6- 8 Problems and Solutions Section 6.3 (6.8 through 6.29) 6.8 Calculate the natural frequencies and mode shapes for a free-free bar. Calculate the temporal solution of the first mode. Solution: Following example 6.31 (with different B.C.s), the spatial r
5- 23 Problems and Solutions Section 5.3 (5.27 through 5.36) 5.27 A motor is mounted on a platform that is observed to vibrate excessively at an operating speed of 6000 rpm producing a 250-N force. Design a vibration absorber (undamped) to add to the plat
5- 3 Problems and Solutions Section 5.1 (5.6 through 5.26) 5.6 A 100-kg machine is supported on an isolator of stiffness 700 103 N/m. The machine causes a vertical disturbance force of 350 N at a revolution of 3000 rpm. The damping ratio of the isolator i
5Problems and Solutions Section 5.1 (5.1 through 5.5) 5.1
1
Using the nomograph of Figure 5.1, determine the frequency range of vibration for which a machine oscillation remains at a satisfactory level under rms acceleration of 1g. Solution: An rms accele
3- 89 Problems and Solutions Section 3.10 (3.65 through 3.71) 3.65*. Compute the response of the system in Figure 3.26 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 system
3Problems and Solutions from Section 3.9 (3.57-3.64)
72
3.57*. 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- 68 Problems and Solutions Section 3.8 (3.53 through 3.56) 3.53 Show that a critically damped system is BIBO stable. Solution: For a critically damped system
h t !" =
(
)
1 !# t !" t ! " e n( ) m
(
)
Let f(t) be bounded by the finite constant M. Using t
3- 61 Problems and Solutions for Section 3.7 (3.45 through 3.52) 3.45 Using complex algebra, derive equation (3.89) from (3.86) with s = j. Solution: From equation (3.86):
Hs=
Substituting s = j! yields
H j! =
()
1 ms + cs + k
2
()
1 m j!
(
)
2
+ c j! + k
3- 57 Problems and Solutions Section 3.6 (3.43 through 3.44) 3.43 A power line pole with a transformer is modeled by
m! + kx = ! ! x 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
3- 54 Problems and Solutions Section 3.5 (3.39 through 3.42) 3.39 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 = S0 and H ! =
()
10 3 +
3- 50 Problems and Solutions for Section 3.4 (3.35 through 3.38) 3.35 Calculate the response of ! m! + cx + kx = F0 !(t) x 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 un
3- 37 Problems and Solutions Section 3.3 (problems 3.26-3.32) 3.26 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:
3- 17 Problems and Solutions for Section 3.2 (3.15 through 3.25) 3.15 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
()
)%
t
()(
( )
3Chapter Three Solutions Problem and Solutions for Section 3.1 (3.1 through 3.14) 3.1 Calculate the solution to
! ! x + 2 x + 2x = ! t " # ! x 0 =1 x 0 = 0
1
()
()
(
)
and plot the response.
! ! Solution: Given: x + 2 x + 2x = ! t " #
(
! ) x (0) = 1, x (
2- 69
Problems and Solutions Section 2.9 (2.87 through 2.93) 2.87*. Compute the response of the system in Figure 2.34 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 system i
5- 35 Problems and Solutions Section 5.4 (5.37 through 5.52) 5.37 A machine, largely made of aluminum, is modeled as a simple mass (of 100 kg) attached to ground through a spring of 2000 N/m. The machine is subjected to a 100-N harmonic force at 20 rad/s.
5- 53 Problems and Solutions Section 5.5 (5.53 through 5.66) 5.53 Design a Houdaille damper for an engine modeled as having an inertia of 1.5 kg.m2 and a natural frequency of 33 Hz. Choose a design such that the maximum dynamic magnification is less than
6-
1
Chapter 6 Problems and Solutions Section 6.2 (6.1 through 6.7) 6.1 Prove the orthogonality condition of equation (6.28). Solution: Calculate the integrals directly. For n = n, let u = n x/l so that du = (n /l)dx and the integral becomes l n
0
1 1 =
Problems and Solutions Section 4.10 (4.91 through 4.98) 4.91* Solve the system of Example 1.7.3 for the vertical suspension system of a car with m = 1361 kg, k = 2.668 x 105 N/m, and c = 3.81 x 104 kg/s subject to the initial conditions of x(0) = 0 and v(
Problems and Solutions for Section 4.9 (4.80 through 4.90) 4.80 Consider the mass matrix "10 !1% M=$ ' # !1 1 & and calculate M-1, M-1/2, and the Cholesky factor of M. Show that LLT = M
M !1/ 2 M !1/ 2 = I M 1/ 2 M 1/ 2 = M
Solution: Given
"10 !1% M=$ ' #
Problems and Solutions for Section 4.7 (4.76 through 4.79) 4.76 Use Lagrange's equation to derive the equations of motion of the lathe of Fig. 4.21 for the undamped case. Solution: Let the generalized coordinates be !1 ,! 2 and ! 3 . The kinetic energy is
Problems and Solutions for Section 4.6 (4.67 through 4.76) 4.67 Calculate the response of the system of Figure 4.16 discussed in Example 4.6.1 if F1(t) = (t) and the initial conditions are set to zero. This might correspond to a two-degree-of-freedom mode
Problems and Solutions for Section 4.5 (4.56 through 4.66) 4.56 Consider the example of the automobile drive train system discussed in Problem 4.44. Add 10% modal damping to each coordinate, calculate and plot the system response. Solution: Let k1 = hub s
Problems and Solutions for Section 4.4 (4.44 through 4.55) 4.44 A vibration model of the drive train of a vehicle is illustrated as the three-degreeof-freedom system of Figure P4.44. Calculate the undamped free response [i.e. ! M(t) = F(t) = 0, c1 = c2 =
Problems and Solutions for Section 4.3 (4.34 through 4.43) 4.34 Solve Problem 4.11 by modal analysis for the case where the rods have equal stiffness
! (i.e., k1 = k2 ), J1 = 3J 2 , and the initial conditions are x(0) = !0 1# and x 0 = 0. " $
T
()
Solutio
Problems and Solutions for Section 4.2 (4.19 through 4.33) 4.19 Calculate the square root of the matrix
" 13 !10 % M=$ ' # !10 8 &
" " a !b % 1/ 2 1/ 2 $ Hint: Let M = $ ' ; calculate M !b c & # #
(
)
2
% and compare to M.' &
Solution: Given:
" 13 !10 % M
Problems and Solutions for Section 4.1 (4.1 through 4.16) 4.1 Consider the system of Figure P4.1. For c1 = c2 = c3 = 0, derive the equation of motion and calculate the mass and stiffness matrices. Note that setting k3 = 0 in your solution should result in
5- 89 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- 86 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 Example 3.2.1 and illustrated in Figure 3.6. Consider a single-degree-of-freedom system with mass m = 2 kg, dam
5- 80 Problems and Solution Section 5.7 (5.74 through 5.80) 5.74 A 100-kg compressor rotor has a shaft stiffness of 1.4 107 N/m. The compressor is designed to operate at a speed of 6000 rpm. The internal damping of the rotor shaft system is measured to be