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: Follo
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. De
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 revo
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 leve
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 sof
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
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
(
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! yiel
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 resp
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
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 La
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), (3.2
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 Eq
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.
! ! Solut
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
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 mach
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 su
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 th
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
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/
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
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
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
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 und
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 co
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 & # #
(
)
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. Not
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 examp
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
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 inter