Update Content - 2022-03-15
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title = "Vibration Simulation using Matlab and ANSYS"
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author = ["Thomas Dehaeze"]
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author = ["Dehaeze Thomas"]
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description = "Nice techniques to analyze resonant systems with Ansys and Matlab."
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keywords = ["Modal Analysis", "FEM"]
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draft = false
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+++
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Tags
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: [Finite Element Model]({{< relref "finite_element_model" >}})
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: [Finite Element Model]({{< relref "finite_element_model.md" >}})
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Reference
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: ([Hatch 2000](#org4036e02))
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: (<a href="#citeproc_bib_item_1">Hatch 2000</a>)
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Author(s)
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: Hatch, M. R.
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@@ -23,17 +23,16 @@ Matlab Code form the book is available [here](https://in.mathworks.com/matlabcen
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## Introduction {#introduction}
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<a id="org96f8e54"></a>
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<span class="org-target" id="org-target--sec:introduction"></span>
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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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### Modal Analysis {#modal-analysis}
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The diagram in Figure [1](#org97c03ca) shows the methodology for analyzing a lightly damped structure using normal modes.
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The diagram in Figure [1](#figure--fig:hatch00-modal-analysis-flowchart) shows the methodology for analyzing a lightly damped structure using normal modes.
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<div class="important">
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<div></div>
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The steps are:
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@@ -48,9 +47,9 @@ The steps are:
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</div>
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<a id="org97c03ca"></a>
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<a id="figure--fig:hatch00-modal-analysis-flowchart"></a>
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{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="Figure 1: Modal analysis method flowchart" >}}
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{{< figure src="/ox-hugo/hatch00_modal_analysis_flowchart.png" caption="<span class=\"figure-number\">Figure 1: </span>Modal analysis method flowchart" >}}
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### Model Size Reduction {#model-size-reduction}
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@@ -58,9 +57,8 @@ The steps are:
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Because finite element models usually have a very large number of states, an important step is the reduction of the number of states while still providing correct responses for the forcing function input and desired output points.
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<div class="important">
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<div></div>
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Figure [2](#orgdbb9ffa) shows such process, the steps are:
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Figure [2](#figure--fig:hatch00-model-reduction-chart) shows such process, the steps are:
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- start with the finite element model
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- compute the eigenvalues and eigenvectors (as many as dof in the model)
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@@ -73,14 +71,14 @@ Figure [2](#orgdbb9ffa) shows such process, the steps are:
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</div>
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<a id="orgdbb9ffa"></a>
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<a id="figure--fig:hatch00-model-reduction-chart"></a>
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{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="Figure 2: Model size reduction flowchart" >}}
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{{< figure src="/ox-hugo/hatch00_model_reduction_chart.png" caption="<span class=\"figure-number\">Figure 2: </span>Model size reduction flowchart" >}}
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### Notations {#notations}
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Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3](#table--tab:notations-stiffness-mass) summarize the notations of this document.
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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.
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<a id="table--tab:notations-modes-nodes"></a>
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<div class="table-caption">
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@@ -129,22 +127,21 @@ Tables [3](#org4819d7f), [2](#table--tab:notations-eigen-vectors-values) and [3]
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## Zeros in SISO Mechanical Systems {#zeros-in-siso-mechanical-systems}
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<a id="orgca1a04d"></a>
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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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Figure [3](#org4819d7f) shows a series arrangement of masses and springs, with a total of \\(n\\) masses and \\(n+1\\) springs.
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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.
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The degrees of freedom are numbered from left to right, \\(z\_1\\) through \\(z\_n\\).
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<a id="org4819d7f"></a>
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<a id="figure--fig:hatch00-n-dof-zeros"></a>
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{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="Figure 3: n dof system showing various SISO input/output configurations" >}}
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{{< figure src="/ox-hugo/hatch00_n_dof_zeros.png" caption="<span class=\"figure-number\">Figure 3: </span>n dof system showing various SISO input/output configurations" >}}
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<div class="important">
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<div></div>
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([Miu 1993](#orgcda3e53)) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
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(<a href="#citeproc_bib_item_2">Miu 1993</a>) shows that the zeros of any particular transfer function are the poles of the constrained system to the left and/or right of the system defined by constraining the one or two dof's defining the transfer function.
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The resonances of the "overhanging appendages" of the constrained system create the zeros.
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@@ -153,17 +150,16 @@ The resonances of the "overhanging appendages" of the constrained system create
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## State Space Analysis {#state-space-analysis}
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<a id="orgc4e6e06"></a>
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<span class="org-target" id="org-target--sec:state_space_analysis"></span>
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## Modal Analysis {#modal-analysis}
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<a id="orge1af07f"></a>
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<span class="org-target" id="org-target--sec:modal_analysis"></span>
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Lightly damped structures are typically analyzed with the "normal mode" method described in this section.
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<div class="important">
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<div></div>
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The modal method allows one to replace the n-coupled differential equations with n-uncoupled equations, where each uncoupled equation represents the motion of the system for that mode of vibration.
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@@ -175,7 +171,6 @@ Heavily damped structures or structures which explicit damping elements, such as
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Thus, the present methods only works for lightly damped structures.
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<div class="important">
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<div></div>
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Summarizing the modal analysis method of analyzing linear mechanical systems and the benefits derived:
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@@ -198,34 +193,34 @@ Summarizing the modal analysis method of analyzing linear mechanical systems and
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#### Equation of Motion {#equation-of-motion}
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Let's consider the model shown in Figure [4](#orgde2ed42) with \\(k\_1 = k\_2 = k\\), \\(m\_1 = m\_2 = m\_3 = m\\) and \\(c\_1 = c\_2 = 0\\).
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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\\).
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<a id="orgde2ed42"></a>
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<a id="figure--fig:hatch00-undamped-tdof-model"></a>
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{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="Figure 4: Undamped tdof model" >}}
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{{< figure src="/ox-hugo/hatch00_undamped_tdof_model.png" caption="<span class=\"figure-number\">Figure 4: </span>Undamped tdof model" >}}
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The equations of motions are:
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\begin{equation}
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\begin{bmatrix}
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m & 0 & 0 \\\\\\
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0 & m & 0 \\\\\\
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m & 0 & 0 \\\\
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0 & m & 0 \\\\
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0 & 0 & m
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\end{bmatrix} \begin{bmatrix}
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\ddot{z}\_1 \\\\\\
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\ddot{z}\_2 \\\\\\
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\ddot{z}\_1 \\\\
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\ddot{z}\_2 \\\\
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\ddot{z}\_3
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\end{bmatrix} + \begin{bmatrix}
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k & -k & 0 \\\\\\
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-k & 2k & -k \\\\\\
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k & -k & 0 \\\\
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-k & 2k & -k \\\\
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0 & -k & k
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\end{bmatrix} \begin{bmatrix}
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z\_1 \\\\\\
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z\_2 \\\\\\
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z\_1 \\\\
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z\_2 \\\\
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z\_3
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\end{bmatrix} = \begin{bmatrix}
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0 \\\\\\
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0 \\\\\\
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0 \\\\
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0 \\\\
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0
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\end{bmatrix} \label{eq:tdof\_eom}
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\end{equation}
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@@ -236,7 +231,6 @@ The equations of motions are:
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Since the system is conservative (it has no damping), normal modes of vibration will exist.
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<div class="important">
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<div></div>
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Having normal modes means that at certain frequencies all points in the system will vibrate at the same frequency and in phase, i.e., **all points in the system will reach their minimum and maximum displacements at the same point in time**.
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@@ -258,7 +252,7 @@ where:
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#### Eigenvalues / Characteristic Equation {#eigenvalues-characteristic-equation}
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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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Re-injecting normal modes <eq:principal_mode> into the equation of motion <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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@@ -285,45 +279,45 @@ One then find:
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\begin{equation}
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\bm{z}\_1 = \begin{bmatrix}
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1 \\\\\\
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1 \\\\\\
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1 \\\\
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1 \\\\
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1
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\end{bmatrix}, \quad \bm{z}\_2 = \begin{bmatrix}
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1 \\\\\\
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0 \\\\\\
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1 \\\\
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0 \\\\
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-1
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\end{bmatrix}, \quad \bm{z}\_3 = \begin{bmatrix}
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1 \\\\\\
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-2 \\\\\\
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1 \\\\
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-2 \\\\
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1
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\end{bmatrix}
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\end{equation}
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Virtual interpretation of the eigenvectors are shown in Figures [5](#orgc0f09b0), [6](#org88e7153) and [7](#org8225e3c).
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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).
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<a id="orgc0f09b0"></a>
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<a id="figure--fig:hatch00-tdof-mode-1"></a>
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{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="Figure 5: Rigid-Body Mode, 0rad/s" >}}
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{{< figure src="/ox-hugo/hatch00_tdof_mode_1.png" caption="<span class=\"figure-number\">Figure 5: </span>Rigid-Body Mode, 0rad/s" >}}
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<a id="org88e7153"></a>
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<a id="figure--fig:hatch00-tdof-mode-2"></a>
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{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="Figure 6: Second Model, Middle Mass Stationary, 1rad/s" >}}
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{{< figure src="/ox-hugo/hatch00_tdof_mode_2.png" caption="<span class=\"figure-number\">Figure 6: </span>Second Model, Middle Mass Stationary, 1rad/s" >}}
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<a id="org8225e3c"></a>
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<a id="figure--fig:hatch00-tdof-mode-3"></a>
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{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="Figure 7: Third Mode, 1.7rad/s" >}}
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{{< figure src="/ox-hugo/hatch00_tdof_mode_3.png" caption="<span class=\"figure-number\">Figure 7: </span>Third Mode, 1.7rad/s" >}}
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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. \eqref{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. <eq:modal_matrix>
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\begin{equation}
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\bm{z}\_m = \begin{bmatrix}
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\bm{z}\_1 & \bm{z}\_2 & \bm{z}\_3
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\end{bmatrix} = \begin{bmatrix}
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z\_{m11} & z\_{m12} & z\_{m13} \\\\\\
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z\_{m21} & z\_{m22} & z\_{m23} \\\\\\
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z\_{m11} & z\_{m12} & z\_{m13} \\\\
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z\_{m21} & z\_{m22} & z\_{m23} \\\\
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z\_{m31} & z\_{m32} & z\_{m33}
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\end{bmatrix} \label{eq:modal\_matrix}
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\end{equation}
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@@ -339,7 +333,6 @@ It is thus useful to **transform the n-coupled second order differential equatio
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In linear algebra terms, the transformation from physical to principal coordinates is known as a **change of basis**.
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<div class="important">
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<div></div>
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There are many options for change of basis, but we will show that **when eigenvectors are used for the transformation, the principal coordinate system has a physical meaning: each of the uncoupled sdof systems represents the motion of a specific mode of vibration**.
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@@ -348,11 +341,11 @@ There are many options for change of basis, but we will show that **when eigenve
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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.
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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.
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This procedure is schematically shown in Figure [8](#org0f0be39).
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This procedure is schematically shown in Figure [8](#figure--fig:hatch00-schematic-modal-solution).
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<a id="org0f0be39"></a>
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<a id="figure--fig:hatch00-schematic-modal-solution"></a>
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{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="Figure 8: Roadmap for Modal Solution" >}}
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{{< figure src="/ox-hugo/hatch00_schematic_modal_solution.png" caption="<span class=\"figure-number\">Figure 8: </span>Roadmap for Modal Solution" >}}
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The condition to guarantee diagonalization is the existence of n-linearly independent eigenvectors, which is always the case if either:
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@@ -407,12 +400,12 @@ One method is to normalize with respect to unity, making the **largest** element
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|
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\begin{equation}
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\bm{z}\_m = \begin{bmatrix}
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||||
1 & 1 & 1 \\\\\\
|
||||
1 & 0 & -2 \\\\\\
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||||
1 & 1 & 1 \\\\
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||||
1 & 0 & -2 \\\\
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||||
1 & -1 & 1
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||||
\end{bmatrix} \Longrightarrow \bm{z}\_n \begin{bmatrix}
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||||
1 & 1 & -0.5 \\\\\\
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||||
1 & 0 & 1 \\\\\\
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||||
1 & 1 & -0.5 \\\\
|
||||
1 & 0 & 1 \\\\
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||||
1 & -1 & -0.5
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||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -423,12 +416,12 @@ Transforming the mass and stiffness matrices give:
|
||||
|
||||
\begin{equation}
|
||||
\bm{m}\_n = \bm{z}\_n^T \bm{m} \bm{z}\_n = \begin{bmatrix}
|
||||
3m & 0 & 0 \\\\\\
|
||||
0 & 2m & 0 \\\\\\
|
||||
3m & 0 & 0 \\\\
|
||||
0 & 2m & 0 \\\\
|
||||
0 & 0 & 1.5m
|
||||
\end{bmatrix}; \quad \bm{k}\_n = \bm{z}\_n^T \bm{k} \bm{z}\_n = \begin{bmatrix}
|
||||
0 & 0 & 0 \\\\\\
|
||||
0 & 2k & 0 \\\\\\
|
||||
0 & 0 & 0 \\\\
|
||||
0 & 2k & 0 \\\\
|
||||
0 & 0 & 4.5k
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -455,12 +448,12 @@ And the normalized mass and stiffness matrices are:
|
||||
|
||||
\begin{equation}
|
||||
\bm{m}\_n = \begin{bmatrix}
|
||||
1 & 0 & 0 \\\\\\
|
||||
0 & 1 & 0 \\\\\\
|
||||
1 & 0 & 0 \\\\
|
||||
0 & 1 & 0 \\\\
|
||||
0 & 0 & 1
|
||||
\end{bmatrix}; \quad \bm{k}\_n = \begin{bmatrix}
|
||||
0 & 0 & 0 \\\\\\
|
||||
0 & 1 & 0 \\\\\\
|
||||
0 & 0 & 0 \\\\
|
||||
0 & 1 & 0 \\\\
|
||||
0 & 0 & 3
|
||||
\end{bmatrix} \frac{k}{m}
|
||||
\end{equation}
|
||||
@@ -471,7 +464,6 @@ The normalized stiffness matrix is known as the **spectral matrix**.
|
||||
Normalizing with respect to mass results in an identify principal mass matrix and squares of the eigenvalues on the diagonal in the principal stiffness matrix, this normalization technique is thus very useful for the following reason.
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
Since we know the form of the principal matrices when normalizing with respect to mass, no multiplying of modal matrices is actually required: **the homogeneous principal equations of motion can be written by inspection knowing only the eigenvalues**.
|
||||
|
||||
@@ -498,7 +490,6 @@ Pre-multiplying by \\(\bm{z}\_n^T\\) and inserting \\(I = \bm{z}\_n \bm{z}\_n^{-
|
||||
Which is re-written in the following form:
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
\begin{equation}
|
||||
\bm{m}\_p \ddot{\bm{z}}\_p + \bm{k}\_p \bm{z}\_p = \bm{F}\_p
|
||||
@@ -517,7 +508,7 @@ where:
|
||||
The vectors of initial displacements \\(\bm{z}\_{op}\\) and velocities \\(\dot{\bm{z}}\_{op}\\) in the principal coordinate system can be expressed as:
|
||||
|
||||
\begin{align}
|
||||
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\\\
|
||||
\bm{z}\_{op} &= \bm{z}\_n^{-1} \bm{z}\_0 \\\\
|
||||
\dot{\bm{z}}\_{op} &= \bm{z}\_n^{-1} \dot{\bm{z}}\_0
|
||||
\end{align}
|
||||
|
||||
@@ -529,7 +520,6 @@ where \\(\bm{z}\_0\\) and \\(\dot{\bm{z}}\_0\\) are the vectors of initial displ
|
||||
We have now everything required to solve the equations in the principal coordinate system.
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
The variables in physical coordinates are the positions and velocities of the masses.
|
||||
The variables in principal coordinates are the displacements and velocities of each mode of vibration.
|
||||
@@ -568,12 +558,12 @@ Let's first examine the force transformation from physical to principal coordina
|
||||
|
||||
\begin{equation}
|
||||
\bm{F}\_p = \bm{z}\_n^T \bm{F} = \begin{bmatrix}
|
||||
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
|
||||
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
|
||||
z\_{n11} & z\_{n12} & z\_{n13} \\\\
|
||||
z\_{n21} & z\_{n22} & z\_{n23} \\\\
|
||||
z\_{n31} & z\_{n32} & z\_{n33}
|
||||
\end{bmatrix}^T \begin{bmatrix}
|
||||
F\_1 \\\\\\
|
||||
F\_2 \\\\\\
|
||||
F\_1 \\\\
|
||||
F\_2 \\\\
|
||||
F\_3
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -584,12 +574,12 @@ Let's now examine the displacement transformation from principal to physical coo
|
||||
|
||||
\begin{equation}
|
||||
\bm{z} = \bm{z}\_n \bm{z}\_p = \begin{bmatrix}
|
||||
z\_{n11} & z\_{n12} & z\_{n13} \\\\\\
|
||||
z\_{n21} & z\_{n22} & z\_{n23} \\\\\\
|
||||
z\_{n11} & z\_{n12} & z\_{n13} \\\\
|
||||
z\_{n21} & z\_{n22} & z\_{n23} \\\\
|
||||
z\_{n31} & z\_{n32} & z\_{n33}
|
||||
\end{bmatrix} \begin{bmatrix}
|
||||
z\_{p1} \\\\\\
|
||||
z\_{p2} \\\\\\
|
||||
z\_{p1} \\\\
|
||||
z\_{p2} \\\\
|
||||
z\_{p3}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -597,7 +587,6 @@ Let's now examine the displacement transformation from principal to physical coo
|
||||
And thus, if we are only interested in the physical displacement of the mass 2 (\\(z\_2 = z\_{n21} z\_{p1} + z\_{n22} z\_{p2} + z\_{n23} z\_{p3}\\)), only the second row of the modal matrix is required to transform the three displacements \\(z\_{p1}\\), \\(z\_{p2}\\), \\(z\_{p3}\\) in principal coordinates to \\(z\_2\\).
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
**Only the rows of the modal matrix that correspond to degrees of freedom to which forces are applied and/or for which displacements are desired are required to complete the model.**
|
||||
|
||||
@@ -698,7 +687,7 @@ Absolute damping is based on making \\(b = 0\\), in which case the percentage of
|
||||
|
||||
## Frequency Response: Modal Form {#frequency-response-modal-form}
|
||||
|
||||
<a id="org027da35"></a>
|
||||
<span class="org-target" id="org-target--sec:frequency_response_modal_form"></span>
|
||||
|
||||
The procedure to obtain the frequency response from a modal form is as follow:
|
||||
|
||||
@@ -706,11 +695,11 @@ The procedure to obtain the frequency response from a modal form is as follow:
|
||||
- use Laplace transform to obtain the transfer functions in principal coordinates
|
||||
- back-transform the transfer functions to physical coordinates where the individual mode contributions will be evident
|
||||
|
||||
This will be applied to the model shown in Figure [9](#orgafc54fa).
|
||||
This will be applied to the model shown in Figure [9](#figure--fig:hatch00-tdof-model).
|
||||
|
||||
<a id="orgafc54fa"></a>
|
||||
<a id="figure--fig:hatch00-tdof-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="Figure 9: tdof undamped model for modal analysis" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_tdof_model.png" caption="<span class=\"figure-number\">Figure 9: </span>tdof undamped model for modal analysis" >}}
|
||||
|
||||
|
||||
### Review from Previous Results {#review-from-previous-results}
|
||||
@@ -725,8 +714,8 @@ From previous analysis, we know the eigenvalues and eigenvectors normalized with
|
||||
|
||||
\begin{equation}
|
||||
\bm{z}\_n = \frac{1}{\sqrt{m}} \begin{bmatrix}
|
||||
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\\\
|
||||
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\\\
|
||||
\frac{1}{\sqrt{3}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} \\\\
|
||||
\frac{1}{\sqrt{3}} & 0 & \frac{-2}{\sqrt{6}} \\\\
|
||||
\frac{1}{\sqrt{3}} & \frac{-1}{\sqrt{2}} & \frac{1}{\sqrt{6}}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -735,13 +724,13 @@ Knowing that in principal coordinates the mass matrix is the identify matrix and
|
||||
|
||||
\begin{equation}
|
||||
\bm{m}\_n = \begin{bmatrix}
|
||||
1 & 0 & 0 \\\\\\
|
||||
0 & 1 & 0 \\\\\\
|
||||
1 & 0 & 0 \\\\
|
||||
0 & 1 & 0 \\\\
|
||||
0 & 0 & 1
|
||||
\end{bmatrix}, \quad
|
||||
\bm{k}\_n = \begin{bmatrix}
|
||||
0 & 0 & 0 \\\\\\
|
||||
0 & 1 & 0 \\\\\\
|
||||
0 & 0 & 0 \\\\
|
||||
0 & 1 & 0 \\\\
|
||||
0 & 0 & 3
|
||||
\end{bmatrix} \frac{k}{m}
|
||||
\end{equation}
|
||||
@@ -761,8 +750,8 @@ The equations of motion in principal coordinates are then:
|
||||
which give:
|
||||
|
||||
\begin{align}
|
||||
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\\\
|
||||
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\\\
|
||||
\ddot{z}\_{p1} &= (F\_1 + F\_2 + F\_3) \frac{1}{\sqrt{3m}} \\\\
|
||||
\ddot{z}\_{p2} + \frac{k}{m} z\_{p2} &= (F\_1 - F\_3) \frac{1}{\sqrt{2m}} \\\\
|
||||
\ddot{z}\_{p3} + \frac{3k}{m} z\_{p3} &= (F\_1 - 2 F\_2 + F\_3) \frac{1}{\sqrt{6m}}
|
||||
\end{align}
|
||||
|
||||
@@ -773,48 +762,48 @@ Taking the Laplace transform of each equation gives:
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix}
|
||||
\frac{z\_{p1}}{F\_{1}} \\\\\\
|
||||
\frac{z\_{p2}}{F\_{1}} \\\\\\
|
||||
\frac{z\_{p1}}{F\_{1}} \\\\
|
||||
\frac{z\_{p2}}{F\_{1}} \\\\
|
||||
\frac{z\_{p3}}{F\_{1}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
|
||||
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\
|
||||
\frac{1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
|
||||
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
z\_{p11} \\\\\\
|
||||
z\_{p21} \\\\\\
|
||||
z\_{p11} \\\\
|
||||
z\_{p21} \\\\
|
||||
z\_{p31}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix}
|
||||
\frac{z\_{p1}}{F\_{2}} \\\\\\
|
||||
\frac{z\_{p2}}{F\_{2}} \\\\\\
|
||||
\frac{z\_{p1}}{F\_{2}} \\\\
|
||||
\frac{z\_{p2}}{F\_{2}} \\\\
|
||||
\frac{z\_{p3}}{F\_{2}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
|
||||
0 \\\\\\
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\
|
||||
0 \\\\
|
||||
\frac{-2}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
z\_{p12} \\\\\\
|
||||
z\_{p22} \\\\\\
|
||||
z\_{p12} \\\\
|
||||
z\_{p22} \\\\
|
||||
z\_{p32}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
\begin{equation}
|
||||
\begin{bmatrix}
|
||||
\frac{z\_{p1}}{F\_{3}} \\\\\\
|
||||
\frac{z\_{p2}}{F\_{3}} \\\\\\
|
||||
\frac{z\_{p1}}{F\_{3}} \\\\
|
||||
\frac{z\_{p2}}{F\_{3}} \\\\
|
||||
\frac{z\_{p3}}{F\_{3}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\\\
|
||||
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\\\
|
||||
\frac{1}{s^{2}\sqrt{3m}} \\\\
|
||||
\frac{-1}{(s^{2} + \omega\_{2}^{2})\sqrt{2m}} \\\\
|
||||
\frac{1}{(s^{2} + \omega\_{3}^{2})\sqrt{6m}}
|
||||
\end{bmatrix} = \begin{bmatrix}
|
||||
z\_{p13} \\\\\\
|
||||
z\_{p23} \\\\\\
|
||||
z\_{p13} \\\\
|
||||
z\_{p23} \\\\
|
||||
z\_{p33}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
@@ -839,7 +828,7 @@ And the transfer functions \\(\frac{z\_i}{F\_j}\\) can be computed.
|
||||
For instance, the contributions to the transfer function \\(\frac{z\_1}{F\_1}\\) are:
|
||||
|
||||
\begin{align}
|
||||
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\\\
|
||||
\frac{z\_1}{F\_1} &= \underbrace{z\_{n11} z\_{p11}}\_{\text{1st mode}} + \underbrace{z\_{n12} z\_{p21}}\_{\text{2nd mode}} + \underbrace{z\_{n13} z\_{p31}}\_{\text{3rd mode}} \\\\
|
||||
& = \frac{\frac{1}{3m}}{s^2} + \frac{\frac{1}{2m}}{s^2 + \omega\_2^2} + \frac{\frac{1}{6m}}{s^2 + \omega\_3^2}
|
||||
\end{align}
|
||||
|
||||
@@ -858,7 +847,6 @@ The forces transform in the principal coordinates using:
|
||||
\end{equation}
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized eigenvector), then \\(\bm{F}\_p\\) will be null except for its i'th term and only the i'th mode will be excited.
|
||||
|
||||
@@ -870,7 +858,6 @@ Thus, if \\(\bm{F}\\) is aligned with \\(\bm{z}\_{ni}\\) (the i'th normalized ei
|
||||
Any transfer function derived from the modal analysis is an additive combination of sdof systems.
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
Each single degree of freedom system has a gain determined by the appropriate eigenvector entries and a resonant frequency given by the appropriate eigenvalue.
|
||||
|
||||
@@ -886,33 +873,33 @@ If modes have some damping:
|
||||
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \label{eq:general\_add\_tf\_damp}
|
||||
\end{equation}
|
||||
|
||||
Equations \eqref{eq:general_add_tf} and \eqref{eq:general_add_tf_damp} shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
|
||||
Equations <eq:general_add_tf> and <eq:general_add_tf_damp> shows that in general every transfer function is made up of **additive combinations of single degree of freedom systems**, with each system having its DC gain determined by the appropriate eigenvector entry product divided by the square of the eigenvalue, \\(z\_{nji} z\_{nki}/\omega\_i^2\\), and with resonant frequency defined by the eigenvalue \\(\omega\_i\\).
|
||||
|
||||
</div>
|
||||
|
||||
Figure [10](#orgf64b6e5) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
|
||||
Figure [10](#figure--fig:hatch00-z11-tf-example) shows the separate contributions of each mode to the total response \\(z\_1/F\_1\\).
|
||||
|
||||
<a id="orgf64b6e5"></a>
|
||||
<a id="figure--fig:hatch00-z11-tf-example"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 10: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 10: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
|
||||
|
||||
The zeros for SISO transfer functions are the roots of the numerator, however, from modal analysis we can see that the zeros arise when modes combine with appropriate phase such that the resulting motion is null.
|
||||
|
||||
|
||||
## SISO State Space Matlab Model from ANSYS Model {#siso-state-space-matlab-model-from-ansys-model}
|
||||
|
||||
<a id="org39bd7f2"></a>
|
||||
<span class="org-target" id="org-target--sec:siso_state_space"></span>
|
||||
|
||||
|
||||
### Introduction {#introduction}
|
||||
|
||||
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#orgc285575).
|
||||
In this section is developed a SISO state space Matlab model from an ANSYS cantilever beam model as shown in Figure [11](#figure--fig:hatch00-cantilever-beam).
|
||||
A z direction force is applied at the midpoint of the beam and z displacement at the tip is the output.
|
||||
The objective is to provide the smallest Matlab state space model that accurately represents the pertinent dynamics.
|
||||
|
||||
<a id="orgc285575"></a>
|
||||
<a id="figure--fig:hatch00-cantilever-beam"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="Figure 11: Cantilever beam with forcing function at midpoint" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_cantilever_beam.png" caption="<span class=\"figure-number\">Figure 11: </span>Cantilever beam with forcing function at midpoint" >}}
|
||||
|
||||
The steps to define the smallest model are:
|
||||
|
||||
@@ -952,7 +939,7 @@ We will discuss in this section two methods of sorting, one which is applicable
|
||||
The general equation for the overall transfer function of undamped and damped systems are:
|
||||
|
||||
\begin{align}
|
||||
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\\\
|
||||
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + \omega\_i^2} \\\\
|
||||
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
|
||||
\end{align}
|
||||
|
||||
@@ -989,7 +976,7 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
|
||||
|
||||
## Ground Acceleration Matlab Model From ANSYS Model {#ground-acceleration-matlab-model-from-ansys-model}
|
||||
|
||||
<a id="org658f39a"></a>
|
||||
<span class="org-target" id="org-target--sec:ground_acceleration"></span>
|
||||
|
||||
|
||||
### Model Description {#model-description}
|
||||
@@ -1003,25 +990,25 @@ If sorting of DC gain values is performed prior to the `truncate` operation, the
|
||||
|
||||
## SISO Disk Drive Actuator Model {#siso-disk-drive-actuator-model}
|
||||
|
||||
<a id="orgcd094f5"></a>
|
||||
<span class="org-target" id="org-target--sec:siso_disk_drive"></span>
|
||||
|
||||
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#org97a4ded)).
|
||||
In this section we wish to extract a SISO state space model from a Finite Element model representing a Disk Drive Actuator (Figure [12](#figure--fig:hatch00-disk-drive-siso-model)).
|
||||
|
||||
|
||||
### Actuator Description {#actuator-description}
|
||||
|
||||
<a id="org97a4ded"></a>
|
||||
<a id="figure--fig:hatch00-disk-drive-siso-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="Figure 12: Drawing of Actuator/Suspension system" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_siso_model.png" caption="<span class=\"figure-number\">Figure 12: </span>Drawing of Actuator/Suspension system" >}}
|
||||
|
||||
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#orga92b66d)).
|
||||
The primary motion of the actuator is rotation about the pivot bearing, therefore the final model has the coordinate system transformed from a Cartesian x,y,z coordinate system to a Cylindrical \\(r\\), \\(\theta\\) and \\(z\\) system, with the two origins coincident (Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model)).
|
||||
|
||||
<a id="orga92b66d"></a>
|
||||
<a id="figure--fig:hatch00-disk-drive-nodes-reduced-model"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="Figure 13: Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_nodes_reduced_model.png" caption="<span class=\"figure-number\">Figure 13: </span>Nodes used for reduced Matlab model. Shown with partial finite element mesh at coil" >}}
|
||||
|
||||
For reduced models, we only require eigenvector information for dof where forces are applied and where displacements are required.
|
||||
Figure [13](#orga92b66d) shows the nodes used for the reduced Matlab model.
|
||||
Figure [13](#figure--fig:hatch00-disk-drive-nodes-reduced-model) shows the nodes used for the reduced Matlab model.
|
||||
The four nodes 24061, 24066, 24082 and 24087 are located in the center of the coil in the z direction and are used for simulating the VCM force.
|
||||
The arrows at the nodes indicate the direction of forces.
|
||||
|
||||
@@ -1045,10 +1032,8 @@ A recommended sequence for analyzing dynamic finite element models is:
|
||||
A small section of the exported `.eig` file from ANSYS is shown bellow..
|
||||
|
||||
<div class="exampl">
|
||||
<div></div>
|
||||
|
||||
<div class="monoblock">
|
||||
<div></div>
|
||||
|
||||
LOAD STEP= 1 SUBSTEP= 1
|
||||
FREQ= 8.1532 LOAD CASE= 0
|
||||
@@ -1089,7 +1074,7 @@ From Ansys, we have the eigenvalues \\(\omega\_i\\) and eigenvectors \\(\bm{z}\\
|
||||
|
||||
## Balanced Reduction {#balanced-reduction}
|
||||
|
||||
<a id="org58a3a47"></a>
|
||||
<span class="org-target" id="org-target--sec:balanced_reduction"></span>
|
||||
|
||||
In this chapter another method of reducing models, “balanced reduction”, will be introduced and compared with the DC and peak gain ranking methods.
|
||||
|
||||
@@ -1117,7 +1102,7 @@ A mode which cannot be excited by the applied force is said to be **uncontrollab
|
||||
For a state space system described by:
|
||||
|
||||
\begin{align\*}
|
||||
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\\\
|
||||
\dot{\bm{x}} &= \bm{A} \bm{x} + \bm{B} u \\\\
|
||||
\bm{y} &= \bm{C} \bm{x}
|
||||
\end{align\*}
|
||||
|
||||
@@ -1159,7 +1144,7 @@ A similar set of definitions can be made for observability:
|
||||
|
||||
\begin{equation}
|
||||
\bm{\mathcal{O}} = \begin{bmatrix}
|
||||
\bm{C} \\ \bm{C} \bm{A} \\ \bm{C} \bm{A}^{2} \\ \vdots \\ \bm{C} \bm{A}^{n-1}
|
||||
\bm{C} \\\ \bm{C} \bm{A} \\\ \bm{C} \bm{A}^{2} \\\ \vdots \\\ \bm{C} \bm{A}^{n-1}
|
||||
\end{bmatrix}
|
||||
\end{equation}
|
||||
|
||||
@@ -1204,16 +1189,16 @@ The **states to be kept are the states with the largest diagonal terms**.
|
||||
|
||||
## MIMO Two Stage Actuator Model {#mimo-two-stage-actuator-model}
|
||||
|
||||
<a id="orgf33e1dd"></a>
|
||||
<span class="org-target" id="org-target--sec:mimo_disk_drive"></span>
|
||||
|
||||
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#org59e7525)).
|
||||
In this section, a MIMO two-stage actuator model is derived from a finite element model (Figure [14](#figure--fig:hatch00-disk-drive-mimo-schematic)).
|
||||
|
||||
|
||||
### Actuator Description {#actuator-description}
|
||||
|
||||
<a id="org59e7525"></a>
|
||||
<a id="figure--fig:hatch00-disk-drive-mimo-schematic"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="Figure 14: Drawing of actuator/suspension system" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_schematic.png" caption="<span class=\"figure-number\">Figure 14: </span>Drawing of actuator/suspension system" >}}
|
||||
|
||||
A piezo-actuator is now bounded into one side of each of the arms.
|
||||
The piezo actuator consists of a ceramic element that changes size when a voltage is applied.
|
||||
@@ -1221,7 +1206,6 @@ The piezo actuator consists of a ceramic element that changes size when a voltag
|
||||
Then the fine positioning motion of the piezo is used in conjunction with VCM's coarse positioning motion, higher servo bandwidth is possible.
|
||||
|
||||
<div class="important">
|
||||
<div></div>
|
||||
|
||||
Instead of applying voltage as the input into the piezo elements, we will assume that we have calculated an equivalent set of forces which can be applied at the ends of the element that will replicate the voltage force function.
|
||||
In this model, we will be applying forces to multiple nodes at the ends of both piezo elements.
|
||||
@@ -1233,11 +1217,11 @@ Since the same forces are being applied to both piezo elements, they represent t
|
||||
|
||||
### Ansys Model Description {#ansys-model-description}
|
||||
|
||||
In Figure [15](#org5f31090) are shown the principal nodes used for the model.
|
||||
In Figure [15](#figure--fig:hatch00-disk-drive-mimo-ansys) are shown the principal nodes used for the model.
|
||||
|
||||
<a id="org5f31090"></a>
|
||||
<a id="figure--fig:hatch00-disk-drive-mimo-ansys"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="Figure 15: Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_disk_drive_mimo_ansys.png" caption="<span class=\"figure-number\">Figure 15: </span>Nodes used for reduced Matlab model, shown with partial mesh at coil and piezo element" >}}
|
||||
|
||||
|
||||
### Matlab Model {#matlab-model}
|
||||
@@ -1354,13 +1338,13 @@ And we note:
|
||||
G = zn * Gp;
|
||||
```
|
||||
|
||||
<a id="orgbe6df95"></a>
|
||||
<a id="figure--fig:hatch00-z13-tf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="Figure 16: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_z13_tf.png" caption="<span class=\"figure-number\">Figure 16: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_3\\)" >}}
|
||||
|
||||
<a id="orgcec939e"></a>
|
||||
<a id="figure--fig:hatch00-z11-tf"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="Figure 17: Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_z11_tf.png" caption="<span class=\"figure-number\">Figure 17: </span>Mode contributions to the transfer function from \\(F\_1\\) to \\(z\_1\\)" >}}
|
||||
|
||||
|
||||
## Matlab with ANSYS {#matlab-with-ansys}
|
||||
@@ -1456,15 +1440,15 @@ State Space Model
|
||||
|
||||
### Simple mode truncation {#simple-mode-truncation}
|
||||
|
||||
Let's plot the frequency of the modes (Figure [18](#org1183b44)).
|
||||
Let's plot the frequency of the modes (Figure [18](#figure--fig:hatch00-cant-beam-modes-freq)).
|
||||
|
||||
<a id="org1183b44"></a>
|
||||
<a id="figure--fig:hatch00-cant-beam-modes-freq"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="Figure 18: Frequency of the modes" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_modes_freq.png" caption="<span class=\"figure-number\">Figure 18: </span>Frequency of the modes" >}}
|
||||
|
||||
<a id="org350c1cb"></a>
|
||||
<a id="figure--fig:hatch00-cant-beam-unsorted-dc-gains"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="Figure 19: Unsorted DC Gains" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_unsorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 19: </span>Unsorted DC Gains" >}}
|
||||
|
||||
Let's keep only the first 10 modes.
|
||||
|
||||
@@ -1531,9 +1515,9 @@ Let's sort the modes by their DC gains and plot their sorted DC gains.
|
||||
[dc_gain_sort, index_sort] = sort(dc_gain, 'descend');
|
||||
```
|
||||
|
||||
<a id="orgd64190f"></a>
|
||||
<a id="figure--fig:hatch00-cant-beam-sorted-dc-gains"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="Figure 20: Sorted DC Gains" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_sorted_dc_gains.png" caption="<span class=\"figure-number\">Figure 20: </span>Sorted DC Gains" >}}
|
||||
|
||||
Let's keep only the first 10 **sorted** modes.
|
||||
|
||||
@@ -1875,9 +1859,9 @@ Then, we compute the controllability and observability gramians.
|
||||
|
||||
And we plot the diagonal terms
|
||||
|
||||
<a id="orgbdc6b3b"></a>
|
||||
<a id="figure--fig:hatch00-gramians"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="Figure 21: Observability and Controllability Gramians" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_gramians.png" caption="<span class=\"figure-number\">Figure 21: </span>Observability and Controllability Gramians" >}}
|
||||
|
||||
We use `balreal` to rank oscillatory states.
|
||||
|
||||
@@ -1893,9 +1877,9 @@ We use `balreal` to rank oscillatory states.
|
||||
[G_b, G, T, Ti] = balreal(G_m);
|
||||
```
|
||||
|
||||
<a id="org2787898"></a>
|
||||
<a id="figure--fig:hatch00-cant-beam-gramian-balanced"></a>
|
||||
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="Figure 22: Sorted values of the Gramian of the balanced realization" >}}
|
||||
{{< figure src="/ox-hugo/hatch00_cant_beam_gramian_balanced.png" caption="<span class=\"figure-number\">Figure 22: </span>Sorted values of the Gramian of the balanced realization" >}}
|
||||
|
||||
Now we can choose the number of states to keep.
|
||||
|
||||
@@ -2136,9 +2120,9 @@ Reduced Mass and Stiffness matrices in the physical coordinates:
|
||||
```
|
||||
|
||||
|
||||
|
||||
## Bibliography {#bibliography}
|
||||
|
||||
<a id="org4036e02"></a>Hatch, Michael R. 2000. _Vibration Simulation Using MATLAB and ANSYS_. CRC Press.
|
||||
|
||||
<a id="orgcda3e53"></a>Miu, Denny K. 1993. _Mechatronics: Electromechanics and Contromechanics_. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.
|
||||
<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_1"></a>Hatch, Michael R. 2000. <i>Vibration Simulation Using Matlab and Ansys</i>. CRC Press.</div>
|
||||
<div class="csl-entry"><a id="citeproc_bib_item_2"></a>Miu, Denny K. 1993. <i>Mechatronics: Electromechanics and Contromechanics</i>. 1st ed. Mechanical Engineering Series. Springer-Verlag New York.</div>
|
||||
</div>
|
||||
|
||||
Reference in New Issue
Block a user