Explanation Given information: The mass of the spring is 2 Kg , the frequency of vibration of mass is 20 Hz , the mass of the suspended mass is 10 Kg . Write the expression for displacement of mass at a distance d y from the top. x = y L A cos ( ω n t ) ...... (I) Here, the displacement of mass in free vibration is x , the amplitude of vibration is A , the angular natural frequency is ω n , distance at which displacement is considered is y , the length of the spring is L . Write the expression for total kinetic energy of the system. KE = 1 2 m ( A ω n ) 2 + ∫ 0 L 1 2 ( m s L ) ( y L ) 2 ( A ω ) 2 d y ...... (II) Here the velocity of suspended mass is A ω n , the mass of the suspended mass is m , the velocity of the spring per unit length is A ω , the mass of the spring per unit length is m s L . Solve Equation (II) for total kinetic energy. KE = 1 2 m ( A ω n ) 2 + m s ( A ω ) 2 2 L 3 [ y 3 3 ] 0 L = 1 2 m ( A ω n ) 2 + m s ( A ω ) 2 2 L 3 [ L 3 3 − 0 ] = 1 2 m ( A ω n ) 2 + m s ( A ω ) 2 2 L 3 ( L 3 3 ) = 1 2 m ( A ω n ) 2 + m s ( A ω ) 2 6 KE = 1 2 ( m + m s 3 ) ( A ω n ) 2

3rd Edition

ISBN: 8220100254963

Author: Palm

Publisher: MCG

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Forced undamped vibrations

In mechanical engineering, vibration is expressed as the term that occurs due to an object's backward and forward movement about a point of equilibrium. Generally, there are three types of vibrations which are,

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Chapter 4, Problem 4.47P

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The spring stiffness.

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Chapter 4, Problem 4.46P

Chapter 4, Problem 4.48P

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Chapter 4 Solutions

EBK 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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