Vibrations

4- A single-cylinder air compressor of mass 100 kg is mounted on rubber mounts. The stiffness and damping constants of the rubber mounts are given by 106 N/m and 2000 N-s/m, respectively. If the unbalance of the compressor is equivalent to a mass 0.1 kg located at the end of the crank (point A), determine the response of the compressor at a crank speed of 3000 rpm. Assume r = 10 cm and /= 40 cm.

For the pendulum mechanism shown below, which is pivoted at O, assume that the mass of the rod, spring, and damper are negligible. What driving frequency will cause resonance? Assume that F(t)= F cost.

What value of c will cause the system to be critically damped? In the figure above, let k = 4000 N/m, 1₁ =1.5m and l₂ = .5m, 1 = 1m, and m = 40kg. This time, consider the beam to have a mass of 40 kg (The length of the beam is 1+4) It is pivoted about O and is assumed to be rigid.

Design the dashpot (i.e., calculate c) so that the damping ratio of the system is 0.2.

Also, determine the amplitude of the steady-state response for 8(t) if a 10 Newton force is applied to the mass as indicated in the figure with a frequency of 10 rad/s; that is, F(t) = 10 cos at where = 10rad / s. A uniform bar of mass mis pivoted at O and supported at the ends by two springs as shown in the figure below. End P of the spring PQ is subjected to a sinusoidal displacement x(t) = x₁ sin cot. Given: 1=1m, k=1000N/m, c=500N-s/m, m=10kg,x=1cm, and = 10rad/s. In addition,

A viscously damped system (spring-mass-damper) has a stiffness of 5000 N/m, critical damping constant of 0.2 N-s/mm, and a log decrement of 2.0. If the system is given an initial velocity of 1m/s, determine the maximum displacement of the system.

A viscously damped system (spring-mass-damper) has a stiffness of 5000 N/m, critical damping constant of 0.2 N-s/mm, and a log decrement of 2.0. If the system is given an initial velocity of 1m/s, determine the maximum displacement of the system.

2. A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0.1 m from the center. The damping ratio of the system is measured to be = 0.06 (viscous damping) and its natural frequency is 7.5 Hz. Calculate the amplitude of the steady-state displacement of the

4.11 Compute the natural frequencies and mode shapes of the following system: \left[\begin{array}{ll} 4 & 0 \\ 0 & 1 \end{array}\right] \ddot{\mathbf{x}}(t)+10\left[\begin{array}{cc} 4 & -2 \\ -2 & 1 \end{array}\right] \mathbf{x}(t)=\mathbf{0} Calculate the response of the system to the initial conditions: Xo = [1 2] and vo= \left[\begin{array}{ll} \sqrt{20} & -2 \sqrt{20} \end{array}\right]^{T}

8. Consider the system in Fig. 2, write the equation of motion,and calculate the response assuming (a) that the system is initially at rest, and (b) that the system has an initial displacement of 0.05 m.