Question

Mechanical vibration subject

3. a. Consider the system of Figure 3. If C1 = C2 = C3 = 0, develops the equation of motion and predict the mass and stiffness matrices. Note that setting k3 = 0 in your solution should result in the stiffness matrix given by [ky + kz -k2 kz b. constructs the characteristics equation from Question 3(a) for the case m1 = 9 kg, m2 = 1 kg, k1 = 24 N/m, k2 = 3 N/m, k3 = 3 N/m, and solve for the system's natural frequencies. ki k2 k3 mi m2 - C2 X2 Figure 3

Answer 1

Given:

C1 = C2 = C3 = 0

ki 24N/m
,  
k2 = 3N/m
,
k3 = 3N/m
,  
mi 9kg
, $m_{2}=1kg$

So this becomes an undamped free vibration.

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$m_{1}\ddot{x}_{1}=-k_{1}x_{1}+k_{2}(x_{2}-x_{1})$

$m_{2}\ddot{x}_{2}=-k_{3}x_{2}-k_{2}(x_{2}-x_{1})$

The equations of motion for the two masses can be written as

$m_{1}\ddot{x}_{1}+(k_{1}+k_{2})x_{1}-k_{2}x_{2}=0$

$m_{2}\ddot{x}_{2}+(k_{2}+k_{3})x_{2}-k_{2}x_{1}=0$

We assume that the motions x1 and x2 are periodic harmonic motions of the form,

$x_{1}=X_{1}sin(\omega _{n}t+\phi )$

$x_{2}=X_{2}sin(\omega _{n}t+\phi )$

where X1 and X2 are amplitudes of the vibration and $\omega _{n}$ is the natural frequency.

Substituting for x1 and x2 in the equations of motion we get,

$X_{1}(k_{1}+k_{2}-m_{1}\omega _{n}^{2})-X_{2}k_{2}=0$

$-X_{1}k_{2}+X_{2}(k_{3}+k_{2}-m_{2}\omega _{n}^{2})=0$

This can be written as

$\begin{vmatrix} k_{1}+k_{2}-m_{1}\omega _{n}^{2} & -k_{2}\\ -k_{2} & k_{3}+k_{2}-m_{2}\omega _{n}^{2} \end{vmatrix}=0$

Expanding this matrix this we get,

$\omega _{n}^{4}-(\frac{k_{2}+k_{3}}{m_{2}}+\frac{k_{1}+k_{2}}{m_{1}})\omega _{n}^{2}+\frac{k_{1}k_{3}+k_{1}k_{2}+k_{2}k_{3}}{m_{1}m_{2}}=0$

$\frac{k_{2}+k_{3}}{m_{2}}+\frac{k_{1}+k_{2}}{m_{1}}=\frac{3+3}{1}+\frac{24+3}{9}=6+3=9$ ,  

$\frac{k_{1}k_{3}+k_{1}k_{2}+k_{2}k_{3}}{m_{1}m_{2}}=\frac{24*3+24*3+3*3}{9*1}=17$

Substitute  $y=\omega _{n}^{2}$ so it becomes a quadratic equation in y

$y^{2}-9y+17=0$

$y_{1,2}=\frac{9\pm \sqrt{9^{2}-4*1*17}}{2*1}=6.3,2.69$

$y_{1}=\omega _{n1}^{2}=6.30$,   $y_{2}=\omega _{n2}^{2}=2.69$

The natural frequencies $\omega _{n1}$ and $\omega _{n2}$ are,

$\omega _{n1}=\sqrt{6.3}=2.5Hz$,   $\omega _{n2}=\sqrt{2.69}=1.64Hz$