@@ -127,22 +127,22 @@ Tables [3](#org02d84e8), [2](#table--tab:notations-eigen-vectors-values) and [3]
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
-
+
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](#org02d84e8) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
+Figure [3](#orgc82b5d8) 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\\).
-
+
{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
-([Miu 1993](#org39eead7)) 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.
+([Miu 1993](#org849cfe4)) 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.
@@ -151,12 +151,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
-
+
## Modal Analysis {#modal-analysis}
-
+
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@@ -196,9 +196,9 @@ 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](#org0c2921d) 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](#orgebf4457) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
-
+
{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
@@ -297,17 +297,17 @@ One then find:
\end{bmatrix}
\end{equation}
-Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc90fe3a), [6](#orgfd8222c) and [7](#orgaf9cc36).
+Virtual interpretation of the eigenvectors are shown in Figures [5](#org520a99d), [6](#org722a9ff) and [7](#org9e25b28).
-
+
{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
-
+
{{< 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_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
@@ -346,9 +346,9 @@ 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](#orgf9a2963).
+This procedure is schematically shown in Figure [8](#orgfbabf08).
-
+
{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
@@ -696,7 +696,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
-
+
The procedure to obtain the frequency response from a modal form is as follow:
@@ -704,9 +704,9 @@ 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](#org48b68a4).
+This will be applied to the model shown in Figure [9](#orge102983).
-
+
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
@@ -888,9 +888,9 @@ Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows tha
-Figure [10](#org87763b9) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
+Figure [10](#org3024448) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
-
+
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@@ -899,16 +899,16 @@ The zeros for SISO transfer functions are the roots of the numerator, however, f
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
-
+
### Introduction {#introduction}
-In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orga66d597).
+In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#org2292476).
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.
-
+
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
@@ -987,7 +987,7 @@ 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}
-
+
### Model Description {#model-description}
@@ -1001,25 +1001,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
-
+
-In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org94e126d)).
+In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org143e4e8)).
### Actuator Description {#actuator-description}
-
+
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: 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](#org4a20950)).
+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](#orgc294fc5)).
-
+
{{< 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" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
-Figure [13](#org4a20950) shows the nodes used for the reduced Matlab model.
+Figure [13](#orgc294fc5) 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.
@@ -1087,7 +1087,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
-
+
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@@ -1202,14 +1202,14 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
-
+
-In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org1453e17)).
+In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org7003388)).
### Actuator Description {#actuator-description}
-
+
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
@@ -1231,9 +1231,9 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
-In Figure [15](#orge94bde1) are shown the principal nodes used for the model.
+In Figure [15](#org472d510) are shown the principal nodes used for the model.
-
+
{{< 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" >}}
@@ -1352,11 +1352,11 @@ And we note:
G = zn * Gp;
```
-
+
{{< 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_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
@@ -1454,13 +1454,13 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
-Let's plot the frequency of the modes (Figure [18](#orga04e866)).
+Let's plot the frequency of the modes (Figure [18](#org6e52a4a)).
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
@@ -1529,7 +1529,7 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
@@ -1873,7 +1873,7 @@ Then, we compute the controllability and observability gramians.
And we plot the diagonal terms
-
+
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
@@ -1891,7 +1891,7 @@ We use `balreal` to rank oscillatory states.
[G_b, G, T, Ti] = balreal(G_m);
```
-
+
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
@@ -2134,8 +2134,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
```
+
## Bibliography {#bibliography}
-
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
+
Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
-
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
+
Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
diff --git a/content/book/leach14_fundam_princ_engin_nanom.md b/content/book/leach14_fundam_princ_engin_nanom.md
index 2af7db2..2c9e6d7 100644
--- a/content/book/leach14_fundam_princ_engin_nanom.md
+++ b/content/book/leach14_fundam_princ_engin_nanom.md
@@ -8,7 +8,7 @@ Tags
: [Metrology]({{< relref "metrology" >}})
Reference
-: ([Leach 2014](#orgc132434))
+: ([Leach 2014](#org023e404))
Author(s)
: Leach, R.
@@ -87,6 +87,7 @@ The measurement of angles is then relative.
This type of angular interferometer is used to measure small angles (less than \\(10deg\\)).
+
## Bibliography {#bibliography}
-
Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier.
.
+Leach, Richard. 2014. _Fundamental Principles of Engineering Nanometrology_. Elsevier. .
diff --git a/content/book/leach18_basic_precis_engin_edition.md b/content/book/leach18_basic_precis_engin_edition.md
index 397cb91..356733f 100644
--- a/content/book/leach18_basic_precis_engin_edition.md
+++ b/content/book/leach18_basic_precis_engin_edition.md
@@ -8,7 +8,7 @@ Tags
: [Precision Engineering]({{< relref "precision_engineering" >}})
Reference
-: ([Leach and Smith 2018](#org50ae2e1))
+: ([Leach and Smith 2018](#orgdc805b5))
Author(s)
: Leach, R., & Smith, S. T.
@@ -17,6 +17,7 @@ Year
: 2018
+
## Bibliography {#bibliography}
-Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
+Leach, Richard, and Stuart T. Smith. 2018. _Basics of Precision Engineering - 1st Edition_. CRC Press.
diff --git a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
index 5c663aa..83b8960 100644
--- a/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
+++ b/content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
@@ -8,7 +8,7 @@ Tags
: [Vibration Isolation]({{< relref "vibration_isolation" >}}), [Reference Books]({{< relref "reference_books" >}}), [Stewart Platforms]({{< relref "stewart_platforms" >}}), [HAC-HAC]({{< relref "hac_hac" >}})
Reference
-: ([Preumont 2018](#orgd83c544))
+: ([Preumont 2018](#org29acb4a))
Author(s)
: Preumont, A.
@@ -61,11 +61,11 @@ There are two radically different approached to disturbance rejection: feedback
#### Feedback {#feedback}
-
+
{{< figure src="/ox-hugo/preumont18_classical_feedback_small.png" caption="Figure 1: Principle of feedback control" >}}
-The principle of feedback is represented on figure [1](#orgda21dda). 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](#orga09f785). 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**.
@@ -87,12 +87,12 @@ The objective is to control a variable \\(y\\) to a desired value \\(r\\) in spi
#### Feedforward {#feedforward}
-
+
{{< figure src="/ox-hugo/preumont18_feedforward_adaptative.png" caption="Figure 2: 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](#orgf75c047).
+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](#org57ee378).
The filter coefficients are adapted in such a way that the error signal at one or several critical points is minimized.
@@ -123,11 +123,11 @@ 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}
-
+
{{< figure src="/ox-hugo/preumont18_design_steps.png" caption="Figure 3: The various steps of the design" >}}
-The various steps of the design of a controlled structure are shown in figure [3](#org1939c0d).
+The various steps of the design of a controlled structure are shown in figure [3](#org8ea735d).
The **starting point** is:
@@ -154,14 +154,14 @@ 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](#orgaf01f6c)):
+From the block diagram of the control system (figure [4](#orga135390)):
\begin{align\*}
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\*}
-
+
{{< figure src="/ox-hugo/preumont18_general_plant.png" caption="Figure 4: Block diagram of the control System" >}}
@@ -186,12 +186,12 @@ 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](#org7ddcf2a).
+A typical plot of \\(\sigma\_z(\omega)\\) is shown figure [5](#orge835b98).
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.
-
+
{{< figure src="/ox-hugo/preumont18_cas_plot.png" caption="Figure 5: Error budget distribution in OL and CL for increasing gains" >}}
@@ -398,11 +398,11 @@ With:
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
\end{equation}
-
+
{{< 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\\)" >}}
-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](#orga618336)).
+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](#orga21e5bb)).
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 \\]
@@ -441,9 +441,9 @@ 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](#orgecdb253)).
+\\(G\_{kk}\\) is a monotonously increasing function of \\(\omega\\) (figure [7](#org4ad84e0)).
-
+
{{< figure src="/ox-hugo/preumont18_collocated_control_frf.png" caption="Figure 7: Open-Loop FRF of an undamped structure with collocated actuator/sensor pair" >}}
@@ -457,9 +457,9 @@ For lightly damped structure, the poles and zeros are just moved a little bit in