Problems and Solutions Section 1.10 (1.91 through 1.103) 1.91 A 2-kg mass connected to a spring of stiffness 103 N/m has a dry sliding friction force (F d) of 3 N. As the mass oscillates, its amplitude decreases 20 cm in 15 cycles. How long does this take
Problems and Solutions Section 1.9 (1.83 through 1.90) 1.83* Reproduce Figure 1.37 for the various time steps indicated. Solution: The code is given here in Mathcad, which can be run repeatedly with different t to see the importance of step size. Matlab a
Problems and Solutions Section 1.8 (1.80 through 1.82) 1.80 Consider the inverted pendulum of Figure 1.36 as discussed in Example 1.8.1. Assume that a dashpot (of damping rate c) also acts on the pendulum parallel to the two springs. How does this affect
Problems and Solutions Section 1.7 (1.73 through 1.79) 1.73 Choose a dashpot's viscous damping value such that when placed in parallel with the spring of Example 1.7.2 reduces the frequency of oscillation to 9 rad/s. Solution: The frequency of oscillation
Problems and Solutions Section 1.6 (1.66 through 1.72)
1.66
Show that the logarithmic decrement is equal to
=
1 x0 ln n xn
where xn is the amplitude of vibration after n cycles have elapsed. Solution:
Ae - n t sin( d t + ) x (t ) ln = ln - n ( t + nt ) s
Problems and Solutions Section 1.5 (1.57 through 1.65) 1.57 A helicopter landing gear consists of a metal framework rather than the coil spring based suspension system used in a fixed-wing aircraft. The vibration of the frame in the vertical direction can
1.47
Calculate the frequency of the compound pendulum of Figure 1.19(b) if a mass mT is added to the tip, by using the energy method. Solution Using the notation and coordinates of Figure 1.19 and adding a tip mass the diagram becomes:
mt
If the mass of t
Problems and Solutions for Section 1.2 and Section 1.3 (1.18 to 1.46) Problems and Solutions Section 1.2 (Numbers 1.18 through 1.26) 1.18* Plot the solution of a linear, spring and mass system with frequency n =2 rad/s, x0 = 1 mm and v0 = 2.34 mm/s, for a
Problems and Solutions Section 1.1 (1.1 through 1.17) 1.1 The spring of Figure 1.2 is successively loaded with mass and the corresponding (static) displacement is recorded below. Plot the data and calculate the spring's stiffness. Note that the data conta
2- 69
Problems and Solutions Section 2.9 (2.78 through 2.83) 2.78*. Compute the response of the system in Figure 2.31 for the case that the damping is linear viscous and the spring is a nonlinear soft spring of the form k ( x ) = kx - k1 x 3 and the syste
2- 60
Problems and Solutions Section 2.8 (2.72 through 2.77) 2.72*. Numerically integrate and plot the response of an underdamped system determined by m = 100 kg, k = 20,000 N/m, and c = 200 kg/s, subject to the initial conditions of x0 = 0.01 m and v0 =
2-44 Problems and Solutions Section 2.7 (2.55 through 2.71) 2.55 Consider a spring-mass sliding along a surface providing Coulomb friction, with stiffness 1.2 10 4 N/m and mass 10 kg, driven harmonically by a force of 50 N at 10 Hz. Calculate the approxim
2-42 Problems and Solutions Section 2.6 (2.51 through 2.54) 2.51 Calculate damping and stiffness coefficients for the accelerometer of Figure 2.15 with moving mass of 0.04 kg such that the accelerometer is able to measure vibration between 2 0 and 50 Hz w
2- 36 Problems and Solutions Section 2.5 (2.43 through 2.50)
2.43
A lathe can be modeled as an electric motor mounted on a steel table. The table plus the motor have a mass of 50 kg. The rotating parts of the lathe have a mass of 5 kg at a distance 0.1 m
2- 27 Problems and Solutions Section 2.4 (2.31 through 2.42) 2.31 A machine weighing 2000 N rests on a support. The support deflects about 5 cm as a result of the weight of the machine. The floor under the support is somewhat flexible and moves, because o
2- 24 Problems and Solutions Section 2.3(2.26 through 2.30) 2.26 Referring to Figure 2.9, draw the solution for the magnitude X for the case m = 100 kg, c = 4000 N s/m, and k = 10,000 N/m. Assume that the system is driven at resonance by a 10-N force. Sol
2- 9 Problems and Solutions Section 2.2 (2.11 through 2.25) 2.11 Calculate the constants A and for arbitrary initial conditions, x 0 and v 0 , in the case of the forced response given by Equation 2.29. Compare this solution to the transient response obtai
Chapter 2 Problems Problems and Solutions Section 2.1 (2.1 through 2.10) 2.1 To familiarize yourself with the nature of the forced response, plot the solution of a forced response of equation (2.2) with = 2 rad/s, given by equation (2.1) for a variety of
3- 70 Problems and Solutions Section 3.10 (3.58 through 3.64) 3.58*. Compute the response of the system in Figure 3.22 for the case that the damping is linear viscous and the spring is a nonlinear soft spring of the form k ( x ) = kx - k1 x 3 and the syst
3Problems and Solutions from Section 3.9 (3.50-3.57)
53
3.50*. Numerically integrate and plot the response of an underdamped system determined by m = 100 kg, k = 1000 N/m, and c = 20 kg/s, subject to the initial conditions of x0 = 0 and v0 = 0, and the ap
3- 49 Problems and Solutions Section 3.8 (3.46 through 3.49) 3.46 Show that a critically damped system is BIBO stable. Solution: For a critically damped system
h(t - ) =
1 (t - )e - n ( t - ) m
Let f(t) be bounded by the finite constant M. Using the inequ
3- 42 Problems and Solutions for Section 3.7 (3.39 through 3.45) 3.39 Using complex algebra, derive Eq. (3.89) from (3.86) with s = j. Solution: From Eq. (3.86):
H (s ) = 1 ms 2 + cs + k
Substituting s = j yields H ( j ) =
m j
( )
1
2
+ c( j ) + k
=
1 k -
3- 39 Problems and Solutions Section 3.6 (3.37 through 3.38) 3.37 A power line pole with a transformer is modeled by
mx + kx = - y
where x and y are as indicated in Figure 3.23. Calculate the response of the relative displacement (x y) if the pole is sub
3- 36 Problems and Solutions Section 3.5 (3.33 through 3.36) 3.33 Calculate the mean-square response of a system to an input force of constant PSD, S0, 10 and frequency response function H ( ) = (3 + 2 j ) Solution: Given: S ff = S 0 and H ( ) =
10 3 + 2
3- 32 Problems and Solutions for Section 3.4 (3.29 through 3.32) 3.29 Calculate the response of mx + cx + kx = F0 (t ) where (t) is the unit step function for the case with x0 = v0 = 0. Use the Laplace transform method and assume that the system is underd
3- 21 Problems and Solutions Section 3.3 (problems 3.22-3.28) 3.22 Derive equations (3.24). (3.25) and (3.26) and hence verify the equations for the Fourier coefficient given by equations (3.21), (3.22) and (3.23). Solution: For n m, integration yields:
0
3- 12 Problems and Solutions for Section 3.2 (3.12 through 3.21) 3.12 Calculate the response of an overdamped single-degree-of-freedom system to an arbitrary non-periodic excitation. Solution: From Equation (3.12): x(t ) = F( )h(t - )d
0 L
For an undamped
3Chapter Three Solutions Problem and Solutions for Section 3.1 (3.1 through 3.11) 3.1 Calculate the solution to
1
x + 2 x + 2 x = (t - ) x (0) = 1 x (0) = 0
and plot the response.
& & x Solution: Given: & + 2 x + 2 x = (t - ) x(0) = 1, x(0 ) = 0 k c = 1.
5- 77 Problems Section 5.9 (5.86 through 5.88) 5.86 Reconsider Example 5.2.1, which describes the design of a vibration isolator to protect an electronic module. Recalculate the solution to this example using Equation (5.92). Solution: If data sheets are
5- 74 Problems and Solutions Section 5.8 (5.81 through 5.85) 5.81 Recall the definitions of settling time, time to peak, and overshoot given in Ex. 3.2.1 and illustrated in Fig. 3.5. Consider a single-degree-of-freedom system with mass m = 2 kg, damping c