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content/book/du19_multi_actuat_system_contr.md
+10 -10
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@@ -75,9 +75,9 @@ which includes the resonance model
and the resonance \\(P\_{ri}(s)\\) can be represented as one of the following forms
\begin{align\*}
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \xi\_i \omega\_i s + \omega\_i^2}
P\_{ri}(s) &= \frac{\omega\_i^2}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2} \\\\
P\_{ri}(s) &= \frac{b\_{2i} s^2 + b\_{1i} \omega\_i s + b\_{0i} \omega\_i^2}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2}
\end{align\*}
@@ -98,11 +98,11 @@ There characteristics are shown on [Table 1](#table--tab:microactuator).
Performance comparison of microactuators
</div>
| | Elect. | PZT | Thermal |
|--------------|-----------------------------------------------|-----------------------------------------------|----------------------------|
| TF | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\xi\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
| \\(\tau\\) | \\(<\SI{0.1}{ms}\\) | \\(<\SI{0.05}{ms}\\) | \\(>\SI{0.1}{ms}\\) |
| \\(\omega\\) | \\(1-\SI{2}{kHz}\\) | \\(20-\SI{25}{kHz}\\) | \\(>\SI{15}{kHz}\\) |
| | Elect. | PZT | Thermal |
|--------------|-------------------------------------------------|-------------------------------------------------|----------------------------|
| TF | \\(\frac{K}{s^2 + 2\zeta\omega s + \omega^2}\\) | \\(\frac{K}{s^2 + 2\zeta\omega s + \omega^2}\\) | \\(\frac{K}{\tau s + 1}\\) |
| \\(\tau\\) | \\(<\SI{0.1}{ms}\\) | \\(<\SI{0.05}{ms}\\) | \\(>\SI{0.1}{ms}\\) |
| \\(\omega\\) | \\(1-\SI{2}{kHz}\\) | \\(20-\SI{25}{kHz}\\) | \\(>\SI{15}{kHz}\\) |
### Single-Stage Actuation Systems {#single-stage-actuation-systems}
@@ -210,10 +210,10 @@ Equation means that \\(S(s)\\) can be shaped similarly to the inverse of the cho
One form of \\(W(s)\\) is taken as
\begin{equation}
W(s) = \frac{\frac{1}{M}s^2 + 2\xi\omega\frac{1}{\sqrt{M}}s + \omega^2}{s^2 + 2\omega\sqrt{\epsilon}s + \omega^2\epsilon}
W(s) = \frac{\frac{1}{M}s^2 + 2\zeta\omega\frac{1}{\sqrt{M}}s + \omega^2}{s^2 + 2\omega\sqrt{\epsilon}s + \omega^2\epsilon}
\end{equation}
where \\(\omega\\) is the desired bandwidth, \\(\epsilon\\) is used to determine the desired low frequency level of sensitivity magnitude and \\(\xi\\) is the damping ratio.
where \\(\omega\\) is the desired bandwidth, \\(\epsilon\\) is used to determine the desired low frequency level of sensitivity magnitude and \\(\zeta\\) is the damping ratio.
The controller can then be synthesis using the linear matrix inequality (LMI) approach.
content/book/ewins00_modal.md
+23 -23
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@@ -379,14 +379,14 @@ which leads to
\begin{align}
s\_{1, 2} &= -\frac{c}{2 m} \pm \frac{\sqrt{c^2 - 4 k m}}{2 m} \\\\
&= - \bar{\omega}\_0 \xi \pm i \bar{\omega}\_0 \sqrt{1 - \xi^2} \nonumber
&= - \bar{\omega}\_0 \zeta \pm i \bar{\omega}\_0 \sqrt{1 - \zeta^2} \nonumber
\end{align}
where
\\[ \bar{\omega}\_0^2 = \frac{k}{m};\quad \xi = \frac{c}{c\_0} = \frac{c}{2 \sqrt{km}} \\]
\\[ \bar{\omega}\_0^2 = \frac{k}{m};\quad \zeta = \frac{c}{c\_0} = \frac{c}{2 \sqrt{km}} \\]
This implies a modal solution of the form
\\[ x(t) = X e^{-\bar{\omega}\_0 \xi t} e^{i (\bar{\omega}\_0 \sqrt{1 - \xi^2})t} = X e^{-a t} e^{i \omega\_0^\prime t} \\]
\\[ x(t) = X e^{-\bar{\omega}\_0 \zeta t} e^{i (\bar{\omega}\_0 \sqrt{1 - \zeta^2})t} = X e^{-a t} e^{i \omega\_0^\prime t} \\]
which is a single mode of vibration with a complex natural frequency having two part:
- **An imaginary or oscillatory part**
@@ -899,7 +899,7 @@ A usual definition of proportional damping is that the damping matrix \\([C]\\)
In this case, the damped system will have eigenvalues and eigenvectors as follows
\begin{align\*}
&\omega\_r^\prime = \bar{\omega}\_r \sqrt{1 - \xi\_r^2} ; \quad \xi\_r = \frac{\beta \bar{\omega}\_r}{2} + \frac{\gamma}{2 \bar{\omega}\_r}\\\\
&\omega\_r^\prime = \bar{\omega}\_r \sqrt{1 - \zeta\_r^2} ; \quad \zeta\_r = \frac{\beta \bar{\omega}\_r}{2} + \frac{\gamma}{2 \bar{\omega}\_r}\\\\
&[\Psi\_\text{damped}] = [\Psi\_\text{undamped}]
\end{align\*}
@@ -1074,8 +1074,8 @@ The eigensolution can thus be described as
\\[ s\_r, s\_r^\*, \quad \\{\psi\\}\_r, \\{\psi\\}\_r^\*;\quad r = 1, N \\]
Each eigenvalue \\(s\_r\\) is expressed in the form
\\[ s\_r = \omega\_r \left( -\xi\_r + i\sqrt{1 - \xi\_r^2} \right) \\]
where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **critical damping ratio** for that mode.
\\[ s\_r = \omega\_r \left( -\zeta\_r + i\sqrt{1 - \zeta\_r^2} \right) \\]
where \\(\omega\_r\\) is the **natural frequency** and \\(\zeta\_r\\) is the **critical damping ratio** for that mode.
**Orthogonality equations** can also be derived:
@@ -1085,12 +1085,12 @@ where \\(\omega\_r\\) is the **natural frequency** and \\(\xi\_r\\) is the **cri
\end{align}
When the modes \\(r\\) and \\(q\\) are a complex conjugate pair:
\\[ s\_r = \omega\_r \left( -\xi\_r - i\sqrt{1 - \xi\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
\\[ s\_r = \omega\_r \left( -\zeta\_r - i\sqrt{1 - \zeta\_r^2} \right); \quad \\{\psi\\}\_q = \\{\psi\\}\_r^\* \\]
From equations \eqref{eq:viscous\_damping\_orthogonality}, we can obtain
\begin{align}
2 \omega\_r \xi\_r &= \frac{\\{\psi\\}\_r^H [C] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{c\_r}{m\_r} \\\\
2 \omega\_r \zeta\_r &= \frac{\\{\psi\\}\_r^H [C] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{c\_r}{m\_r} \\\\
\omega\_r^2 &= \frac{\\{\psi\\}\_r^H [K] \\{\psi\\}\_r}{\\{\psi\\}\_r^H [M] \\{\psi\\}\_r} = \frac{k\_r}{m\_r}
\end{align}
@@ -1104,11 +1104,11 @@ but this expression is not particularly convenient for numerical applications.
We then seek a similar series expansion that was found for the undamped, proportionally-damped and hysteretically damped systems.
The obtain result is
\\[ \alpha\_{jk}(\omega) = \sum\_{r=1}^N \frac{({}\_rR\_{jk}) + i (\omega/\omega\_r)({}\_rS\_{jk})}{\omega\_r^2 - \omega^2 + 2 i \omega \omega\_r \xi\_r} \\]
\\[ \alpha\_{jk}(\omega) = \sum\_{r=1}^N \frac{({}\_rR\_{jk}) + i (\omega/\omega\_r)({}\_rS\_{jk})}{\omega\_r^2 - \omega^2 + 2 i \omega \omega\_r \zeta\_r} \\]
where the coefficients \\(R\\) and \\(S\\) are obtained from:
\begin{align\*}
\\{{}\_rR\_k\\} &= 2 \left( \xi\_r \text{Re}\\{{}\_rG\_k\\} - \text{Im}\\{{}\_rG\_k\\} \sqrt{1 - \xi\_r^2} \right)\\\\
\\{{}\_rR\_k\\} &= 2 \left( \zeta\_r \text{Re}\\{{}\_rG\_k\\} - \text{Im}\\{{}\_rG\_k\\} \sqrt{1 - \zeta\_r^2} \right)\\\\
\\{{}\_rS\_k\\} &= 2 \text{Re}\\{{}\_rG\_k\\}\\\\
\\{{}\_rG\_k\\} &= (\theta\_{kr}/a\_r) \\{\theta\\}\_r
\end{align\*}
@@ -1221,13 +1221,13 @@ Here \\(C\_r\\) may be complex whereas \\(D\_r\\) is real.
FRF Formulae and Natural Frequencies
</div>
| **Case** | **C** | **D** | **Free** \\(\omega\_r^\prime\\) | **Forced** \\(\omega\_r\\) |
|-------------|---------------------------------|---------------------------------|----------------------------------|----------------------------|
| Undamped | \\(\mathbb{R}\\) | 0 | \\(\bar{\omega}\_r\\) | \\(\bar{\omega}\_r\\) |
| Prop. Hyst. | \\(\mathbb{R}\\) | \\(\mathbb{R}\\) | \\(\bar{\omega}\_r\\) | \\(\bar{\omega}\_r\\) |
| Prop. Visc. | \\(\mathbb{R}\\) | \\(\mathbb{R}\\) (\\(\omega\\)) | \\(\omega\_r\sqrt{1-\xi\_r^2}\\) | \\(\bar{\omega}\_r\\) |
| Gen. Hyst. | \\(\mathbb{C}\\) | \\(\mathbb{R}\\) | \\(\omega\_r\\) | \\(\omega\_r\\) |
| Gen. Visc. | \\(\mathbb{C}\\) (\\(\omega\\)) | \\(\mathbb{R}\\) (\\(\omega\\)) | \\(\omega\_r\sqrt{1-\xi\_r^2}\\) | \\(\omega\_r\\) |
| **Case** | **C** | **D** | **Free** \\(\omega\_r^\prime\\) | **Forced** \\(\omega\_r\\) |
|-------------|---------------------------------|---------------------------------|------------------------------------|----------------------------|
| Undamped | \\(\mathbb{R}\\) | 0 | \\(\bar{\omega}\_r\\) | \\(\bar{\omega}\_r\\) |
| Prop. Hyst. | \\(\mathbb{R}\\) | \\(\mathbb{R}\\) | \\(\bar{\omega}\_r\\) | \\(\bar{\omega}\_r\\) |
| Prop. Visc. | \\(\mathbb{R}\\) | \\(\mathbb{R}\\) (\\(\omega\\)) | \\(\omega\_r\sqrt{1-\zeta\_r^2}\\) | \\(\bar{\omega}\_r\\) |
| Gen. Hyst. | \\(\mathbb{C}\\) | \\(\mathbb{R}\\) | \\(\omega\_r\\) | \\(\omega\_r\\) |
| Gen. Visc. | \\(\mathbb{C}\\) (\\(\omega\\)) | \\(\mathbb{R}\\) (\\(\omega\\)) | \\(\omega\_r\sqrt{1-\zeta\_r^2}\\) | \\(\omega\_r\\) |
#### Mobility and Impedance FRF Parameters {#mobility-and-impedance-frf-parameters}
@@ -3265,7 +3265,7 @@ The peak-picking method is applied as follows (illustrated on [Figure 46](#figur
\begin{equation}
\begin{aligned}
\eta\_r &= \frac{\omega\_a^2 - \omega\_b^2}{2 \omega\_r^2} \approx \frac{\Delta\omega}{\omega\_r} \\\\
2\xi\_r &= \eta\_r
2\zeta\_r &= \eta\_r
\end{aligned}
\end{equation}
4. We now obtain an estimate for the **modal constant** of the mode being analyzed by assuming that the total response in this resonant region is attributed to a single term in the general FRF series:
@@ -3390,14 +3390,14 @@ For SDOF system with **viscous** damping, rather than structural damping, the **
And we have
\begin{equation}
\tan\left(\frac{\theta}{2}\right) = \frac{1 - (\omega/\omega\_r)^2}{2 \xi \omega/\omega\_r}
\tan\left(\frac{\theta}{2}\right) = \frac{1 - (\omega/\omega\_r)^2}{2 \zeta \omega/\omega\_r}
\end{equation}
From points at \\(\omega\_a\\) and \\(\omega\_b\\), we obtain
\begin{equation}
\begin{aligned}
\xi &= \frac{\omega\_a^2 - \omega\_b^2}{2 \omega\_r \left( \omega\_a \tan(\theta\_a/2) + \omega\_b \tan(\theta\_b/2) \right)}\\\\
\zeta &= \frac{\omega\_a^2 - \omega\_b^2}{2 \omega\_r \left( \omega\_a \tan(\theta\_a/2) + \omega\_b \tan(\theta\_b/2) \right)}\\\\
&= \frac{\omega\_a - \omega\_b}{\omega\_r \left( \tan(\theta\_a/2) + \tan(\theta\_b/2) \right)} \text{ for light damping}
\end{aligned}
\end{equation}
@@ -3405,7 +3405,7 @@ From points at \\(\omega\_a\\) and \\(\omega\_b\\), we obtain
Finally, selecting two points for which \\(\theta\_a = \theta\_b = \SI{90}{\degree}\\):
\begin{equation}
\xi = \frac{\omega\_2 - \omega\_1}{2 \omega\_r}
\zeta = \frac{\omega\_2 - \omega\_1}{2 \omega\_r}
\end{equation}
@@ -3736,7 +3736,7 @@ This method is a special version of the general curve fitting approach but is ba
Rational fraction FRF formulation:
\begin{equation}
H(\omega) = \sum\_{r=1}^N \frac{A\_r}{\omega\_r^2 - \omega^2 + 2 i \omega \omega\_r \xi\_r} \label{eq:frf\_clasic}
H(\omega) = \sum\_{r=1}^N \frac{A\_r}{\omega\_r^2 - \omega^2 + 2 i \omega \omega\_r \zeta\_r} \label{eq:frf\_clasic}
\end{equation}
\begin{equation}
@@ -3823,7 +3823,7 @@ where \\([X], [Y], [Z], \\{G\\}\\) and \\(\\{F\\}\\) are known measured quantiti
Once the solution has been obtained for the coefficients \\(a\_k, \dots , b\_k, \dots\\) then the second stage of the modal analysis can be performed in which the required **modal parameters are derived**.
This is usually done by solving the two polynomial expressions which form the numerator and denominator of equations \eqref{eq:frf\_clasic} and \eqref{eq:frf\_rational:}
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\xi\_r\\)
- the denominator is used to obtain the natural frequencies \\(\omega\_r\\) and damping factors \\(\zeta\_r\\)
- the numerator is used to determine the complex modal constants \\(A\_r\\)
In order to determine the order, the analysis is repeated using different assumed values for the order \\(m\\) and are compared.
content/book/fleming14_desig_model_contr_nanop_system.md
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@@ -39,13 +39,13 @@ Typical value of \\(R\_s\\) is between \\(10\\) and \\(100 \Omega\\).
When considering the effects of both output impedance and cable inductance, the transfer function from source voltage \\(V\_s\\) to load voltage \\(V\_L\\) is second-order low pass filter:
\begin{equation}
\frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \xi \frac{s}{\omega\_r} + 1}
\frac{V\_L(s)}{V\_s(s)} = \frac{1}{\frac{s^2}{\omega\_r^2} + 2 \zeta \frac{s}{\omega\_r} + 1}
\end{equation}
with:
- \\(\omega\_r = \frac{1}{\sqrt{L C\_p}}\\)
- \\(\xi = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)
- \\(\zeta = \frac{R\_s \sqrt{L C\_p}}{2 L}\\)
## Amplifier small-signal Bandwidth {#amplifier-small-signal-bandwidth}
content/book/hatch00_vibrat_matlab_ansys.md
+16 -16
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@@ -622,19 +622,19 @@ And the damping matrix expressed in the principal coordinates is:
We note:
\begin{equation}
c\_{pi} = a + b \omega\_i^2 = 2 \xi\_i \omega\_i
c\_{pi} = a + b \omega\_i^2 = 2 \zeta\_i \omega\_i
\end{equation}
where \\(\xi\_i\\) is the percentage of critical damping for the i'th mode:
where \\(\zeta\_i\\) is the percentage of critical damping for the i'th mode:
\begin{equation}
\xi\_i = \frac{c\_i}{2 \sqrt{k\_{pi} m\_{pi}}} = \frac{c\_i}{2 m\_{pi} \sqrt{\omega\_i^2}} = \frac{a + b \omega\_i^2}{2 \omega\_i}
\zeta\_i = \frac{c\_i}{2 \sqrt{k\_{pi} m\_{pi}}} = \frac{c\_i}{2 m\_{pi} \sqrt{\omega\_i^2}} = \frac{a + b \omega\_i^2}{2 \omega\_i}
\end{equation}
And the equation in principal coordinates becomes:
\begin{equation}
\ddot{z}\_{pi} + 2 \xi\_i \omega\_i \dot{z}\_{pi} + \omega\_i^2 z\_{pi} = F\_{pi}
\ddot{z}\_{pi} + 2 \zeta\_i \omega\_i \dot{z}\_{pi} + \omega\_i^2 z\_{pi} = F\_{pi}
\end{equation}
This type of damping is known as **proportional damping**, where the damping for each mode is proportional to the critical damping for that mode.
@@ -642,10 +642,10 @@ This type of damping is known as **proportional damping**, where the damping for
#### Simple Proportional Damping {#simple-proportional-damping}
Viscous damping in each mode is taken to be an arbitrary percentage \\(\xi\\) of the critical damping \\(c\_{cr}\\):
Viscous damping in each mode is taken to be an arbitrary percentage \\(\zeta\\) of the critical damping \\(c\_{cr}\\):
\begin{equation}
c = \frac{\xi}{2 \sqrt{k m}} = \xi \cdot c\_{cr}
c = \frac{\zeta}{2 \sqrt{k m}} = \zeta \cdot c\_{cr}
\end{equation}
@@ -654,19 +654,19 @@ Viscous damping in each mode is taken to be an arbitrary percentage \\(\xi\\) of
Recognizing that the higher modes of vibration damp out quickly, "relative" damping yields damping in proportion to frequencies in normal modes:
\begin{equation}
\xi\_i = \frac{a + b\omega\_i^2}{2 \omega\_i}
\zeta\_i = \frac{a + b\omega\_i^2}{2 \omega\_i}
\end{equation}
If a value of \\(\xi\_1\\) for the first mode, is assumed:
If a value of \\(\zeta\_1\\) for the first mode, is assumed:
\begin{equation}
b = \frac{2 \xi\_1}{\omega\_1}
b = \frac{2 \zeta\_1}{\omega\_1}
\end{equation}
The value of the damping for any other mode is:
\begin{equation}
\xi\_i = \xi\_1 \frac{\omega\_i}{\omega\_1}
\zeta\_i = \zeta\_1 \frac{\omega\_i}{\omega\_1}
\end{equation}
@@ -861,7 +861,7 @@ It can be shown that for a general system with \\(m\\) undamped modes:
If modes have some damping:
\begin{equation}
\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}
\frac{z\_j}{F\_k} = \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \zeta\_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\\).
@@ -923,13 +923,13 @@ The DC gain contributions of the eliminated modes are not included in the overal
In order to eliminate this error, the Matlab function `modred` is introduced.
Using `modred` is analogous to using Guyan reduction to reduce some less important degrees of freedom.
We will discuss in this section two methods of sorting, one which is applicable for models with the same value of damping for all modes \\(\xi\_i = \xi\\) ("uniform" damping), and another which is applicable for models with different damping values for each mode ("non-uniform" damping).
We will discuss in this section two methods of sorting, one which is applicable for models with the same value of damping for all modes \\(\zeta\_i = \zeta\\) ("uniform" damping), and another which is applicable for models with different damping values for each mode ("non-uniform" damping).
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 + 2 \xi\_i \omega\_i s + \omega\_i^2}
\frac{z\_j}{F\_k} &= \sum\_{i = 1}^m \frac{z\_{nji} z\_{nki}}{s^2 + 2 \zeta\_i \omega\_i s + \omega\_i^2}
\end{align}
The **DC gain** of the i'th mode can be obtained by substituting \\(s = j\omega = 0\\):
@@ -943,12 +943,12 @@ where \\(z\_{nji} z\_{nki}\\) is the product of the j'th (output) row and k'th (
At resonance, the **peak gain** amplitude of each mode is given by substituting \\(s = j \omega\_i\\):
\begin{equation}
\text{Peak gain}\_i = \frac{z\_{ji}}{F\_{ki}} = \frac{-j}{2 \xi\_i} \frac{z\_{nji} z\_{nki}}{\omega\_i^2}
\text{Peak gain}\_i = \frac{z\_{ji}}{F\_{ki}} = \frac{-j}{2 \zeta\_i} \frac{z\_{nji} z\_{nki}}{\omega\_i^2}
\end{equation}
And we see that the peak gain for a mode is the DC gain of the same mode divided by \\(2 \xi\_i\\).
And we see that the peak gain for a mode is the DC gain of the same mode divided by \\(2 \zeta\_i\\).
If the same value of \\(\xi\\) is used for all modes, then all the DC gain terms are divided by the same \\(2 \xi\\) terms and the relative amplitude of the DC gains and peak gains are the same, so there is no difference between sorting a uniform damping model using DC gain or peak gain.
If the same value of \\(\zeta\\) is used for all modes, then all the DC gain terms are divided by the same \\(2 \zeta\\) terms and the relative amplitude of the DC gains and peak gains are the same, so there is no difference between sorting a uniform damping model using DC gain or peak gain.
However, if the modes have different damping the relationship between the DC gain and peak gain for all the modes is not a constant value and peak gain must be used to rank modes for importance.
content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md
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@@ -345,21 +345,21 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
\\[ diag(\mu\_i) \ddot{z} + \Phi^T C \Phi + diag(\mu\_i \omega\_i^2) z = \Phi^T f \\]
If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
\\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
\\[ \Phi^T C \Phi = diag(2 \zeta\_i \mu\_i \omega\_i) \\]
One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\zeta\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
And we obtain decoupled modal equations \eqref{eq:modal\_eom}.
<div class="cbox">
\begin{equation}\label{eq:modal\_eom}
\ddot{z} + 2 \xi \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
\ddot{z} + 2 \zeta \Omega \dot{z} + \Omega^2 z = z^{-1} \Phi^T f
\end{equation}
with:
- \\(\xi = diag(\xi\_i)\\)
- \\(\zeta = diag(\zeta\_i)\\)
- \\(\Omega = diag(\omega\_i)\\)
- \\(\mu = diag(\mu\_i)\\)
@@ -373,12 +373,12 @@ Typical values of the modal damping ratio are summarized on [Table 2](#table--ta
Typical Damping ratio
</div>
| **Damping Ratio** | **Application** |
|--------------------------------|-------------------------|
| \\(\xi \simeq 0.001 - 0.005\\) | Space structures |
| \\(\xi \simeq 0.01 - 0.02\\) | Mechanical engineering |
| \\(\xi \simeq 0.05\\) | Civil engineering |
| \\(\xi \simeq 0.2\\) | When ground is involved |
| **Damping Ratio** | **Application** |
|----------------------------------|-------------------------|
| \\(\zeta \simeq 0.001 - 0.005\\) | Space structures |
| \\(\zeta \simeq 0.01 - 0.02\\) | Mechanical engineering |
| \\(\zeta \simeq 0.05\\) | Civil engineering |
| \\(\zeta \simeq 0.2\\) | When ground is involved |
The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
@@ -407,7 +407,7 @@ With:
- \\(D\_i(\omega)\\) is the dynamic amplification factor of mode \\(i\\) given by
\begin{equation}\label{eq:dynamic\_amplification}
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \xi\_i \omega/\omega\_i}
D\_i(\omega) = \frac{1}{1 - \omega^2/\omega\_i^2 + 2 j \zeta\_i \omega/\omega\_i}
\end{equation}
<a id="figure--fig:neglected-modes"></a>
@@ -489,7 +489,7 @@ By looking at [Figure 7](#figure--fig:collocated-control-frf), we see that negle
The open-loop transfer function of a lighly damped structure with a collocated actuator/sensor pair can be written:
\begin{equation}\label{eq:frf\_lighly\_damped\_collocated}
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \xi\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \xi\_j s /\omega\_j + 1)}
G(s) = G\_0 \frac{\Pi\_i(s^2/z\_i^2 + 2 \zeta\_i s/z\_i + 1)}{\Pi\_j(s^2/\omega\_j^2 + 2 \zeta\_j s /\omega\_j + 1)}
\end{equation}
The corresponding Bode plot is represented in [Figure 9](#figure--fig:alternating-p-z). Every imaginary pole at \\(\pm j\omega\_i\\) introduces a \\(\SI{180}{\degree}\\) phase lag and every imaginary zero at \\(\pm jz\_i\\) introduces a phase lead of \\(\SI{180}{\degree}\\).
@@ -576,14 +576,14 @@ The transfer function between the total force and the current \\(i\\) applied to
<div class="cbox">
\begin{equation}\label{eq:proof\_mass\_eq}
\frac{F}{i} = \frac{-s^2 T}{s^2 + 2\xi\_p \omega\_p s + \omega\_p^2}
\frac{F}{i} = \frac{-s^2 T}{s^2 + 2\zeta\_p \omega\_p s + \omega\_p^2}
\end{equation}
with:
- \\(T\\) is the transducer constant
- \\(\omega\_p = \frac{k}{m}\\) is the natural frequency of the spring-mass system
- \\(\xi\_p\\) is the damping ratio
- \\(\zeta\_p\\) is the damping ratio
</div>
@@ -609,7 +609,7 @@ If \\(x\_0\\) is the displacement of the support and if the voice coil is open (
By using the two equations, we obtain:
\begin{equation}\label{eq:geophone}
\frac{e}{\dot{x\_0}} = \frac{-s^2 T}{s^2 + 2\xi\_p\omega\_p s + \omega\_p^2}
\frac{e}{\dot{x\_0}} = \frac{-s^2 T}{s^2 + 2\zeta\_p\omega\_p s + \omega\_p^2}
\end{equation}
Above the corner frequency, the gain of the geophone is equal to the transducer constant \\(T\\).
content/book/schmidt20_desig_high_perfor_mechat_third_revis_edition.md
+8 -8
View File
@@ -428,7 +428,7 @@ In this chapter some important lessons have been learned, which are summarised a
- Stiffness, whether it is created mechanically or by means of a control system, is determinative for precision
- Every mechanical structure can be modelled as a combination of bodies, springs and dampers, either as separate bodies or as finite elements within a body
- Phase is prominent factor regarding the possibility to control the motion of a mechanical system
- The quality factor \\(Q\\) and damping ratio \\(\xi\\) are inverse proportional. Each have their practical value.
- The quality factor \\(Q\\) and damping ratio \\(\zeta\\) are inverse proportional. Each have their practical value.
- A damping placed in parallel with a lumped mass-spring system limits the magnitude of the resonance at the natural frequency.
However, it also increases the transmissibility at frequencies above the native frequency.
- The dynamic behavior of a complex system and its response to a stimulus can be derived and understood by viewing it as a superposition of contributions of its eigenmodes, each with its own mode-shape, eigenfrequency, modal parameters and damping.
@@ -454,7 +454,7 @@ The core of the control system is the _plant_, which is the physical system that
<a id="table--tab:walk-control-loop"></a>
<div class="table-caption">
<span class="table-number"><a href="#table--tab:walk-control-loop">Table 3</a>:</span>
Symbols used in <a href="#org0e13ef0">3</a>
Symbols used in <a href="#org9f480e2">3</a>
</div>
| Symbol | Meaning | Unit |
@@ -633,7 +633,7 @@ The measured frequency-response of the scanning unit taken as as an example is s
A mathematical model of a seconder-order mass-spring system with a force input is fitted to this measured response:
\begin{equation}
G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\xi\_f \omega\_0 s + \omega\_0^2}
G(s) = \frac{C\_f \omega\_0^2}{s^2 + 2\zeta\_f \omega\_0 s + \omega\_0^2}
\end{equation}
When positioning at high scanning frequency, the resonance at the first mode-shape of this scanning unit causes oscillations, which adversely affect the tracking accuracy.
@@ -641,7 +641,7 @@ In order to solve this problem, a feedforward controller is designed that compen
This means that the transfer function of the feedforward controller is:
\begin{equation}
C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{\omega\_0^2}
C\_{ff}(s) = \frac{s^2 + 2 \zeta\_f \omega\_0 s + \omega\_0^2}{\omega\_0^2}
\end{equation}
However, such controller needs to be modified in such a way that it becomes _realizable_.
@@ -649,23 +649,23 @@ In this case, it is decided to create a resulting overall transfer function of t
In order to realize this controller, first two poles have to be added:
\begin{equation}
C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{s^2 + 2\cdot 1 \cdot \omega\_0 s + \omega\_0^2}
C\_{ff}(s) = \frac{s^2 + 2 \zeta\_f \omega\_0 s + \omega\_0^2}{s^2 + 2\cdot 1 \cdot \omega\_0 s + \omega\_0^2}
\end{equation}
There the damping ratio of the resulting dynamics is chosen to be \\(\xi = 1\\).
There the damping ratio of the resulting dynamics is chosen to be \\(\zeta = 1\\).
This feedforward controller is basically a _notch filter_.
In order to create an additional attenuation of higher frequency resonances due to flexible-body mode-shapes, another first-order pole is added at the first eigenfrequency:
\begin{equation}
C\_{ff}(s) = \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0)(s^2 + 2\omega\_0 s + \omega\_0^2)}
C\_{ff}(s) = \frac{s^2 + 2 \zeta\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0)(s^2 + 2\omega\_0 s + \omega\_0^2)}
\end{equation}
Then this controller is connected in series with the scanning unit, the anti-resonance of the controller and the resonance of the piezo-scanner cancel each other out:
\begin{align}
G\_{t,ff}(s) &= G(s)G\_{ff}(s) \\\\
&= \frac{C\_f}{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \xi\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\
&= \frac{C\_f}{s^2 + 2 \zeta\_f \omega\_0 s + \omega\_0^2} \frac{s^2 + 2 \zeta\_f \omega\_0 s + \omega\_0^2}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}} \\\\
&= \frac{C\_f}{(s + \omega\_0){s^2 + 2 \omega\_0 s + \omega\_0^2}}
\end{align}