Explanation Given information: The inertia of rod about Pivot is I , input displacement is z , the angular displacement is small, the distance of spring from pivot is L 1 , the damping coefficient is c , spring constant is k , distance of damper from pivot is L 2 and the distance of mass from pivot is L 3 . Write the expression for torque due to spring force. T k = k x ( L 1 cos θ ) ........ (I) Here, torque due to spring force is T k , displacement in spring is x , the distance of spring from pivot is L 1 and angular displacement is θ . Write the expression for displacement in spring. x = L 1 θ ...... (II) Substitute L 1 θ for x in Equation (I). T k = k L 1 θ ( L 1 cos θ ) = k L 1 2 θ ( cos θ ) ........ (III) Write the expression for centre of gravity of the system. d = L 1 + L 3 2 ...... (IV) Here, center of gravity is d and the distance of mass from pivot is L 3 . Write the expression for moment due to rod. T r = m r g d sin θ ........ (V) Here, moment due to rod is T r and acceleration due to gravity is g . Write the expression for moment due to damping force. T d = c ( L 2 θ ˙ ) L 2 cos θ ...... (VI) Here, moment due to damping force is T d , damping coefficient is c and the distance of damper from pivot is L 2 . Write the expression for moment due to mass. T m = m g L 3 sin θ ...... (VII) Here, moment due to mass is T m . Write the expression for moment balancing about the pivot. I θ ¨ = − T k − T r − T d − T m ........ (VIII) Here, moment of inertia of the rod is I and the angular acceleration is θ ¨ . Substitute k L 1 2 θ ( cos θ ) for T k , m r g d sin θ for T r , c ( L 2 θ ˙ ) L 2 cos θ for T d and m g L 3 sin θ for T m in Equation (VIII)

3rd Edition

ISBN: 9780073398068

Author: III William J. Palm

Publisher: MCG

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Chapter 4, Problem 4.62P

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The equation of motion.

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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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