Chapter 13, Problem 13.23P

To determine

(a)

The expression for $\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}$.

The expression for $\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}$.

Answer to Problem 13.23P

The expression for $\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}$ is $\frac{\sqrt{{\left(\lambda -\mu r^2\right)}^2+{\left(2\varsigma r\right)}^2}}{\sqrt{\left|{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2+{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2\right|}}$.

The expression for $\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}$ is $\frac{\sqrt{\left|\left({\lambda }^2+{\left(2\varsigma rj\right)}^2\right)\right|}}{\left|\sqrt{{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2+{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2}\right|}$.

Explanation of Solution

Given information:

The main mass is $m_1$.

Write the expression for force of mass 1.

$\begin{array}{l}f=m_1{\ddot{x}}_1+c\left({\dot{x}}_1-{\dot{x}}_2\right)+\frac{k_1}{2}{x}_1+\frac{k_1}{2}{x}_1+k_2\left(x_1-x_2\right)\\ f=m_1{\ddot{x}}_1+c{\dot{x}}_1+k_1{x}_1+k_2{x}_2-k_2{x}_2-c{\dot{x}}_2\end{array}$           ...... (I)

Here, mass of body 1 is $m_1$, damping coefficient is $c$, velocity of mass 1 is ${\dot{x}}_1$, velocity of mass 2 is ${\dot{x}}_2$, stiffness is $k$, displacement is $x$, acceleration of mass 1 is ${\ddot{x}}_1$ and the coefficient of friction is $f$.

Write the expression for absorber mass.

$\begin{array}{l}{m}_2{\ddot{x}}_2+c{\dot{x}}_2-c{\dot{x}}_1+k_2\left(x_2-x_1\right)\\ m_2{\ddot{x}}_2=-c{\dot{x}}_2+c{\dot{x}}_1-k_2{x}_2+k_2{x}_1\end{array}$          . (II)

Here, mass of the absorber is $m_2$. and the acceleration of absorber is ${\ddot{x}}_2$.

Apply Laplace transform on Equation (I).

$\begin{array}{l}L\left(f\right)=L\left[m_1{\ddot{x}}_1+c{\dot{x}}_1+k_1{x}_1+k_2{x}_2-k_2{x}_2-c{\dot{x}}_2\right]\\ F\left(s\right)=\left(m_1{s}^2+cs+k_1+k_2\right)X_1\left(s\right)-\left(cs+k_2\right)X_2\left(s\right)\end{array}$           (III)

Take Laplace transform of equation (II).

$\begin{array}{l}L\left[m_2{\ddot{x}}_2\right]=L\left[-c{\dot{x}}_2+c{\dot{x}}_1-k_2{x}_2+k_2{x}_1\right]\\ m_2{s}^2{X}_2\left(s\right)=-cs{X}_2\left(s\right)+cs{X}_1\left(s\right)-k_2{X}_2\left(s\right)+k_2{X}_1\left(s\right)=0\\ -\left(cs+k_2\right)X_1\left(s\right)+\left(m_2{s}^2+cs+k_2\right)X_2\left(s\right)=0\end{array}$           .. (IV)

Apply Cramer rule in Equation (III) and (IV).

$\left(\begin{array}{cc}{m}_1{s}^2+cs+k_1+k_2& -\left(cs+k_2\right)\\ -\left(cs+k_2\right)& m_2{s}^2+cs+k_2\end{array}\right)\left(\begin{array}{l}{X}_1\left(s\right)\\ X_2\left(s\right)\end{array}\right)=\left(\begin{array}{l}F\left(s\right)\\ 0\end{array}\right)$           ...... .. (V)

Further solve Equation (V).

$\begin{array}{l}D\left(s\right)=\left[\begin{array}{l}{m}_1{m}_2{s}^4+c{m}_2{s}^3+k_1{m}_2{s}^2+k_2{m}_2{s}^2+m_{1}c{s}^3+{\left(cs\right)}^2+\\ c{k}_{1}s+k_{2}cs+m_1{k}_2{s}^2+k_{2}cs+k_1{k}_2+k_2{k}_2-\left(c^2{s}^2+k_2^2+2c{k}_{2}s\right)\end{array}\right]\\ D\left(s\right)=m_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3+\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2\end{array}$           (VI)

Write the expression for parameter $\mu$.

$\mu =\frac{m_2}{m_1}$           ...... ... (VII)

Write the expression for parameter ${\omega }_1^2$.

${\omega }_1^2=\frac{k_1}{m_1}$           ...... .. (VIII)

Write the expression for parameter $\varsigma$.

$\varsigma =\frac{c}{2\sqrt{m_{1}k}}$          . (IX)

Write the expression for parameter $r$.

$r=\frac{\omega }{{\omega }_1}$           .. (X)

Here, angular speed of main mass is ${\omega }_1$.

Write the expression for parameter ${\omega }_2^2$.

${\omega }_2^2=\frac{k_2}{m_2}$           ...... .. (XI)

Write the expression for parameter $\lambda$.

$\lambda =\frac{k_2}{k_1}$          . (XII)

Write the expression for parameter $\alpha$.

$\alpha =\frac{{\omega }_1}{{\omega }_2}$          . (XIII)

Substitute $j\omega$ for $s$ in Equation (VI).

$\begin{array}{c}D\left(j\omega \right)=\left[\left(j\omega \right)\left(\begin{array}{l}{m}_1{m}_2{\left(j\omega \right)}^3+c\left(m_2+m_1\right){\left(j\omega \right)}^2\\ +\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)\left(j\omega \right)+c{k}_1+k_1{k}_2\end{array}\right)\right]\\ =\left|\begin{array}{l}{m}_1{m}_2{\omega }^4-c\left(m_2+m_1\right){\omega }^{3}j+\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)\\ -{\omega }^2+c{k}_{1}j\omega +k_1{k}_2\end{array}\right|\end{array}$           ...... (XIV)

Substitute $\frac{m_2}{m_1}$ for $\mu$, $\frac{k}{m_1}$ for ${\omega }_1^2$, $\varsigma$ for $\frac{c}{2\sqrt{m_{1}k}}$, $r$ for $\frac{\omega }{{\omega }_1}$, ${\omega }_2^2$ for $\frac{k_2}{m_2}$, $\lambda$ for $\frac{k_2}{k_1}$ and $\frac{{\omega }_1}{{\omega }_2}$ for $\alpha$ in Equation (XIV).

$D\left(j\omega \right)=\left|m_1^2{\omega }_1^4\left[\mu r^4-\left(1+\gamma +\mu \gamma \right)+\mu {\alpha }^2\right]+\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\right|$           (XV)

Apply Cramer rule for Equation (VI) to obtain $D_1\left(s\right)$.

$\begin{array}{c}{D}_1\left(s\right)=\left(\begin{array}{cc}F\left(s\right)& -\left(cs+k_2\right)\\ 0& m_2{s}^2+cs+k_2\end{array}\right)\\ =F\left(s\right)\left(m_2{s}^2+cs+k_2\right)+0\times \left(-\left(cs+k_2\right)\right)\\ =F\left(s\right)\left(m_2{s}^2+cs+k_2\right)\end{array}$          . (XVI)

Apply Cramer rule for Equation (VI) to obtain $D_2\left(s\right)$.

$\begin{array}{c}{D}_2\left(s\right)=\left(\begin{array}{cc}{m}_2{s}^2+cs+k_2+k_1& F\left(s\right)\\ -\left(cs+k_2\right)& 0\end{array}\right)\\ =0\times \left(m_2{s}^2+cs+k_2+k_1\right)-F\left(s\right)\times \left(-\left(cs+k_2\right)\right)\\ =F\left(s\right)\left(cs+k_2\right)\end{array}$          . (XVII)

Write the expression for $X_1\left(s\right)$.

$X_1\left(s\right)=\frac{D_1\left(s\right)}{D\left(s\right)}$           ...... .. (XVIII)

Substitute $F\left(s\right)\left(m_2{s}^2+cs+k_2\right)$ for $D_1\left(s\right)$ and $m_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3+\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2$ for $D\left(s\right)$ in equation (XVIII).

$X_1\left(s\right)=\frac{F\left(s\right)\left(m_2{s}^2+cs+k_2\right)}{m_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3+\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2}$          . (XIX)

Substitute $j\omega$ for $s$ in Equation (XIX).

$\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}=\frac{\left(m_2{\left(j\omega \right)}^2+cj\omega +k_2\right)}{\left[\begin{array}{l}{m}_1{m}_2{\left(j\omega \right)}^4+c\left(m_1+m_2\right){\left(j\omega \right)}^3\\ +\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right){\left(j\omega \right)}^2+c{k}_{1}j\omega +k_1{k}_2\end{array}\right]}$           (XX)

Substitute $\frac{m_2}{m_1}$ for $\mu$, $\frac{k}{m_1}$ for ${\omega }_1^2$, $\varsigma$ for $\frac{c}{2\sqrt{m_{1}k}}$, $r$ for $\frac{\omega }{{\omega }_1}$, ${\omega }_2^2$ for $\frac{k_2}{m_2}$, $\lambda$ for $\frac{k_2}{k_1}$ and $\frac{{\omega }_1}{{\omega }_2}$ for $\alpha$ in Equation (XIX).

$k_1\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}=\frac{{\left(k_1\right)}^2\sqrt{{\left(\lambda -\mu r^2\right)}^2+{\left(2\varsigma r\right)}^2}}{m_1^2{\omega }_1^4\left|\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]+\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\right|}$ .... (XXI)

Substitute $m_1{\omega }_1^2$ for $k$ in Equation (XXI).

$\begin{array}{l}{k}_1\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}=\frac{k_1^2\sqrt{{\left(\lambda -\mu r^2\right)}^2+{\left(2\varsigma r\right)}^2}}{k_1^2\left|\begin{array}{l}\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]\\ +\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\end{array}\right|}\\ k_1\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}=\frac{\sqrt{{\left(\lambda -\mu r^2\right)}^2+{\left(2\varsigma r\right)}^2}}{\sqrt{\left|\begin{array}{l}{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2\\ +{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2\end{array}\right|}}\end{array}$ ......... (XXII)

Write the expression for $X_2\left(s\right)$.

$X_2\left(s\right)=\frac{D_2\left(s\right)}{D\left(s\right)}$           ....... (XXIII)

Substitute $F\left(s\right)\left(cs+k_2\right)$ for $D_2\left(s\right)$ and $m_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3+\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2$ for $D\left(s\right)$ in Equation (XXIII).

$\begin{array}{l}{X}_2\left(s\right)=\frac{F\left(s\right)\left(cs+k_2\right)}{\left[\begin{array}{l}{m}_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3\\ +\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2\end{array}\right]}\\ \frac{X_2\left(s\right)}{F\left(s\right)}=\frac{\left(cs+k_2\right)}{\left[\begin{array}{l}{m}_1{m}_2{s}^4+c\left(m_1+m_2\right)s^3\\ +\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right)s^2+c{k}_{1}s+k_1{k}_2\end{array}\right]}\end{array}$ ........ (XXIV)

Substitute $j\omega$ for, $\frac{m_2}{m_1}$ for $\mu$, $\frac{k}{m_1}$ for ${\omega }_1^2$, $\varsigma$ for $\frac{c}{2\sqrt{m_{1}k}}$, $r$ for $\frac{\omega }{{\omega }_1}$, ${\omega }_2^2$ for $\frac{k_2}{m_2}$, $\lambda$ for $\frac{k_2}{k_1}$ and $\frac{{\omega }_1}{{\omega }_2}$ for $\alpha$ in Equation (XXIV).

$k_1\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}=\frac{k_1{k}_1\left|\left(\lambda +2\varsigma rj\right)\right|}{\left[\begin{array}{l}{m}_1{m}_2{\omega }^4-c\left(m_1+m_2\right)j{\omega }^3\\ -\left(m_2{k}_1+m_2{k}_2+m_1{k}_2\right){\omega }^2+c{k}_{1}j\omega +k_1{k}_2\end{array}\right]}$           ...... (XXV)

Solve Equation (XV) and (XXV).

$k_1\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}=\frac{k_1{k}_1\left|\left(\lambda +2\varsigma rj\right)\right|}{m_1^2{\omega }_1^4\left|\left[\begin{array}{l}\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2\\ +\mu {\alpha }^2\end{array}\right]+\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\right|}$ ....... (XXVI)

Substitute $m_1{\omega }_1^2$ for $k_1$ in RHS of Equation (XXVI).

$\begin{array}{l}{k}_1\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}=\frac{m_1{\omega }_1^2{m}_1{\omega }_1^2\left|\left(\lambda +2\varsigma rj\right)\right|}{m_1^2{\omega }_1^4\left|\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]+\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\right|}\\ =\frac{\left|\left(\lambda +2\varsigma rj\right)\right|}{\left|\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]+\left[2\varsigma r-2\left(1+\mu \right)r^3\right]j\right|}\\ =\frac{\sqrt{\left|\left({\lambda }^2+{\left(2\varsigma rj\right)}^2\right)\right|}}{\left|\sqrt{{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2+{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2}\right|}\end{array}$.

Conclusion:

The expression for $\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}$ is $\frac{\sqrt{{\left(\lambda -\mu r^2\right)}^2+{\left(2\varsigma r\right)}^2}}{\sqrt{\left|{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2+{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2\right|}}$.

The expression for $\frac{X_2\left(j\omega \right)}{F\left(j\omega \right)}$ is $\frac{\sqrt{\left|\left({\lambda }^2+{\left(2\varsigma rj\right)}^2\right)\right|}}{\left|\sqrt{{\left[\mu r^4-\left(1+\lambda +\mu \lambda \right)r^2+\mu {\alpha }^2\right]}^2+{\left[{\left(2\varsigma r\right)}^2\left(1-\left(1+\mu \right)r^2\right)\right]}^2}\right|}$.

To determine

(b)

The plot between $\frac{k{X}_1}{F}$ and $\frac{\omega }{\sqrt{\frac{k}{m_1}}}$.

Answer to Problem 13.23P

The plot between $\frac{k{X}_1}{F}$ and $\frac{\omega }{\sqrt{\frac{k}{m_1}}}$ is shown in Figure-(1).

Explanation of Solution

Given information:

The main mass is $m_1$ and the absorber mass is $m_2$.

To plot the graph, consider some constant values such as $10$ for $m_1$, $30$ for $m_2$, ${10}^4$ for $k$ and $\left[0,2\right]$ for $r$.

Calculation:

Substitute $10$ for $m_1$ and $30$ for $m_2$ in equation (VII).

$\begin{array}{c}\mu =\frac{30}{10}\\ =0.3333\end{array}$

Substitute ${10}^4$ for $k$ and $10$ for $m_1$ in Equation (VIII).

$\begin{array}{l}{\omega }_1^2=\frac{{10}^4}{10}\\ {\omega }_1^2=1000\\ {\omega }_1=\sqrt{1000}\\ {\omega }_1=31.6\end{array}$

Substitute $2\times {10}^4$ for $k_2$ in Equation (XI).

$\begin{array}{l}{\omega }_2^2=\frac{2\times {10}^4}{30}\\ {\omega }_2^2=666.666\\ \omega =25.8\end{array}$

Substitute $2\times {10}^4$ for $k_2$ and ${10}^4$ in equation (XII).

$\begin{array}{c}\lambda =\frac{2\times {10}^4}{{10}^4}\\ =2\end{array}$

Substitute $31.6$ for ${\omega }_1$ and $25.8$ for $\omega$ in equation (XIII).

$\begin{array}{c}\alpha =\frac{31.6}{25.8}\\ =1.2\end{array}$

Assume $0$ for $\omega$ and $0.1$ for $\varsigma$.

Substitute $0$ for $\omega$ and $31.6$ for ${\omega }_1$ in Equation (X).

$\begin{array}{c}r=\frac{0}{31.6}\\ =0\end{array}$

Substitute $0$ for $r$, $2$ for $\lambda$, $0.3333$ for $\mu$, $1.2$ for $\alpha$ and $0.1$ for $\varsigma$ in Equation (XXII).

$\begin{array}{c}{k}_1\frac{X_1\left(j\omega \right)}{F\left(j\omega \right)}=\frac{\sqrt{{\left(2-\left(0.3333\right){\left(0\right)}^2\right)}^2+{\left(2\times 0.1\times 0\right)}^2}}{\left|\left[\begin{array}{l}{\left[\mu \times 0^4-\left(1+2+0.3333\times 2\right)\times 0^2+0.3333\times {\left(1.2\right)}^2\right]}^2+\\ {\left[{\left(2\times 0.1\times 0\right)}^2\left(\left(1-\left(1+0.3333\right)\right)0^2\right)\right]}^2\end{array}\right]\right|}\\ =\frac{2}{\sqrt{0.23}}\\ \approx 4.16\end{array}$

Substitute $10$ for $m$, $0.3$ for $\varsigma$ and ${10}^4$ in Equation (IX).

$\begin{array}{l}0.3=\frac{2}{2\sqrt{10\times {10}^4}}\\ c=2\sqrt{10\times {10}^4}\times 0.3\\ c=1789.7\end{array}$

Prepare a table for various value of $\frac{k_1{X}_1}{F_1}$, $\frac{\omega }{\sqrt{\frac{k}{m_1}}}$ and $\varsigma$.

$\omega$	$\varsigma$	$\frac{k{X}_1\left(j\omega \right)}{F\left(j\omega \right)}$	$\frac{\omega }{\sqrt{\frac{k}{m_1}}}$
$0$	$0.1$	$4.2$	$0$
$10$	$0.1$	$0.61$	$0.316$
$20$	$0.1$	$1.99$	$0.633$
$30$	$0.1$	$0.67$	$0.949$
$0$	$0.3$	$4.16$	$0$
$10$	$0.3$	$0.61$	$0.316$
$20$	$0.3$	$2$	$0.633$
$30$	$0.3$	$0.7$	$0.949$
$0$	$0.5$	$4.16$	$0$
$10$	$0.5$	$0.62$	$0.316$
$20$	$0.5$	$2.01$	$0.633$
$30$	$0.5$	$0.76$	$0.949$
$0$	$1$	$4.16$	$0$
$10$	$1$	$0.64$	$0.316$
$20$	$1$	$2.04$	$0.633$
$30$	$1$	$0.98$	$0.949$

The figure below shows the plot between $\frac{k_1{X}_1}{F_1}$ and $\frac{\omega }{\sqrt{\frac{k}{m_1}}}$.

        Figure-(1)

Conclusion:

The plot between $\frac{k_1{X}_1}{F_1}$ and $\frac{\omega }{\sqrt{\frac{k}{m_1}}}$ is shown below.

        Figure-(1).