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@@ -21,9 +21,7 @@ Year
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Matlab Code form the book is available [here](https://in.mathworks.com/matlabcentral/fileexchange/2186-vibration-simulation-using-matlab-and-ansys).
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## Introduction {#introduction}
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<span class="org-target" id="org-target--sec-introduction"></span>
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## Introduction {#sec:introduction}
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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.
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@@ -125,9 +123,8 @@ Because finite element models usually have a very large number of states, an imp
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| Principal Force Vector | \\(\bm{F}\_p\\) | \\(m \times 1\\) |
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## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
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## Zeros in SISO Mechanical Systems {#sec:zeros_siso_systems}
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<span class="org-target" id="org-target--sec-zeros-siso-systems"></span>
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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.
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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**.
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@@ -148,14 +145,10 @@ The resonances of the "overhanging appendages" of the constrained system create
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</div>
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## State Space Analysis {#state-space-analysis}
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<span class="org-target" id="org-target--sec-state-space-analysis"></span>
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## State Space Analysis {#sec:state_space_analysis}
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## Modal Analysis {#modal-analysis}
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<span class="org-target" id="org-target--sec-modal-analysis"></span>
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## Modal Analysis {#sec:modal_analysis}
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Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
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@@ -252,7 +245,7 @@ where:
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#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
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Re-injecting normal modes \ref{eq:principal\_mode} into the equation of motion \ref{eq:tdof\_eom} gives the eigenvalue problem:
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Re-injecting normal modes \eqref{eq:principal\_mode} into the equation of motion \eqref{eq:tdof\_eom} gives the eigenvalue problem:
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\begin{equation}
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(\bm{k} - \omega\_i^2 \bm{m}) \bm{z}\_{mi} = 0
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@@ -310,7 +303,7 @@ Virtual interpretation of the eigenvectors are shown in [Figure 5](#figure--fig:
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#### Modal Matrix {#modal-matrix}
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The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \ref{eq:modal\_matrix}
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The modal matrix is an \\(n \times m\\) matrix with columns corresponding to the \\(m\\) system eigenvectors as shown in Eq. \eqref{eq:modal\_matrix}
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\begin{equation}
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\bm{z}\_m = \begin{bmatrix}
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@@ -685,9 +678,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
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## Modal Analysis: State Space Form {#modal-analysis-state-space-form}
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## Frequency Response: Modal Form {#frequency-response-modal-form}
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<span class="org-target" id="org-target--sec-frequency-response-modal-form"></span>
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## Frequency Response: Modal Form {#sec:frequency_response_modal_form}
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The procedure to obtain the frequency response from a modal form is as follow:
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@@ -873,7 +864,7 @@ If modes have some damping:
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\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
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\end{equation}
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Equations \ref{eq:general\_add\_tf} and \ref{eq:general\_add\_tf\_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
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Equations \eqref{eq:general\_add\_tf} and \eqref{eq:general\_add\_tf\_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
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</div>
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@@ -886,9 +877,7 @@ Equations \ref{eq:general\_add\_tf} and \ref{eq:general\_add\_tf\_damp} shows th
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The zeros for SISO transfer functions are the roots of the numerator, however, from modal analysis we can see that the zeros arise when modes combine with appropriate phase such that the resulting motion is null.
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## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
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<span class="org-target" id="org-target--sec-siso-state-space"></span>
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## SISO State Space Matlab Model from ANSYS Model {#sec:siso_state_space}
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### Introduction {#introduction}
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@@ -974,9 +963,7 @@ However, the `truncate` method has the advantage that it does not exhibit the un
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If sorting of DC gain values is performed prior to the `truncate` operation, the system DC gain error may be acceptable while maintaining better high frequency performance.
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## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
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<span class="org-target" id="org-target--sec-ground-acceleration"></span>
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## Ground Acceleration Matlab Model From ANSYS Model {#sec:ground_acceleration}
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### Model Description {#model-description}
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@@ -988,9 +975,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
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### Matlab State Space Model from Eigenvalue Run {#matlab-state-space-model-from-eigenvalue-run}
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## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
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<span class="org-target" id="org-target--sec-siso-disk-drive"></span>
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## SISO Disk Drive Actuator Model {#sec:siso_disk_drive}
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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)).
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@@ -1072,9 +1057,7 @@ The dimensions of this operation are \\((1 \times n) \times (n \times m) = (1 \t
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From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\).
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## Balanced Reduction {#balanced-reduction}
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<span class="org-target" id="org-target--sec-balanced-reduction"></span>
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## Balanced Reduction {#sec:balanced_reduction}
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In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
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@@ -1187,9 +1170,7 @@ Because the controllability and observability gramians are identical, there is n
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The **states to be kept are the states with the largest diagonal terms**.
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## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
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<span class="org-target" id="org-target--sec-mimo-disk-drive"></span>
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## MIMO Two Stage Actuator Model {#sec:mimo_disk_drive}
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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)).
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