System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 4, Problem 4.64P
To determine

The equation of motion for the system.

Expert Solution & Answer
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Answer to Problem 4.64P

The equation of motion for the system is x+k1cx¨+(k1+k2)mx˙+k1k2mcx=ky˙m.

Explanation of Solution

Figure (1) shows the free body diagram of the system.

System Dynamics, Chapter 4, Problem 4.64P

Figure-(1)

Consider a point A on the system and its displacement xA as shown in figure (1), the mass of the system is m, the damping coefficient is c, the stiffness of the spring (1) is k1 and of the spring (2) is k2, the displacement at different point are x, y, and xA

Write the expression for horizontal force on mass m.

mx¨=c(x˙Ax˙)k2x ..... (I)

Here, the acceleration is x¨, velocity at point A is x˙A.

Write the expression for horizontal force at point A.

k1(yxA)=c(x˙Ax˙) ...... (II)

Write Laplace transform for Equation (I).

ms2X(s)=[c(sXA(s)sX(s))k2X(s)]ms2X(s)+k2X(s)+csX(s)=csXA(s)X(s)[ms2+k2+cs]=csXA(s)X(s)[ms2+k2+cs]cs=XA(s) ...... (III)

Write Laplace transform for Equation (II)

k1[Y(s)XA(s)]=c[(s(XA(s)sX(s)))]k1Y(s)k1XA(s)=csXA(s)csX(s)k1Y(s)+csX(s)=csXA(s)+k1XA(s) ..... (IV)

Substitute X(s)[ms2+k2+cs]cs for XA(s) in Equation (IV)

k1Y(s)+csX(s)=csX(s)[ms2+k2+cs]cs+k1X(s)[ms2+k2+cs]csk1Y(s)+csX(s)=(cs+k1)[ms2+k2+cscs]X(s)k1Y(s)=(cs+k1)[ms2+k2+cscs]X(s)csX(s)k1Y(s)=[(cs+k1)(ms2+k2+cs)cscs]X(s)

k1Y(s)=[(cs+k1)(ms2+k2+cs)cscs]X(s)k1Y(s)=[mcs3+c2s2+k2cs+k1ms2+k1cs+k1k2c2s2cs]X(s)k1Y(s)=[mcs3+k2cs+k1ms2+k1cs+k1k2cs]X(s)

k1csY(s)=(mcs3+k2cs+k1ms2+k1cs+k1k2)X(s)sk1cY(s)=mcs3X(s)+k1ms2X(s)+s(k1c+k2c)X(s)+k1k2X(s) ...... (V)

Take inverse Laplace of Equation (V).

k1cy˙=mcx+k1mx¨+(k1c+k2c)x˙+k1k2xx+k1mmcx¨+(k1c+k2c)mcx˙+k1k2mcx=k1cy˙mcx+k1cx¨+(k1+k2)mx˙+k1k2mcx=k1y˙m.

Conclusion:

The equation of motion for the system is x+k1cx¨+(k1+k2)mx˙+k1k2mcx=ky˙m.

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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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