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14
content/phdthesis/afzali-far16_vibrat_dynam_isotr_hexap_analy_studies.md
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@ -2,13 +2,15 @@
title = "Vibrations and dynamic isotropy in hexapods-analytical studies"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Afzali-Far, B."
ref_year = 2016
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Isotropy of Parallel Manipulator]({{<relref "isotropy_of_parallel_manipulator.md#" >}})
Reference
: ([Afzali-Far 2016](#orge2f1c73))
: ([Afzali-Far 2016](#org28bc5f9))
Author(s)
: Afzali-Far, B.
@ -95,7 +97,7 @@ Dynamic isotropy for the Stewart platform leads to a series of restrictive condi
When considering inertia of the struts, conditions are becoming more complex.
<a id="org51c1dc1"></a>
<a id="org75888b1"></a>
{{< figure src="/ox-hugo/afzali-far16_isotropic_hexapod_example.png" caption="Figure 1: Architecture of the obtained dynamically isotropic hexapod" >}}
@ -115,9 +117,9 @@ where \\(\sigma I\\) is a scaled identity matrix.
The isotropic constrain of the standard hexapod imposes special inertia of the top platform which may not be wanted in practice (\\(I\_{zz} = 4 I\_{yy} = 4 I\_{xx}\\)).
A class of generalized Gough-Stewart platforms are proposed to eliminate the above constrains.
Figure [2](#org14fbbb3) shows a schematic of proposed generalized hexapod.
Figure [2](#org09a0134) shows a schematic of proposed generalized hexapod.
<a id="org14fbbb3"></a>
<a id="org09a0134"></a>
{{< figure src="/ox-hugo/afzali-far16_proposed_generalized_hexapod.png" caption="Figure 2: Parametrization of the proposed generalized hexapod" >}}
@ -132,11 +134,11 @@ The main findings of this dissertation are:
- Comprehensive and fully parametric model of the hexapod for symmetric configurations are established both in the Cartesian and joint space.
- Inertia of the struts are taken into account to refine the model.
- A novel approach in order to obtain dynamically isotropic hexapods is proposed.
- A novel architecture of hexapod is introduced (Figure [2](#org14fbbb3)) which is dynamically isotropic for a wide range of inertia properties.
- A novel architecture of hexapod is introduced (Figure [2](#org09a0134)) which is dynamically isotropic for a wide range of inertia properties.
</div>
## Bibliography {#bibliography}
<a id="orge2f1c73"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.
<a id="org28bc5f9"></a>Afzali-Far, Behrouz. 2016. “Vibrations and Dynamic Isotropy in Hexapods-Analytical Studies.” Lund University.

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content/phdthesis/babakhani12_activ_dampin_vibrat_high_precis_motion_system.md
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@ -2,13 +2,15 @@
title = "Active damping of vibrations in high-precision motion systems"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Babakhani, B."
ref_year = 2012
+++
Tags
: [Active Damping]({{<relref "active_damping.md#" >}})
Reference
: ([Babakhani 2012](#org0b93bb2))
: ([Babakhani 2012](#orgf88b4c7))
Author(s)
: Babakhani, B.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org0b93bb2"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. <https://doi.org/10.3990/1.9789036534642>.
<a id="orgf88b4c7"></a>Babakhani, Bayan. 2012. “Active Damping of Vibrations in High-Precision Motion Systems.” University of Twente. <https://doi.org/10.3990/1.9789036534642>.

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content/phdthesis/bishop02_devel_precis_point_contr_vibrat.md
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@ -2,6 +2,8 @@
title = "Development of precision pointing controllers with and without vibration suppression for the NPS precision pointing hexapod"
author = ["Thomas Dehaeze"]
draft = true
ref_author = "Bishop Jr, R. M."
ref_year = 2002
+++
Tags
@ -9,7 +11,7 @@ Tags
Reference
: ([Bishop Jr 2002](#org6e5ba62))
: ([Bishop Jr 2002](#org19f0b30))
Author(s)
: Bishop Jr, R. M.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org6e5ba62"></a>Bishop Jr, Ronald M. 2002. “Development of Precision Pointing Controllers with and without Vibration Suppression for the NPS Precision Pointing Hexapod.” Naval Postgraduate School, Monterey, California.
<a id="org19f0b30"></a>Bishop Jr, Ronald M. 2002. “Development of Precision Pointing Controllers with and without Vibration Suppression for the NPS Precision Pointing Hexapod.” Naval Postgraduate School, Monterey, California.

6
content/phdthesis/hanieh03_activ_stewar.md
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@ -2,13 +2,15 @@
title = "Active isolation and damping of vibrations via stewart platform"
author = ["Thomas Dehaeze"]
draft = true
ref_author = "Hanieh, A. A."
ref_year = 2003
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Active Damping]({{<relref "active_damping.md#" >}})
Reference
: ([Hanieh 2003](#org68b0cb0))
: ([Hanieh 2003](#orgf310fe8))
Author(s)
: Hanieh, A. A.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org68b0cb0"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.
<a id="orgf310fe8"></a>Hanieh, Ahmed Abu. 2003. “Active Isolation and Damping of Vibrations via Stewart Platform.” Université Libre de Bruxelles, Brussels, Belgium.

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content/phdthesis/jabben07_mechat.md
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@ -2,10 +2,15 @@
title = "Mechatronic design of a magnetically suspended rotating platform"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Jabben, L."
ref_year = 2007
+++
Tags
: [Dynamic Error Budgeting]({{< relref "dynamic_error_budgeting" >}})
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
Reference
: ([Jabben 2007](#org6250919))
Author
: Jabben, L.
@ -13,9 +18,6 @@ Author
Year
: 2007
DOI
:
## Dynamic Error Budgeting {#dynamic-error-budgeting}
@ -161,21 +163,21 @@ Three factors influence the performance:
The DEB helps identifying which disturbance is the limiting factor, and it should be investigated if the controller can deal with this disturbance before re-designing the plant.
The modelling of disturbance as stochastic variables, is by excellence suitable for the optimal stochastic control framework.
In Figure [1](#orgbf22b5e), the generalized plant maps the disturbances to the performance channels.
In Figure [1](#orgcc56194), the generalized plant maps the disturbances to the performance channels.
By minimizing the \\(\mathcal{H}\_2\\) system norm of the generalized plant, the variance of the performance channels is minimized.
<a id="orgbf22b5e"></a>
<a id="orgcc56194"></a>
{{< figure src="/ox-hugo/jabben07_general_plant.png" caption="Figure 1: Control system with the generalized plant \\(G\\). The performance channels are stacked in \\(z\\), while the controller input is denoted with \\(y\\)" >}}
#### Using Weighting Filters for Disturbance Modelling {#using-weighting-filters-for-disturbance-modelling}
Since disturbances are generally not white, the system of Figure [1](#orgbf22b5e) needs to be augmented with so called **disturbance weighting filters**.
Since disturbances are generally not white, the system of Figure [1](#orgcc56194) needs to be augmented with so called **disturbance weighting filters**.
A disturbance weighting filter gives the disturbance PSD when white noise as input is applied.
This is illustrated in Figure [2](#org27e9aeb) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
This is illustrated in Figure [2](#org772dfb7) where a vector of white noise time signals \\(\underbar{w}(t)\\) is filtered through a weighting filter to obtain the colored physical disturbances \\(w(t)\\) with the desired PSD \\(S\_w\\) .
The generalized plant framework also allows to include **weighting filters for the performance channels**.
This is useful for three reasons:
@ -184,7 +186,7 @@ This is useful for three reasons:
- some performance channels may be of more importance than others
- by using dynamic weighting filters, one can emphasize the performance in a certain frequency range
<a id="org27e9aeb"></a>
<a id="org772dfb7"></a>
{{< figure src="/ox-hugo/jabben07_weighting_functions.png" caption="Figure 2: Control system with the generalized plant \\(G\\) and weighting functions" >}}
@ -209,9 +211,9 @@ So, to obtain feasible controllers, the performance channel is a combination of
By choosing suitable weighting filters for \\(y\\) and \\(u\\), the performance can be optimized while keeping the controller effort limited:
\\[ \\|z\\|\_{rms}^2 = \left\\| \begin{bmatrix} y \\ \alpha u \end{bmatrix} \right\\|\_{rms}^2 = \\|y\\|\_{rms}^2 + \alpha^2 \\|u\\|\_{rms}^2 \\]
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#org5ae58f0) is obtained.
By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha\\) and plotting the performance \\(\\|y\\|\\) vs the controller effort \\(\\|u\\|\\), the curve as depicted in Figure [3](#orgeab38dd) is obtained.
<a id="org5ae58f0"></a>
<a id="orgeab38dd"></a>
{{< figure src="/ox-hugo/jabben07_pareto_curve_H2.png" caption="Figure 3: An illustration of a Pareto curve. Each point of the curve represents the performance obtained with an optimal controller. The curve is obtained by varying \\(\alpha\\) and calculating an \\(\mathcal{H}\_2\\) optimal controller for each \\(\alpha\\)." >}}
@ -235,3 +237,8 @@ By calculation \\(\mathcal{H}\_2\\) optimal controllers for increasing \\(\alpha
> To use the measured PSDs in an optimal control design, such as H2-control, the disturbances must be modelled using linear time invariant models with multiple white noise input.
> To derive such models, spectral factorization is used.
> It is recommended to investigate which methods for spectral factorization are currently available and numerically robust.
## Bibliography {#bibliography}
<a id="org6250919"></a>Jabben, Leon. 2007. “Mechatronic Design of a Magnetically Suspended Rotating Platform.” Delft University.

74
content/phdthesis/li01_simul_fault_vibrat_isolat_point.md
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@ -2,13 +2,15 @@
title = "Simultaneous, fault-tolerant vibration isolation and pointing control of flexure jointed hexapods"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Li, X."
ref_year = 2001
+++
Tags
: [Stewart Platforms]({{<relref "stewart_platforms.md#" >}}), [Vibration Isolation]({{<relref "vibration_isolation.md#" >}}), [Cubic Architecture]({{<relref "cubic_architecture.md#" >}}), [Flexible Joints]({{<relref "flexible_joints.md#" >}}), [Multivariable Control]({{<relref "multivariable_control.md#" >}})
Reference
: ([Li 2001](#orgc147fe0))
: ([Li 2001](#org8036ec7))
Author(s)
: Li, X.
@ -22,15 +24,15 @@ Year
### Flexure Jointed Hexapods {#flexure-jointed-hexapods}
A general flexible jointed hexapod is shown in Figure [1](#orge84e431).
A general flexible jointed hexapod is shown in Figure [1](#orgd9d105c).
<a id="orge84e431"></a>
<a id="orgd9d105c"></a>
{{< figure src="/ox-hugo/li01_flexure_hexapod_model.png" caption="Figure 1: A flexure jointed hexapod. {P} is a cartesian coordinate frame located at, and rigidly attached to the payload's center of mass. {B} is the frame attached to the base, and {U} is a universal inertial frame of reference" >}}
Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orga3eb26a).
Flexure jointed hexapods have been developed to meet two needs illustrated in Figure [2](#orgaa02e76).
<a id="orga3eb26a"></a>
<a id="orgaa02e76"></a>
{{< figure src="/ox-hugo/li01_quet_dirty_box.png" caption="Figure 2: (left) Vibration machinery must be isolated from a precision bus. (right) A precision paylaod must be manipulated in the presence of base vibrations and/or exogenous forces." >}}
@ -41,12 +43,12 @@ On the other hand, the flexures add some complexity to the hexapod dynamics.
Although the flexure joints do eliminate friction and backlash, they add spring dynamics and severely limit the workspace.
Moreover, base and/or payload vibrations become significant contributors to the motion.
The University of Wyoming hexapods (example in Figure [3](#org051e360)) are:
The University of Wyoming hexapods (example in Figure [3](#orgf80b696)) are:
- Cubic (mutually orthogonal)
- Flexure Jointed
<a id="org051e360"></a>
<a id="orgf80b696"></a>
{{< figure src="/ox-hugo/li01_stewart_platform.png" caption="Figure 3: Flexure jointed Stewart platform used for analysis and control" >}}
@ -85,7 +87,7 @@ J = \begin{bmatrix}
\end{bmatrix}
\end{equation}
where (see Figure [1](#orge84e431)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
where (see Figure [1](#orgd9d105c)) \\(p\_i\\) denotes the payload attachment point of strut \\(i\\), the prescripts denote the frame of reference, and \\(\hat{u}\_i\\) denotes a unit vector along strut \\(i\\).
To make the dynamic model as simple as possible, the origin of {P} is located at the payload's center of mass.
Thus all \\({}^Pp\_i\\) are found with respect to the center of mass.
@ -138,9 +140,9 @@ Equation \eqref{eq:hexapod_eq_motion} can be rewritten as:
\end{split}
\end{equation}
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#org8deb4db)).
If the hexapod is designed such that the payload mass/inertia matrix written in the base frame (\\(^BM\_x = {}^B\_PR \cdot {}^PM\_x \cdot {}^B\_PR\_T\\)) and \\(J^T J\\) are diagonal, the dynamics from \\(u\_1\\) to \\(y\\) are decoupled (Figure [4](#org493f606)).
<a id="org8deb4db"></a>
<a id="org493f606"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf.png" caption="Figure 4: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
@ -150,7 +152,7 @@ Alternatively, a new set of inputs and outputs can be defined:
u\_2 = J^{-1} f\_m, \quad y = J^{-1} (l - l\_r)
\end{equation}
And another decoupled plant is found (Figure [5](#org7a23a21)):
And another decoupled plant is found (Figure [5](#orgbeff72d)):
\begin{equation} \label{eq:hexapod\_eq\_motion\_decoup\_2}
\begin{split}
@ -159,7 +161,7 @@ And another decoupled plant is found (Figure [5](#org7a23a21)):
\end{split}
\end{equation}
<a id="org7a23a21"></a>
<a id="orgbeff72d"></a>
{{< figure src="/ox-hugo/li01_decoupling_conf_bis.png" caption="Figure 5: Decoupling the dynamics of the Stewart Platform using the Jacobians" >}}
@ -199,15 +201,15 @@ The control bandwidth is divided as follows:
### Vibration Isolation {#vibration-isolation}
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#org0dc1d11).
The system is decoupled into six independent SISO subsystems using the architecture shown in Figure [6](#orgd7c310d).
<a id="org0dc1d11"></a>
<a id="orgd7c310d"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control.png" caption="Figure 6: Vibration isolation control strategy" >}}
One of the subsystem plant transfer function is shown in Figure [6](#org0dc1d11)
One of the subsystem plant transfer function is shown in Figure [6](#orgd7c310d)
<a id="orgcd4b06b"></a>
<a id="org1d9e762"></a>
{{< figure src="/ox-hugo/li01_vibration_control_plant.png" caption="Figure 7: Plant transfer function of one of the SISO subsystem for Vibration Control" >}}
@ -244,18 +246,18 @@ The reason is not explained.
### Pointing Control Techniques {#pointing-control-techniques}
A block diagram of the pointing control system is shown in Figure [8](#orgec13571).
A block diagram of the pointing control system is shown in Figure [8](#orge6a2624).
<a id="orgec13571"></a>
<a id="orge6a2624"></a>
{{< figure src="/ox-hugo/li01_pointing_control.png" caption="Figure 8: Figure caption" >}}
The plant is decoupled into two independent SISO subsystems.
The decoupling matrix consists of the columns of \\(J\\) corresponding to the pointing DoFs.
Figure [9](#org23ec3f5) shows the measured transfer function of the \\(\theta\_x\\) axis.
Figure [9](#org54b4cd4) shows the measured transfer function of the \\(\theta\_x\\) axis.
<a id="org23ec3f5"></a>
<a id="org54b4cd4"></a>
{{< figure src="/ox-hugo/li01_transfer_function_angle.png" caption="Figure 9: Experimentally measured plant transfer function of \\(\theta\_x/\theta\_{x\_d}\\)" >}}
@ -269,11 +271,11 @@ A typical compensator consists of the following elements:
The unity control bandwidth of the pointing loop is designed to be from **0Hz to 20Hz**.
A feedforward control is added as shown in Figure [10](#org68adfa5).
A feedforward control is added as shown in Figure [10](#orga527171).
\\(C\_f\\) is the feedforward compensator which is a 2x2 diagonal matrix.
Ideally, the feedforward compensator is an invert of the plant dynamics.
<a id="org68adfa5"></a>
<a id="orga527171"></a>
{{< figure src="/ox-hugo/li01_feedforward_control.png" caption="Figure 10: Feedforward control" >}}
@ -285,9 +287,9 @@ The simultaneous vibration isolation and pointing control is approached in two w
1. **Closing the vibration isolation loop first**: Design and implement the vibration isolation control first, identify the pointing plant when the isolation loops are closed, then implement the pointing compensators.
2. **Closing the pointing loop first**: Reverse order.
Figure [11](#orgedfc92b) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
Figure [11](#orge85d506) shows a parallel control structure where \\(G\_1(s)\\) is the dynamics from input force to output strut length.
<a id="orgedfc92b"></a>
<a id="orge85d506"></a>
{{< figure src="/ox-hugo/li01_parallel_control.png" caption="Figure 11: A parallel scheme" >}}
@ -304,19 +306,19 @@ However, the interaction between loops may affect the transfer functions of the
The dynamic interaction effect:
- Only happens in the unity bandwidth of the loop transmission of the first closed loop.
- Affect the closed loop transmission of the loop first closed (see Figures [12](#orgfc5ad76) and [13](#org8dcf497))
- Affect the closed loop transmission of the loop first closed (see Figures [12](#org1065b18) and [13](#orgba389c3))
As shown in Figure [12](#orgfc5ad76), the peak resonance of the pointing loop increase after the isolation loop is closed.
As shown in Figure [12](#org1065b18), the peak resonance of the pointing loop increase after the isolation loop is closed.
The resonances happen at both crossovers of the isolation loop (15Hz and 50Hz) and they may show of loss of robustness.
<a id="orgfc5ad76"></a>
<a id="org1065b18"></a>
{{< figure src="/ox-hugo/li01_closed_loop_pointing.png" caption="Figure 12: Closed-loop transfer functions \\(\theta\_y/\theta\_{y\_d}\\) of the pointing loop before and after the vibration isolation loop is closed" >}}
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#org8dcf497)).
The same happens when first closing the vibration isolation loop and after the pointing loop (Figure [13](#orgba389c3)).
The first peak resonance of the vibration isolation loop at 15Hz is increased when closing the pointing loop.
<a id="org8dcf497"></a>
<a id="orgba389c3"></a>
{{< figure src="/ox-hugo/li01_closed_loop_vibration.png" caption="Figure 13: Closed-loop transfer functions of the vibration isolation loop before and after the pointing control loop is closed" >}}
@ -331,19 +333,19 @@ Thus, it is recommended to design and implement the isolation control system fir
### Experimental results {#experimental-results}
Two hexapods are stacked (Figure [14](#org66cdd5c)):
Two hexapods are stacked (Figure [14](#orgc3b1ba9)):
- the bottom hexapod is used to generate disturbances matching candidate applications
- the top hexapod provide simultaneous vibration isolation and pointing control
<a id="org66cdd5c"></a>
<a id="orgc3b1ba9"></a>
{{< figure src="/ox-hugo/li01_test_bench.png" caption="Figure 14: Stacked Hexapods" >}}
First, the vibration isolation and pointing controls were implemented separately.
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org3b66ca1).
Using the vibration isolation control alone, no attenuation is achieved below 1Hz as shown in figure [15](#org933bc12).
<a id="org3b66ca1"></a>
<a id="org933bc12"></a>
{{< figure src="/ox-hugo/li01_vibration_isolation_control_results.png" caption="Figure 15: Vibration isolation control: open-loop (solid) vs. closed-loop (dashed)" >}}
@ -352,9 +354,9 @@ The simultaneous control is of dual use:
- it provide simultaneous pointing and isolation control
- it can also be used to expand the bandwidth of the isolation control to low frequencies because the pointing loops suppress pointing errors due to both base vibrations and tracking
The results of simultaneous control is shown in Figure [16](#orgb25318f) where the bandwidth of the isolation control is expanded to very low frequency.
The results of simultaneous control is shown in Figure [16](#org3618406) where the bandwidth of the isolation control is expanded to very low frequency.
<a id="orgb25318f"></a>
<a id="org3618406"></a>
{{< figure src="/ox-hugo/li01_simultaneous_control_results.png" caption="Figure 16: Simultaneous control: open-loop (solid) vs. closed-loop (dashed)" >}}
@ -407,4 +409,4 @@ Proposed future research areas include:
## Bibliography {#bibliography}
<a id="orgc147fe0"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.
<a id="org8036ec7"></a>Li, Xiaochun. 2001. “Simultaneous, Fault-Tolerant Vibration Isolation and Pointing Control of Flexure Jointed Hexapods.” University of Wyoming.

18
content/phdthesis/monkhorst04_dynam_error_budget.md
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@ -2,13 +2,15 @@
title = "Dynamic error budgeting, a design approach"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Monkhorst, W."
ref_year = 2004
+++
Tags
: [Dynamic Error Budgeting]({{<relref "dynamic_error_budgeting.md#" >}})
Reference
: ([Monkhorst 2004](#org0da5be0))
: ([Monkhorst 2004](#org114939a))
Author(s)
: Monkhorst, W.
@ -95,9 +97,9 @@ Find a controller \\(C\_{\mathcal{H}\_2}\\) which minimizes the \\(\mathcal{H}\_
In order to synthesize an \\(\mathcal{H}\_2\\) controller that will minimize the output error, the total system including disturbances needs to be modeled as a system with zero mean white noise inputs.
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#org4676e24)).
This is done by using weighting filter \\(V\_w\\), of which the output signal has a PSD \\(S\_w(f)\\) when the input is zero mean white noise (Figure [1](#orgf4eeaee)).
<a id="org4676e24"></a>
<a id="orgf4eeaee"></a>
{{< figure src="/ox-hugo/monkhorst04_weighting_filter.png" caption="Figure 1: The use of a weighting filter \\(V\_w(f)\,[SI]\\) to give the weighted signal \\(\bar{w}(t)\\) a certain PSD \\(S\_w(f)\\)." >}}
@ -108,23 +110,23 @@ The PSD \\(S\_w(f)\\) of the weighted signal is:
Given \\(S\_w(f)\\), \\(V\_w(f)\\) can be obtained using a technique called _spectral factorization_.
However, this can be avoided if the modelling of the disturbances is directly done in terms of weighting filters.
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org7706a36)).
Output weighting filters can also be used to scale different outputs relative to each other (Figure [2](#org9ce3aeb)).
<a id="org7706a36"></a>
<a id="org9ce3aeb"></a>
{{< figure src="/ox-hugo/monkhorst04_general_weighted_plant.png" caption="Figure 2: The open loop system \\(\bar{G}\\) in series with the diagonal input weightin filter \\(V\_w\\) and diagonal output scaling iflter \\(W\_z\\) defining the generalized plant \\(G\\)" >}}
#### Output scaling and the Pareto curve {#output-scaling-and-the-pareto-curve}
In this research, the outputs of the closed loop system (Figure [3](#org8166dc2)) are:
In this research, the outputs of the closed loop system (Figure [3](#orged901e4)) are:
- the performance (error) signal \\(e\\)
- the controller output \\(u\\)
In this way, the designer can analyze how much control effort is used to achieve the performance level at the performance output.
<a id="org8166dc2"></a>
<a id="orged901e4"></a>
{{< figure src="/ox-hugo/monkhorst04_closed_loop_H2.png" caption="Figure 3: The closed loop system with weighting filters included. The system has \\(n\\) disturbance inputs and two outputs: the error \\(e\\) and the control signal \\(u\\). The \\(\mathcal{H}\_2\\) minimized the \\(\mathcal{H}\_2\\) norm of this system." >}}
@ -151,4 +153,4 @@ Drawbacks however are, that no robustness guarantees can be given and that the o
## Bibliography {#bibliography}
<a id="org0da5be0"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.
<a id="org114939a"></a>Monkhorst, Wouter. 2004. “Dynamic Error Budgeting, a Design Approach.” Delft University.

8
content/phdthesis/poel10_explor_activ_hard_mount_vibrat.md
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@ -2,13 +2,15 @@
title = "An exploration of active hard mount vibration isolation for precision equipment"
author = ["Thomas Dehaeze"]
draft = true
ref_author = "van der Poel, G. W."
ref_year = 2010
+++
Tags
: [Vibration Isolation](vibration_isolation.md)
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
Reference
: ([Poel 2010](#orgf254685))
: ([Poel 2010](#org4dd001c))
Author(s)
: van der Poel, G. W.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="orgf254685"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.
<a id="org4dd001c"></a>Poel, Gerrit Wijnand van der. 2010. “An Exploration of Active Hard Mount Vibration Isolation for Precision Equipment.” University of Twente. <https://doi.org/10.3990/1.9789036530163>.

216
content/phdthesis/rankers98_machin.md
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@ -2,13 +2,15 @@
title = "Machine dynamics in mechatronic systems: an engineering approach."
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Rankers, A. M."
ref_year = 1998
+++
Tags
: [Finite Element Model]({{<relref "finite_element_model.md#" >}})
Reference
: ([Rankers 1998](#org5830c60))
: ([Rankers 1998](#orgefff92e))
Author(s)
: Rankers, A. M.
@ -172,13 +174,13 @@ The basic questions that are addressed in this thesis are:
### Basic Control Aspects {#basic-control-aspects}
A block diagram representation of a typical servo-system is shown in Figure [1](#orgb58f3ae).
A block diagram representation of a typical servo-system is shown in Figure [1](#orga3c7082).
The main task of the system is to achieve a desired positional relation between two or more components of the system.
Therefore, a sensor measures the position which is then compared to the desired value, and the resulting error is used to generate correcting forces.
In most systems, the "actual output" (e.g. position of end-effector) cannot be measured directly, and the feedback will therefore be based on a "measured output" (e.g. encoder signal at the motor).
It is important to realize that these two outputs can differ, first due to resilience in the mechanical system, and second because of geometrical imperfections in the mechanical transmission between motor and end-effector.
<a id="orgb58f3ae"></a>
<a id="orga3c7082"></a>
{{< figure src="/ox-hugo/rankers98_basic_el_mech_servo.png" caption="Figure 1: Basic elements of mechanical servo system" >}}
@ -189,10 +191,10 @@ The correction force \\(F\\) is defined by:
F = k\_p \epsilon + k\_d \dot{\epsilon} + k\_i \int \epsilon dt
\end{equation}
It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#orge942ce1)).
It is illustrative to see that basically the proportional and derivative part of such a position control loop is very similar to a mechanical spring and damper that connect two points (Figure [2](#org427ddca)).
If \\(c\\) and \\(d\\) represent the constant mechanical stiffness and damping between points \\(A\\) and \\(B\\), and a reference position profile \\(h(t)\\) is applied at \\(A\\), then an opposing force \\(F\\) is generated as soon as the position \\(x\\) and speed \\(\dot{x}\\) of point \\(B\\) does not correspond to \\(h(t)\\) and \\(\dot{h}(t)\\).
<a id="orge942ce1"></a>
<a id="org427ddca"></a>
{{< figure src="/ox-hugo/rankers98_basic_elastic_struct.png" caption="Figure 2: Basic Elastic Structure" >}}
@ -208,9 +210,9 @@ These properties are very essential since they introduce the issue of **servo st
An important aspect of a feedback controller is the fact that control forces can only result from an error signal.
Thus any desired set-point profile first leads to a position error before the corresponding driving forces are generated.
Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in Figure [3](#orgaea0875).
Most modern servo-systems have not only a feedback section, but also a **feedforward** section, as indicated in Figure [3](#org1b1236e).
<a id="orgaea0875"></a>
<a id="org1b1236e"></a>
{{< figure src="/ox-hugo/rankers98_feedforward_example.png" caption="Figure 3: Mechanical servo system with feedback and feedforward control" >}}
@ -255,9 +257,9 @@ Basically, machine dynamics can have two deterioration effects in mechanical ser
#### Actuator Flexibility {#actuator-flexibility}
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#orgb4b4c74)) due to **compliance between the motor and the load**.
The basic characteristics of what is called "actuator flexibility" is the fact that in the frequency range of interest (usually \\(0-10\times \text{bandwidth}\\)) the driven system no longer behaves as one rigid body (Figure [4](#org5d43342)) due to **compliance between the motor and the load**.
<a id="orgb4b4c74"></a>
<a id="org5d43342"></a>
{{< figure src="/ox-hugo/rankers98_actuator_flexibility.png" caption="Figure 4: Actuator Flexibility" >}}
@ -267,9 +269,9 @@ The basic characteristics of what is called "actuator flexibility" is the fact t
The second category of dynamic phenomena results from the **limited stiffness of the guiding system** in combination with the fact the the device is driven in such a way that it has to rely on the guiding system to suppress motion in an undesired direction (in case of a linear direct drive system this occurs if the driving force is not applied at the center of gravity).
In general, a rigid actuator possesses six degrees of freedom, five of which need to be suppressed by the guiding system in order to leave one mobile degree of freedom.
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org3bdc7e7)).
In the present discussion, a planar actuator with three degrees of freedom will be considered (Figure [5](#org97374b8)).
<a id="org3bdc7e7"></a>
<a id="org97374b8"></a>
{{< figure src="/ox-hugo/rankers98_guiding_flexibility_planar.png" caption="Figure 5: Planar actuator with guiding system flexibility" >}}
@ -289,9 +291,9 @@ The last category of dynamic phenomena results from the **limited mass and stiff
In contrast to many textbooks on mechanics and machine dynamics, it is good practice always to look at the combination of driving force on the moving part, and **reaction force** on the stationary part, of a positioning device.
When doing so, one has to consider what the effect of the reaction force on the systems performance will be.
In the discussion of the previous two dynamic phenomena, the stationary part of the machine was assumed to be infinitely stiff and heavy, and therefore the effect of the reaction force was negligible.
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#org01aa9a3)).
However, in general the stationary part is neither infinitely heavy, nor is it connected to its environment with infinite stiffness, so the stationary part will exhibit a resonance that is excited by the reaction forces (Figure [6](#orgf5be3d6)).
<a id="org01aa9a3"></a>
<a id="orgf5be3d6"></a>
{{< figure src="/ox-hugo/rankers98_limited_m_k_stationary_machine_part.png" caption="Figure 6: Limited Mass and Stiffness of Stationary Machine Part" >}}
@ -304,9 +306,9 @@ The effect of frame vibrations is even worse where the quality of positioning of
To understand and describe the behaviour of a mechanical system in a quantitative way, one usually sets up a model of the system.
The mathematical description of such a model with a finite number of DoF consists of a set of ordinary differential equations.
Although in the case of simple systems, such as illustrated in Figure [7](#org10e5a34) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
Although in the case of simple systems, such as illustrated in Figure [7](#orgdefa7a1) these equations may be very understandable, in the case of complex systems, the set of differential equations itself gives only limited insight, and mainly serves as a basis for numerical simulations.
<a id="org10e5a34"></a>
<a id="orgdefa7a1"></a>
{{< figure src="/ox-hugo/rankers98_1dof_system.png" caption="Figure 7: Elementary dynamic system" >}}
@ -489,14 +491,14 @@ The overall transfer function can be found by summation of the individual modal
Due to the equivalence with the differential equations of a single mass spring system, equation \eqref{eq:eoq_modal_i} is often represented by a single mass spring system on which a force \\(f^\prime = \phi\_i^T f\\) acts.
However, this representation implies an important loss of information because it neglects all information about the mode-shape vector.
Consider the system in Figure [8](#org494d682) for which the three mode shapes are depicted in the traditional graphical representation.
Consider the system in Figure [8](#org6a9ce2a) for which the three mode shapes are depicted in the traditional graphical representation.
In this representation, the physical DoF are located at fixed positions and the mode shapes displacement is indicated by the length of an arrow.
<a id="org494d682"></a>
<a id="org6a9ce2a"></a>
{{< figure src="/ox-hugo/rankers98_mode_trad_representation.png" caption="Figure 8: System and traditional graphical representation of modes" >}}
Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** (Figure [9](#orgbaf5c78)).
Alternatively, considering that for each mode the mode shape vector defined a constant relation between the various physical DoF, one could also **represent a mode shape by a lever** (Figure [9](#org40e9316)).
<div class="important">
<div></div>
@ -509,13 +511,13 @@ System with no, very little, or proportional damping exhibit real mode shape vec
Consequently, the respective DoF can only be in phase or in opposite phase.
All DoF on the same side of the rotation point have identical phases, whereas DoF on opposite sides have opposite phases.
The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#orgbaf5c78)).
The modal DoF \\(q\_i\\) can be interpreted as the displacement at a distance "1" from the pivot point (Figure [9](#org40e9316)).
<a id="orgbaf5c78"></a>
<a id="org40e9316"></a>
{{< figure src="/ox-hugo/rankers98_mode_new_representation.png" caption="Figure 9: System and new graphical representation of mode-shape" >}}
In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in Figure [10](#orge97f125) (a).
In the case of a lumped mass model, as in the previous example, it is possible to indicate at each physical DoF on the modal lever the corresponding physical mass, as shown in Figure [10](#org19300d0) (a).
The resulting moment of inertia \\(J\_i\\) of the i-th modal lever then is:
\begin{equation}
@ -530,20 +532,20 @@ m\_i = \phi\_j^T M \phi\_j = \sum\_{k=1}^n m\_k \phi\_{ik}^2
As a result of this, the **modal mass** \\(m\_i\\) could be interpreted as the resulting mass moment of inertia of the modal lever, or alternatively as a **mass located at a distance "1" from the pivot point**.
The transition from physical masses to modal masses is illustrated in Figure [10](#orge97f125) for the mode 2 of the example system.
The transition from physical masses to modal masses is illustrated in Figure [10](#org19300d0) for the mode 2 of the example system.
The modal stiffness \\(k\_2\\) is simply calculated via the relation between natural frequency, mass and stiffness:
\begin{equation}
k\_i = \omega\_i^2 m\_i
\end{equation}
<a id="orge97f125"></a>
<a id="org19300d0"></a>
{{< figure src="/ox-hugo/rankers98_mode_2_lumped_masses.png" caption="Figure 10: Graphical representation of mode 2 with (a.) lumped masses and (b.) modal mass and stiffness" >}}
Let's now consider the effect of excitation forces that act on the physical DoF.
The scalar product \\(\phi\_{ik}f\_k\\) of each force component with the corresponding element of the mode shape vector can be seen as the moment that acts on the modal level, or as an equivalent force that acts at the location of \\(q\_i\\) on the lever.
Based on the graphical representation in Figure [11](#org3899e03), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
Based on the graphical representation in Figure [11](#org4b58275), it is not difficult to understand the contribution of mode i to the transfer function \\(x\_l/f\_k\\):
\begin{equation}
\boxed{\left( \frac{x\_l}{f\_k} \right)\_i = \frac{\phi\_{ik}\phi\_{il}}{m\_i s^2 + k\_i}}
@ -551,7 +553,7 @@ Based on the graphical representation in Figure [11](#org3899e03), it is not dif
Hence, the force \\(f\_k\\) must be multiplied by the distance \\(\phi\_{ik}\\) in order to find the equivalent excitation force at the location of \\(q\_i\\) on the lever, whereas the resulting modal displacement \\(q\_i\\) must be multiplied by the distance \\(\phi\_{il}\\) in order to obtain the displacement of the physical DoF \\(x\_l\\).
<a id="org3899e03"></a>
<a id="org4b58275"></a>
{{< figure src="/ox-hugo/rankers98_lever_representation_with_force.png" caption="Figure 11: Graphical representation of mode \\(i\\), including the proper location of a force component \\(f\_k\\) that acts on physical DoF \\(x\_k\\)" >}}
@ -562,14 +564,14 @@ This linear combination of physical DoF, which will be called "User DoF" can be
x\_u = b\_1 x\_1 + \dots + b\_n x\_n = b^T x
\end{equation}
User DoF can be indicated on the modal lever, as illustrated in Figure [12](#org69af888) for a user DoF \\(x\_u = x\_3 - x\_2\\).
User DoF can be indicated on the modal lever, as illustrated in Figure [12](#org47e1b6e) for a user DoF \\(x\_u = x\_3 - x\_2\\).
The location of this user DoF \\(x\_u\\) with respect to the pivot point of modal lever \\(i\\) is defined by \\(\phi\_{iu}\\):
\begin{equation}
\phi\_{iu} = b^T \phi\_i
\end{equation}
<a id="org69af888"></a>
<a id="org47e1b6e"></a>
{{< figure src="/ox-hugo/rankers98_representation_user_dof.png" caption="Figure 12: Graphical representation of mode including user DoF \\(x\_u = x\_3 - x\_2\\)" >}}
@ -582,13 +584,13 @@ Even though the dimension mode vector can be very large, only **three user DoF**
<div class="exampl">
<div></div>
To illustrate this, a servo controlled positioning device is shown in Figure [13](#org8d21910).
To illustrate this, a servo controlled positioning device is shown in Figure [13](#org97bdfca).
The task of the device is to position the payload with respect to a tool that is mounted to the machine frame.
The actual accuracy of the machine is determined by the relative motion of these two components (actual output).
However, direct measurement of the distance between the tool and the payload is not possible and therefore the control action is based on the measured distance between a sensor and the slide on which the payload is mounted (measured output).
The slide is driven by a linear motor which transforms the output of the controller into a force on the slide and a reaction force on the stator (input).
<a id="org8d21910"></a>
<a id="org97bdfca"></a>
{{< figure src="/ox-hugo/rankers98_servo_system.png" caption="Figure 13: Schematic representation of a servo system" >}}
@ -613,9 +615,9 @@ These effective modal parameters can be used very effectively in understanding t
<div class="exampl">
<div></div>
The eigenvalue analysis of the two mass spring system in Figure [14](#orgccdbdfe) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#org866202d).
The eigenvalue analysis of the two mass spring system in Figure [14](#org7e6e047) leads to the modal results summarized in Table [1](#table--tab:2dof-example-modal-params) and which are graphically represented in Figure [15](#org4026259).
<a id="orgccdbdfe"></a>
<a id="org7e6e047"></a>
{{< figure src="/ox-hugo/rankers98_example_2dof.png" caption="Figure 14: Two mass spring system" >}}
@ -641,7 +643,7 @@ whereas the modal stiffnesses follow from \\(k\_i = \omega\_i^2 m\_i\\).
| Modal Mass [kg] | \\(m\_1 = 50.8\\) | \\(m\_2 = 11.1\\) |
| Modal Stiff [N/m] | \\(k\_1 = 0.46\cdot 10^7\\) | \\(k\_2 = 1.23\cdot 10^7\\) |
<a id="org866202d"></a>
<a id="org4026259"></a>
{{< figure src="/ox-hugo/rankers98_example_2dof_modal.png" caption="Figure 15: Graphical representation of modes and modal parameters of the two mass spring system" >}}
@ -661,18 +663,18 @@ The results are summarized in Table [2](#table--tab:2dof-example-modal-params-ef
| Effective stiff - DoF 1 | \\(k\_{\text{eff},11} = 1.02 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},21} = 1.02 \cdot 10^9 N/m\\) |
| Effective stiff - DoF 2 | \\(k\_{\text{eff},12} = 0.84 \cdot 10^7 N/m\\) | \\(k\_{\text{eff},22} = 1.25 \cdot 10^7 N/m\\) |
The effective modal parameters can then be used in the graphical representation of Figure [16](#orgabaa73a).
Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system (Figure [17](#org493aaea)).
The effective modal parameters can then be used in the graphical representation of Figure [16](#orge5c667a).
Based on this representation, it is now very easy to construct the individual modal contributions to the frequency response function \\(x\_1/F\_1\\) of the example system (Figure [17](#org48e2572)).
<a id="orgabaa73a"></a>
<a id="orge5c667a"></a>
{{< figure src="/ox-hugo/rankers98_example_2dof_effective_modal.png" caption="Figure 16: Alternative graphical representation of modes of two mass spring system based on the effective modal mass and stiffnesses in DoF \\(x\_1\\)" >}}
One can observe that the low frequency part of each modal contribution corresponds to the inverse of the calculated effective modal mass stiffness at DoF \\(x\_1\\) whereas the high frequency contribution is defined by the effective modal mass.
In the final Bode diagram (Figure [17](#org493aaea), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
In the final Bode diagram (Figure [17](#org48e2572), below) one can observe an interference of the two modal contributions in the frequency range of the second natural frequency, which in this example leads to a combination of an anti-resonance an a resonance.
<a id="org493aaea"></a>
<a id="org48e2572"></a>
{{< figure src="/ox-hugo/rankers98_2dof_example_frf.png" caption="Figure 17: Frequency Response Function \\(x\_1/f\_1\\)" >}}
@ -700,10 +702,10 @@ The technique furthermore gives an indication of the amount of frequency shift t
<div class="exampl">
<div></div>
Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#org7cab420) by attaching a linear spring \\(k\\) between two of the three represented DoF.
Assuming that one is asked to increase the natural frequency of the mode corresponding to Figure [18](#org0568cc5) by attaching a linear spring \\(k\\) between two of the three represented DoF.
As the relative motion between \\(x\_A\\) and \\(x\_B\\) is the largest of all possible combinations, this is the choice that will maximize the natural frequency of the mode.
<a id="org7cab420"></a>
<a id="org0568cc5"></a>
{{< figure src="/ox-hugo/rankers98_example_3dof_sensitivity.png" caption="Figure 18: Graphical representation of a mod with 3 DoF" >}}
@ -724,13 +726,13 @@ f\_{\text{new},i}(\Delta k) &= \frac{1}{2\pi}\sqrt{\frac{k\_{\text{eff},i} + \De
<div class="exampl">
<div></div>
Let's use the two mass spring system in Figure [14](#orgccdbdfe) as an example.
Let's use the two mass spring system in Figure [14](#org7e6e047) as an example.
In order to analyze the effect of an extra mass at \\(x\_2\\), the effective modal mass at that DoF needs to be known for both modes (see Table [2](#table--tab:2dof-example-modal-params-eff)).
Then using equation \eqref{eq:sensitivity_add_m}, one can estimate the effect of an extra mass \\(\Delta m = 1 kg\\) added to \\(m\_2\\).
To estimate the influence of extra stiffness between the two DoF, one needs to calculate the effective modal stiffness that corresponds to the relative motion between \\(x\_2\\) and \\(x\_1\\).
This can be graphically done as shown in Figure [19](#org2cdc396):
This can be graphically done as shown in Figure [19](#orgc4d5cdc):
\begin{align}
k\_{\text{eff},1,(2-1)} &= 0.46 \cdot 10^7 / 0.07^2 = 93.9 \cdot 10^7 N/m \\\\\\
@ -753,7 +755,7 @@ The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-res
| \\(\Delta m = 1 kg\\) added to \\(m\_2\\) | 47.5 | 160.7 |
| \\(\Delta k = 10^7 N/m\\) added between \\(x\_2\\) and \\(x\_1\\) | 48.1 | 237.2 |
<a id="org2cdc396"></a>
<a id="orgc4d5cdc"></a>
{{< figure src="/ox-hugo/rankers98_example_sensitivity_2dof.png" caption="Figure 19: Graphical representation of modes and modal parameters of two mass spring system" >}}
@ -763,9 +765,9 @@ The results are summarized in Table [3](#table--tab:example-sensitivity-2dof-res
### Modal Superposition {#modal-superposition}
Previously, the lever representation was used only to represent the individual mode shapes.
In the mechanism shown in Figure [20](#org1066c2d), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
In the mechanism shown in Figure [20](#org94e8405), the motion of the output \\(y\\) is equals to the sum of the motion of the two inputs \\(x\_1\\) and \\(x\_2\\).
<a id="org1066c2d"></a>
<a id="org94e8405"></a>
{{< figure src="/ox-hugo/rankers98_addition_of_motion.png" caption="Figure 20: Addition of motion" >}}
@ -775,9 +777,9 @@ This approach can be applied to the concept of modal superposition, which expres
x\_k(t) = \sum\_{i=1}^n \phi\_{ik} q\_i(t) = \sum\_{i=1}^n x\_{ki}(t)
\end{equation}
Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to Figure [21](#org8de0f99), which is a visualization of the transformation between the modal and the physical domains.
Combining the concept of summation of modal contribution with the lever representation of mode shapes leads to Figure [21](#org0db9eeb), which is a visualization of the transformation between the modal and the physical domains.
<a id="org8de0f99"></a>
<a id="org0db9eeb"></a>
{{< figure src="/ox-hugo/rankers98_conversion_modal_to_physical.png" caption="Figure 21: Conversion between modal DoF to physical DoF" >}}
@ -788,9 +790,9 @@ The "rigid body modes" usually refer to the lower natural frequencies of a machi
This is misleading at it suggests that the structure exhibits no internal deformation.
A better term for such a mode would be **suspension mode**.
To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#org7eacb7b)).
To illustrate the important of the internal deformation, a very simplified physical model of a precision machine is considered (Figure [22](#orga668da4)).
<a id="org7eacb7b"></a>
<a id="orga668da4"></a>
{{< figure src="/ox-hugo/rankers98_suspension_mode_machine.png" caption="Figure 22: Simplified physical model of a precision machine" >}}
@ -841,9 +843,9 @@ The interaction between the desired (rigid body) motion and the dynamics of one
### Basic Characteristics of Mechanical FRF {#basic-characteristics-of-mechanical-frf}
Consider the position control loop of Figure [23](#orga7c3e4d).
Consider the position control loop of Figure [23](#orga09dfda).
<a id="orga7c3e4d"></a>
<a id="orga09dfda"></a>
{{< figure src="/ox-hugo/rankers98_mechanical_servo_system.png" caption="Figure 23: Mechanical position servo-system" >}}
@ -853,9 +855,9 @@ In the ideal situation the mechanical system behaves as one rigid body with mass
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{m s^2}
\end{equation}
The corresponding Bode and Nyquist plots and shown in Figure [24](#org4eddb5f).
The corresponding Bode and Nyquist plots and shown in Figure [24](#orga6589a0).
<a id="org4eddb5f"></a>
<a id="orga6589a0"></a>
{{< figure src="/ox-hugo/rankers98_ideal_bode_nyquist.png" caption="Figure 24: FRF of an ideal system with no resonances" >}}
@ -881,7 +883,7 @@ which simplifies equation \eqref{eq:effect_one_mode} to:
Equation \eqref{eq:effect_one_mode_simplified} will be the basis for the discussion of the various patterns that can be observe in the frequency response functions and the effect of resonances on servo stability.
Three different types of intersection pattern can be found in the amplitude plot as shown in Figure [25](#org4d0be94).
Three different types of intersection pattern can be found in the amplitude plot as shown in Figure [25](#orgf0d0a35).
Depending on the absolute value of \\(\alpha\\) one can observe:
- \\(|\alpha| < 1\\): two intersections
@ -890,22 +892,22 @@ Depending on the absolute value of \\(\alpha\\) one can observe:
The interaction between the rigid body motion and the additional mode will not only depend on \\(|\alpha|\\) but also on the **sign** of \\(\alpha\\), which determined the **phase relation between the two contributions**.
<a id="org4d0be94"></a>
<a id="orgf0d0a35"></a>
{{< figure src="/ox-hugo/rankers98_frf_effect_alpha.png" caption="Figure 25: Contribution of rigid-body motion and modal dynamics to the amplitude and phase of FRF for various values of \\(\alpha\\)" >}}
The general shape of the overall FRF can be constructed for all cases (Figure [26](#org7226848)).
The general shape of the overall FRF can be constructed for all cases (Figure [26](#org0cad86f)).
Interesting points are the interaction of the two parts at the frequency that corresponds to an intersection in the amplitude plot.
At this frequency the magnitudes are equal, so it depends on the phase of the two contributions whether they cancel each other, thus leading to a zero, or just add up.
<a id="org7226848"></a>
<a id="org0cad86f"></a>
{{< figure src="/ox-hugo/rankers98_final_frf_alpha.png" caption="Figure 26: Bode diagram of final FRF (\\(x\_{\text{servo}}/F\_{\text{servo}}\\)) for six values of \\(\alpha\\)" >}}
<div class="important">
<div></div>
When analyzing the plots of Figure [26](#org7226848), four different types of FRF can be found:
When analyzing the plots of Figure [26](#org0cad86f), four different types of FRF can be found:
- -2 slope / zero / pole / -2 slope (\\(\alpha > 0\\))
- -2 slope / pole / zero / -2 slope (\\(-1 < \alpha < 0\\))
@ -914,9 +916,9 @@ When analyzing the plots of Figure [26](#org7226848), four different types of FR
</div>
All cases are shown in Figure [27](#org8b59092).
All cases are shown in Figure [27](#org90ed06a).
<a id="org8b59092"></a>
<a id="org90ed06a"></a>
{{< figure src="/ox-hugo/rankers98_interaction_shapes.png" caption="Figure 27: Bode plot of the different types of FRF" >}}
@ -935,15 +937,15 @@ f\_{lp} &= 4 \cdot f\_b
\end{align\*}
with \\(f\_b\\) the bandwidth frequency.
The asymptotic amplitude plot is shown in Figure [28](#org57d75c7).
The asymptotic amplitude plot is shown in Figure [28](#org8238373).
<a id="org57d75c7"></a>
<a id="org8238373"></a>
{{< figure src="/ox-hugo/rankers98_pid_amplitude.png" caption="Figure 28: Typical crossover frequencies of a PID controller with 2nd order low pass filtering" >}}
With these settings, the open loop response of the position loop (controller + mechanics) looks like Figure [29](#orgcc5f9dc).
With these settings, the open loop response of the position loop (controller + mechanics) looks like Figure [29](#org027c7fb).
<a id="orgcc5f9dc"></a>
<a id="org027c7fb"></a>
{{< figure src="/ox-hugo/rankers98_ideal_frf_pid.png" caption="Figure 29: Ideal open loop FRF of a position servo without mechanical resonances (\\(f\_b = 30\text{ Hz}\\))" >}}
@ -952,37 +954,37 @@ With these settings, the open loop response of the position loop (controller + m
Conclusions are:
- A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency (Figure [30](#org9f4ad31)) and high frequency (Figure [32](#org7659335)) regions.
In the medium frequency region (Figure [31](#org2fc068b)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
- A "-2 slope / zero / pole / -2 slope" characteristic leads to a phase lead, and is therefore potentially destabilizing in the low frequency (Figure [30](#orgc50054c)) and high frequency (Figure [32](#org4bca113)) regions.
In the medium frequency region (Figure [31](#orgf5dadc6)), it adds an extra phase lead to the already existing margin, which does not harm the stability.
- A "-2 slope / pole / zero / -2 slope" combination has the reverse effect.
It is potentially destabilizing in the medium frequency range (Figure [34](#orgbce6fc9)) and is harmless in the low (Figure [33](#orgb85562f)) and high frequency (Figure [35](#orgc43335a)) ranges.
It is potentially destabilizing in the medium frequency range (Figure [34](#orgd1068f3)) and is harmless in the low (Figure [33](#org3baa684)) and high frequency (Figure [35](#org5e8c4cc)) ranges.
- The "-2 slope / pole / -4 slope" behavior always has a devastating effect on the stability of the loop if located in the low of medium frequency ranges.
These conclusions may differ for different mass ratio \\(\alpha\\).
</div>
<a id="org9f4ad31"></a>
<a id="orgc50054c"></a>
{{< figure src="/ox-hugo/rankers98_zero_pole_low_freq.png" caption="Figure 30: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with low frequency resonance" >}}
<a id="org2fc068b"></a>
<a id="orgf5dadc6"></a>
{{< figure src="/ox-hugo/rankers98_zero_pole_medium_freq.png" caption="Figure 31: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with medium frequency resonance" >}}
<a id="org7659335"></a>
<a id="org4bca113"></a>
{{< figure src="/ox-hugo/rankers98_zero_pole_high_freq.png" caption="Figure 32: Open Loop FRF of type \"-2 slope / zero / pole / -2 slope\" with high frequency resonance" >}}
<a id="orgb85562f"></a>
<a id="org3baa684"></a>
{{< figure src="/ox-hugo/rankers98_pole_zero_low_freq.png" caption="Figure 33: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with low frequency resonance" >}}
<a id="orgbce6fc9"></a>
<a id="orgd1068f3"></a>
{{< figure src="/ox-hugo/rankers98_pole_zero_medium_freq.png" caption="Figure 34: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with medium frequency resonance" >}}
<a id="orgc43335a"></a>
<a id="org5e8c4cc"></a>
{{< figure src="/ox-hugo/rankers98_pole_zero_high_freq.png" caption="Figure 35: Open Loop FRF of type \"-2 slope / pole / zero / -2 slope\" with high frequency resonance" >}}
@ -992,15 +994,15 @@ These conclusions may differ for different mass ratio \\(\alpha\\).
#### Actuator Flexibility {#actuator-flexibility}
Figure [36](#orgeddfc41) shows the schematic representation of a system with a certain compliance between the motor and the load.
Figure [36](#org76618cc) shows the schematic representation of a system with a certain compliance between the motor and the load.
<a id="orgeddfc41"></a>
<a id="org76618cc"></a>
{{< figure src="/ox-hugo/rankers98_2dof_actuator_flexibility.png" caption="Figure 36: Servo system with actuator flexibility - Schematic representation" >}}
The corresponding modes are shown in Figure [37](#org28f6c88).
The corresponding modes are shown in Figure [37](#org22997f9).
<a id="org28f6c88"></a>
<a id="org22997f9"></a>
{{< figure src="/ox-hugo/rankers98_2dof_modes_act_flex.png" caption="Figure 37: Servo System with Actuator Flexibility - Modes" >}}
@ -1013,9 +1015,9 @@ The following transfer function must be considered:
\end{align}
with \\(\alpha = m\_2/m\_1\\) (mass ratio) relates the "mass" of the additional modal contribution to the mass of the rigid body motion.
The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" (Figure [38](#orgeb05620)).
The resulting FRF exhibit a "-2 slope / zero / pole / -2 slope" (Figure [38](#orgd26d173)).
<a id="orgeb05620"></a>
<a id="orgd26d173"></a>
{{< figure src="/ox-hugo/rankers98_2dof_act_flex_frf.png" caption="Figure 38: Mechanical FRF of a system with actuator flexibility and position measurement at motor" >}}
@ -1045,9 +1047,9 @@ Now we are interested by the following transfer function:
\frac{x\_2}{F\_{\text{servo}}} = = \frac{1}{m\_1 + m\_2} \left( \frac{1}{s^2} - \frac{1}{s^2 + \omega\_2^2} \right)
\end{equation}
The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" (Figure [39](#org476f71c)).
The mass ratio \\(\alpha\\) equal -1, and thus the FRF will be of type "-2 slope / pole / -4 slope" (Figure [39](#org6fb421e)).
<a id="org476f71c"></a>
<a id="org6fb421e"></a>
{{< figure src="/ox-hugo/rankers98_2dof_act_flex_meas_load_frf.png" caption="Figure 39: FRF \\(k\_p \cdot (x\_{\text{servo}}/F\_{\text{servo}})\\) of a system with actuator flexibility and position measurement at the load" >}}
@ -1063,26 +1065,26 @@ Guideline in presence of actuator flexibility with measurement at the load posit
#### Guiding System Flexibility {#guiding-system-flexibility}
Here, the influence of a limited guiding stiffness (Figure [40](#org5654036)) on the FRF of such an actuator system will be analyzed.
Here, the influence of a limited guiding stiffness (Figure [40](#orgdc96e95)) on the FRF of such an actuator system will be analyzed.
The servo force \\(F\_{\text{servo}}\\) acts at a certain distance \\(a\_F\\) with respect to the center of gravity, and the servo position is measured at a distance \\(a\_s\\) with respect to the center of gravity.
Due to the symmetry of the system, the Y motion is decoupled from the X and \\(\phi\\) motions and can therefore be omitted in this analysis.
<a id="org5654036"></a>
<a id="orgdc96e95"></a>
{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex.png" caption="Figure 40: 2DoF rigid body model of actuator with flexibility of the guiding system" >}}
Considering the two relevant modes (Figures [41](#org18f041e) and [42](#orgb340007)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
Considering the two relevant modes (Figures [41](#org4e90a02) and [42](#org61da72f)), the resulting transfer function \\(x\_{\text{servo}}/F\_{\text{servo}}\\) can be constructed from the contributions of the individual modes:
\begin{equation}
\frac{x\_{\text{servo}}}{F\_{\text{servo}}} = \frac{1}{ms^2} + \frac{a\_s a\_F}{Js^2 + 2cb^2}
\end{equation}
<a id="org18f041e"></a>
<a id="org4e90a02"></a>
{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex_x_mode.png" caption="Figure 41: Graphical representation of desired X-motion" >}}
<a id="orgb340007"></a>
<a id="org61da72f"></a>
{{< figure src="/ox-hugo/rankers98_2dof_guiding_flex_rock_mode.png" caption="Figure 42: Graphical representation of parasitic rocking mode" >}}
@ -1121,9 +1123,9 @@ As this point, the resonance will not be present in the FRF.
#### Limited Mass and Stiffness of Stationary Machine Part {#limited-mass-and-stiffness-of-stationary-machine-part}
Figure [43](#org56e126a) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
Figure [43](#orgfdfcf1c) shows a simple model of a translational direct drive motion on a frame with limited mass and stiffness, and in which the control system operates on the measured position \\(x\_{\text{servo}} = x\_2 - x\_1\\).
<a id="org56e126a"></a>
<a id="orgfdfcf1c"></a>
{{< figure src="/ox-hugo/rankers98_frame_dynamics_2dof.png" caption="Figure 43: Model of a servo system including frame dynamics" >}}
@ -1158,9 +1160,9 @@ Guidelines regarding frame motion:
<div class="sum">
<div></div>
The amount of contribution of a certain mode (Figure [44](#orge7882a4)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
The amount of contribution of a certain mode (Figure [44](#org5c18c54)) to the open loop response and its interaction with the desired motion is determined by the modal mass and stiffness, but also by the location of the driving force and the location of the response DoF.
<a id="orge7882a4"></a>
<a id="org5c18c54"></a>
{{< figure src="/ox-hugo/rankers98_mode_representation_guideline.png" caption="Figure 44: Graphical representation of mode i" >}}
@ -1187,26 +1189,26 @@ If such a modification is not required and the modes are not excited by some oth
### Steps in a Modelling Activity {#steps-in-a-modelling-activity}
One can distinguish at least four steps in any modelling activity (Figure [45](#org60f12a2)).
One can distinguish at least four steps in any modelling activity (Figure [45](#org48b9139)).
<a id="org60f12a2"></a>
<a id="org48b9139"></a>
{{< figure src="/ox-hugo/rankers98_steps_modelling.png" caption="Figure 45: Steps in a modelling activity" >}}
1. The first step consists of a translation of the real structure or initial design drawing of a structure in a **physical model**.
Such a physical model is a simplification of the reality, but contains all relations that are considered to be important to describe the investigated phenomenon.
This step requires experience and engineering judgment in order to determine which simplifications are valid.
See for example Figure [46](#orgf77c197).
See for example Figure [46](#org2abb458).
2. Once a physical model has been derived, the second step consists of translating this physical model into a **mathematical model**.
The real world is now represented by a set of differential equations.
This step is fairly straightforward, because it is based and existing approaches and rules (Example in Figure [46](#orgf77c197)).
This step is fairly straightforward, because it is based and existing approaches and rules (Example in Figure [46](#org2abb458)).
3. The third step consists of actuator **simulation run**, the outcome of which is the value of some quantity (for instance stress of some part, resonance frequency, FRF, etc.).
4. The final step is the **interpretation** of results.
Here, the calculated results and previously defined specifications are compared.
On the basis of this comparison, design decisions are taken.
It is important to realize that the design decisions taken in this step are the actual outcome of the modelling process.
<a id="orgf77c197"></a>
<a id="org2abb458"></a>
{{< figure src="/ox-hugo/rankers98_illustration_first_two_steps.png" caption="Figure 46: Illustration of the first two steps in the modelling process" >}}
@ -1256,9 +1258,9 @@ Therefore, computer simulations should be regarded as a means to guide the desig
<div class="exampl">
<div></div>
This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece (Figure [47](#orgc7e396f)).
This three step modelling approach is now illustrated by the example of a fast and accurate pattern generator in which an optical unit has to move in X and Y directions with respect to a work piece (Figure [47](#org288ca2c)).
<a id="orgc7e396f"></a>
<a id="org288ca2c"></a>
{{< figure src="/ox-hugo/rankers98_pattern_generator.png" caption="Figure 47: The basic elements of the pattern generator" >}}
@ -1275,11 +1277,11 @@ Based on the required throughput of the machine, an acceleration level of \\(1m/
<div></div>
One of the most crucial step in the modelling process is the **definition of proper criteria on the basis of which the simulation results can be judged**.
In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** (Figure [48](#orgf322acd)).
In most cases, this implies that functional system-specifications in combination with **assumed imperfections and disturbances** need to be translated into **dynamics and control specifications** (Figure [48](#orge17647e)).
</div>
<a id="orgf322acd"></a>
<a id="orge17647e"></a>
{{< figure src="/ox-hugo/rankers98_system_performance_spec.png" caption="Figure 48: System performance specifications need to be translated into criteria on the basis of which simulation results can be judged" >}}
@ -1318,16 +1320,16 @@ In this stage, the designer only has a rough idea about the outlines of the mach
<div class="exampl">
<div></div>
One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions (Figure [49](#orgda306b4)).
One of the potential concepts for this machine consists of a stationary work piece with an optical unit that moves in both the X and Y directions (Figure [49](#org45d70da)).
In the X direction, two driving forces are applied to the slides, whereas the position is measured by two linear encoders mounted between the slide and the granite frame.
<a id="orgda306b4"></a>
<a id="org45d70da"></a>
{{< figure src="/ox-hugo/rankers98_pattern_generator_concept.png" caption="Figure 49: One of the possible concepts of the pattern generator" >}}
In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility (Figure [50](#orgdefe8fe)).
In this stage of the design, a simple model of the dynamic effects in the X direction could consist of the base, the slides, the guiding rail, the optical housing and intermediate flexibility (Figure [50](#orgac882b7)).
<a id="orgdefe8fe"></a>
<a id="orgac882b7"></a>
{{< figure src="/ox-hugo/rankers98_concept_1dof_evaluation.png" caption="Figure 50: Simple 1D model for the analysis of the dynamic behaviour in the X direction" >}}
@ -1351,9 +1353,9 @@ Typically, such a model contains 5-10 rigid bodies connected by suitable connect
<div class="exampl">
<div></div>
Figure [51](#orgd35f0b3) shows such a 3D model of a different concept for the pattern generator.
Figure [51](#org075947e) shows such a 3D model of a different concept for the pattern generator.
<a id="orgd35f0b3"></a>
<a id="org075947e"></a>
{{< figure src="/ox-hugo/rankers98_pattern_generator_rigid_body.png" caption="Figure 51: Rigid body model of a concept based on a movement of the work-piece in X direction, and a movement of the optical unit in Y direction" >}}
@ -1401,11 +1403,11 @@ Due to the complexity of the structures it is normally not very practical to bui
- The resulting mass and stiffness matrices can easily have many thousands degrees of freedom, which puts high demands on the required computing capacity.
A technique which overcomes these disadvantages is the co-called **sub-structuring technique**.
In this approach, illustrated in Figure [52](#org475956c), the system is divided into substructures or components, which are analyzed separately.
In this approach, illustrated in Figure [52](#orgc59fe2e), the system is divided into substructures or components, which are analyzed separately.
Then, the (reduced) models of the components are assembled to form the overall system.
By doing so, the size of the final system model is significantly reduced.
<a id="org475956c"></a>
<a id="orgc59fe2e"></a>
{{< figure src="/ox-hugo/rankers98_substructuring_technique.png" caption="Figure 52: Steps in the creation of an overall system model based on detailed FE models of the components" >}}
@ -1449,4 +1451,4 @@ It has static solution capacity, and the frequency of the highest fixed-interfac
## Bibliography {#bibliography}
<a id="org5830c60"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.
<a id="orgefff92e"></a>Rankers, Adrian Mathias. 1998. “Machine Dynamics in Mechatronic Systems: An Engineering Approach.” University of Twente.

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@ -2,6 +2,8 @@
title = "Modeling and robust adaptive tracking control of a planar precision positioning system"
author = ["Thomas Dehaeze"]
draft = true
ref_author = "Treichel, K."
ref_year = 2017
+++
Tags
@ -9,7 +11,7 @@ Tags
Reference
: ([Treichel 2017](#org1662bdf))
: ([Treichel 2017](#org35de13e))
Author(s)
: Treichel, K.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org1662bdf"></a>Treichel, Kai. 2017. “Modeling and Robust Adaptive Tracking Control of a Planar Precision Positioning System.” Ilmenau University of Technology.
<a id="org35de13e"></a>Treichel, Kai. 2017. “Modeling and Robust Adaptive Tracking Control of a Planar Precision Positioning System.” Ilmenau University of Technology.

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@ -2,6 +2,8 @@
title = "Dynamic modeling, experimental identification, and active vibration control design of a smart parallel manipulator."
author = ["Thomas Dehaeze"]
draft = true
ref_author = "Wang, X."
ref_year = 2007
+++
Tags
@ -9,7 +11,7 @@ Tags
Reference
: ([Wang 2007](#org006aaaa))
: ([Wang 2007](#orgdaa802c))
Author(s)
: Wang, X.
@ -20,4 +22,4 @@ Year
## Bibliography {#bibliography}
<a id="org006aaaa"></a>Wang, Xiaoyun. 2007. “Dynamic Modeling, Experimental Identification, and Active Vibration Control Design of a Smart Parallel Manipulator.” University of Toronto.
<a id="orgdaa802c"></a>Wang, Xiaoyun. 2007. “Dynamic Modeling, Experimental Identification, and Active Vibration Control Design of a Smart Parallel Manipulator.” University of Toronto.

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@ -2,13 +2,15 @@
title = "Element and system design for active and passive vibration isolation"
author = ["Thomas Dehaeze"]
draft = false
ref_author = "Zuo, L."
ref_year = 2004
+++
Tags
: [Vibration Isolation]({{<relref "vibration_isolation.md#" >}})
Reference
: ([Zuo 2004](#org700ab89))
: ([Zuo 2004](#org05cd1c8))
Author(s)
: Zuo, L.
@ -16,20 +18,6 @@ Author(s)
Year
: 2004
<div style="display: none;">
\(
\newcommand{\eatLabel}[2]{}
\newenvironment{subequations}{\eatLabel}{}
\)
</div>
\begin{equation}
\begin{align}
\left[ H\_{xf}(\omega) \right]\_{n \times n} &= \left[ S\_{x^\prime v}(\omega) \right]\_{n \times n} \left[ S\_{f^\prime v}(\omega) \right]\_{n \times n}^{-1} \\\\\\
\left[ H\_{xf}(\omega) \right]\_{n \times n} &= \left[ S\_{f^\prime f^\prime}(\omega) \right]\_{n \times n}^{-1} \left[ S\_{x^\prime f^\prime}(\omega) \right]\_{n \times n}
\end{align}
\end{equation}
> Vibration isolation systems can have various system architectures.
> When we configure an active isolation system, we can use compliant actuators (such as voice coils) or stiff actuators (such as PZT stacks).
> We also need to consider how to **combine the active actuation with passive elements**: we can place the actuator in parallel or in series with the passive elements.
@ -40,19 +28,19 @@ Year
> They found that coupling from flexible modes is much smaller than in soft active mounts in the load (force) feedback.
> Note that reaction force actuators can also work with soft mounts or hard mounts.
<a id="org44c9181"></a>
<a id="orgdaec88b"></a>
{{< figure src="/ox-hugo/zuo04_piezo_spring_series.png" caption="Figure 1: PZT actuator and spring in series" >}}
<a id="org631f004"></a>
<a id="org84417be"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_spring_parallel.png" caption="Figure 2: Voice coil actuator and spring in parallel" >}}
<a id="orgc4102de"></a>
<a id="orge3c9205"></a>
{{< figure src="/ox-hugo/zuo04_piezo_plant.png" caption="Figure 3: Transmission from PZT voltage to geophone output" >}}
<a id="org063d6bb"></a>
<a id="orge26e6a6"></a>
{{< figure src="/ox-hugo/zuo04_voice_coil_plant.png" caption="Figure 4: Transmission from voice coil voltage to geophone output" >}}
@ -60,4 +48,4 @@ Year
## Bibliography {#bibliography}
<a id="org700ab89"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.
<a id="org05cd1c8"></a>Zuo, Lei. 2004. “Element and System Design for Active and Passive Vibration Isolation.” Massachusetts Institute of Technology.

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