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content/book/ewins00_modal.md
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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.