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 a single degree of freedom using Guyan reduction. Solut
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 lumped-mass matrix computed in Section 8.5. Solution: U
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. Since the beam is tapered, be careful how you divide up th
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( x ,t ) = 1 () 3 + 2 () 2 + 3 () + 4 () u(0, t ) = 1 (
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 Problem 8.3. Assume material properties of aluminum, a
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 free at both ends rather than clamped. Solution: The finit
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 using the systems mode shapes. 2 0 Solution: 2 0 0 3 1
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 verify equations (7.15) and (7.16). Solution: From Window
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: The system looks to have 8 modes with approximate nat
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 plots are shown. The one at the top far right is the exac
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 of the accelerometer (i.e. the smallest magnitude signal
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 ) = 10. Assume a solution of the form: wn ( 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 modes at x = l/2, l/4, and 3l/4. How many modes are neede
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 for a square membrane is wtt + = with boundary condition
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 called the cantilevered beam problem. Solution: Clamped-fr
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 J0 = 10 kg m2/rad is attached to the end of the shaft at
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
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