The inner loop uses a set of collocated actuator/sensor pairs for decentralized active damping with guaranteed stability ; the outer loop consists of a non-collocated HAC based on a model of the actively damped structure.
This approach has the following advantages:
@@ -1576,9 +1560,9 @@ This approach has the following advantages:
- The active damping makes it easier to gain-stabilize the modes outside the bandwidth of the output loop (improved gain margin)
- The larger damping of the modes within the controller bandwidth makes them more robust to the parmetric uncertainty (improved phase margin)
<a id="org0c9fed0"></a>
<a id="figure--fig:hac-lac-control"></a>
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="Figure 21: Principle of the dual-loop HAC/LAC control" >}}
{{< figure src="/ox-hugo/preumont18_hac_lac_control.png" caption="<span class=\"figure-number\">Figure 21: </span>Principle of the dual-loop HAC/LAC control" >}}
#### Wide-Band Position Control {#wide-band-position-control}
@@ -1818,7 +1802,8 @@ This approach has the following advantages:
### Problems {#problems}
## Bibliography {#bibliography}
<a id="orgf75c814"></a>Preumont, Andre. 2018. _Vibration Control of Active Structures - Fourth Edition_. Solid Mechanics and Its Applications. Springer International Publishing. <https://doi.org/10.1007/978-3-319-72296-2>.
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Preumont, Andre. 2018. <i>Vibration Control of Active Structures - Fourth Edition</i>. Solid Mechanics and Its Applications. Springer International Publishing. doi:<a href="https://doi.org/10.1007/978-3-319-72296-2">10.1007/978-3-319-72296-2</a>.</div>
</div>

1164
content/book/skogestad07_multiv_feedb_contr.md
View File

File diff suppressed because it is too large Load Diff

592
content/book/taghirad13_paral.md
View File

File diff suppressed because it is too large Load Diff