System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 4, Problem 4.52P
To determine

(a)

The equation of motion of the system.

Expert Solution
Check Mark

Answer to Problem 4.52P

The Equation of motion of the system are m1x1=cx1+k(x2x1) and m2x2=fk(x2x1).

Explanation of Solution

Given information:

The damping coefficient is c and friction is negligible.

The figure below shows the free body diagram.

System Dynamics, Chapter 4, Problem 4.52P , additional homework tip  1

Figure-(1)

Write the equation of motion for mass 1.

m1x1+cx1+k(x1x2)=0m1x1=cx1k(x1x2)m1x1=cx1+k(x2x1) ...... (I)

Here, mass of body 1 is m1, damping coefficient is c, spring constant is k, displacement of mass1 is x1, and the displacement of mass 2 is x2.

Write the Equation of motion for mass 2.

m2x2+k(x2x1)=fm2x2=fk(x2x1) ...... (II)

Here, mass of body 2 is m2 and the input force is f.

Conclusion:

The equation of motion of the system is m1x1=cx1+k(x2x1) and m2x2=fk(x2x1).

To determine

(b)

The equation of motion of the system.

Expert Solution
Check Mark

Answer to Problem 4.52P

The equation of motion of the system is m1x1=kx1+c(x2x1)=0 and m2x2=fc(x2x1).

Explanation of Solution

Given information:

The damping coefficient is c and friction is negligible.

The figure below shows the free body diagram of the block 1 and 2.

System Dynamics, Chapter 4, Problem 4.52P , additional homework tip  2

Figure-(2)

Write the Equation of motion for mass 1.

m1x1+kx1+c(x1x2)=0m1x1=kx1c(x1x2)m1x1=kx1+c(x2x1)=0 ........ (III)

Here, mass of body 1 is m1, damping coefficient is c, spring constant is k, displacement of mass1 is x1, and the displacement of mass 2 is x2.

Write the equation of motion for mass 2.

m2x2+c(x2x1)=fm2x2=fc(x2x1) ...... (IV)

Here, mass of body 2 is m2 and the input force is f.

Conclusion:

The equation of motion of the system is m1x1=kx1+c(x2x1)=0 and m2x2=fc(x2x1).

To determine

(c)

The equation of motion.

Expert Solution
Check Mark

Answer to Problem 4.52P

The equation of motion of the system is m1x1=cx1k1x1+k2(x2x1)=0 and m2x2=fk2(x2x1).

Explanation of Solution

Given information:

The damping coefficient is c and friction is negligible.

The Figure below shows the free body diagram of the system.

System Dynamics, Chapter 4, Problem 4.52P , additional homework tip  3

Figure-(3)

Write the equation of motion for mass 1.

m1x1+cx1+k1x1+k2(x1x2)=0m1x1=cx1k1x1k2(x1x2)m1x1=cx1k1x1+k2(x2x1)=0 ........ (V)

Here, mass of body 1 is m1, damping coefficient is c, spring constant for first spring is k1, the spring constant for spring 2 is k2 displacement of mass1 is x1, and the displacement of mass 2 is x2.

Write the Equation of motion for mass 2.

m2x2+k2(x2x1)=fm2x2=fk2(x2x1) ...... (VI)

Here, mass of body 2 is m2 and the input force is f.

Conclusion:

The equation of motion of the system is m1x1=cx1k1x1+k2(x2x1)=0 and m2x2=fk2(x2x1).

To determine

(d)

The Equation of motion.

Expert Solution
Check Mark

Answer to Problem 4.52P

The Equation of motion of the system is m1x1=c1x1k1x1+c2(x2x1)+k2(x2x1) and m2x2=fc2(x2x1)k2(x2x1).

Explanation of Solution

Given information:

The damping coefficient of damper 1 is c1, the damping coefficient of damper 2 is c2.

The figure below shows the free diagram of the system.

System Dynamics, Chapter 4, Problem 4.52P , additional homework tip  4

Figure-(4)

Write the Equation of motion for mass 1.

m1x1+c1x1+k1x1+c2(x1x2)+k2(x1x2)=0m1x1=c1x1k1x1c2(x1x2)k2(x1x2)m1x1=c1x1k1x1+c2(x2x1)+k2(x2x1) ........ (VII)

Here, mass of body 1 is m1, damping coefficient of first damper is c1, damping coefficient of second damper is c2 spring constant for first spring is k1, the spring constant for spring 2 is k2 displacement of mass1 is x1, displacement of mass 2 is x2.

Write the Equation of motion for mass 2.

m2x2+c2(x2x1)+k2(x2x1)=fm2x2=fc2(x2x1)k2(x2x1) ...... (VIII)

Here, mass of body 2 is m2 and the input force is f.

Conclusion:

The equation of motion of the system is m1x1=c1x1k1x1+c2(x2x1)+k2(x2x1) and m2x2=fc2(x2x1)k2(x2x1).

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Chapter 4 Solutions

System Dynamics

Ch. 4 - Plot the spring force felt by the mass shown in...Ch. 4 - Calculate the expression for the natural frequency...Ch. 4 - Prob. 4.13PCh. 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. 4.20PCh. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 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. 4.27PCh. 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. 4.30PCh. 4 - For Figure P4.31, the equilibrium position...Ch. 4 - Prob. 4.32PCh. 4 - Prob. 4.33PCh. 4 - 4.34 For Figure P4.34, assume that the cylinder...Ch. 4 - Use the Rayleigh method to obtain an expression...Ch. 4 - Prob. 4.36PCh. 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. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Prob. 4.42PCh. 4 - The vibration of a motor mounted on the end of a...Ch. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - A certain cantilever beam vibrates at a frequency...Ch. 4 - Prob. 4.47PCh. 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. 4.50PCh. 4 - Prob. 4.51PCh. 4 - Prob. 4.52PCh. 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. 4.55PCh. 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. 4.60PCh. 4 - Find the transfer function YsXs for the system...Ch. 4 - Prob. 4.62PCh. 4 - 4.63 In the system shown in Figure P4.63, the...Ch. 4 - Prob. 4.64PCh. 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. 4.67PCh. 4 - Prob. 4.68PCh. 4 - Prob. 4.69PCh. 4 - Figure P4.70 shows a quarter-car model that...Ch. 4 - Prob. 4.71PCh. 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. 4.81PCh. 4 - Prob. 4.82PCh. 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. 4.86PCh. 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. 4.90PCh. 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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