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content/book/hatch00_vibrat_matlab_ansys.md
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@@ -23,14 +23,14 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
## Introduction {#introduction}
<span class="org-target" id="org-target--sec:introduction"></span>
<span class="org-target" id="org-target--sec-introduction"></span>
The main goal of this book is to show how to take results of large dynamic finite element models and build small Matlab state space dynamic mechanical models for use in control system models.
### Modal Analysis {#modal-analysis}
The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
The diagram in [Figure 1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
<div class="important">
@@ -58,7 +58,7 @@ Because finite element models usually have a very large number of states, an imp
<div class="important">
Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
[Figure 2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
- start with the finite element model
- compute the eigenvalues and eigenvectors (as many as dof in the model)
@@ -78,11 +78,11 @@ Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the s
### Notations {#notations}
Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
[Figure 3](#figure--fig:hatch00-n-dof-zeros), [Table 2](#table--tab:notations-eigen-vectors-values) and [Table 3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
<a id="table--tab:notations-modes-nodes"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a></span>:
<span class="table-number"><a href="#table--tab:notations-modes-nodes">Table 1</a>:</span>
Notation for the modes and nodes
</div>
@@ -97,7 +97,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
<a id="table--tab:notations-eigen-vectors-values"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a></span>:
<span class="table-number"><a href="#table--tab:notations-eigen-vectors-values">Table 2</a>:</span>
Notation for the dofs, eigenvectors and eigenvalues
</div>
@@ -112,7 +112,7 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
<a id="table--tab:notations-stiffness-mass"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a></span>:
<span class="table-number"><a href="#table--tab:notations-stiffness-mass">Table 3</a>:</span>
Notation for the mass and stiffness matrices
</div>
@@ -127,12 +127,12 @@ Tables [3](#figure--fig:hatch00-n-dof-zeros), [2](#table--tab:notations-eigen-ve
## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
<span class="org-target" id="org-target--sec:zeros_siso_systems"></span>
<span class="org-target" id="org-target--sec-zeros-siso-systems"></span>
The origin and influence of poles are clear: they represent the resonant frequencies of the system, and for each resonance frequency, a mode shape can be defined to describe the motion at that frequency.
We here which to give an intuitive understanding for **when to expect zeros in SISO mechanical systems** and **how to predict the frequencies at which they will occur**.
Figure [3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
[Figure 3](#figure--fig:hatch00-n-dof-zeros) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
<a id="figure--fig:hatch00-n-dof-zeros"></a>
@@ -150,12 +150,12 @@ The resonances of the "overhanging appendages" of the constrained system create
## State Space Analysis {#state-space-analysis}
<span class="org-target" id="org-target--sec:state_space_analysis"></span>
<span class="org-target" id="org-target--sec-state-space-analysis"></span>
## Modal Analysis {#modal-analysis}
<span class="org-target" id="org-target--sec:modal_analysis"></span>
<span class="org-target" id="org-target--sec-modal-analysis"></span>
Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
@@ -193,7 +193,7 @@ 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](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
Let's consider the model shown in [Figure 4](#figure--fig:hatch00-undamped-tdof-model) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
<a id="figure--fig:hatch00-undamped-tdof-model"></a>
@@ -293,7 +293,7 @@ One then find:
\end{bmatrix}
\end{equation}
Virtual interpretation of the eigenvectors are shown in Figures [5](#figure--fig:hatch00-tdof-mode-1), [6](#figure--fig:hatch00-tdof-mode-2) and [7](#figure--fig:hatch00-tdof-mode-3).
Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:hatch00-tdof-mode-1), [Figure 6](#figure--fig:hatch00-tdof-mode-2) and [Figure 7](#figure--fig:hatch00-tdof-mode-3).
<a id="figure--fig:hatch00-tdof-mode-1"></a>
@@ -341,7 +341,7 @@ 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](#figure--fig:hatch00-schematic-modal-solution).
This procedure is schematically shown in [Figure 8](#figure--fig:hatch00-schematic-modal-solution).
<a id="figure--fig:hatch00-schematic-modal-solution"></a>
@@ -687,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
## Frequency Response: Modal Form {#frequency-response-modal-form}
<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
<span class="org-target" id="org-target--sec-frequency-response-modal-form"></span>
The procedure to obtain the frequency response from a modal form is as follow:
@@ -695,7 +695,7 @@ 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](#figure--fig:hatch00-tdof-model).
This will be applied to the model shown in [Figure 9](#figure--fig:hatch00-tdof-model).
<a id="figure--fig:hatch00-tdof-model"></a>
@@ -877,7 +877,7 @@ Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general
</div>
Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
[Figure 10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
<a id="figure--fig:hatch00-z11-tf-example"></a>
@@ -888,12 +888,12 @@ 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}
<span class="org-target" id="org-target--sec:siso_state_space"></span>
<span class="org-target" id="org-target--sec-siso-state-space"></span>
### Introduction {#introduction}
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in [Figure 11](#figure--fig:hatch00-cantilever-beam).
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
@@ -976,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
<span class="org-target" id="org-target--sec:ground_acceleration"></span>
<span class="org-target" id="org-target--sec-ground-acceleration"></span>
### Model Description {#model-description}
@@ -990,9 +990,9 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
<span class="org-target" id="org-target--sec-siso-disk-drive"></span>
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-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](#figure--fig:hatch00-disk-drive-siso-model)).
### Actuator Description {#actuator-description}
@@ -1001,14 +1001,14 @@ In this section we wish to extract a SISO state space model from a Finite Elemen
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident ([Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
<a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
[Figure 13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
The arrows at the nodes indicate the direction of forces.
@@ -1074,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
## Balanced Reduction {#balanced-reduction}
<span class="org-target" id="org-target--sec:balanced_reduction"></span>
<span class="org-target" id="org-target--sec-balanced-reduction"></span>
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
@@ -1189,9 +1189,9 @@ The **states to be kept are the states with the largest diagonal terms**.
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
<span class="org-target" id="org-target--sec-mimo-disk-drive"></span>
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
In this section, a MIMO two-stage actuator model is derived from a finite element model ([Figure 14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
### Actuator Description {#actuator-description}
@@ -1217,7 +1217,7 @@ Since the same forces are being applied to both piezo elements, they represent t
### Ansys Model Description {#ansys-model-description}
In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
In [Figure 15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
<a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
@@ -1440,7 +1440,7 @@ State Space Model
### Simple mode truncation {#simple-mode-truncation}
Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
Let's plot the frequency of the modes ([Figure 18](#figure--fig:hatch00-cant-beam-modes-freq)).
<a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
@@ -2123,6 +2123,6 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
## Bibliography {#bibliography}
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using MATLAB and ANSYS</i>. CRC Press.</div>
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
</div>