Damped Forced Vibration Exercises

Example 1.1

The spring system is connected to a crosshead that oscillates vertically when the wheel rotates with a constant angular velocity of ω. If the amplitude of the steady-state vibration is observed to be 400 mm, and the springs each have a stiffness k=2500 N/m, determine the two possible values of ω at which the wheel must rotate. The block has a mass of 50 kg.

$$\begin{align} & Y=0.4m \\ & k=2500N/m \\ & \underline{m=50kg} \\ & {{k}_{eq}}=2k=2\left( 2500 \right)=5000 \\ & {{\omega }_{n}}=\sqrt{\frac{{{k}_{eq}}}{m}}=\sqrt{\frac{5000}{50}}=10\text{ rad/s} \\ & {{\left( {{Y}_{P}} \right)}_{\max }}=\frac{{{\delta }_{0}}}{1-{{\left( \frac{\omega }{{{\omega }_{n}}} \right)}^{2}}} \\ & \pm 0.4=\frac{0.2}{1-{{\left( \frac{\omega }{10} \right)}^{2}}} \\ & {{\left( \frac{\omega }{10} \right)}^{2}}=1\pm 0.5 \\ & \frac{{{\omega }_{1}}^{2}}{100}=1.5\Rightarrow 12.2\text{ rad/s} \\ & \frac{{{\omega }_{2}}^{2}}{100}=0.5\Rightarrow 7.07\text{ rad/s} \\ \end{align}$$

Example 1.2

If the 30 kg block is subjected to a periodic force of 300 sin 5*t, k=1500 N/m, and c=300 Ns/m

Determine the equation that describes the steady-state vibration as a function of time.

$$\begin{align} & {{k}_{eq}}=2k=2\left( 1500 \right)=3000\text{ N/m} \\ & {{\omega }_{eq}}=\sqrt{\frac{{{k}_{eq}}}{m}}=\sqrt{\frac{3000}{30}}=10\text{ rad/s} \\ & {{\text{c}}_{c}}=2m{{\omega }_{n}}=2\cdot 30\cdot 10=600\text{ Ns/m} \\ & \frac{c}{{{c}_{c}}}=\frac{300}{600}=0.5 \\ & Y=\frac{\frac{{{F}_{0}}}{{{k}_{eq}}}}{\sqrt{{{\left[ 1-{{\left( \frac{\omega }{{{\omega }_{n}}} \right)}^{2}} \right]}^{2}}+{{\left[ \left( 2\frac{c}{{{c}_{c}}} \right)\left( \frac{\omega }{{{\omega }_{n}}} \right) \right]}^{2}}}} \\ & Y=\frac{\frac{300}{3000}}{\sqrt{{{\left[ 1-{{\left( \frac{5}{10} \right)}^{2}} \right]}^{2}}+{{\left[ \frac{2\cdot 0.5\cdot 5}{10} \right]}^{2}}}} \\ & Y=0.1109\text{ m} \\ & {{\text{y}}_{P}}=0.111\sin \left( 5t-0.588 \right)\text{ m} \\ \end{align}$$

Example 1.3

Determine the differential equation of motion for the damped vibratory system shown. What type of motion occurs? 

$$\begin{align} & \sum{{{F}_{y}}=m{{a}_{y}};} \\ & mg-k\left( y+{{y}_{st}} \right)-2c\dot{y}=m\ddot{y}, \\ & m\ddot{y}+ky+2c\dot{y}+k{{y}_{st}}-mg=0, \\ & k{{y}_{st}}-mg=0, \\ & m\ddot{y}+2c\dot{y}+ky=0 \\ & 25\ddot{y}+400\dot{y}+100y=0 \\ & \ddot{y}+16\dot{y}+4y=0 \\ & {{\omega }_{n}}=\sqrt{\frac{k}{m}}=\sqrt{\frac{4}{1}}=2\text{ rad/s} \\ & {{\text{c}}_{c}}=2m{{\omega }_{n}}=2\cdot 25\cdot 2=100\text{ Ns/m} \\ \end{align}$$

Since damping of the system is larger than the critical damping the system will not vibrate. In conclusion the system is overdamped.