Problems and Solutions Section 8.6 (8.50 through 8.54) 8.50 Consider the machine punch of Figure P8.15. Recalculate the fundamental natural frequency by reducing the model obtained in Problem 8.16 to
Problems and Solutions Section 8.5 (8.44 through 8.49) 8.44 Derive a consistent-mass matrix for the system of Figure 8.9. Compare the natural frequencies of this system with those calculated with the
Problems and Solutions Section 8.4 (8.34 through 8.43) 8.34 Refer to the tapered bar of Figure P8.13. Calculate a lumped-mass matrix for this system and compare it to the solution of Problem 8.13. Sin
Problems and Solutions Section 8.3 (8.21 through 8.33) 8.21 Use equations (8.47) and (8.46) to derive equation (8.48) and hence make sure that the author and reviewer have not cheated you. Solution: u
Problems and Solutions Section 8.2 (8.8 through 8.20) 8.8 Consider the bar of Figure P8.3 and model the bar with two elements. Calculate the frequencies and compare them with the solution obtained in
Chapter 8 Problems and Solutions Section 8.1 (8.1 through 8.7) 8.1 Consider the one-element model of a bar discussed in Section 8.1. Calculate the finite element of the bar for the case that it is fre
Problems and Solutions Section 7.6 (7.25-7.31) 7.25 Referring to Section 5.4 and Window 5.3, calculate the receptance matrix of equation (7.25) for the following two-degree-of-freedom system, without
Problems and Solutions Section 7.5 (7.20-7.24) 7.20 Using the definition of the mobility transfer function of Window 7.4, calculate the Re and Im parts of the frequency response function and hence ver
Problems and Solutions for Section 7.4 (7.10-7.19) 7.10 Consider the magnitude plot of Figure P7.10. How many natural frequencies does this system have, and what are their approximate values?
Solution
Problems and Solutions Section 7.3 (7.6-7.9) 7.6 Represent 5 sin 3t as a digital signal by sampling the signal at /3, /6 and /12 seconds. Compare these three digital representations. Solution: Four pl
Problems and Solutions Section 7.2 (7.1-7.5) 7.1 A low-frequency signal is to be measured by using an accelerometer. The signal is physically a displacement of the form 5 sin (0.2t). The noise floor o
6- 64 Problems and Solutions Section 6.8 (6.64 through 6.68) 6.64 Calculate the response of the damped string of Example 6.8.1 to a disturbance force of f(x,t) = (sin x/l) sin10t. Solution: f ( x , t
6- 52 Problems and Solutions Section 6.7 (6.53 through 6.63) 6.53 Calculate the response of Example 6.7.1 for l = 1 m, E = 2.6 1010 N/m2 and = 8.5 103 kg/m3. Plot the response using the first three mo
6- 45 Problems and Solutions Section 6.6 (6.48 through 6.52) 6.48 Calculate the natural frequencies of the membrane of Example 6.6.1 for the case that one edge x = 1 is free. Solution: The equation fo
6- 37
Problems and Solutions Section 6.5 (6.40 through 6.47) 6.40 Calculate the natural frequencies and mode shapes of a clamped-free beam. Express your solution in terms of E, I, , and l. This is cal
6- 27
Problems and Solutions Section 6.4 (6.30 through 6.39) 6.30 Calculate the first three natural frequencies of torsional vibration of a shaft of Figure 6.7 clamped at x = 0, if a disk of inertia J
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
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