# Vibrations

1- A vibrating mass of 300 kg, mounted on a massless support by a spring of stiffness 40,000 N/m and a damper of unknown damping coefficient, is observed to vibrate with a 10-mm amplitude while the support vibration has a maximum amplitude of only 2.5 mm (at resonance). Calculate the damping constant and the amplitude of the force on the base.

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) = x, sinwt. Given: 1=1m, k=1000N/m, c=500N-s/m,m=10kg, x, = 1cm, and w= 10rad/s. In addition, 5. What is the frequency of the steady-state output in rad/s? 6. What is the steady-state amplitude in radians? 7. Determine the steady-state amplitude if c = 0 and compare this value to the one determined with damping. 8. For the case when there is damping, c ≠ 0, what is the phase angle ∅ if the displacement is of the form A* sin(wt+∅)?

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.

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.

In Problem 1.71, using the given data as reference, find the variation of the damping constant E when a. r is varied from 0.5 cm to 1.0 cm. b. h is varied from 0.05 cm to 0.10 cm. c. a is varied from 2 cm to 4 cm.

A mass, m is supported on a light beam which is free to rotate about the pivot at O as

4. Two horizontal frictionless rails make an angle with each other, as shown in the figure. Each rail has a bead of mass m on it, and the beads are connected by a spring with spring constant k and relaxed length zero. Assume that one of the rails is positioned a tiny distance above the other, so that the beads can pass freely through the crossing. Find the general solution for the positions of each mass as functions of time.