58 Resonance

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58 Resonance - Math

232 ω = 4 occurs, because of the amplitude factor t If ω 6= 4, then xp(t) is a pure harmonic oscillation, hence bounded If ω = 4, then xp(t) equals a time–varying amplitude Ct times a pure harmonic oscillation, hence it

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5.8 Resonance231

5.8 Resonance

The study of vibrating mechanical systems ends here with thetheory of pure and practical resonance.

Pure Resonance

The notion ofpure resonancein the differential equation x

??(t) +ω20x(t) =F0cos(ωt)(1)is the existence of a solution that is unbounded ast→ ∞. We already

know (page 224) that forω?=ω0, the general solution of (1) is the sum of two harmonic oscillations, hence it is bounded. Equation (1) for ω=ω0has by the method of undetermined coefficients the unbounded oscillatory solutionx(t) =F0

2ω0tsin(ω0t). To summarize:

Pure resonance occurs exactly when the natural internal frequencyω0matches the natural external frequencyω, in which case all solutions of the differential equation are un- bounded. In Figure 20, this is illustrated forx??(t)+16x(t) = 8cos4t, which in (1) corresponds toω=ω0= 4 andF0= 8. x tFigure 20. Pure resonance forx??(t) + 16x(t) = 8cosωt. Graphed are the solutionx(t) =tsin4tforω= 4and the envelope curvesx=±t. Resonance and undetermined coefficients.An explanation of resonance can be based upon the theory of undetermined coefficients.

An initial trial solution of

x ??(t) + 16x(t) = 8cosωt isx=d1cosωt+d2sinωt. The homogeneous solutionxh=c1cos4t+ c

2sin4tconsidered in thefixup rulehas duplicate terms exactly when

the natural frequencies match:ω= 4. Then the final trial solution is x(t) =?d

1cosωt+d2sinωt ω?= 4,

t(d1cosωt+d2sinωt)ω= 4.(2)Even before the undetermined coefficientsd1,d2are evaluated, we can

decide that unbounded solutions occur exactly when frequency matching 232
ω= 4 occurs, because of the amplitude factort. Ifω?= 4, thenxp(t) is a pure harmonic oscillation, hence bounded. Ifω= 4, thenxp(t) equals a time-varying amplitudeCttimes a pure harmonic oscillation, hence it is unbounded. The Wine Glass Experiment.Equation (1) is advertised as the basis for a physics experiment which has appeared often on Public Tele- vision, called thewine glass experiment. A famous physicist, in front of an audience of physics students, equips a lab table with a frequency generator, an amplifier and an audio speaker. Thevaluablewine glass is replaced by a glass beaker. The frequency generator is tuned to the natural frequency of the glass beaker (ω≈ω0), then the volume knob on the amplifier is suddenly turned up (F0adjusted larger), whereupon the sound waves emitted from the speaker break the glass beaker. The glass itself will vibrate at a certain frequency, as can be determined experimentally bypingingthe glass rim. This vibration operates within elastic limits of the glass and the glass will not break underthese cir- cumstances. A physical explanation for the breakage is thatan incoming sound wave from the speaker is timed to add to the glass rim excursion. After enough amplitude additions, the glass rim moves beyond the elas- tic limit and the glass breaks. The explanation implies thatthe external frequency from the speaker has to match the natural frequency of the glass. But there is more to it: the glass has some natural damping that nullifies feeble attempts to increase the glass rim amplitude. The physi- cist uses to great advantage this natural damping totunethe external frequency to the glass. The reason for turning up the volume on the amplifier is to nullify the damping effects of the glass. The amplitude additions then build rapidly and the glass breaks. Soldiers Breaking Cadence.The collapse of the Broughton bridge near Manchester, England in 1831 is blamed for the now-standard prac- tise of breaking cadence when soldiers cross a bridge. Bridges like the Broughton bridge have many natural low frequencies of vibration, so it is possible for a column of soldiers to vibrate the bridge at one of the bridge"s natural frequencies. The bridge locks onto the frequency while the soldiers continue to add to the excursions with every step, causing larger and larger bridge oscillations.

Practical Resonance

The notion of pure resonance is easy to understand both mathematically and physically, because frequency matching characterizesthe event. This ideal situation never happens in the physical world, becausedamping is

5.8 Resonance233

always present. In the presence of dampingc >0, it will be established below thatonly bounded solutions existfor the forced spring-mass system mx

??(t) +cx?(t) +kx(t) =F0cosωt.(3)Our intuition about resonance seems to vaporize in the presence of damp-

ing effects. But not completely. Most would agree that the undamped intuition is correct when the damping effects are nearly zero. Practical resonanceis said to occur when the external frequencyω has been tuned to produce the largest possible solution (a more precise definition appears below). It will be shown that this happensfor the condition k/m-c2/(2m2), k/m-c2/(2m2)>0.(4)Pure resonanceω=ω0≡? k/mis the limiting case obtained by set- ting the damping constantcto zero in condition (4). This strange but predictable interaction exists between the damping constantcand the size of solutions, relative to the external frequencyω, even though all solutions remain bounded. The decomposition ofx(t) into homogeneous solutionxh(t) and partic- ular solutionxp(t) gives some intuition into the complex relationship between the input frequencyωand the size of the solutionx(t). The homogeneous solution. For positive damping,c >0, equation (3) has homogeneous solutionxh(t) =c1x1(t)+c2x2(t) where according to therecipethe basis elementsx1andx2are given in terms of the roots of the characteristic equationmr2+cr+k= 0, as classified by the discriminantD=c2-4mk, as follows:

Case 1,D >0x1=er1t,x2=er2twithr1andr2negative.

Case 2,D= 0x1=er1t,x2=ter1twithr1negative.

Case 3,D <0x1=eαtcosβt,x2=eαtsinβtwithβ >0 andαnegative. It follows thatxh(t) contains a negative exponential factor, regardless of the positive values ofm,c,k. A solutionx(t) is called atransient solutionprovided it satisfies the relation limt→∞x(t) = 0. The conclu- sion: The homogeneous solutionxh(t)of the equationmx??(t) + cx ?(t) +kx(t) = 0is a transient solution for all positive values ofm,c,k. A transient solution graphx(t) for largetlies atop the axisx= 0, as in

Figure 21, because lim

t→∞x(t) = 0. 234

6πtx

2 0

Figure 21. Transient oscillatory solution

x= 2e-t(cost+ sint)of the differential equationx??+ 2x?+ 2x= 0. The particular solution. The method of undetermined coefficients gives a trial solution of the formx(t) =Acosωt+Bsinωtwith coeffi- cientsA,Bsatisfying the equations (k-mω2)A+ (cω)B=F0, (-cω)A+ (k-mω2)B= 0.(5)Solving (5) with Cramer"s rule or elimination produces the solution

A=(k-mω2)F0

(k-mω2)2+ (cω)2, B=cωF0(k-mω2)2+ (cω)2.(6)Thesteady-statesolution, periodic of period 2π/ω, is given by

x p(t) =F0 (k-mω2)2+ (cω)2? (k-mω2)cosωt+ (cω)sinωt? F0

?(k-mω2)2+ (cω)2cos(ωt-α),(7)whereαis defined by the phase-amplitude relations (see page 216)

Ccosα=k-mω2, Csinα=cω,

C=F0/?

(k-mω2)2+ (cω)2.(8)The terminologysteady-staterefers to that partxss(t) of the solution x(t) that remains when the transient portion is removed, that is, when all terms containing negative exponentials are removed. Asa result, for largeT, the graphs ofx(t) andxss(t) ont≥Tare the same. This feature ofxss(t) allows us to find its graph directly from the graph of x(t). We say thatxss(t) isobservable, because it is the solution visible in the graph after thetransients(negative exponential terms) die out. Readers may be mislead by the method of undetermined coefficients, in which it turns out thatxp(t) andxss(t) are the same. Alternatively, a particular solutionxp(t) can be calculated by variation of parameters, a method which produces inxp(t) extra terms containing negative expo- nentials. These extra terms come from the homogeneous solution - their appearance cannot always be avoided. This justifies the careful defini- tion of steady-state solution, in which the transient termsare removed fromxp(t) to producexss(t). Practical resonanceis said to occur when the external frequencyω has been tuned to produce the largest possible steady-stateamplitude.

5.8 Resonance235

Mathematically, this happens exactly when the amplitude functionC= C(ω) defined in (8) has a maximum. If a maximum exists on 0< ω <∞, thenC?(ω) = 0 at the maximum. The derivative is computed by the power rule: C ?(ω) =-F0

22(k-mω2)(-2mω) + 2c2ω((k-mω2)2+ (cω)2)3/2

2mk-c2-2m2ω2?C(ω)3

there is no maximum. If 2km-c2>0, then 2km-c2-2m2ω2= 0 has exactly one rootω=? k/m-c2/(2m2) in 0< ω <∞and byC(∞) = 0 it follows thatC(ω) is a maximum. In summary: Practical resonance formx??(t)+cx?(t)+kx(t) =F0cosωt occurs precisely when the external frequencyωis tuned to k/m-c2/(2m2)andk/m-c2/(2m2)>0. In Figure 22, the amplitude of the steady-state periodic solution is graphed against the external natural frequencyω, for the differential equationx??+cx?+ 26x= 10cosωtand damping constantsc= 1,2,3.

The practical resonance condition isω=?

26-c2/2. Ascincreases

from 1 to 3, the maximum point (ω,C(ω)) satisfies a monotonicity con- dition: bothωandC(ω) decrease ascincreases. The maxima for the three curves in the figure occur atω=⎷

25.5,⎷24,⎷21.5. Pure reso-

nance occurs whenc= 0 andω=⎷ 26.
C c= 2c= 3c= 1Figure 22. Practical resonance forx??+cx?+ 26x= 10cosωt: amplitude

C= 10/?

(26-ω2)2+ (cω)2versus external frequencyωforc= 1,2,3. Uniqueness of the Steady-State Periodic Solution.Any two solutions of the nonhomogeneous differential equation (3) which are pe- riodic of period 2π/ωmust be identical. The vehicle of proof is to show that their differencex(t) is zero. The differencex(t) is a solution of the homogeneous equation, it is 2π/ω-periodic and it has limit zero at infin- ity. A periodic function with limit zero must be zero, therefore the two solutions are identical. A more general statement is true: Consider the equationmx??(t)+cx?(t)+kx(t) =f(t)with f(t+T) =f(t)andm,c,kpositive. Then aT-periodic solution is unique. 236
In Figure 23, the unique steady-state periodic solution is graphed for the differential equationx??+2x?+2x= sint+2cost. The transient solution of the homogeneous equation and the steady-state solution appear in Figure 24. In Figure 25, several solutions are shown for the differential equationx??+2x?+2x= sint+2cost, all of which reproduce eventually the steady-state solutionx= sint. x 0

6πt1

Figure 23. Steady-state periodic

solutionx(t) = sintof the differential equationx??+ 2x?+ 2x= sint+ 2cost. 2 1 0

06πtx

Figure 24. Transient solution of

x ??+ 2x?+ 2x= 0and the steady-state solution ofx??+ 2x?+ 2x= sint+ 2cost. x3 2 1

00t6πFigure 25. Solutions ofx

??+ 2x?+ 2x= sint+ 2costwithx?(0) = 1 andx(0) = 1,2,3, all of which graphically coincide with the steady-state solutionx= sintfort≥π. Pseudo-Periodic Solution.Resonance gives rise to solutions of the formx(t) =A(t)sin(ωt-α) whereA(t) is a time-varying amplitude. Figure 26 shows such a solution, which is called apseudo-periodic solutionbecause it has a natural period 2π/ωarising from the trigono- metric factor sin(ωt-α). The only requirement onA(t) is that it be non-vanishing, so that it acts like an amplitude. Thepseudo-periodof a pseudo-periodic solution can be determined graphically,by computing the length of time it takes forx(t) to vanish three times. t 200x
4 eFigure 26. The pseudo-periodicsolutionx=te-t/4sin(3t)of

16x??+ 8x?+ 145x= 96e-t/4cos3tand its

envelope curvesx=±te-t/4.

Exercises 5.8

Resonance and Beats. Classify for

resonance or beats. In the case of res- onance, find an unbounded solution.

In the case of beats, find the solution

forx(0) = 0,x?(0) = 0, then graph it through a full period of the slowly- varying envelope.

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