ASE 365 - Test 3 Merged

ASE 365 - Test 3 Merged - ASE 365Structural Dynamics Dr J K...

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ASE 365—Structural Dynamics Name: Dr. J. K. Bennighof April 30, 2009 TEST 3 This test is closed-book, except that you are allowed to use one two-sided sheet of notes that you have prepared beforehand. Do all of your work neatly , with each problem beginning on a separate page. Clearly designate each of your answers with a box or circle. All numerical answers should either be exact or accurate to four decimal places. When you hand in your test, place the test handout, with your name on it, on top, followed by your work, with your sheet of notes on the bottom. Staple all pages together. Problem 1 (30%) The two degree of freedom system shown below has two masses, two springs, two dashpots, two displacements, and is excited by one force. (a) Show with two sketches the non-conservative virtual work δW nc done as virtual displacements δx 1 and δx 2 take place. Find the kinetic energy T and potential energy V for this system in terms of generalized coordinates x 1 and x 2 and their time derivatives. Write δW nc , T , and V in terms of matrices and vectors. (b) Obtain equations of motion for this system, working from the quantities found in part (a). Write these in the standard matrix-vector form. x 1 ( t ) x 2 ( t ) k 1 k 2 c 2 F ( t ) m 1 c 1 m 2 θ x I k m r Prob. 1 Prob. 2 Problem 2 (30%) A disk of mass moment of inertia I rotates about its center without friction. Mass m is attached to the disk through a flexible rod whose length equals the disk radius r , and whose effective stiffness is k . The angular displacement of the disk is θ , and the displacement of m due to the flexing of the rod is x . The kinetic and potential energies, assuming x is small, are T = 1 2 I ˙ θ 2 + 1 2 m (2 r ˙ θ + ˙ x ) 2 and V = 1 2 kx 2 , so the equations of motion are bracketleftbigg I + 4 mr 2 2 mr 2 mr m bracketrightbiggbraceleftbigg ¨ θ ¨ x bracerightbigg + bracketleftbigg 0 0 0 k bracketrightbiggbraceleftbigg θ x bracerightbigg = braceleftbigg 0 0 bracerightbigg (a) Obtain the natural frequencies and modes of vibration. Scale each eigenvector so that one of its entries is equal to one. Describe the motion in each mode, in words, as completely as you can. (b) What is the total angular momentum of this system about the disk center in terms of the modal velocities ˙ η 1 and ˙ η 2 , assuming that the motion is represented in terms of your eigenvectors from part (a)? (Remember angular momentum is r × m ˙ r for a particle, and I ω for a rigid body.) 1
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ASE 365—Structural Dynamics Test 3 April 30, 2009 Problem 3 (40%) The system shown below is excited by the exponentially decaying step input F 2 ( t ) = F 0 e - αt u ( t ), which is applied to the second mass. The equations of motion are given by bracketleftbigg 2 m 0 0 m bracketrightbiggbraceleftbigg ¨ x 1 ¨ x 2 bracerightbigg + bracketleftbigg 4 k - 2 k - 2 k 3 k bracketrightbiggbraceleftbigg x 1 x 2 bracerightbigg = braceleftbigg 0 F 0 e - αt u ( t ) bracerightbigg .
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