Explanation Given information: The inner radius of the pulley is R 1 , the outer radius of the pulley is R 2 , spring constant is k , mass is m , displacement of the mass is x and the acceleration due to gravity is g . Write the expression for tension in the string. T = − m x ¨ ....... (1) Here, tension is T , mass is m and the displacement of the mass is x ¨ . Write the expression for the Torque. Σ Torque = I θ ¨ T R 2 − k x R 1 R 2 R 1 = I θ ¨ ....... (2) Here, outer radius of pulley is R 2 , spring constant is k , inner radius of the pulley is R 1 and the moment of inertia is I the angular acceleration is θ ¨ . Substitute − m x ¨ for T in Equation (2). − m x ¨ R 2 − k x R 1 R 2 R 1 = I θ ¨ ....... (3) Write the expression for displacement. x = R 2 θ ....... (4) Substitute R 2 θ for x in Equation (3). − m R 2 R 2 θ ¨ − k R 2 θ R 1 R 2 R 1 = I θ ¨ − m R 2 2 θ ¨ − k R 2 θ R 1 R 2 R 1 − I θ ¨ I θ ¨ + m R 2 2 θ ¨ + k R 2 θ R 1 R 2 R 1 = 0 I + m R 2 2 θ ¨ + k R 1 2 θ = 0

3rd Edition

ISBN: 9780073398068

Author: III William J. Palm

Publisher: MCG

See similar textbooks

Question

Chapter 4, Problem 4.13P

To determine

The expression for natural frequency of the system.

Expert Solution & Answer

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...

Knowledge Booster

Recommended textbooks for you

Elements Of Electromagnetics
Mechanical Engineering
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Oxford University Press
Mechanics of Materials (10th Edition)
Mechanical Engineering
ISBN:9780134319650
Author:Russell C. Hibbeler
Publisher:PEARSON
Thermodynamics: An Engineering Approach
Mechanical Engineering
ISBN:9781259822674
Author:Yunus A. Cengel Dr., Michael A. Boles
Publisher:McGraw-Hill Education
Control Systems Engineering
Mechanical Engineering
ISBN:9781118170519
Author:Norman S. Nise
Publisher:WILEY
Mechanics of Materials (MindTap Course List)
Mechanical Engineering
ISBN:9781337093347
Author:Barry J. Goodno, James M. Gere
Publisher:Cengage Learning
Engineering Mechanics: Statics
Mechanical Engineering
ISBN:9781118807330
Author:James L. Meriam, L. G. Kraige, J. N. Bolton
Publisher:WILEY