Lore of wide use. Enjoy the great illustrative power of mathematics!
Why do sinusoidal variations with time seem to crop up everywhere in Physics? By sinusoidal I mean any variation that is a linear combination A sin 2πft + B cos 2πft, where A and B are constants, and f (sometimes ν)is the frequency in per second, or hertz (Hz). The angular frequency ω = 2πf, in radians per second, is often convenient. The unit Hz should not be used with angular frequency, to avoid confusion. Our sinusoidal variation can also be written C cos (ωt + ε), where C = (A2 + B2)1/2 is the amplitude and ε = tan-1(A/B) is the phase constant. The quantity (ωt + ε) is the phase. A variation of this type is called harmonic, or, more precisely, simple harmonic, since it involves only one frequency f.
The Figure at the right shows a graphic representation of simple harmonic motion. The vector of length A rotates anticlockwise from its initial position at time 0 as shown by the dotted vector. The projection of the terminus of the rotating vector on the x-axis represents a simple harmonic motion (as does the projection on the y-axis). A vector (x,y) can also be expressed as a complex number in the form x + iy = Aeiωt = A (cos ωt + i sin ωt). It is very convenient to express a harmonic variation in complex exponential form, Aei(ωt + ε), because then all the rules of manipulation of exponents are available. The complex constant Aeiε is called a phasor, and is very useful for the specification of a harmonic variation. Engineers use j instead of i for the imaginary unit to avoid confusion with the symbol for electrical current. Physicists often take the time variation as e-iωt. In any case, two parameters are necessary for the complete specification of a simple harmonic motion.
With these definitions out of the way, we can return to the question. The small-amplitude natural vibrations of bodies are observed to be simple harmonic, or else a sum of several simple harmonic variations. Ohm's Law (of acoustics) asserts that the fundamental aural perception is that of a simple harmonic variation. In fact, this is the origin of the word. Fourier's Theorem shows that an arbitrary periodic time variation can be expressed as a sum of simple harmonic variations, usually an infinite number. These are only a few examples of the ubiquity of harmonic motion. They all are consequences of the mathematical behaviour of simple harmonic motion, often combined with the principle of superposition.
The two essential mathematical properties of simple harmonic motion are: (1)the sum of any number of such motions is also a harmonic motion of the same frequency, with at most a difference of amplitude and phase constant, and (2) the derivative (or integral) of a harmonic motion is also a harmonic motion of the same frequency, again with at most a difference of amplitude and phase constant. The exponential function eax also has these properties, but increases without bound for large positive or negative x (depending on the sign of a), so it is unsuitable for the description of a steady motion. The harmonic vibration, on the other hand, is periodic with period T = 1/f. Also, the two linearly independent functions e+ax and e-ax cannot be turned into one another by differentiation, as the sine and cosine can. The exponential and trigonometric functions are closely related, as we know, and the sine and cosine are linear combinations of complex exponentials.
The differential equation d2y/dx2 + a2y = 0 has solutions e-iax and e+iax, or sin ax and cos ax, as can be found by substitution in the equation. Each pair of solutions is such that no linear combination of them is a constant, so they are called linearly independent. The general solution of the differential equation is y(x) = A sin ax + B cos ax, and it can be made to satisfy any two algebraic conditions, such as having specified values at two different points, or for having a specified value and derivative at a certain point. These conditions are variously called initial or boundary conditions. For example, B = y(0) and A = y'(0)/a, if the initial position and slope are specified at the origin. This differential equation occurs over and over, so it is good to remember its solutions and their properties.
For example, consider a mass M hanging from a fixed point O by a flexible, inextensible, weightless cord of length L. The motion can be described in terms of the distance x that M moves to the right or left of its equilibrium position. When it is displaced by x, the weight Mg acts directly downward, but the force of the cord on M is angled a bit. The net horizontal force on M is then -Mg(x/L), if we assume that the displacement x is always small compared to L (say, less than 0.1L). By Newton's Second Law, Md2x/dt2 = -Mg(x/L), or d2x/dt2 + (g/L)x = 0. This is just the equation we were talking about, so M moves harmonically with angular frequency ω = (g/L)1/2. The frequency depends on the strength of gravity, and on the length of the cord, but remarkably not on the mass M, nor on the amplitude of vibration. The isochronous and harmonic nature of the motion depend on the smallness of the motion. For large swings, the pendulum no longer moves with simple harmonic motion! Finite motion is a different and more difficult problem (but one easily soluble).
The way to solve more difficult problems of this sort was shown by J. L. Lagrange (1736-1813; Mécanique Analytique, 1788). One begins by selecting any convenient parameters, called generalized coordinates, that define the state of the system. If only one such parameter q is necessary, we have a system with one degree of freedom. For the pendulum, q could be the angle θ that the line OM makes with the vertical. The velocity of M can then be expressed in terms of the rate of change of θ, as v = Ldθ/dt. The kinetic energy T of M is then M(Ldθ/dt)2/2. Lagrange showed that the partial derivative of T with respect to the rate of change of the coordinate is a generalized momentum. In this case, p = ML2(dθ/dt), which is the angular momentum about O, but the result also holds in general. Lagrange now expresses the potential energy V in terms of the coordinate, which in this case is MgL cos θ. The negative of the derivative of the potential energy with respect to the coordinate is the generalized force. In this case, it is -MgL sin θ, which is the torque about O. Newton's Second Law then gives MLd2θ/dt2 + Mg sin θ = 0, or d2θ/dt2 + (g/L)sin θ = 0. This is not our previous equation of motion, because the sine appears instead of just the angle. We won't solve it, since we just wanted to show how to use generalized coordinates to get equations of motion. The momenta and forces that are associated with a generalized coordinate are appropriate to the type of coordinate. Since we chose an angle, we get angular momenta and torques in place of linear momenta and forces.
In our example, T was independent of q, and V was independent of q' = dq/dt. Lagrange also showed how to handle more complex cases by defining the Lagrangian function L = T - V, and then writing the equation of motion as d(δL/δq')/dt = δL/δq (the δ's represent partial differentiation). This is Lagrange's equation, and further details are to be found in books on advanced mechanics. There is one such equation for each generalized coordinate, in systems with more than one degree of freedom. It is a very powerful way to find equations of motion. With very special choices of the generalized coordinates, each of Lagrange's equations may contain only a single generalized coordinate, and so the system is easily solved. These special generalized coordinates are called normal coordinates.
Thus far, the pendulum does not move unless it is given an initial disturbance, say by being held away from the equilibrium position and released, or by being given a knock when hanging still. Our equations imply that the pendulum will continue moving with the same amplitude from then on, but we know from experience that the amplitude slowly decreases, and the pendulum eventually comes to rest, due to air resistance and friction in the support. To make the pendulum move continuously, it must be driven or forced. Suppose we do this by moving the support from side to side a distance y = D cos ω't. We suppose D << L, so the added force on M is +Mg(y/L). Now, the equation of motion is d2x/dt2 + (g/L)x = (g/L)D cos ω't, after removing the common factor M. We require any solution that will give the new right-hand term, called a particular solution, to which we can add the solution we already know that describes the natural motion and contains two constants that we can choose so that boundary conditions are satisfied./P>
Let us try x = C cos ω't. When it is substituted in the equation, we get -Cω'2 + Cω2 = Dω2, omitting the common factor cos ω't. We have a solution, provided that C = Dω2/(ω2 - ω'2). This motion is at the frequency of the disturbance, and proportional to the magnitude of the disturbance. If the forcing frequency ω' is less than the natural frequency, the forced motion is in phase with the force, and as ω' approaches zero, it is also of the same amplitude. If ω' is greater than ω, the forced motion is 180° out of phase with the force, and its amplitude becomes less and less as the forcing frequency is raised. When ω' and ω are about equal, the amplitude of the forced motion becomes very large, which is called resonance.
This is all borne out very nicely by the pendulum, as illustrated in the Figure. In the natural motion, the pendulum swings about the point of support, and the effective length of the cord is L. Forced at a low frequency, the pendulum swings about a higher support, and the effective length L' is greater than L. It is simply the length of a pendulum that would swing at the forcing frequency. At a high forcing freqency, the pendulum swings about a lower support with effective length L", again corresponding to a pendulum whose natural frequency is the forcing frequency. The phase relations are very clearly shown. At zero forcing frequency, the pendulum hangs vertical as the support slowly moves from side to side. At a very high forcing frequency, M remains practically fixed, as the support moves rapidly from side to side. As another example, consider a vehicle of mass M supported by springs of spring constant K between body and wheels. Its natural frequency will be (K/M)1/2. If the springs compress a distance d when the body is put on them, the natural frequency will be (g/d)1/2. When the wheels experience a short sharp vertical shock, they will move upwards while the body remains largely undisturbed. The mass M will experience an upward impulse, so the body will begin to move upward in a harmonic motion at its natural frequency that will die out quickly if sufficient frictional resistance is provided. If the vertical disturbance is a rapid oscillation, the body will hardly be disturbed. On the other hand, disturbances near and below the natural frequency will give rise to considerable motion, especially in the case of resonance.
It is possible to find the response of a system to an arbitrary force f(t), for example by the method of variation of parameters. The solution is then expressed as an integral of f(t) times sine and cosine functions. A force f(t) = (μ/πτ)/(t2 + τ2) can represent the application of a force for a short time, where μ is the impulse, or integral of force times time, and τ is a time parameter. The force is peaked, with a width at half-maximum of 2τ, as shown in the Figure. The motion of the oscillator, assumed to be intially at rest in its equilibrium position, is then x(t) = (μ/ω)e-ωτsin ωt. This shows the smoothing effect clearly, when τ is comparable with the natural period of oscillation.
Suppose we apply D cos ω't to a pendulum initially at rest, and then ask what happens when ω' = ω. The conditions x = dx/dt = 0 at t = 0 then give the solution x = D[ω2/(ω2 - ω'2)](cos ω't - cos ωt). We have chosen the arbitrary constants A and B in the free motion to satisfy the boundary conditions. The difference of the cosines is 2 sin[(ω' - ω)/2]sin[(ω' + ω)/2]. As ω' approaches ω, the first sine can be replaced by its argument, which cancels the difference in frequencies in one factor of the denominator. Finally, when ω' = ω, the limit is x = (D/2) ωt sin ωt. The amplitude of oscillation increases linearly with time. This can only go on for a short time, however, for our approximations become invalid with large amplitudes. A pendulum 1 m long has a period quite close to 2 s. If we drive the pendulum by shaking the support by the small amplitude of 0.1 mm at 0.5 Hz, it will take 10.5 minutes to reach an amplitude of 10 cm, which is about the limit at which our approximations are valid. Since losses increase with amplitude, the amplitude will eventually be limited at a value for which the energy contributed by the exciting force balances the losses.
All this analysis applies not only to mechanical systems, but also to electrical circuits. Here, the definitions of inductance L and capacitance C are v = -L(di/dt) and q = Cv, or i = C(dv/dt). In the RLC circuit shown below, the positive direction of the current, and the polarity of the voltages are shown. In the present case, R = 0, so dv/dt = -L(d2i/dt2) = i/C, or d2i/dt2 + (1/LC)i = 0, which is the familiar equation of motion of a vibrating system. The natural angular frequency is ω = (LC)-1/2. One analogy between mechanical and electrical systems is i : dx/dt, v : -F, L : M, C : 1/K. Another is i : F, v : dx/dt, L : 1/K, C : M. In either case, one can imagine electrical and mechanical systems that will behave identically. All our results for forced vibrations and impulsive excitation still hold for electric circuits.
So far we have not considered the forces that bring a vibrating system to rest in the absence of continuous excitation. In an electric circuit, resistance provides such a force, and it is easily handled mathematically. If a capacitor C, an inductance L and a resistance R are connected in series, the equation of motion is d2i/dt2 + (R/L)di/dt + (1/LC)i = 0, found by summing the voltages around the circuit and equating the sum to zero, then differentiating with respect to t. If a solution eat is assumed, and substituted in the equation, we find that it is a solution provided the equation a2 + (R/L)a + (1/LC) = 0 is satisfied. This means that a = -(R/2L) + [(R/L)2 - 1/LC]1/2, or a = -(R/2L) + jω[1 - 1/Q2]1/2, where ω = (1/LC)1/2 and Q = (L/C)1/2/R. If Q >> 1, the resonant frequency is little changed, but there is an exponential decay e-Rt/2L, or an energy decay time constant (for the square of the current) τ = L/R = Q/ω. Q can also be expressed as ωL/R, a more familiar form. Every inductance L has some R associated with it (which includes magnetic losses as well as resistance of the wire).
As Q becomes smaller, the decay becomes more rapid, and the frequency of vibration decreases. Even at Q = 2, the frequency is still 87% of its value without damping. However, at Q = 1, the vibration ceases to be oscillatory, and the motion last only a fraction of the natural period. This case is called critical damping. In the RLC circuit, it occurs when R = (L/C)1/2. If R is proportional to L, halving L will increase Q by the square root of 2.
The presence of a little dissipation makes forced motion easier to understand. Let's introduce some dissipation, and write the equation of motion as q" + 2rq' + ω2q = Dejω't (primes on the q's indicate time derivatives). The forcing term is expressed as an exponential, to make it easier to handle phase shifts, since they will now not simply be π or -π. Assuming a solution Cejω't, we find C = D/[(ω2 - ω'2) + 2jrω']. The first thing to do is to express this amplitude in polar form, so we can see the magnitude and phase directly. The result is C = De-jε/[(ω2 - ω'2)2 + 4(rω')2]1/2. The phase constant ε = tan-1[2rω'/(ω2 - ω'2)]. We recover our earlier results when r = 0, so we are on the right track. Exactly at resonance, ω' = ω, C = e-jεD/2rω with ε = π/2. The amplitude is finite, and the phase shift is π/2. This means the external force is in phase with the velocity, so it does the maximum work on the oscillator. The phase shift increases continuously from 0 to π as ω' goes from 0 to infinity, with most of the change close to resonance.
Near resonance, the expressions can be thrown into a more symmetrical approximate form. We write ω' - ω = Δω, ω' + ω = 2ω, and elsewhere replace ω' by ω. The result is: C = e-jε(Dr/ω)/(2r2 + Δω2), with ε = tan-1 (r/Δω). The symmetrical response curve we have met before; it is called Lorentzian, and its Fouier transform is the exponential. The amplitude of oscillation decreases exponentially in the time domain, which is a consequence of the Lorenzian response in the frequency domain (but this is another story).
Composed by J. B. Calvert
Created 5 June 2000