Vibration Control

5/28/2018 Vibration Control

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5 Vibration control

perfectly regular. The response will always have minor transients affectingit, so you are reminded again that the theoretical steady-state solution is

5 1 Total vibration and transientsIn explaining the forced vibration response I have not been telling thewhole story. Even for a single-degree-of-freedom model there can be anatural (free) vibration and a forced response at the same time forming thetotal vibration. For example, if a component vibrating in its steady-stateresponse to a forcing vibration were hit by a hammer then it would alsoperform a natural vibration. In time, this natural vibration would bedamped out to leave only the steady-state vibration.Consider a case in which the mounting is vibrating steadily, but the massIbl in our model is being held stationary. It is then released. What happens? Ifby sheer chance it was released having the appropriate speed anddisplacement and the correct phase to agree with its steady-state forced

only an approx imation to th e true behaviour of a real system. The steady-state solution does, however, indicate the importanc e of resonance an d itspotential dangers.

I I

You should also note in this context that in certain circumstances thedamping in a system can lead to an increasing transient vibration ratherthan one which dies away. This situation is more likely in systems withmore than the one degree of freedom, to which I have restricted thisintrod uctio n to vibration so far. An increasing transient is associated withinstability in a vibrating system. Typical engineering examples are a ircraftwing flutter and wheel shimmy. We do not, however, have the time todiscuss instability further in this U nit.

response then it would proceed to perform just that. However, normallythe release conditions will not agree with the steady-state requirement. InFigure 35, say that it is released with zero speed and zero displacement att =0, when the steady-state response wanted it to be at X -A,Figure 35(a). The result is that it performs a natural vibration of initialigure 5 amplitude A Figure 35(b), superimposed on the steady-state vibration,Figure 35(c). Because this natural vibration is damped and dies away it is

cni c called the transient In due course there remains only the steady-stateforced response. This is a particularly simple example of a transient, but itis intended to illustrate to you that a transient can arise whenever theresponding object is disturbed from equilibrium or from a steady-statevibration. This can happen either because it has been acted on byadditional forces, for example a n impact, o r because the forcing functionchanges in some way (amplitude, frequency o r phase). very commoncase is when the forcing vibration comes from a starting machine.However. even when the vibration source is runnine steadilv it will not be


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5 2 Vlbratlon toleranceBecause of the dangers of vibration in general and resonance in particular,the mechanical engineering des~gnermust always pay close attention topossible vibration problems. The acceptable level of vibration depends onthe detailed application. delicate instrument may need to he mounted insuch a way as to prevent dangerous vibration reaching it. structure maye damaged by the stresses associated with vibration even if the yield

stress is not reached, the cyclical stress variations can lead to fatigue failureas was mentioned in Unit 10 The tolerance of an instrument o r structure E..L.thfo vibration is amenable to analysis, so in such~caseshe designer will beable to specify acceptable levels of vibration.In the case of human beings such analysis is not practicable. In this casethe limits are not so well defined because they vary considerably from oneindividual to another. Also the limits are not those of mechanical failure ofthe body so much as subjective factors such as concentration, headachesand deterioration of visual acuity. Extensive tests have given rise tostandards for limits of human exposure to vibration. The acceptable levelsdepend upon the period for which they must be tolerated. Figure 36 showsa very simple vibration model of the human body, indicating thepossibility of a number of resonance frequencies. F~gure37 shows thevibration limits for an 8-hour period approximately a working day. Byusing logarithmic scales the limits conveniently fonn straight lines. Theworst frequencies are those in the region 4 to 8 Hz orresponding to anatural frequency of the internal organs. The three curves represent therather subjective variables of reduced comfort, decreased proficiency andexposure l i t . The decreased proficiency limit is a suitable general guideto acceptable vibration for vehicle drivers or factory workers. Figure 36

Figure 37 Human vibration limits vertic al whole body vibration, 8-hour period


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Exceeding the limit would be liable to lead to increased accident ratesthrough loss of concentration. Exceeding the exposure limit can lead to adirect health risk from the vibration itself. Note that the vibration isexpressed here in terms of the peak acce leration, giving a more convenientgraph than displacement amplitude would. This amplitude can b f o u n d bydividing by the frequency (in rad s - l ) squared because for sinusoidalvibration X Oax.

If a bus driver is subject to an average of 4 Hz vibration from the enginefor a full working day, an d is not to have decreased proficiency, what is themaximum vibration to which the driver may be subjected, expressed as(a) peak acceleration, (b) amplitude?

If the bus driver is mainly subject to 8 z vibration from the engine attick-over, what is the maximum amplitude acceptable for (a) comfort,(b) proficiency, (c) exposure limit?

5 3 ontrol metho sThe main message of the Unit so far has been that vibration can bedamaging, especially if resonance occurs. What can the engineeringdesigner do to minimize vibration problems? This area of study could becalled vibration control , and you have now covered sufficient of the basictheory of vibration to look at the methods used to achieve this.The first idea that you need to recognize is tha t there is always a sou rce ofvibration, and there is also an object which is subjected to tha t vibration . Ishall call the latter the responder. The source and responder are no t nor-mally the same, so there is a transmission path between the two ( F i r e 38).Examples would be: (1) a lorry (source) shaking the windows of yourhouse (responder); 2) an engine (source) shaking a carburettor (re-sponder) mounted on it. causing fuel feed problems; (3) a rough mad(source) shaking a car driver (responder). In practice one source mayvibrate many responders, and equally one responder may be affected bymany sources. Sometimes the source and responder are not readilydistinguishable, for example a lathe tool vibrating because of the forcesexerted on it by the workpiece he source is really thec uttingop eratio n, acombination o tool and workpiece. Equally, an engine is its own sou= ofvibration. However, in most cases the source and responder are readilyidentifiable. A good example typical of those met in practice would be aproblem where a m p ro ca ti n g engine mounted o n a structure is liable tocause damage or other adverse effects to delicate instruments mountedelsewhere on the structure. The structure might be a building, motor car, aship, or an aeroplane. (If your only experience of flying is in a modern jetaircraft, I can assure you that they are incomparably smoother than theolder piston-engined aircraft used to be.)In principle there are throe possible methods of vibration control we canmodiiy the s o w , he transmission path o r the responder.

odification of the somce of vikationThis is often the most effective and direct solution . Improving the balanceof rotating m achinery is one such procedure. A classic example is the wheelof a car. A badly unbalanced wheel can produce some very unpleasanteffects on the m u p a n t s of the car. Sometimes the resonant responseof thebonnet is clearly visible too. T he sensible answer is to improve the wheelbalance, which involves add ing small pieces of lead to move the centre of


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mass to the centre of the axle, reducing the f o r a s that cause the vibrationto negligible amplitude. Balancing is, however, not always possible. Mostreciprocating engine designs cannot be balanced perfectly even in prin-ciple, although the use of a larger number of cylinders generally reducesthe residual unbalanceable forces. A rough road is another source ofvibration that cannot normally be considered alterable. Within limits adriver can avoid pot-holes and choose a smooth route, but the vehicledesign must take account of the fact that the road is by no means perfectlysmooth.Modifkrtion of the responderThis is often a satisfactory means to obviate vibration damage. heforcing vibration has a substantially constant frequency then the naturalfrequency of the responder can be adjusted to ensure that resonance isavoided. This may be done by adjustment of s t i hc s s o r mass. However,this is more difficult if the vibration source COVCTS a range of frequencies,for example an engine covering a range of speeds or a car being shaken bya road whose surface undulates in an irregular manner. In such casesa d q u a t e damping must be incorporated to minimize the e k t of theunavoidable rcsonances (remember the effect of damping on resonance).Car body panels often have damping material such s felt applied to them.Another possible modification to the responder isto add an auxiliary massand spring. If tuned to the appropriate frequency the motion of theauxiliary mass will be large,but the vibration of the original responder willbe reduced. A device of this kind is often called a vibration absorber.Many rR . brbcar engine crankshaftsare fitted with torsional vibration devices of thistype.Modl .dm of the vihtioD tn lm*r ioo p thBy introducing a flexible member between the source and responder it maybe possible to reduce the transmission of vibration. In practice this meansmounting either the source or responder on a flexible support. This iscalled vfbratbn isolation because the machine is 'isolated' against vibra- riLn(*.IrlrtL.tion transmission The springs of a car suspension re a good example,isolating the responder (the car body) fmm the s o w the road . TheRcxibity ofthe t y m also plays an important role here The passengers a nlurther isolated from the c r body by the kxibility of the seats Therem inder of this d o n s conccrncd with the performance of vibrationisolating mountings.5 4 Vlbratlon lsolatlonIf attention to reducing the vibration at source and attention t o the abilityof the responder to withstand vibration prove inadquate, then thenmaining possiity is t o i n t e m p t the trammission path in some wayT he most convenient p k o d o this is nonnally a t the source mountingor the respondex mounting, A and B rcapeaiwly in Rgun 38. It turns outthat these am very similar problcma.Rrs t consider a responder, shown in Figure 39on a m ounting modelled bya simple spring and dam per in parallel. In the majority of practica l cases asimple mounting such s the one shown is adequate. If the ground isvibrating with amplitude X, what is the amplitude A of the responder?You already know how to find that. It is A TXo where T is thetransmissibility of the mounting for the mass that it cames. What mPcharacterist ics a n required of the mounting to give vibration isolation? 9For a sm llA obviously we want a small T nd referring to Figure 34 it isclear that a high frequency ratio is the main thing to a for.1t is not soobvious what to d o in the way of dunping U the frequency ratio is highthen light damping smallC) would be better, but this might be dangerousif resonance occurs because of the high peak value of T for small C


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Figure 4

Flgiue l mplitudes m

Before saying any more about the practical implications of these curves letme turn to the problem of isolating a vibration source. Figure 40 shows asource with mass m again on a mounting that can be modelled by thespring of stiffness k and damper of coefficient c in parallel. Thece is a forceF = F, sin Rt acting on it, usuaUy as a result of the motion of internalcomponents, where F is the force amplitude. It is analogous to thedisplacement amplitude, the peak value of a sinusoidally varying displace-ment. Amplitude just means the peak value. Here I shall useamplitude to mean the peak value of a varying displacement and forceamplitude to mean the peak value of a varying force.We are trying to reduce the transm~ssion f vibration from the source intothe supporting structure. We need to know the force exerted by the springand damper on the structure (Figure 41). U we neglect the displacement ofthe structure as smaU relative to that of the source, then the spring lengthchange is just X and the relative velocity of the two ends of the damper isjust i The spring exerts a force kx and the damper a force c2 We derivedan expression for this total force P kx c i in Section 4.1. The total forceis still sinusoidal with frequency 6 rad S- , as was the original vibratingforce. The amplitude of the transmitted sinusoldal force is P,, which isgiven by

P, TF,where T is the transmissibility. Hence you can use Figure 34 for estimatingthe transmitted force, or else use the transmissibility formula

The numerical evaluation of T from this formula is quite straightforwardwith a scientific calculator, and is preferable. The graph permits a simplecheck of your calculation. If you decide to use the graph, then when is notequal to one of the graph line values you can estimate a position inbetween the lines (i.e. interpolate). In using the graph, please rememberthat the scales are logarithmic. If the frequency range is beyond the graphthen you w l l have to use the formula.

xampleA 150 kg car engine on rubber mountings of stiffness 60 kN m- anddamping ratio 0.1 is subject to a sinusoidal force of amplitude 200 N andfrequency 25 Hz Estimate the force amplitude transmitted through themountings.

W m=60 10 /150)1i2 20 rad S-R/m 7.85T .0307 (by calculator)Po=TFo=0.0307X 200=6.1N

My estimate of the transmitted force amplitude is 6.1 N.

SA 22Estimate the transmitted force amplitude for the engine of the example if(a) the damping is negligible, (b) 0.1 and k 120 kN m- l, c) C 0.1and k-180kNm- ,(d) 5=0.5 and k=180kNm- l.


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Whether you are trying t o isolate a structure from a vibration source, or aresponder from a vibrating support structure, it is the transmissibility ofthe mounting that matt- as A TX, nd P, TF,. We nee to makethe transmissibility small. I have a ln ad y said tha t this means that we wanta large frequency ratio. We certainly want to keep away from resonance.The value of f is usually 6xed by existing operating conditions, forexample an engine speed, so for large f2 wwe must make the naturalfrequency W small. Now o m o we want s d , that is a soRspring. Also you can see from Figure 34 that provided Q w s beyond thecrossover point it is better to have small C.In principle it is possible to achieve any required degree of isolation bymaking o and very small. However, there re practical limitations. Thefirst one is that th e spring must support the weight of the object. A verysoft spring will have a large de kc tio n due t o this e mglk . Also if isvery small any extra forces will cause further large ddicctioas, forexample, if somcone leans on it. A fnquency ratio of Q/o= 10 willhowever, give a considerable d e p f isolation especially for small C sounless the requirements are very stringent it is usually possible to meetthem with a simple isolating system of this type. More complicatedsystems could be used in extreme Note that the more d i c u l t casestend to be when Q is small, because this requires extremely small UThe other practical problem is tha t of start-up . If say an engine is beingisolated, the design may be quite satisfactory at the full operating speed,but during starting an d stopping Q is much sm aller. In fact R must go fromzero to groater than 0 80 it must p s through the resonance conditionwhen R W This is often readily observable, for example on sp indr iers ,and also on diesel engines at very slow tick-over. If resonance is traversedrapidly then the amplitude does not have time to build up to reallydangerous values. However, this does depend to some degree on thedamping. More damping helps to prevent resonana problems (smallerpeak T t the cost of poorer isolation at full operating conditions. Theoptimum values obviously depend upon the exact application. In somecases for example a car suspension, the manufacturers go to a good deal oftrouble and expensc to include special dampers. In many o ther cases thedegree of damping needed is quite modest and not critical. This is onereason why rubber is popular for vibration isolation mountings. No t onlyd m t provide the stiffness but l o a usdul small amo unt of dampingresulting from its own internal friction. Hence it is especially useful forsmall machine mountings, engine8 in can being a classic example.Figure 42 shows two t y p i d rubber mo untings in which the rubber isbonded to convenient metal mounting plates. The inherent damping insuch mountings varies widely depending upon the shape r u b k com-pound, frequency and amplitude but is usually in the range0 01 to 0 1withC 0 05being a typical average value. This is, of course, not just a propertyof the material but also depends on the mass W i g supported.


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BAA 20 kg single-cylinder motorcycle engine requ ires a vibration isolatingmounting. As a result of the motion of its internal components at a typicalrunning speed of3000 rev min - the engine is subject to an approximatelysinusoidal force of amplitude 4 N and frequency 50 Hz The m ountingproposed has stihess 20kN m - and negligible damping.(a) Estimate the natural frequency.(b) Estima te the transmissibility.(c) Estima te the transmitted force.BA 4For the engine of the prev ious SAQ if the specification for the transm ittedf o r a was that it be lesp than 10N stimate(a) The maximum acceptable transmissibility.(b ) The acceptable natural frequency.(C) The acceptab le moun ting stiffness.(d) If 0 1 what would be the acceptable natural frequency?(C) What would e the acceptab le mounting stiffness?

In many practical cases of mechanical vibration, the force amplitude isproportional to the square of the running speed. This is typical ofreciprocating engines and poorly balanced rotating machines.

SA PSIn these circumstances estimate the vibration force amplitude acting onthe motorcycle engine at the following speeds.(a) 750 rev min - , (b ) 2000 rev min-l, (c) 4000 rev min- l,(d) 6000 rev min- l

SAOWIf the mount ings were rubber b locks of stiffness 100kN m- and dampingratio 0.1, estimate a) the resonance speed, b) the transmissibility atresonance, (c) the transmitted fora at resonance.

SA 7Estimate the engine amplitude at resonance.

BA WIf the resonance speed lies within the operating speed range of the enginethen the results can be very unpleasant, and even disastrous. In practicethe resonance speed is kept below the lowest speed, and if possible belowabout 0 4 times the lowest speed. The planned tick-over speed for themotorcycle engine is 600 rev min e .(a) What undamped natural frequency should be aimed for?(b) What mounting st ih es s is required?(c) The damping ratio is =0 1 Estimate the transmitted force at tick-over.(d) Estimate the transmitted fo r a at 6 ev min- .


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SA gAn air compressor of mass 260 kg normally runs at 900 rev min-l, when itis subject to a vibratory force of amplitude 2 kN and frequency 15Hz trequires a vibration isolating mounting to reduce the force amplitudetransmitted to the floor, which should not exceed 6 N. It is intended touse rubber mountings.(a) Neglecting damping, estimate the necessary transmissibility, naturalfrequency and mounting stiffness.(b) Taking 0.1 estimate the necessary natural frequency and stiffness.(C) Tests indicate that more damping is required to reduce the resonanceamplitude, so extra dampers are added giving a total dampingcoefficient of 2 kN s m- , with the same stiffness as calculated in (b).Estimate the transmitted force in normal running.(d) Estimate the amplitude and phase lag at resonance (the force isproportional to speed squared).

ummaryIn this Unit you have been introduced to single-degree-of-freedomvibratory systems in translation with sinusoidal forces and displacements.This is not as restrictive as it might seem because it covers many practicalcases, and is a necessary foundation for any further analysis coveringrotation, multiple degrees of freedom and non-sinusoidal force variation.The vibration o fa one-degree-of-freedom system, like the one in Figure lis in many cases analogous to the motion ofa more complex system in oneof its modes of vibration. The results of this Unit, therefore, at least givesome insight into the vibratory behaviour of multiple-degree-of-freedomsystems.Vibration is divided into two main types natural and forced. Naturalvibration occurs at the damped natural frequency, usually with a decreas-ing amplitude. The undamped natural frequency is a theoretical notion,being the vibration frequency that would occur if there was zero damping.In many cases it is also a good approximation to the damped naturalfrequency.The response of a system to forced harmonic vibration depends upon thefrequency ratio he forcing frequency divided by the undamped naturalfrequency. If this is close to 1 then very large response amplitudes canoccur. The system is then in resonance.In principle the total vibration ofa system is a combination of natural andforced vibration. In practice, however, the natural vibration usually diesaway leaving forced vibration as the dominant component, unless thesystem is unstable.The effects of vibration can be very damaging, so vibration isolation isoften necessary. The force amplitude transmitted from a vibration sourceto its supporting structure is given by P = T F where T is the trans-missibility. The displacement amplitude transmitted to a responder from avibrating support is A = T X . (The amplitude of vibration of a source dueto a directly applied force is A = M F / k where M is the magnificationratio.) To achieve vibration isolation a small value of T is needed. Thisrequires a high frequency ratio. small value of damping ratio improvesisolation at high frequency ratios but permits a larger resonance amplitude.


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lossaryTermAmplitudeBalancingCritical dampingCycleDamped forced vibrationDamped natural frequency

Textreference Explanation

Damped natural vibration 2Damped period 2DampingDamping coefficientDamping factor

Damping ratio

DashpotForce amplitude 3.2Forced vibration 1Forced harmonic vihrationFree vihrationFrequencyFrequency ratioGround motionhertz (Hz)InstabilityLightly damped systems 2Natural vibrationNatural (vibration) frequency 1Period

A magnitude of peak displacement from mean position duringvihrationProcedure for reducing vibration in rotating machinery byadjusting the mass distributionValue of damping, given by a damping ratio 1 below whichdamped natural vibration can occur. (See Figure 12)The motion of the system from any given position until the systemreturns to the same position and is moving in the same directionForced vibration including the effects of dampingFrequency of natural vibration when damping is allowed for:

w , (in rad S- )or / o , / 2 n (in Hz)Natural vibration when damping is presentTime for one cycle in damped natural vibration,

7 l fd (in S)Property of system, such as friction, which causes loss of energyand amplitude reductionCoefficient c (in N s m- ) in expression c i for damping forceExponent a in exponential decay term in damped free vibration,defined asa c/2m m (in s - l )Ratio of damping to critical damping, defined by

c=2

Device used to represent source of damping in a system. (SeeFigure 16)F, (or P,), the peak value of an harmonic (sinusoidal) forceVibration in response to continuous external forcingVibration of system due to external excitation arising from eitherground motion or an applied force, which varies harmonicallySee natural vibrationNumber of cycles of vibration performed in one second (fin Hz).Also quantified as o (orQ in rad S- , where 2xfRatio of frequency of applied force, or ground motion, to(undamped) natural frequency, Q / oMotion of previously steady mounting - sometimes referred to asdisplacement excitationUnit of frequency of one cycle per secondCondition in system when natural (free) vibration increases ratherthan decays, following a disturbanceSystems with low damping, usually damping ratio 0.1Vibration of a system after a disturbance with no further externalforc~ngalso free vibration)Frequency of natural, or free, vihration o n rad S - lT i e or one cycle of vibration r n S) for undamped system


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atarn reference E W

Phase lag 4 1 Factor I which helps to specify forced harmonic vibrationrelative to the applied fora, defined bytan 6= Y Qlw)1 fl1wl2- 6 is the phasc angle). Scc lso gurc 30

ResonanceResonance frequency 4Simple harmonic motion 1TransientTransient vibrationTransmissibility

Vibration absorberVibration isolation

Condition in forced harmonic vibration when amplitude ofmotion is large when =WFrequency D=W)a t which resonance occurs in forced harmonicvibrationMotion in which the acceleration is proportional to the dis-placement, and in the opposite directionNatural free) vibration of a system which dies away due todamping, sometimes called a transient. See 5.1Factor relating inpu t on one side of a sp ring to the reaponse onthe other either relates input force amplitude to transmitted forceamplitude or input displacement amplitude to output motionamplitude). Scc also 4.1 and 4.2)Auxiliary mass and spring which is dded to a vibrating system tod u c e vibrationA technique for modifying the transmission path o fa vibra tion toreduce the vibration outpu t

Vibration) tolerance 5.2 Aooeptable level of v ibration


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Answers to Self Assessment QuestionsSA 1(a) J(k/m) has units of

J(N m-'/kg) J(kgm S-'m-'/kg)=J(s-')=S 1

m h u units rad S- , which agrm because radium aredimensionless.b) k/m)'12

(10/o.035)1i'16.9 rad s -l

f w/2n- .7 z

(c) The comparison is good.(d) z- IN

0.37 S

SA 2(a) W -(k/2m)'i1 =(k/m) '/tA14(b) 0-(10/0.07)' = 11.95 rad S -

f 1.9 z(C) hecomparison is good.

X-Bsinmt+Ccaswtwhen. from SAQ 1.

16.9 rad S-At t=0,

x=C=lOmmDiflcrcntiating the expression for x with mpc to time,

=Bo,w~mt-Crusinwtand at 1-0

l-&u=SOmma-'Therefore

C-10mm50 mm S-'B=--- 2.959 mm16.9 rad S-

The amplitudeA (IO)' (2.959)'l '

SA 4Damping force F =c l, so c F/%.Un i t s anN/m ~- ' -Ns rn -~ .Substituting kg m S- for N gives kg S-'. his is a morefundamental e x p d o n d he units, but it is morecommon pr to use N S m-'.

SA 5(a) r d S-'(h) lads-'(C) N s m-', or kg 9 - l(d) s-l(c) C has no units.

SA 6(a) (l ') ' =

l - I2=I[ 0.75

0.866b) (1 -C2) ' 0.95

C=0.31


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igure44

o=7.906rads-P 1.26 Z

a - 5.625 S-0.71

m4 5.56 rad S-fd 0.884 z1/u 0.178 S

/,/a = 0.157 cyclesIn this c gc it ise siest to sketch the vibration ofkoqucncy o vith no dun pin s then sket h tbe decliningamplitude, a d hmwmbim tbe two.

systm 4o= 51.6 rad S-==.2 z

a=OM)Ss-c 0.oool

In1~tencydaarepedomud sothedunpdnatural frequen y is fd = 10 z ad 62.8 rad S- ). Theamplitude hlh by the factor e in 2 S, soa = 1/(2 S) .5 B -o 62.8nd B -=0.008

Over one cyde ths amplitude ratio is 0.1.

(a) m u - / n d U- so that P/o hu o units.(b) m/m so that A/Xo b s no units.(c) Both aides of the equation haw numbers only h marc no m, I. kg or other units Equations arranged inthis way arc very uadul to engineers; they arc oftensaid to be nondimensional.

m = 51.6 rad S-h= 8.2 Zl/a = 200 S

/./a= 1640 cyclsr


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(a ) A /X ,= I, 1.19, 2.78, 5.26, m, 4.76, 227 , 1.04, 0.45,0.33, 0.125, 0.01, 0.0001

Figure 47(c) AIX, is infinite at O/ o = l .(d) Trou ble, possibly breakageSA 12(a) From W = k m and W W, (Little damping)

k = m o ' = 5 0 D N m - 1(b) O/w = 30125= 1.2(C) T = 2.27(d) A = TX,= 2.27 2 mm = 4.54 mm

(a) X, = 2 mm, A = 0.4 mm maximum.From A = TX,, the maximum transmissibility is

(b ) The frequency ratio must be higher than the onegiving T=0.2. Fro m the graph the minimumfrequency ratio is D/w 2.5. If you calculate O/wfrom T using the formula you will h d O/w = 2.45,allowing for the transmissibility formula giving anegative value when D /o > (remember you ignoredthe negative sign when plotting the amplitude inS Q 11).For O/w = 2.5, and 2n 30 = 188.5 rad S- '

The natural frequency must be less than this, to givea frequency ratio higher than 2.5(C) U = k/m, so k mmz 2.3 kN m-' is the maximumacceptable stiffness.

SA0 14(a) O = 2 n ~ M = 3 1 4 r a d s - ~

w = kfm)' = (6 X 104/140)111= 20.7 rad S-n/o= 15.17Neglecting damping, M = 0.0044A= MF,/k = 0.06 mmNote: M = 0.0044 even if a n allowance is mad e fordamping with = 0.02.

(b) P, = kA = 3.6 N amplitudeSA0 15(a) 1/2(= 4.2(b) l / X = 10(C) 1 Z=M(d) 1/21= 250

3 Q @= 0.5916T .538A = 7.7 mm

(b) O/w= 1.0057T = 80.06A = 4 0 0 m mThis amplitude wu ld not be re lised in practice. Thistheoretical amplitude is far beyond the possible limitsof the spring extension.

(c) O/w = 5.9161T - .0294

In bot h cases (a) an d (c), neglecting the damping wouldnot have signi~icantly ffected the results.SA 17(a) O = 2 n x 5 0 = 3 1 4 ra d s - I

w =(k/m) ' = 12.54 rad S- 'n / ~25M = 0.0016A MF,/k=0.12 mm@ = 179.9as we would expect for a very high frequencyratio. Note that tan @ = -0.0002, so that6 = -0.01' 180 = 179.9 .

(b) For Q/w = lM = 1 / 2 ( = 2 0


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(C) = MFo/k= 2.7 mmT = ( I + ~ ~ ) ~ / ~ M - M o z s. . PO= 20.025 X 3 0.1 N

(d) Folk= 0.136 mmMaximum M p.136 = 7.3

(C) M = IIZ. = 112M=0.0680 WithI) /w=ZSa~dF-0 .068M = 0.W16=0.12 mm (n o significsnt chsngc)

The frequency ratio is muoh too high for the inonass indamping ratio to alter the magnification ratio.

S A 0 18(a) = 6 X 9.81 = 58.86 N

F = ke (linear spring)

d) D = 2 x X 1 2 = 7 5 A n d s - 'I)/m = 4.17T= 0.0671

= TX, =0.134 mmC) Maximum T at nso nan w mm/l mm = 4

when T= z

(g) I)-2n X 10-62.83 red S-'I /W = 3.475T=0.121A TXo .145 mm

Ranembering t h a t9 s th phase lag and -9 is the phasean&

tan 4 0 3 , 1 - ( n z / ~ ) 2a) tan 155 l - I ) , I W ) ~where I), -50 q I , = 55Hz; O,/I), = 0.9091mce- .7995 [I -(I),/UI)~] -o.909i[i - I),/OI)~]

andw2[0.9O91 + 1.799Sl= 1.7995 (50)' +0.9091 (55)'

: m= 51.732 z(b) n , / ~ 9665

~ , I W 0.9665. .MI 1.628At mmmI)/o= I, so that

M, = 1/X = 17.483MRAmplitude at rsroau~lc 15mm =226 mmM1

SA 4a ) From th gmph the M o ur protickmy limit is and m t i o n mp litude of 2 m S- t 40 z

(b ) I )=2nx40-251rads - '= 2/(251) '= 0.03 mm

SA mQ = 2 n x 8 = 5 0 , 3 r e d s - '(a) Gnufort pal f =0.18 m S- , A = 0.07 mm(b) Rofioicaey: pal f=0.42 m S-, = 0.17 mm(C) Exposum pak 3=0.83 m S-,, A = 0.33 mm


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SA(a ) O/w 7.85, C 0

T = 0.0165P , = T F o = 3 . 3 N

(h) m= =28 .3 r ads - '157/28.3 5.55

T 0.05P,= TF,= ION

(C) 0= =34 .64ra ds - IR/o 4.53T 0.07P o = T F o = 1 4 N

(d) R/ o =4.53, c 0.5T = 0.23P , = T F , = 4 6 N

( a ) o = J k l m = 3 1 . 6 2 r a d s - '/= 5.03 Hz

Damping is negligible, soW, 31.6 rad S - , J 5.03 z

3000(b) R= - 2a=314 .2 rad S - '60O/w = 9.94(o r 50/5.03 9.94)T 0.0102

(C) P, TF o 4.1 N

SA 4(a) F, 400 N, P, l 0 N maximum

T 10/400 0.025 maxim um(b ) With 0 (negligible)

D/ o 6.403 minimumR = 3 1 4 . 2 r ad s - lo 314,216,403= 49.07 rad s-l(Remember here that strictly T = -0.025 if R/ o > 1.)

(c) k = m o Z=48.2 kN m- '(d) With C 0.1

R /o 9.21 minimumm = 314.219.2 34.12 rad S-

( e ) k = m w 1 = 2 3 . 3 k N m - l

SA 5(a ) 400 (750/3000)1 25 N(b ) 400 (2000/3000)' = 178 N(C) 400 (4000/300 0)2= 711 N(d) 400 X (6000/3000)' 1600 N

(a) The resonan- speed is approximately the undampednatural hqu en cy

(b) 0.1, so at resonance

SA 8( a ) 6 0 0 r ev m i n - ' = 1 0 r ev s - l

R = Z n x 1 0 = 6 2 . 8 3 r ad s - 'We require J= 0.4 10 4 zand o = 2 r r / = 2 5 .1 3 r ad s - '

(b) mm' 12.63 kN m - ' maximum(C ) At tick-over O/w 2.5 (110.4 by design)

0.1T = 0 . 2 1F, = 400 X (600/3000)' 16 NP , = T F o = 3 . 4 N

(d) O rev min -'F, 400 (6000/3000)' = 1600 NR 2rr 6000160 628 .3 rad S -l210 628.3125.13 25[Alternatively R/o = 10014 251C=0.1T 0.0082P o = T F o = 1 3 N


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O W3 T 60/2WO 0.03 maximum

0/0=5.86; 0=900/60-15revs-'o 015.86 2n 1515.86= 16.08 rad s-lk =m 67.3kNm-L[Again remember here that strictly T = -0.03.1

(b) T = 0.03, 0.1a/o 7.99o=0/7.99= 11.8rads-'k m 36.2 kNm-'

c) P O =TFoc=ZOOONsm-'

c/2m 3.846 S-= alw 3.84611 1.8 0.326a10 7.99

T 0.084PO 168N(The extra damping has increased the transmittedforcc amplitude to beyond the spified value.)

(d) Resonance speed is approximately the undampednatural frequency:o = 11.8rads-l, f 1.88 zFo=2W0 (1.8a/is)a= 3 1 . 4 ~

MFo/kFor 0.326, M 1.53 at resonance. . A 1.33mm

Phase lag at resonance is given by tan m so + 90 .


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Index to Blockamplitudeamplitude ratiobalancingcritical dam pingcycledamped forced vibrationdamped natural frequencydamped natural vibrationda m m pe r i oddamped transmissibilitydampingdamping coefficientdamping factordamping ratiodashpotfatigueforce amplitudeforced harmonic vibrationforced vibrationfree vibrationfrequencyfrequency ratioground motionhertz Hz)Hertz, H.human body vibration modelhuman vibration toleranceinstability

isolation mountinglightly damped systemmagnification rationanual (vibration) kequencynatural vibrationNewton s Second a wnotationperiodphase anglephase lagwonancensonance fequencyresponderresponse amplitudeS1 unitssimple harmonic motionspring-mass systemtolerancetotal vibrationtransienttransient vibrationvansmissibilitytransmission pathvibration absorbervibration controlvibration isolationvibration tolerance

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Vibration Control