Explanation Given Information: The disk is the solid cylinder and angle of oscillation is small. Write the expression for the potential energy for the system. P E = 1 2 k 1 4 x 2 + δ 2 ...... (1) Here, the spring constant is k , deflection at the equilibrium state is δ , and the deflection of the system is x . For maximum potential energy, x = A and for minimum potential energy, x = 0 . Substitute A for x and P E max for P E in Equation (1). P E max = 1 2 k 1 4 A 2 + δ 2 Substitute 0 for x and P E min for P E in Equation (1). P E min = 1 2 k 1 4 0 2 + δ 2 = 1 2 k δ 2 Write the expression for the kinetic energy of the system. K E = 1 2 m 1 x ˙ 2 + 1 2 I θ ˙ 2 + 1 2 m 2 x ˙ 2 4 ...... (2) Here, the velocity of the system is x ˙ , mass of the pulley is m 2 , mass of the block is m 1 angular velocity is θ ˙ , and moment of inertia of pulley is I . Write the expression for the angular velocity. θ ˙ = x ˙ 2 R Substitute x ˙ 2 R for θ ˙ in Equation (2). K E = 1 2 m 1 x ˙ 2 + 1 2 I x ˙ 2 R 2 + 1 2 m 2 x ˙ 2 4 = 1 2 m 1 x ˙ 2 + 1 2 I x ˙ 2 4 R 2 + 1 2 m 2 x ˙ 2 4 = 1 2 m 1 + I 4 R 2 + m 2 4 x ˙ 2 Write the expression of energy balance by applying Rayleigh’s method. K E = P E max − P E min ...... (3) Substitute 1 2 k δ 2 for P E min , 1 2 k 1 4 A 2 + δ 2 for P E max , and 1 2 m 1 + I 4 R 2 + m 2 4 x ˙ 2 for K E in Equation (3)

3rd Edition

ISBN: 9780073398068

Author: III William J. Palm

Publisher: MCG

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Question

Chapter 4, Problem 4.36P

To determine

The expression for the natural frequency of the system.

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Chapter 4, Problem 4.35P

Chapter 4, Problem 4.37P

Chapter 4 Solutions

System Dynamics

Ch. 4 - Prob. 4.1P Ch. 4 - In the spring arrangement shown in Figure P4.2....Ch. 4 - In the arrangement shown in Figure P4.3, a cable...Ch. 4 - In the spring arrangement shown in Figure P4.4,...Ch. 4 - For the system shown in Figure P4.5, assume that...Ch. 4 - The two stepped solid cylinders in Figure P4.6...Ch. 4 - A table with four identical legs supports a...Ch. 4 - The beam shown in Figure P4.8 has been stiffened...Ch. 4 - Determine the equivalent spring constant of the...Ch. 4 - Compute the equivalent torsional spring constant...

Ch. 4 - Plot the spring force felt by the mass shown in...Ch. 4 - Calculate the expression for the natural frequency...Ch. 4 - Prob. 4.13P Ch. 4 - Obtain the expression for the natural frequency of...Ch. 4 - 4.15 A connecting rod having a mass of 3.6 kg is...Ch. 4 - Calculate the expression for the natural frequency...Ch. 4 - For each of the systems shown in Figure P4.17, the...Ch. 4 - The mass m in Figure P4.18 is attached to a rigid...Ch. 4 - In the pulley system shown in Figure P4.19, the...Ch. 4 - Prob. 4.20P Ch. 4 - Prob. 4.21P Ch. 4 - Prob. 4.22P Ch. 4 - In Figure P4.23, assume that the cylinder rolls...Ch. 4 - In Figure P4.24 when x1=x2=0 the springs are at...Ch. 4 - 4.25 In Figure P4.25 model the three shafts as...Ch. 4 - In Figure P4.26 when 1=2=0 the spring is at its...Ch. 4 - Prob. 4.27P Ch. 4 - For the system shown in Figure P4.28, suppose that...Ch. 4 - For the system shown in Figure P4.29, suppose that...Ch. 4 - Prob. 4.30P Ch. 4 - For Figure P4.31, the equilibrium position...Ch. 4 - Prob. 4.32P Ch. 4 - Prob. 4.33P Ch. 4 - 4.34 For Figure P4.34, assume that the cylinder...Ch. 4 - Use the Rayleigh method to obtain an expression...Ch. 4 - Prob. 4.36P Ch. 4 - 4.37 Determine the natural frequency of the system...Ch. 4 - Determine the natural frequency of the system...Ch. 4 - Use Rayleigh's method to calculate the expression...Ch. 4 - Prob. 4.40P Ch. 4 - Prob. 4.41P Ch. 4 - Prob. 4.42P Ch. 4 - The vibration of a motor mounted on the end of a...Ch. 4 - Prob. 4.44P Ch. 4 - Prob. 4.45P Ch. 4 - A certain cantilever beam vibrates at a frequency...Ch. 4 - Prob. 4.47P Ch. 4 - 4.48 The static deflection of a cantilever beam is...Ch. 4 - Figure P4.49 shows a winch supported by a...Ch. 4 - Prob. 4.50P Ch. 4 - Prob. 4.51P Ch. 4 - Prob. 4.52P Ch. 4 - 4.53 In Figure P4.53 a motor supplies a torque T...Ch. 4 - Derive the equation of motion for the lever system...Ch. 4 - Prob. 4.55P Ch. 4 - Figure P4.56a shows a Houdaille damper, which is a...Ch. 4 - 4.57 Refer to Figure P4.57. Determine the...Ch. 4 - For the system shown in Figure P4.58, obtain the...Ch. 4 - Find the transfer function ZsXs for the system...Ch. 4 - Prob. 4.60P Ch. 4 - Find the transfer function YsXs for the system...Ch. 4 - Prob. 4.62P Ch. 4 - 4.63 In the system shown in Figure P4.63, the...Ch. 4 - Prob. 4.64P Ch. 4 - Figure P4.65 shows a rack-and-pinion gear in which...Ch. 4 - Figure P4.66 shows a drive train with a spur-gear...Ch. 4 - Prob. 4.67P Ch. 4 - Prob. 4.68P Ch. 4 - Prob. 4.69P Ch. 4 - Figure P4.70 shows a quarter-car model that...Ch. 4 - Prob. 4.71P Ch. 4 - 4.72 Derive the equation of motion for the system...Ch. 4 - A boxcar moving at 1.3 m/s hits the shock absorber...Ch. 4 - For the systems shown in Figure P4.74, assume that...Ch. 4 - Refer to Figure P4.75a, which shows a ship’s...Ch. 4 - In this problem, we make all the same assumptions...Ch. 4 - Refer to Figure P4.79a, which shows a water tank...Ch. 4 - The “sky crane” shown on the text cover was a...Ch. 4 - Prob. 4.81P Ch. 4 - Prob. 4.82P Ch. 4 - Suppose a mass in moving with a speed 1 becomes...Ch. 4 - Consider the system shown in Figure 4.6.3. Suppose...Ch. 4 - Prob. 4.86P Ch. 4 - Figure P4.87 shows a mass m with an attached...Ch. 4 - Figure P4.88 represents a drop forging process....Ch. 4 - Refer to Figure P4.89. A mass m drops from a...Ch. 4 - Prob. 4.90P Ch. 4 - (a) Obtain the equations of motion of the system...Ch. 4 - Refer to part (a) of Problem 4.90. Use MATLAB to...Ch. 4 - Refer to Problem 4.91. Use MATLAB to obtain the...Ch. 4 - 4.94 (a) Obtain the equations of motion of the...Ch. 4 - 4.95 (a) Obtain the equations of motion of the...

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