Part IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Examples paper 3

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1 ENGINEERING Part IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Exaples paper 3 IRST YEAR Straightforward questions are ared with a Tripos standard questions are ared *. Systes with two or ore degrees of freedo Throughout this exaples paper, assue that displaceents are sall and neglect the effects of daping. ree vibration 1. How any degrees of freedo have each of the dynaic systes shown in ig. 1? (a) Planar double pendulu (b) U-tube anoeter (c) Point ass in rigid frae (d) Spring-ounted engine bloc (e) Planar articulated-vehicle odel with pin joint ig. 1 (f) Tacoa Narrows bridge. or the syste shown in ig., show that the equation of free otion can be written in the for M ÿ + K y = where M = 1, K = 1 + and y = y. If the displaceents in the j th noral ode of free vibration are given by y j = Y j cosω j t, deduce that each natural frequency ω j and its corresponding noral ode Y j ust satisfy [ K ω j M ] Y j =. Hence find the natural frequencies and noral odes for the case 1 = = and 1 = =. Setch the two ode shapes. 1

2 1 1 ig. 3. Two rigid discs with oents of inertia J and J are ounted on three light elastic shafts of torsional stiffness,, and as shown in ig. 3. The angles of rotations of the discs fro their equilibriu positions are θ 1 and θ respectively. Show that the equation of free torsional vibration of the syste can be written in the for Mθ _ + K _ θ = where M = J J, K = 3 and θ 1 _ θ = θ. Using the ethod outlined in Q above, find the natural frequencies and noral odes. Setch the ode shapes. Type in the following Matlab progra to chec your answers: K=[ -1;-1 3]; M=[1 ; ]; [V,D]=eig(K,M); for n=1: ode=v(:,n); ode=ode/ode(1); freq=d(n,n); disp(sprintf('mode %i has squared frequency %g and ode [%g, %g]',n,freq,ode)) end Do you understand the progra? eig calculates eigenvalues and vectors: type help eig in the Matlab coand window to see details. Save your progra, to use again in question 5. θ 1 θ J J ig Deterine by inspection the noral odes of vibration of the systes shown in ig. 4. Hence calculate the natural frequencies of each syste. Describe the otion of the two asses in ig. 4(b) for the following sets of initial conditions: (i) = =1 and! =! = ; (ii) =1, = and! =! = ; (iii)* = = and! =1,! = ;

3 / (a) ig. 4 (b) *5. Two equal asses are attached to a third ass M by two equal springs each of stiffness as shown in ig. 5. Derive the equations of otion for the syste in ters of the three coordinates, and y 3 as shown. Calculate the three natural frequencies and corresponding noral odes, and setch the ode shapes. (Hint: two of the natural frequencies and ode shapes can be deterined easily by inspection.) What is the significance of a zero natural frequency? Modify the progra fro question 3 to chec your answers. Try different ratios M/. y 3 M ig. 5 orced haronic vibration (Ignore any transient response throughout) 6. Show that the equation of otion for the syste shown in ig. 6 ay be written in the for M ÿ + K y = f where M =, K =, y = y and f = f. or free otion (f = ) find the natural frequencies and noral odes (by inspection). When f = cosωt, the forced haronic response is given by y = Y cosωt. Deduce that Y = " # K ω M Hence derive expressions for the response aplitudes Y 1 and Y and setch their variation with frequency ω. (It is easiest to plot the non-diensional quantities Y 1 / and Y / against nondiensional-frequency squared ω /. Give due regard to signs.) $ % 1 " & # $ '. % 3

4 f ig. 6 *7. igure 7 shows a odel of a caravan suspension. The ass of the caravan body is 1 and that of the axle assebly is. The corresponding vertical displaceents are and. The stiffness of the suspension springs and of the tyres are 1 and respectively. Road roughness is represented by the displaceent x of the botto of the tyre spring as shown. Show that the equation of otion ay be written in the for 1 ÿ 1 ÿ y = x. A particular caravan has 1 = 5 g, 1 = N/ and = 4 g, = 16 N/. It is towed at a constant velocity V along a bupy road whose surface profile varies sinusoidally with an aplitude X of 5 and a wavelength L of 1.5 so that the tyre displaceent input is x = X cosωt, where ω = πv L. The caravan displaceent and that of its axle are = Y 1 cosωt and = Y cosωt. ind the aplitude of caravan vibration Y 1 when the velocity V = 5 /h. ind also the aplitude of the axle vibration Y. Why is the axle otion so large? [Hint: calculate the two natural frequencies.] Coent on the use of shoc absorbers. y 1 1 V y x 1 L ig. 7 4

5 *8. The vibration of a pedestrian footbridge in its fundaental bending ode ay be odelled as a ass 1 = 1, g supported on a spring of stiffness 1 =.7 MN/ so that the natural frequency of the bridge in this ode is.6 Hz. A vibration absorber is attached at the centre of the bridge to prevent excessive aplitudes of vibration when hooligans jup up and down on the bridge. The absorber coprises a ass supported on a spring of stiffness as shown in ig. 8. Derive the atrix equation for vertical otion when the hooligan exerts a force f on the bridge. If f = cosωt, derive an expression for the aplitude of vibration of the bridge Y 1 as a function of ω. It is desired that there is no bridge otion when it is excited at ω = Ω. Deterine the required absorber stiffness in ters of the absorber ass and Ω. or the case = 1 g and = 7 N/, calculate the two natural frequencies of the syste. Setch the variation of 1 Y 1 / with ω, both with and without the absorber. What is the effect of adding daping to the absorber? f 1 1 ig. 8 Eight such absorbers are fitted to the cycle bridge over the Cabridge railway station. Each absorber coprises a 1 g sprung ass in an oil-filled enclosure the oil provides daping. 5

6 Answers 1., 1, 3, 6, 4, Y 1. ω 1 =.38, Y = ω =.618, Y = Use the Matlab progra! 4. (a) ω 1 = 4, ω = 5. Y 1 (b) ω 1 =, ω = (i) = = cos ω 1 t ; (ii) =.5 ( cos ω 1 t + cos ω t ) ; (iii) =.5 ( 1 ω 1 sin ω 1 t + 1 ω sin ω t ) ; =.5 ( 1 ω 1 sin ω 1 t 1 ω sin ω t ) M ÿ 1 ÿ + ÿ = y 3 Use the Matlab progra to chec answers. Zero natural frequency corresponds to rigid-body otion. Mode shape is always [1, 1, 1,...] 6. ω 1 =, Y 1 = 1 1 (rigid body otion) ω =, Y = 1-1 Y 1 = 1 ω / ω / [ ω /] Y = 1 ω / [ ω /] input at 11.1 Hz is just above the 'wheel-hop' frequency (1.7 Hz) 1 8. ÿ 1 ÿ y = f Y 1 = ( ω ) where Δ = Δ 1 ω 4 [ ( 1 + ) + 1 ]ω + 1 = Ω ω 1 = 14. rad/s ω = 19. rad/s 1 Y 1 = (Ω ω )Ω (ω 1 ω )(ω ω ) Daping reduces peainess of the two resonances. Instead of large bridge otion, we have large otion of the absorber ass. It is often easier to add daping to an absorber than to a bridge. or further practice, the following Tripos questions fro Paper 1 are suitable: 13 Q1; 1 Q11; 11 Q1; 1 Q1; 9 Q1; 8 Q9; 7 Q1 J Woodhouse Easter 14 6