Page 59 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

46 Free Vibration Chap. 2

2-36 A vibrating system is started under the following initial conditions: jc = 0 and
X = Uq. Determine the equation of motion when (a) ^ = 2.0, (b) = 0.50, and (c)
i = 1.0. Plot nondimensional curves for the three cases with co^t as abscissa and

X(o^/Vqas ordinate.
2-37 In Prob. 2-36, compare the peak values for the three dampings specified.
2-38 A vibrating system consisting of a mass of 2.267 kg and a spring of stiffness 17.5 N/cm
is viscously damped such that the ratio of any two consecutive amplitudes is 1.00 and
0.98. Determine (a) the natural frequency of the damped system, (b) the logarithmic
decrement, (c) the damping factor, and (d) the damping coefficient.
2-39 A vibrating system consists of a mass of 4.534 kg, a spring of stiffness 35.0 N/cm, and a
dashpot with a damping coefficient of 0.1243 N /cm /s. Find (a) the damping factor, (b)
the logarithmic decrement, and (c) the ratio of any two consecutive amplitudes.
2-40 A vibrating system has the following constants: m = 17.5 kg, k = 70.0 N/cm, and
c = 0.70 N /cm /s. Determine (a) the damping factor, (b) the natural frequency of
damped oscillation, (c) the logarithmic decrement, and (d) the ratip of any two
consecutive amplitudes.
2-41 Set up the differential equation of motion for the system shown in Fig. P2-41.
Determine the expression for (a) the critical damping coefficient, and (b) the natural
frequency of damped oscillation.

Figure P2-41.

2-42 Write the differential equation of motion for the system shown in Fig. P2-42 and
determine the natural frequency of damped oscillation and the critical damping
coefficient.

Figure P2-42.
2-43 A spring-mass system with viscous damping is displaced from the equilibrium position
and released. If the amplitude diminished by 5% each cycle, what fraction of the
critical damping does the system have?
2-44 A rigid uniform bar of mass m and length / is pinned at O and supported by a spring
and viscous damper, as shown in Fig. P2-44. Measuring 6 from the static equilibrium
position, determine (a) the equation for small 6 (the moment of inertia of the bar