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Problems

1. A mass is suspended from a spring system as shown in figure Q1. Determine

the natural frequency of the system if k1 = k2 = 4000 N/m, k3 = 7000 N/mand m = 50 kg.

2. Find the mass m the system shown in figure Q2 such that its natural frequency

is 5Hz when k1 = 1200 N/m, k2 = 1000 N/m, k3 = 1500 N/m, k4 = k5 = 300N/m.

3. A belt is placed over a pulley of mass 12kg and radius of gyration 100 mm as

shown in Figure Q3. It is attached to a mass of 4kg at one end and to a springof constant k = 500 N/m at the other end. If the mass is moved 75 mm down

from its equilibrium position and released, determine;

(a) the natural frequency and period of vibration

(b) the maximum velocity of the cylinder during oscillations.

Assume friction is sufficient to prevent the belt from slipping on the pulley.

xk

3

m

k1

k2

Figure Q1 Figure Q2

k4

m

k2

k3

k5

k1

150 mm

4kg

B C

A

Figure Q3 Figure Q4

x

k

m

c

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4. A mass weighing 45 N is supported by a spring and a damper as shown in

figure Q4. The time for 20 complete oscillations is found to be 8 s and the

ratio of first downward displacement to the sixth is found to be 2.25. Find the

stiffness of the spring in kN/m and damping coefficient in Ns/m

5. A mass of 10 kg is supported on springs which deflects 20 mm under the

weight of the mass. The vibrations of the mass are constrained to be linear andvertical and are damped by a dashpot which reduces the amplitude to one-

quarter of its initial value in two complete oscillations. Find;a. the magnitude of the damping force at unit speed

b. the periodic time of damped vibration

6. A machine of mass 100kg is mounted on springs and is fitted with a dashpot

(shock absorber) to dampen out vibrations. There are three springs each of

stiffness 15N/mm and it is observed that the amplitude (measured in the samesense of direction from the equilibrium level) of free vibration diminishes from

24.6mm to 4.1mm in two complete oscillations. Determine (a) the damping ratioof the system, (b) the damping constant of the dashpot, (c) the natural

frequency of damped vibration, and (d) the required stiffness of a spring for

non-oscillatory response with the minimum damping ratio if the number ofsprings, damping constant and the mass of the machine were unchanged.

(Answer:

0.1412; 600Ns/m; 3.34Hz; 300N/m)

7. A machine of mass 150kg is supported on layers of rubber and felt. The rubberlayer and the felt layer have spring properties with stiffnesses of 15N/mm and

20N/mm respectively. The rubber layer has an effective viscous dampingconstant of 600Ns/m. It is observed that the amplitude of free vibration of the

machine diminishes from 19.2mm to 3.2mm in two complete oscillations.

Determine (a)the damping ratio of the system, (b) the effective dampingconstant of the felt layer, and

(c) the natural frequency of damped vibrations.