SolSec 4.5

# SolSec 4.5 - Problems and Solutions for Section 4.5(4.51...

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Problems and Solutions for Section 4.5 (4.51 through 4.60) 4.51 Consider the example of the automobile drive train system discussed in Problem 4.43. Add 10% modal damping to each coordinate, calculate and plot the system response. Solution: Let k 1 = hub stiffness and k 2 = axle and suspension stiffness. From Problem 4.41, the equation of motion with damping is 75 0 0 0 100 0 0 0 3000 10 000 11 0 13 2 02 2 0 00 0 0 1 + −− = () = () = [] ˙˙ , ˙ xx x0 x and m/s T Other calculations from Problem 4.41 yield: λω 2 3 12 3 77 951 8 8290 362 05 19 028 0 1537 0 1775 0 9721 0 8803 0 4222 0 2163 0 4488 0 8890 0 0913 == = = =− rad/s rad/s rad/s 2 3 .. . . . . . . . . . vv v Use the summation method to find the solution. Transform the initial conditions: qx 0 0 0 0 0 54 7723 M M T / / ˙ ˙ . Also, . 1 . 0 3 2 1 = = = ζ rad/s 932 . 19 rad/s 7848 . 8 3 2 = = d d ω The solution is given by qv v tc c t d e t i t di ii () =+ ++ 14 1 3 3 2 ζω ωφ sin where φ i di i T i T iii T i = () + () = tan ˙ , 1 0 23 vq (Eq. (4.114)) di i i T di i i = = ˙ cos sin , 0 Thus, 2642 . 0 3485 . 1 0 3 2 3 2 = = = = d d Now, v v 0 0 2 3 41 2 3 + = = cd d i i i i di i i sin ˙ sin cos

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Pre-multiply by v 1 T : vq 11 14 00 0 53 2414 T T c c () == ˙ . So, qv v v te t t e t tt =+ + −− 53 2414 1 3485 8 7848 0 2648 18 932 1 0 8829 2 1 9028 3 . . sin . . sin . .. The solution is given by xq x tM t e t e t () = () () = + +− 12 0 8829 1 9028 0 9449 1 1 1 0 1371 0 05693 0 005325 8 7848 0 01369 0 002349 0 0004407 18 932 / . . . . sin . . . . sin . m The following Mathcad session illustrates the solution without the rigid body mode The read solid line is the first mode with the rigid body mode included.
4.52 Consider the model of an airplane discussed in problem 4.44, Figure 4.36. (a) Resolve the problem assuming that the damping provided by the wing rotation is ζ i = 0.01 in each mode and recalculate the response. (b) If the aircraft is in flight, the damping forces may increase dramatically to ζ i = 0.1. Recalculate the response and compare it to the more lightly damped case of part (a). Solution: From Problem 4.44, with damping 3000 0 0 0 12 000 0 0 0 3 000 53 820 53 820 0 53 820 107 640 53 820 0 53 820 53 820 , , ˙˙ ˙ ,, , ++ −− = xx x 0 C q q0 0 0 0 200 00 17 94 4 2356 26 91 5 1875 11 2 33 () = [] == . ˙ .. T m 0 rad/s rad/s rad/s 2 λω vvv 123 0 4082 0 8165 0 4082 0 7071 0 0 7071 0 5774 0 5774 0 5774 = = =− .

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SolSec 4.5 - Problems and Solutions for Section 4.5(4.51...

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