&vjl/i f i ~ l,j~j( V;' '14 V;;; =

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1 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currenti When a 20-Ib weight is suspended from a spring. the spring is stretched a distance of 4 in. Determine the natural frequency and the. period of vibration for a 10-1b weight attached to the same spring. 20 k == T = 60 Iblfl T2 V;;; = w. = fk~ 10 n.2 = radls 21T r =..., =0.452 s ru. J f = - =2.21 Hz r AIls AIls A spring has a stiffness of 600 N/m. If a 4-kg, block is attached to the spring, pushed 50 mm above its equilibrium position, and released from rest, determine the equation which describes the block's motion. Assume that positive displacement is measured downward. V;' '14 WI' = ~ == (WiS = rddls l' = O. X = m at, = 0 x = A sin lon' + Bcosw.' = 0+ B B = V = Aw" cos wnl - sin (1).1 o= A(12.25) - 0 A=O &vjl/i f i ~ l,j~j( f Thus. x = cos(l2.2r) mans When a 3-kg block is suspended from a spring. the spring is stretched a distance of 60 mm. Detennine the natural frequency and the period of vibration for a O.2-kg block attached to the same spring. F k = 6x = 3(9.8]) = N/m g /490.5 ((I. = - = -- = fad/s m 0.2 f = lon = = 7.88 Hz 2.."1' 2)T I I r = 7 = 7.88 = s Ans Ans 617

2 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currel *22-8. If the block in Prob :7 is given an upward velocity of 4 mls when it is displaced downward a distance of 60 mm from its equilibrium position. determine the equation which describes the motion. What is the amplitude of the motion? Assume that positive displacement is measured downward. Since F = tax, where Ax = 175 mm ~ = 8(9.81) "' N/m Ax If ~ Wn = V;;;" -8- = 7.49 radls v = -4mJs, x 0.06m att = B B = "' A(7,49) - 0 A Thus, x,. [-Q.534sin(7.49~ Ocos(7.49t)] m Ans c = IAZ + SZ,. {(-Q.534)2 + {O.(6)l = m Ans Determine the frequency ofvibration for the block. The springs are originally compressed fl. 4k x+-x =0 m -4kx =mx f=~ ~ 1r V;;,Ans 620

3 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently * The square plate has a mass m and is suspended at its corner by the pin O. Determine the natural period of vibration if it is displaced a small amount and released. Va 2 21r fa,=- =6.10 1= <u. Vg w The disk has a weight of 10 lb and rolls without slipping on the horizontal surface as it oscil1ates about its equilibrium position. If the disk: is displaced, by rolling it counterclockwise 0.4 cad, determine the equation which describes its oscillatory motion when it is released. to). 11(;=- I ( - (1)2+ 10 (/)2 O.465Sslug fr k = 100 lblft w. "" ';429.3 =: nulls 10lb Ift~T N f.i) =0, fj :: 0.4, I =0 0.4 ::0+ B B=0.4 (J) "" U= Aw. COSW"I- Bw,. sin"""l O=A"'n -0 Thus. (j =0.4oos(20.7t) ADs

4 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently 22-4L Use a block-and-spring model like that shown in Fig a but suspended from a vertical position and subjected to a periodic support displacement of 8 80 cos wet, determine the equation of motion for the system, and obtain its general solution. Define the displacement y measured from the static equilibrium position of the block when t O. Since W = ko,,; k kf)o Y +-Y -cosujol (1) m m Yc = A sin wny + B cos wny(general sol) YP C cos wot (Particular 801.) Substitute YP into Eq.(l) k C( _Wfj2 + -)cos Wot cos wot m m c Thus, Y = Yc + YP ~ Y A sin w,l + B cos W n! + k m 2 cos Wol Cn; -wo) AIlS The 20-1b block is attached to a spring having a stiffness of 20 lb/fl A force F =(6cos2t) lb, where t is in seconds, is applied to the block. Determine the maximum speed of the block after frictional forces cause the free vibrations to dampen out. Fu 1-(:r T c=., cv" = [f. = rjf = radls V'iii vl1 6cos2t 6 C = :iii,== 0343 ft 1- (,.6;4l r XI' == Ccos2t Xp == -C(2) sin 21 Maximum velocity is =C(2) =: 0.343(2) =0.685 rtis AIlS 636

5 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currel The instrument is centered uniformly on a platformp, which in turn is supported by four springs, each spring having a stiffness k = 130 N/m. If the floor is subjected to a vibration Wo = 7 Hz, having a vertical displacement amplitude Bo = 0.17 ft, determine the vertical displacement amplitude of the platfonn and instrument:the instrument and the platform have a total weight of 18 lb. k = 4(130) = 520lb/fl So = Wo= 7 Hz = 7('br) ::: mdls Using Eq , the arnpliti1d<'l is 1- (;~r Smce. (f)" = If- == fl20 Is =30.50 mdls m len, (xp ),,,,,. = I 1 = ft! (::~) (x p ),,,,,, = 1.89 in. 641

6 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they currently The 450-kg trailer is pulled with a constant speed over the surfaee ofa bumpy road, which may be approximated by a cosine curve having an amplitude of 50 mrn and wave length of 4 m. If the two springs s which support the trailer each have a stiffness of 800 N/m, determine the speed V which will cause the greatest vibration (resonanee) of the trailer. Neglect the weight of the wheels. The amplitude is 00 ~ 50 rnrn 0.05 m The wave length is it = 4 m k = 2(800) :: 1600 N/m w = II. == V1600 = 1.89 radls n V; = :: 'For maximum vibration of the trailel; resonance must occur, i.e., Thus, the trailer must travel it == 4 m, in.. = 3.33 s, so that it 4 v R "" - :=: - = 1.20 m/s ADs., Determine the amplitude of vibration of the trailer in Prob if the speed v 15 km/h. 15(1000) v = 15 krnjh:=: 3600 mls == 4.17 mis 8 0 = 0.05 m As shown in Prob , the velocity is inversely proponional to the peciod. 1 Since - =fthen 1I1e velocity is proportional off, w" and w() r. Hence, the amplitude of mooon is 642.

7 2007 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected under all copyright laws as they curren Detennine the angular velocity of the flywbeel in Prob which will produce an amplitude of vibration of 0.25 in. TIle constant value Fo of the periodic force is due to the centrifugal force of the unbalanced mass. 0.25) (10) Fe =lltan == 1IIr<»02 = ( wol =O.OO6470wJ F = O.00647Oc.!j sin("'ri' k f "" tl~ = 1800 Iblft Wn = If = Ii;::;; ::::: From Eq , the amplitlide of the steady stall:; motion is c= = Folk I-(:r O.OO6<no ( ~) l-c9~~57r wo = 19.7 cadis Ans The engine is mounted on a foundation block which is spring-supported. Describe the steady-state vibration of the system if the block and engine have a total weight of ]500 Ib and the engine, when running, creates an impressed force F = (50sin2t) lb. where t is in seconds.' Assume that the system vibrates only in the vertical direction, whit the positive displacement measured downward, and that the total stiffness of the springs can be represented as k = 2000 Ib/ft. The steady-slate vibration is defined by Eq Fo I-G:::) xi' = k 2 sinatu/ Since F = 50 sin 2t Then Fit =50 lb, Il>Q 2 I1IdIs k = 2000 lb/ft Ji ooo f!: =1500 = 6.55 radls UJ. == V;;; Hence, xi' ;; 2000 gin2! 2 )2 J - ( 6,55 XI' = ( sin 21) ft AM 644

8 2010 Pearson Education, Inc., Upper Saddle River. NJ. All rights reserved. This material is protected under all copyright laws as they currently * Determine the differential equation of motion for the damped vibratory system shown. What type of motion occurs? mg k(y + Yst) - 2ey = my loon/m my + ky + 2ey + kyst - mg 0 Equilibrium kysr - mg = 0 my + 2cy + ky 0 Here m = 25 kg k 100 Nlm e 2ooN slm 25y + 400y + 100y 0 (1) Y+ 16y + 4y = 0 Ans. By comparing Eq. (1) to Eq c 200N s/m m = 25 k 100 e = 400 Wn =.Ji = 2 rad/s Cc 2mw" 2(25)(2) = 100 N slm Since C > C e, the system will not vibrate. Therefore, it is overdamped. Ans.

Section 3.7: Mechanical and Electrical Vibrations

Section 3.7: Mechanical and Electrical Vibrations Second order linear equations with constant coefficients serve as mathematical models for mechanical and electrical oscillations. For example, the motion