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 mass is attached at one end of a uniform bar of mass whose other end is pivote dat point O as shown in the following figure. Determine the natural frequency of vibration of the resulting pendulum for small angular displacements. (10 marks)2. I_{1 \text { aroundo }}=m_{1} l^{2} I_{2 \text { coundmiddle point }}=\frac{1}{12} m_{2} I^{2}

6. The free response of a 1000-kg automobile with stiffness ofk= 400,000N/m is observed to be of the form give in Figure 6. Modeling the automobile as a single-degree-of-freedom oscillation in the vertical direction, determine the damping coefficient if the displacement at is measured to be 2cm and0.22 cm at

A variable speed electric motor, having an unbalance, is mounted on an isolator. As the speed of the motor is increased from zero, the amplitudes of the vibration of the motor have been observed to be 0.55 in. at resonance and 0.15 in. beyond resonance (i.e. faraway from resonance). Find the damping ratio of the suspension system.

A simplified model of a washing machine is illustrated in the figure below. A bundle of wet clothes forms a mass of 1 kg (m,) in the machine and causes a rotating unbalance. The total mass of the washer includes the clothes is 20 kg (M) and the radius of the washer basket (e)is 0.25 m. Assume that the spin cycle rotates at 300 rpm. Let K be 1000 N/m and (=0.01. a.) Calculate the magnitude of the unbalance force causing the vibration of the machine. b.) What percent of the above force is transmitted to the foundation? c.) Calculate the magnitude of the force transmitted into the foundation? d.) Suggest and EXPLAIN two ways of reducing the force transmission to the foundation without altering the speed of the spin cycle.i.)

6. Solve for the steady-state response of the system illustrated in Fig. 4. y = 0.02 sin (150t), k,=10,000 N/m, k=5000 N/m,m=0.5 kg, c=8.66 N•s/m.

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.

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 machine is subjected to two harmonic motions, and the resultant motion, as displayed by an oscilloscope, is shown in Fig. 1.113. Find the amplitudes and frequencies of the two motions.

1. Two springs each have spring constant k and equilibrium length/. They are both stretched a distance and attached to a mass m and two walls, as shown in figure below. At a given instant, the right spring constant is somehow magically changed to 3k (the relaxed length remains/). What is the resulting x(t)? Take the initial position to be x = 0.