EN40: ynaics and Vibrations Final Exaination Tuesday May 15, 011 School of Engineering rown University NME: General Instructions No collaboration of any ind is peritted on this exaination. You ay use double sided pages of reference notes. No other aterial ay be consulted Write all your solutions in the space provided. No sheets should be added to the exa. Mae diagras and setches as clear as possible, and show all your derivations clearly. Incoplete solutions will receive only partial credit, even if the answer is correct. If you find you are unable to coplete part of a question, proceed to the next part. Please initial the stateent below to show that you have read it `y affixing y nae to this paper, I affir that I have executed the exaination in accordance with the cadeic Honor ode of rown University. PLESE WITE YOU NME OVE LSO! 1-10: (0 PTS) 11: (10 PTS) 1: (10 PTS) 13: (10 PTS) 14: (10 PTS) TOTL (60 PTS)
FO POLEMS 1-10 WITE YOU NSWE IN THE SPE POVIE. ONLY THE NSWE PPEING IN THE SPE POVIE WILL E GE. ILLEGILE NSWES WILL NOT EEIVE EIT. 1. Two cylinders with equal ass density start at rest, and roll without slipping down an incline. ylinder 1 has a radius and cylinder has a radius. Which cylinder will have a higher velocity when it arrives at point? (a) ylinder 1 (b) ylinder (c) oth the sae (d) This is one of life s unanswerable questions. (1) () For a cylinder with general radius r the KE is T ( I v ) / and since the cylinder rolls without sliding v/ r. Energy conservation gives gh ( I / r ) v / and finally recall I r / so that gh 3 v / 4 v gh / 3. This is independent of the radius or ass of the cylinder. Or just reeber Prof Franc s class deo!. The unit g /s is used for: (a) otational Kinetic Energy (b) Power (c) ngular Moentu (d) ll of the above (e) None of the above The units are ( a) I ( g ) / s ( b) P Fv ( g / s )( / s) ( c) h rv. g. / s 3. vibration isolation table can be idealized as shown in the figure. It is subjected to a haronic force F(t) with aplitude 0.1N and angular frequency 10 rad/s. The aplitude of vertical vibration is (a) 0.1 (b) 0. (c) 1 (d) 10 (e) 0 (f) None of the above nswer: (c) nswer: (c) The natural frequency is / 10 rad / s. The syste is therefore at resonance. The daping coefficient is 1/ ( ) 1/ 00. t resonance the aplitude is ( F / ) / ( ) 0.1 00 /1000 / 0.01 10. nswer: (d) 0 1000N/ F(t) 10g 1Ns/ 1
4. bead of ass slides on frictionless ring of radius in a vertical plane. The bloc is subjected to a vertical gravitational force g as well as a force P=g that is always oriented along the direction of sliding. The bloc starts fro rest at point. What is the velocity of the bloc when it reaches point? (a) v g (b) v ( 1) g (c) v 4g (d) v 1 g (e) None of the above The change in KE is equal to the wor done by the forces acting on the bead. The wor done by gravity is g, the wor done by P is P( / ) g. Therefore v /g g nswer: (b) 5. bird lands near the tip of a branch, and is observed to oscillate up and down about once a second. The bird on the branch can be idealized as a lightly daped spring-ass syste. When the vibration stops, the (static) deflection of the tip of the branch is approxiately equal to (a) 0 (b) 0.05 (c) 0.5 (d) 0.50 (e) 0.75 (f) 1 The bird and branch behave lie a lightly daped spring-ass syste. The static deflection is related to the natural frequency by g / g / 9.8 / ( ) 0.5 n n 6. vehicle ounted on its suspension syste is idealized as a rigid body supported by three springs as shown in the figure. How any vibration frequencies does the syste have, assuing that otion is confined to the plane of the figure? a) b) 3 c) 4 d) 6 e) None of the above L nswer: (c) h The syste has 3 OF otion horizontally, vertically and rotation. There are therefore 3 natural frequencies. nswer (b)
7. The figure shows a collision between two identical spheres. The restitution coefficient for the collision e=0. efore the collision, oves with speed v 0 and is stationary. uring the collision (a) Moentu and energy are both conserved (b) Moentu is conserved, and the energy increases (c) Moentu is conserved, and the energy decreases (d) Energy is conserved and oentu increases (e) Energy is conserved and oentu decreases v 0 ollision Moentu is conserved during a collision. Since e=0 the two spheres have the sae velocity after collision this eans that after collision v v 0 /. The total energy after collision is ( v / ) / / 4. This is half the energy before collision. 0 v0 8. ritically aped vibrating syste (a) Vibrates forever if it is disturbed fro equilibriu (b) Vibrates if disturbed fro equilibriu but the vibrations decay quicly (c) eturns to equilibriu following a disturbance without vibration (d) Never returns to its equilibriu configuration if disturbed (e) Feels wet and insulted. nswer (c) The critically daped solution decays exponentially to equilibriu. There is no vibration. nswer (c) 9. eats are heard when two sounds have ) nearly the sae aplitude ) nearly the sae frequencies ) twice the aplitude ) exactly twice the wavelength eats occur when two signals with siilar frequencies cobine because of the difference in frequencies the signals alternately interfere constructively and destructively to give a slowly varying aplitude odulation. nswer (b) 10. The viscous daping factor for the syste shown in the figure is (a) c / (b) c / (c) c / (d) c / c The two springs are in parallel, so the effective stiffness is. The standard forula for daping coefficient for a spring-ass syste gives c / c / eff eff nswer (b) 3
11. The figure shows two identical asses that are connected to springs with stiffness. The asses vibrate with displaceents x1( t), x( t ) fro their equilibriu positions. When x x there is no force in the springs. 1 0 x 1 x 11.1 raw a free body diagra for each ass on the figure provided below F S1 F S F S F S3 11. Write down the changes in length of each of the three springs in ters of x1, x. Hence, use Newton s laws to show that x1( t), x( t ) satisfy the equations of otion 1 d x x1 x 0 d x x1 x 0 The increase in spring lengths (fro left to right) are x1, x x1, x The spring forces follow as FS 1 x1 FS ( x x1 ) FS 3 x Newton s law for the two asses gives d x1 F S1 FS x1 ( x x1 ) d x F F ( x x ) x These can be rearranged into the for given. S S3 1 (3POINTS) 11.3 dd and subtract the equations of otion to show that the noral odes q x x q x x satisfy equations of the for 1 1 1 1 d q 1 q1 q d q 0 0 Hence, deterine forulas for the two natural frequencies 1,. dding the equations gives Subtracting gives dq1 1 1 1 d x x ( x x ) 0 q 0 d dq1 3 x 1 x 3 ( x1 x) 0 q 1 0 The natural frequencies follow as 1 / 3 / 4
(4 POINTS) 1. The figure shows a robot ar. Point on the ar is required to ove horizontally with constant speed 1/s. This is accoplished by rotating lins and with appropriate angular speeds, and angular accelerations,. The goal of this proble is to calculate values for,,, at the instant shown. 1 45 0 j i 1 V x = 1/s a = 0 1.1 eterine forulas for the velocity vectors v, v of points and, in ters of,. (You do not need to solve for, until 1.3). pplying the rigid body ineatics forula gives v v ω r / ( i j) / ( i j ) / v v i ( i j) / j 1. eterine forulas for the acceleration vectors a, a of points and in ters of,,,. (You do not need to solve for, until 1.3) a a ( i j) / ( i j) / ( i j) / ( i j) / a a i i ( ) / ( ) / i j i j j i 1.3 Hence, calculate the required values of,,, We now that v i. Using the i and j coponents of 1.1 gives two equations for, We also now that a rad / s / 0 1 rad / s 0 which gives ( ) / ( ) / i j i j j i 0 ( i j) / j ( i j) / i 0 rad / s 1 1 1 rad / s 4 (4 POINTS) 5
13. The figure shows a bar with ass and length L that is pivoted about point. The bar is stabilized by a torsional spring with stiffness, which exerts a restoring oent with agnitude at. The goal of this proble is to deterine the natural frequency of sall aplitude vibrations of the bar. 13.1 State, or derive, the ass oent of inertia of the bar about point, in ters of and L. L The ass oent of inertia about the center is fro scratch as L/ 1 3 1 ) L 1 L 1 L/ I x dx L L L /1 (you can derive this The parallel axis theore then gives L 1 1 I L L 1 3 ( POINTS) 13. Write down the total potential energy of the syste, in ters of,g,l,., The potential energy is L 1 g cos 13.3 Write down the total inetic energy of the syste, in ters of,, d L. ( POINTS) The inetic energy is 1 d I = 1 L 6 d 13.4 Hence, use energy conservation to show that satisfies L d L g sin 0 3 d L 1 1 The total energy is constant so L g cos 6 ifferentiate with respect to tie 1 d d L d d L g sin 0 3 1 d L L g 3 sin 0 13.5 Finally, deterine a forula for the natural frequency of vibration. ecall that d g g sin 3 0 3 n L L L L ( POINTS) ( POINTS) ( POINTS) 6
L j i O r F 14. n inerter is a suspension eleent that exerts a force F that is related to its length L by d L F The figure shows a proposed design for such a device. It consists of a dis with ass, radius and ass oent of inertia / which is rigidly connected to an axle with radius r. The axle rolls without slip between platens and (which have negligible ass). The objective of this proble is to derive an equation for the coefficient 14.1 raw a free body diagra showing the forces acting on the dis/axle and platen on the figures provided below. Gravity ay be neglected. N O r T T N F T 1 N 1 (The noral forces are optional) 14. Suppose is stationary and the dis and axle rotate with angular velocity. Find the velocity vectors v and v of the center O of the axle and point in ters of r and O O j i The wheel rolls without slip on the eber. The center therefore has velocity vo ri. There is no slip at the contact between the axle and. therefore oves with the sae velocity as the axle at the contact point this is v ri ( POINTS) 7
14.3 Write down the equations of linear (F=a) and rotational ( M Iα ) otion for the dis. Newton s law gives ( T T1 ) i ( N1 N) j aoi (the j coponent should be consistent with F) The rotational equation gives ( T1 T) r I 14.4 Hence, show that (1 ) 4 r Fro proble 14. v r a r a / ( r) v r a r a 0 0 / ( POINTS) Fro proble 14.3 T T r a 1 1 Finally the F for the bar shows that T for I) / T T I / r a I / ( r ) 1 T a ( I / r ) F and of course F (1 ) d L 4 r a d L. Therefore (substituting 8