ORDINARY DIFFERENTIAL EQUATIONS POWERTOOLLesson 28 -- Application: Unforced Spring-Mass OscillatorsProf. Douglas B. MeadeIndustrial Mathematics InstituteDepartment of MathematicsUniversity of South CarolinaColumbia, SC 29208 URL: http://www.math.sc.edu/~meade/E-mail: meade@math.sc.edu Copyright \251 2001 by Douglas B. MeadeAll rights reserved-------------------------------------------------------------------

28.A Unforced Oscillatory Motion - Undamped 28.A-1 Example 1 28.A-2 Example 228.B Unforced Oscillatory Motion - Damped 28.B-1 Case 1: Underdamped Motion ( NiMsJiokJSJiRyIiIyIiIiooIiIlRiclImFHRiclImNHRichIiI= < 0 ) 28.B-2 Case 2: Critically Damped Motion ( NiMsJiokJSJiRyIiIyIiIiooIiIlRiclImFHRiclImNHRichIiI= = 0 ) 28.B-3 Case 3: Overdamped Motion ( NiMsJiokJSJiRyIiIyIiIiooIiIlRiclImFHRiclImNHRichIiI= > 0 )28.C Resonance 28.C-1 Realistic Resonance 28.C-2 Unrealistic Resonance

restart;with( DEtools ):with( plots ):

A spring that obeys Hooke's law - the displacement is proportional to the applied force - is called a linear spring. Suppose that such a spring whose natural (unstretched or compressed) length is NiMlIkxH is suspended vertically so that its upper end is attached to a fixed support, and its lower end is attached to a mass NiMlIm1H. The attached mass will stretch the spring until it attains an equilibrium position. In a coordinate system NiMlInhH measured positive upward from the equilibrium position of the mass, measure displacements of the mass with the variable NiMtJSJ4RzYjJSJ0Rw==. Again, the origin of this coordinate system is the equilibrium position of the suspended mass, and displacements upward are positive.The force of gravity acting downward on the mass is NiMsJComJSJtRyIiIiUiZ0dGJiEiIg== j, where NiMlImdH is the gravitational constant, is counterbalanced by the sag in the spring past its natural length NiMlIkxH when the mass is first attached. This downward gravitational force is always in balance with the upward force generated by the spring as it sags from its natural length to the equilibrium position that establishes the origin NiMvJSJ4RyIiIQ== for the coordinate system. This gravitational force never appears in the equation of motion for the spring-mass system described above.According to Hooke's law, the force exerted by the spring on the mass is NiMsJComJSJrRyIiIiUieEdGJiEiIg== j, where NiMlImtH is the spring constant expressed in units of force divided by distance. The acceleration of the mass under the action of the spring is a = NiMqKCUiZEciIiMlInhHIiIiKiQlI2R0R0YlISIi j. Newton's second law, namely, F = NiMlIm1H a, then gives the equation of motion asNiMlIm1H NiMvKiglImRHIiIjJSJ4RyIiIiokJSNkdEdGJiEiIiwkKiYlImtHRigtRic2IyUidEdGKEYr orNiMvLCYqJiUibUciIiItJSR4JydHNiMlInRHRidGJyomJSJrR0YnLSUieEdGKkYnRiciIiE= Initial conditions would be the initial displacement of the mass from equilibrium, and any initial velocity given to the mass. Thus, the mass could be lifted initially, in which case NiMqKC0lInhHNiMiIiEiIiIlIj5HRihGJ0Yo. Alternatively, the mass could be pulled down below equilibrium so that NiMyLSUieEc2IyIiIUYn. Of course, if the mass is not displaced from equilibrium initially, NiMvLSUieEc2IyIiIUYn.If the mass is thrown upward initially, it would have a positive initial velocity, and hence, NiMqKC0lI3gnRzYjIiIhIiIiJSI+R0YoRidGKA==. Alternatively, the mass could be tossed downward initially, in which case NiMyLSUjeCdHNiMiIiFGJw==. If the mass is released from rest, then NiMvLSUjeCdHNiMiIiFGJw==.These considerations lead to the IVPode1 := m*diff(x(t),t,t)+k*x(t) = 0; ic1 := x(0)=alpha, D(x)(0)=beta;for NiMtJSJ4RzYjJSJ0Rw==, the displacement of the mass from equilibrium. The solution of the differential equation itself, namely,dsolve(ode1, x(t));follows from the exponential guess NiMvLSUieEc2IyUidEctJSRleHBHNiMlI3J0Rw== and the characteristic equation NiMvLCYqJiUibUciIiIqJCUickciIiNGJ0YnJSJrR0YnIiIh Since the characteristic roots are thenNiMlInJH = + NiMqJiUiaUciIiItJSVzcXJ0RzYjKiYlImtHRiUlIm1HISIiRiU= the general solution will be a linear combination of the members of the (real) fundamental set NiM8JC0lJGNvc0c2IyomJSZvbWVnYUciIiIlInRHRiktJSRzaW5HRiY=, where NiMvJSZvbWVnYUctJSVzcXJ0RzYjKiYlImtHIiIiJSJtRyEiIg== is called the angular frequency.The trigonometric formNiMsJiomJSJhRyIiIi0lJGNvc0c2IyomJSZvbWVnYUdGJiUidEdGJkYmRiYqJiUiYkdGJi0lJHNpbkdGKUYmRiY= can be transformed toNiMqJiUiQUciIiItJSRjb3NHNiMsJiomJSZvbWVnYUdGJSUidEdGJUYlJSRwaGlHISIiRiU= whereNiMvJSJBRy0lJXNxcnRHNiMsJiokJSJhRyIiIyIiIiokJSJiR0YrRiw= and the phase angle NiMlJHBoaUc= is determined by the two equationsNiMvLSUkY29zRzYjJSRwaGlHKiYlImFHIiIiJSJBRyEiIg== and NiMvLSUkc2luRzYjJSRwaGlHKiYlImJHIiIiJSJBRyEiIg== In this form, the general solution of the undamped spring-mass system clearly undergoes periodic motion with angular frequency NiMvJSZvbWVnYUctJSVzcXJ0RzYjKiYlImtHIiIiJSJtRyEiIg== and amplitude NiMlIkFH. Such motion is called harmonic motion, and without damping, would persist undiminished forever. Undamped spring-mass systems are therefore called natural harmonic oscillators.

Maple does not have a single command for converting the trigonometric formq := a*cos(omega*t) + b*sin(omega*t);to the formQ := A*cos(omega*t - phi);Much of the management of this transition remains in the hands of the user. For example, the equationNiMsJiomJSJhRyIiIi0lJGNvc0c2IyomJSZvbWVnYUdGJiUidEdGJkYmRiYqJiUiYkdGJi0lJHNpbkdGKUYmRiY= = NiMqJiUiQUciIiItJSRjb3NHNiMsJiomJSZvbWVnYUdGJSUidEdGJUYlJSRwaGlHISIiRiU= could be treated as an identity in NiMlInRH, so that Maple can compute the appropriate values of NiMlIkFH and NiMlJHBoaUc= asqq := solve(identity(q=Q,t), {A,phi});It is typical to take the amplitude as positive, so the first solution would be the preferred one.Note also the usage of the two-argument arctangent function. The single-argument arctangent function, NiMtJSdhcmN0YW5HNiMqJiUieUciIiIlInhHISIi, returns an angle in the range NiMtJSFHNiQsJComJSNQaUciIiIiIiMhIiJGK0Yn, which is equivalent to the angle being in the first or fourth quadrants. This is the function one finds on the typical scientific calculator. Use of this function with a pointNiMtJSFHNiQlInhHJSJ5Rw== loses sign information when either NiMlInhH or NiMlInlH is negative.Most scientific programming languages have some form of the two-argument arctangent function, NiMtJSdhcmN0YW5HNiQlInlHJSJ4Rw==, which returns an angle in the range (NiQsJCUjUGlHISIiRiQ=]. This is equivalent to preserving the signs on the coordinates of points NiMtJSFHNiQlInhHJSJ5Rw== so that the angle returned is in the proper one of the four quadrants.

Convert the trigonometric formq := 3*cos(2*t) - 5*sin(2*t);to the formQ := A*cos(2*t - phi);Again treating the equationq = Q;as an identity in NiMlInRH, Maple determines two solutions for the constants NiMlIkFH and NiMlJHBoaUc= viaqq := solve(identity(q = Q,t), {A,phi});Selecting the positive amplitude, the desired form is theneval(Q,qq[1]);The amplitude and phase angle are, respectively,A = eval(A,qq[1]); phi = eval(phi,qq[1]);

Linear damping is resistance proportional to NiMqJiUjZHhHIiIiJSNkdEchIiI=, the velocity, and opposing the motion. Air resistance against the moving mass would be an example of such a force. If linear damping is added to the spring-mass system of Section 28.A, the differential equation governing displacements from equilibrium becomesode1 := m * diff( x(t), t$2 ) + b * diff( x(t), t ) + k * x(t) = 0;where NiMlImJH is the coefficient of linear damping. The oscillatory motion experienced by such a system is called unforced, or free motion, and corresponds to the absence of a driving, or forcing term on the right-hand side of the equation. Hence, the DE for unforced motion is homogeneous.Surprisingly, the very similar DE NiMlIkxH NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKCUiUkdGKA== NiMsJiomJSNkeEciIiIlI2R0RyEiIkYmKiZGJkYmJSJDR0YoRiY= NiMvJSJ4RyIiIQ== governs the unforced RLC electric circuit (see Lesson 29) where NiMtJSJ4RzYjJSJ0Rw== represents the charge, NiMqJiUjZHhHIiIiJSNkdEchIiI= represents the current, NiMlIlJH is the resistance, NiMlIkxH is the inductance, and NiMlIkNH is the capacitance (NiMqJiIiIkYkJSJDRyEiIg== is the elastance).The solution procedure for a second-order linear ODE with constant coefficients calls for a search for exponential solutions, so the exponential guess leads toq1 := factor( eval( ode1, x(t)=exp(lambda*t) ) );and the characteristic equation chareqn := q1 / exp( lambda*t );The values of the characteristic roots are thencharvals := solve( chareqn, {lambda} );Since the characteristic equation is quadratic, the discriminantdiscrim(lhs(chareqn),lambda);determines the nature of the characteristic roots which, for NiMlIm1H positive, and NiMlImJH and NiMlImtH nonnegative, can be negative, zero, or positive. In each case, the system experiences a distinctive motion called respective, underdamped, critically damped, or overdamped.These three cases are examined individually in the next three subsections.

When the discriminant is negative, that is, in the case NiMyLCYqJCUiYkciIiMiIiIqKCIiJUYoJSJtR0YoJSJrR0YoISIiIiIh, the characteristic values are complex conjugates that can be written asNiMlJmFscGhhRw== + NiMqJiUmb21lZ2FHIiIiJSJpR0Yl whereNiMvJSZhbHBoYUcsJCooJSJiRyIiIiIiIyEiIiUibUdGKkYq and NiMvJSZvbWVnYUcqKC0lJXNxcnRHNiMsJiooIiIlIiIiJSJtR0YsJSJrR0YsRiwqJCUiYkciIiMhIiJGLEYxRjJGLUYy = NiMtJSVzcXJ0RzYjLCYqJiUia0ciIiIlIm1HISIiRikqJCooJSJiR0YpIiIjRitGKkYrRi9GKw== A fundamental set contains the independent solutionsNiMvJiUieEc2IyIiIiomLSUkZXhwRzYjKiYlJmFscGhhR0YnJSJ0R0YnRictJSRjb3NHNiMqJiUmb21lZ2FHRidGLkYnRic= NiMvJiUieEc2IyIiIyomLSUkZXhwRzYjKiYlJmFscGhhRyIiIiUidEdGLkYuLSUkc2luRzYjKiYlJm9tZWdhR0YuRi9GLkYu which are both exponentially damped sinusoids. (The literature refers to both the sine and cosine functions as sinusoids.) The general solution, a linear combination of these functions, can be written either asNiMqJi0lJGV4cEc2IyomJSZhbHBoYUciIiIlInRHRilGKSwmKiYmJSJjRzYjRilGKS0lJGNvc0c2IyomJSZvbWVnYUdGKUYqRilGKUYpKiYmRi42IyIiI0YpLSUkc2luR0YyRilGKUYp or asNiMqKCUiQUciIiItJSRleHBHNiMqJiUmYWxwaGFHRiUlInRHRiVGJS0lJGNvc0c2IywmKiYlJm9tZWdhR0YlRitGJUYlJSRwaGlHISIiRiU= where NiMvJSJBRy0lJXNxcnRHNiMsJiokJiUiY0c2IyIiIiIiI0YtKiQmRis2I0YuRi5GLQ==, NiMvLSUkY29zRzYjJSRwaGlHKiYmJSJjRzYjIiIiRiwlIkFHISIi, and NiMvLSUkc2luRzYjJSRwaGlHKiYmJSJjRzYjIiIjIiIiJSJBRyEiIg==. In this latter form, NiMqJiUiQUciIiItJSRleHBHNiMqJiUmYWxwaGFHRiUlInRHRiVGJQ== is the damped amplitude, NiMlJm9tZWdhRw== is the angular frequency, NiMqKCUmb21lZ2FHIiIiIiIjISIiJSNQaUdGJw== is the quasi-frequency, NiMqKCIiIyIiIiUjUGlHRiUlJWJldGFHISIi is the quasi-period, and NiMlJHBoaUc= is the phase angle.The decaying exponential term forces all solutions to decay to the zero function as NiMlInRH increases.A typical example of underdamped motion is generated by the differential equationq1 := eval(ode1, {m=1,b=4,k=16});whose solution for the initial conditionsic1 := x(0)=2, D(x)(0)=2;isX1 := collect(rhs(dsolve({q1,ic1},x(t))),exp);Alternatively, this solution can be expressed in the formQQ := A*cos(2*sqrt(3)*t-phi)*exp(-2*t);where NiMlIkFH is the positive solution inparams := solve(identity(X1=QQ,t), {A,phi});Hence, the equivalent form of the solution isX2 := eval(QQ,params[1]);Figure 28.1 shows the solution in black, and the exponential envelopes + NiMqJi0lJXNxcnRHNiMiIigiIiItJSRleHBHNiMsJComIiIjRiglInRHRighIiJGKA== in red.plot([X1, sqrt(7)*exp(-2*t), -sqrt(7)*exp(-2*t)], t=0..3, color=[black,red,red], title="Figure 28.1");

When the discriminant is zero, that is, in the case NiMvLCYqJCUiYkciIiMiIiIqKCIiJUYoJSJtR0YoJSJrR0YoISIiIiIh, the characteristic values are the repeated root NiMvJSZhbHBoYUcsJCooJSJiRyIiIiIiIyEiIiUibUdGKkYq. In this case, two linearly independent solutions to the ODE areNiMvJiUieEc2IyIiIi0lJGV4cEc2IyomJSZhbHBoYUdGJyUidEdGJw== and NiMvJiUieEc2IyIiIyomJSJ0RyIiIi0lJGV4cEc2IyomJSZhbHBoYUdGKkYpRipGKg== so that the general solution isX := (c[1]+c[2]*t)*exp(-b/2/m*t);The motion that results from the case of the repeated characteristic root is called critically damped motion. It stands at the interface between underdamped motion and overdamped motion, the third case to be studied below. However, engineers consider that critically damped motion never occurs in actual practice. This is because the parameters in a physical system are never determined so precisely as to cause the discriminant to vanish. Even if the perfect system were built so that the oscillator experienced critically damped motion, the very motion of the spring would warm the metal and change the spring constant, and the system would no longer be truely critically damped. Thus, critical damping is really a mathematical concept.Initial values for NiMtJSJ4RzYjJSJ0Rw== and NiMtJSN4J0c2IyUidEc= lead to the equationseq1 := eval(X,t=0) = x(0); eq2 := eval(diff(X,t),t=0) = `x'`(0);whose solution isexpand(solve({eq1,eq2},{c[1],c[2]}));Surprisingly, the initial velocity NiMvLSUjeCdHNiMiIiEsJCoqJSJiRyIiIi0lInhHRiZGKyIiIyEiIiUibUdGL0Yv implies NiMvJiUiY0c2IyIiIyIiIQ==.The IVPode2 := eval(ode1, {m=1,b=8,k=16}); ic2 := x(0)=2, D(x)(0)=0;governs the critically damped motion exhibited by the solutionXC1 := rhs(dsolve({ode2, ic2}, x(t)));that is graphed in Figure 28.2.plot(XC1, t=0..3, title="Figure 28.2");

So long as NiMqKCUiYkciIiIlIj5HRiUiIiFGJQ==, the common exponential factor overwhelms the linear factor and forces all solutions to zero. The decay to zero is not necessarily monotonic. The general solution of the critically damped equation has a critical point att[crit] = expand(solve( diff(X,t)=0, t));which is positive if and only if NiMwJiUiY0c2IyIiIyIiIQ== and NiMqJiYlImNHNiMiIiJGJyZGJTYjIiIjISIi < NiMqKCIiIyIiIiUibUdGJSUiYkchIiI=.The IVPode2; ic3 := x(0)=2, D(x)(0)=5;governs the critically damped motion exhibited by the solutionXC2 := rhs(dsolve({ode2, ic3}, x(t)));and graphed in Figure 28.3.plot(XC2, t=0..3, title="Figure 28.3");Lifting the mass and tossing it upward to set it into motion causes it to rise above the initial height, but then to descend exponentially to equilibrium.

Note that if NiMlImJH = 0, then NiMqKCIiJSIiIiUibUdGJSUia0dGJQ== = 0 and NiMqKCUibUciIiIlIj5HRiUiIiFGJQ== imply that NiMvJSJjRyIiIQ==. In this case of undamped motion, the ODE simplifies toeval( ode1, [m=1,b=0,k=0] );a fundamental set of solutions is NiM8JCIiIiUidEc=, and the general solution is a linear function. Hence, solutions do not tend to zero as NiMlInRH increases. In fact, except in special cases where the solution is a constant function, the solutions become unbounded as NiMlInRH increases.

The third case is the most straightforward. If the discriminant is positive, that is, if NiMsJiokJSJiRyIiIyIiIiooIiIlRiclIm1HRiclImtHRichIiI= > 0, then there are two distinct, real characteristic roots, namely,NiMlJmFscGhhRw== + NiMlJWJldGFH where againNiMvJSZhbHBoYUcsJCooJSJiRyIiIiIiIyEiIiUibUdGKkYq butNiMvJSViZXRhRyooLSUlc3FydEc2IywmKiQlImJHIiIjIiIiKigiIiVGLSUibUdGLSUia0dGLSEiIkYtRixGMkYwRjI= = NiMtJSVzcXJ0RzYjLCYqJCooJSJiRyIiIiIiIyEiIiUibUdGLEYrRioqJiUia0dGKkYtRixGLA== Writing these characteristic roots aslambda[1] := eval( lambda, charvals[1] ); lambda[2] := eval( lambda, charvals[2] );a fundamental set of solutions is thenX[1] := exp( lambda[1]*t ); X[2] := exp( lambda[2]*t );so the general solution isX[h] := c[1]*X[1] + c[2]*X[2];The system giving rise to such a solution is called overdamped, and by a stretch of notation, the differential equation itself, its solution, or the resulting motion, can be modified by the adjective "overdamped."Observe that, because 0 < NiMsJiokJSJiRyIiIyIiIiooIiIlRiclIm1HRiclImtHRichIiI=NiMxJSFHKiQlImJHIiIj, both characteristic values are negative. In fact, NiMmJSdsYW1iZGFHNiMiIiM= < NiMmJSdsYW1iZGFHNiMiIiI= < 0. All solutions of an overdamped equation ultimately decay to zero as NiMlInRH increases. But, just as with the critically damped case, the solution can have one critical point:dX[h] := diff( X[h], t ):tcrit := solve( dX[h] = 0, t );To determine when the critical point occurs with NiMqKCUidEciIiIlIj5HRiUiIiFGJQ==, observe that the location of the critical point can be expressed as NiMmJSJ0RzYjJSVjcml0Rw== = NiMsJComJSJtRyIiIi0lJXNxcnRHNiMsJiokJSJiRyIiI0YmKigiIiVGJkYlRiYlImtHRiYhIiJGMUYx ln( NiMqJiYlImNHNiMiIiJGJyZGJTYjIiIjISIi NiMqJiYlJ2xhbWJkYUc2IyIiIkYnJkYlNiMiIiMhIiI= ) This time is positive precisely when 0 < NiMqJiYlImNHNiMiIiJGJyZGJTYjIiIjISIi NiMqJiYlJ2xhbWJkYUc2IyIiIkYnJkYlNiMiIiMhIiI= < 1.

To conclude, note that except for the trivial case with NiMlImJH = NiMlImtH = 0, all solutions to the ODE for unforced oscillatory motion are transient solutions that eventually "die out" as time passes. (The first generation of television sets were built with vacuum tubes that had to "warm up" to reach their operating points. During this warm-up phase, the TV screen would generally show a gradually brightening, wavering and distorted image that was the physical realization of the transients in the circuits.)The driven or forced oscillator is governed by the nonhomogeneous ODENiMlIm1H NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKCUiYkdGKA== NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomJSJrR0YnJSJ4R0YnRictJSJmRzYjJSJ0Rw== where NiMtJSJmRzYjJSJ0Rw== is called the forcing term. The general solution of this equation has the formNiMvJiUieEc2IyUiZ0csJiZGJTYjJSJoRyIiIiZGJTYjJSJwR0Ys where NiMmJSJ4RzYjJSJoRw== is the homogeneous solution built as a linear combination of the members of the fundamental set. This solution contains the decaying exponentials discussed in this worksheet. The particular solution NiMmJSJ4RzYjJSJwRw== is determined by the driving term, and any part of it that persists over time constitutes the steady-state solution, the solution that is seen after all transients die out.

Anthropomorphically speaking, resonance can be described as a temper tantrum thrown by Mother Nature. It is an exaggerated response to a stimulus.Analytically speaking, resonance in a driven oscillator is the phenomenon whereby the steady-state response to a sinusoidal forcing term attains maximal amplitude. Thus, the amplitude of the steady-state solution of the driven damped oscillator described by the ODE NiMlIm1H NiMsJiooJSJkRyIiIyUieEciIiIqJCUjZHRHRiYhIiJGKCUiYkdGKA== NiMvLCYqJiUjZHhHIiIiJSNkdEchIiJGJyomJSJrR0YnJSJ4R0YnRictJSJmRzYjJSJ0Rw== whereNiMvLSUiZkc2IyUidEcqJiUiQUciIiItJSRjb3NHNiMqJiUmb21lZ2FHRipGJ0YqRio= achieves a maximal value at some angular frequency NiMvJSZvbWVnYUcmRiQ2IyUiUkc=.Familiar occurrences of resonance might include the sound-box on a stringed musical instrument, the tuning circuit in an AM radio, the annoying rattle of a motor driven at certain speeds, or the collapse of the Tacoma Narrows Bridge. The body of an acoustic guitar is a resonator that amplifies the sound of the vibrating strings. The tuning circuit of an AM radio is essentially a driven RLC circuit whose resonant oscillations are used to oppose those of the carrier signal, leaving behind the broadcast information. Less useful instances of resonance include the irritation of a noisy motor whose resonance is causing nearby objects to vibrate noisily or destructively. The rythmic driving of a bridge under the footfalls of an army marching in step can cause destructive resonance. Supposedly, to avoid resonance, marching armies were instructed to break step when crossing bridges.The collapse in 1940 of the Tacoma Narrows Bridge over the Puget Sound requires more discussion. This enormous (almost 6000 feet long) engineering structure was designed to withstand winds of up to 120 mph. But, due to its tendency to undulate in light winds, the bridge was fondly known as Galloping Gertie. On November 7, 1940, only four months after being completed, the bridge was completely destroyed in 42 mph winds. The full incident is preserved on a famous four-minute film clip; a short excerpt is available at http://www.physics.bcit.ca/netshow/tacoma.avi. For many years the accepted explanation of this tragedy was resonance. In recent years, however, an alternate theory based on aerodynamically induced condition of self-excitation or "negative damping" has gained wide acceptance. Additional details can be found at many sites on the WWW, including http://www.urbanlegends.com/science/bridge_resonance.html http://www.vibrationdata.com/Tacoma.htm http://www.ketchum.org/wind.html http://www.eng.uab.edu/cee/reu_nsf99/tacoma.htm http://www.math.uconn.edu/~kmoore/tacoma.html http://www.madsci.org/posts/archives/may98/892678504.Eg.r.html http://www.nationmaster.com/encyclopedia/Tacoma-Narrows-Bridge http://en2.wikipedia.org/wiki/Tacoma_Narrows_Bridge This is not the place to debate these theories, but both require a solid understanding of resonance and damping. The following discussion of resonance proceeds in two parts. First, the realistic case of the driven damped oscillator is considered. Second, the unrealistic case of the undamped oscillator is discussed. Since all real systems have damping of some sort, undamped oscillators are actually only met as an approximation to real systems.

Without loss of generality, let the equation for the driven damped oscillator be ODE := diff(x(t),t,t) + B*diff(x(t),t) + K*x(t) = A*cos(omega*t);where the driving (angular) frequency NiMlJm9tZWdhRw== is as yet undetermined. Treat it as a parameter that can be adjusted, much like the speed on an electric fan can be changed by turning a knob. The standard equation for the damped oscillator has been divided by the mass NiMlIm1H, so NiMvJSJCRyomJSJiRyIiIiUibUchIiI= and NiMvJSJLRyomJSJrRyIiIiUibUchIiI=.The steady-state response is essentially the particular solution since the homogeneous solution, containing exponentials whose real parts are negative, will be a transient of the system and become negligible after a certain time. This particular solution isXP := dsolve(ODE,x(t),output=basis)[2];The formQ := a*cos(omega*t-phi);is more convenient for determining the amplitude NiMlImFH, which is found viasol := solve(identity(XP=Q,t),{a,phi});Again selecting the positive amplitude, and ignoring the phase angle, writeAMP := eval(a,sol[1]);This amplitude is a function of the driving frequency NiMlJm9tZWdhRw==. Resonance occurs at the frequency that causes this function to be a maximum. Simple calculus reveals that this resonant frequency isSOL := solve(diff(AMP,omega)=0,omega);The positive solution is the resonant frequency, written asomega[R] = eval(SOL[2], {B=b/m, K=k/m});An alternate form for this expression isNiMvJiUmb21lZ2FHNiMlIlJHLSUlc3FydEc2IywmKiYlImtHIiIiJSJtRyEiIkYuKiglImJHIiIjRjNGMCokRi9GM0YwRjA= where NiMmJSZvbWVnYUc2IyUiUkc= is real and positive if NiMyKiQlImJHIiIjKihGJiIiIiUia0dGKCUibUdGKA==. Thus, if the damping coefficient NiMlImJH is nto less than NiMtJSVzcXJ0RzYjKigiIiMiIiIlIm1HRiglImtHRig=, no value of the driving frequency will cause the system to resonate. Alternatively, the threshold of damping for resonance is NiMyKiQlImJHIiIjKihGJiIiIiUibUdGKCUia0dGKA==, whereas the threshold of damping for the system to be underdamped is NiMyKiQlImJHIiIjKigiIiUiIiIlIm1HRiklImtHRik=. Loosely speaking, it might be said that half the systems that can oscillate can be made to resonate. An underdamped system that is made to resonate will also have a natural frequencyNiMvJiUmb21lZ2FHNiMlIk5HLSUlc3FydEc2IywmKiYlImtHIiIiJSJtRyEiIkYuKiglImJHIiIjIiIlRjAqJEYvRjNGMEYw Thus, for the damped oscillator, the natural frequency is slightly larger than the resonant frequency, whereas for the undamped oscillator, the natural and resonant frequencies will shortly be seen to be the same.If NiMvJSJtRyIiIg==, the expression for the amplitude of the steady-state response isAMP;Choosing the parameter valuesparams1 := {A=1, B=1/10, K=1};gives the amplitude functionF := eval(AMP, params1);whose graph is seen in Figure 28.4.plot(F,omega=0..3, title="Figure 28.4");Figure 28.4 supports the initial claim that resonance can be thought of as a temper tantrum by Mother Nature. The amplitude of the driving function is just NiMvJSJBRyIiIg==, but the magnitude of the response at steady-state is on the order of 10. For very little provocation, the oscillator has over-responded, much like a child having a temper tantrum.The response to the driving term clearly depends on the damping parameter. Choose the system parameters to be NiMvJSJtRyIiIg== andparams2 := {A=1, B=b, K=1};so that the amplitude function isG := eval(AMP, params2);The dependence of the amplitude on both the damping coefficient and the driving frequency can be seen in Figure 28.5.plot3d(G, omega=0..2, b=0..1, axes=box, title="Figure 28.5");As damping increases upward from NiMvJSJiRyIiIQ==, the magnitude of the steady-state amplitude diminishes. Thus, the greater the damping, the less the effect of resonance. As NiMlImJH decreases, the spike in amplitude at resonance clearly increases. The plane sections NiMvJSJiRyUpY29uc3RhbnRH shown in Figure 28.6 again capture the dependence of the resonant peak on damping. The tallest peak corresponds to the smallest value of NiMlImJH, and the smallest peak, to the largest.plot([seq(eval(G,b=j/10),j=1..9)],omega=0..2, title="Figure 28.6");

Without damping, the driven oscillator equation would beODE1 := eval(ODE,B=0);in which case a particular solution would bedsolve(ODE1, x(t), output=basis)[2];Clearly, as NiMlJm9tZWdhRw== approaches NiMvLSUlc3FydEc2IyUiS0ctRiU2IyomJSJrRyIiIiUibUchIiI=, the natural (angular) frequency of the system, the amplitude of the steady-state solution will become infinite. In reality, there is always some damping that prevents this solution from actually existing forever. In physical terms usually one or more system components break while the amplitude is still finite. Because these unbounded solutions are primarily of theoretical interest, this is called unrealistic resonance.Another approach to unrealistic resonance is to drive the system initially at the natural angular frequency. The differential equation for the system then becomesODE2 := eval(ODE1, {K=k/m, omega=sqrt(k/m)});A particular solution is thenXP := dsolve(ODE2, x(t), output=basis)[2];where, in spite of the complicated form, it should be clear that one of the trigonometric terms is multiplied by NiMlInRH. This is because, by the method of undetermined coefficients, the homogeneous solution and the driving term share a common term. Hence, the particular solution requires multiplication by NiMlInRH, and the trigonometric term in the particular solution without the factor NiMlInRH is actually redundant - it also appears in the homogeneous solution.To within a constant factor, the particular solution is actuallyXp := t*sin(sqrt(k/m)*t);Taking NiMvJSJrRyUibUc= = 1, this becomesxt := eval(Xp, {k=1,m=1});whose graph is then seen in Figure 28.7, along with the lines NiMlInhH = + NiMlInRH, the envelopes on the oscillations generated by NiMtJSRzaW5HNiMlInRH.plot([xt, t,-t], t=0..25, color=[black,red,red], title="Figure 28.7");The oscillations would become unbounded were it not for the constraints of physical reality.

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