System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 4, Problem 4.41P
To determine

The natural frequency of the system by using Rayleigh’s method.

Expert Solution & Answer
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Answer to Problem 4.41P

The natural frequency of the system is 12π(2kL12+mgL2)mL12.

Explanation of Solution

Write the expression for equality condition of potential energy and kinetic energy for Rayleigh’s criterion if the system is vibrating at natural frequency.

(KE)max=(PE)max ...... (I)

Here, the kinetic energy of the system is KE and the potential energy of the system is PE Write the law of conservation of energy for simple harmonic motion.

KEmax+PEmin=PEmax+KEmin ...... (II)

For simple harmonic motion potential energy is at maximum and kinetic energy is zero when the displacement of the system is at maximum.

Substitute 0 for KEmin in Equation (II).

KEmax+PEmin=PEmax+0KEmax+PEmin=PEmaxKEmax=PEmaxPEmin ...... (III)

Write the expression of equation of motion for simple harmonic motion.

x(t)=Aosin(ωnt+ϕ) ...... (IV)

Here, the displacement of the pendulum with respect to time is x(t), the amplitude is A, the natural frequency is ωn, the initial phase angle is ϕ.

Differentiate equation (IV) with respect to t.

x˙(t)=Aoωncos(ωnt+ϕ) ...... (V)

Substitute 0 for ϕ and x(t)max for x˙(t) in Equation (V).

x˙(t)max=Aocos(ωnt+0)x˙(t)max=Aoωn

Write the expression of moment of inertia of reverse pendulum.

Io=mL22

Here, the moment of inertia is Io, the mass of the pendulum is m and length of the pendulum is L2.

Write the expression of maximum kinetic energy of the system.

KEmax=12Io(x˙(t)maxL2)2 ...... (VI)

Substitute Aoωn for x˙(t)max in Equation (VII)

KEmax=12Io(AoωnL2)2

Write the expression of potential energy of the system.

PE=k(L1θ)2+mgL2(1cosθ) ...... (VII)

Here, the angle of deviation is θ.

Since the angle is very small so cosθ1θ22.

Substitute 1θ22 for cosθ in Equation (VIII)

PE=k(L1θ)2+mgL2(1(1θ22))=k(L1θ)2+mgL2θ22=(kL12+mgL22)θ2 ..... (VIII)

Substitute AoL1 for θ and PEmax for PE in Equation (VIII).

PEmax=(kL12+mgL22)(AoL1)2

Write the expression for fundamental natural frequency of the system in terms of frequency of the vibration of the cantilever beam.

ωn=2πfn ...... (IX)

Substitute (kL12+mgL22)(AoL1)2 for PEmax and 12Io(AoωnL2)2 for KEmax in Equation (I).

12mL22(AoωnL2)2=(kL12+mgL22)(AoL1)2mL22Ao2(ωnL2)2=(2kL12+mgL2)(Ao2L12)mL22(ωnL2)2=(2kL12+mgL2)(1L12)(ωn)2=(2kL12+mgL2)mL12

ωn=(2kL12+mgL2)mL12

Substitute (2kL12+mgL2)mL12 for ωn in Equation (IX).

12π(2kL12+mgL2)mL12=2πfnfn=12π(2kL12+mgL2)mL12.

Conclusion:

The natural frequency of the system is 12π(2kL12+mgL2)mL12.

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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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