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Section 3.8 – Forced Vibrations

Directions. The following are review problems for the section. It is recommended you work the problems yourself, and then click "Answer" to check your answer. If you do not understand a problem, you can click "Video" to learn how to solve it. 
  1. A spring is stretched 6 in by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a damping constant of 0.25 lb\(\cdot\)s/ft and is acted by an external force of \( 4 \cos 2t\) lb.
    1. Find the steady-state response of this system.
    2. If the given mass is replaced by a mass \(m\), determine the value of \(m\) for which the amplitude of the steady-state response is maximum.
    3. If the mass is the same as in the problem, determine the value \(\omega\) of the frequency of the external force \( 4 \cos \omega t\) lb at which "practical resonance" occurs, i.e., the amplitude of the steady-state response is maximized.​

      1. \(u_p=u_{st}=\dfrac{120}{450.5}\cos(2t)+\dfrac{4}{450.5}\sin(2t)\)
      2. \(m=4\) slugs
      3. \(w_{res}=\sqrt{63.5}\ \dfrac{\mbox{rad}}{\mbox{sec}}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V47


  2. A mass weighing 4 lb stretches a spring 1.5 in. The mass is given a positive displacement 2 in from its equilibrium position and released with no initial velocity. Assuming that there is no damping and the mass is acted on by an external force of \(2\cos 3t\) lb,
    1. Formulate the initial value problem describing the motion of mass
    2. Solve the initial value problem.
    3. If the given external force is replaced by a force \(4\cos\omega t\) of frequency \(\omega\), find the value of \(\omega\) for which resonance occurs.​

      1. \(
              \begin{cases}
                \dfrac{1}{8}x''+32x=2\cos(3t) \\
                \\
                x(0)=\dfrac{1}{6}\ \mbox{ft} ,\ x'(0)=0\\
              \end{cases}
            \)
      2. \(x=\dfrac{151}{247(6)}\cos(16t)+\dfrac{16}{247}\cos(3t)\)
      3. \(w=16\ \dfrac{\mbox{rad}}{\mbox{s}}\)

      To see the full video page and find related videos, click the following link.
      MATH 308 WIR22A V48


  3. A 3 kg object is attached to a spring and will stretch the spring 392 mm by itself. There is no damping in the system and a forcing function of the form \(F(t)=10\cos(\omega t)\) is attached to the object and the system will experience resonance. If the object is initially displaced 20 cm downward from its equilibrium position and given a velocity of 10 cm/sec upward find the displacement at any time \(t\).​

    \(x=0.2\cos(5t)-0.02\sin(5t)+\dfrac{1}{3}t\sin(5t)\)

    To see the full video page and find related videos, click the following link.
    MATH 308 WIR22A V49