1.2 Complex Form of Solution

One may obtain any number of particular solutions of Equation (1.2) by simply inserting different values of A and f into Equation (1.5). Let x₁(t) and x₂(t) be two of these particular solutions. Since they are both solutions, one must have

            d²x₁/dt² + w_o² x₁ = 0
and

            d²x₂/dt² + w_o² x₂ = 0

If the second of these is multiplied by i = Ö (-1) and added to the first, one obtains

(1.6)       d²x₁/dt² + i d²x₂/dt² + w_o² (x₁ + i x₂) = 0

Let x(t) be defined as follows (symbols written in bold face represent complex quantities)

(1.7)       x(t) = x₁(t) + i x₂(t)

Functions such as x(t) which consist of this simple arrangement of two real functions are called complex functions. Differentiation or integration of these functions is defined as indicated below:

            dx/dt = dx₁/dt + i dx₂/dt

           d²x/dt² = d²x₁/dt² +i d²x₂/dt²

            òx dt = ò x₁ dt + i ò x₂ dt

Using these rules, it is possible to write Equation (1.6) as

(1.8)        d²x/dt²  + w_o² x = 0

This complex differential equation is identical in form to Equation (1.2). A solution of this complex differential equationis any complex function of the form (1.7) which satisfies it. It can be easily shown, if it is not already apparent, that the function

(1.9)        x(t) = A cos(w_o t + f ) + i A sin ((w_o t + f )

is a solution of Equation (1.9). Using Euler's theorem, one can write this in the form

(1.10)          x(t) = A exp(iw_o t )

where A = Aexp (if ). Now the real part of Equation (1.10) corresponds exactly to Equation (1.5), the general solution of the equation of motion. For reasons that will become apparent later on, one prefers to work with (1.10) and to regard it as the equation which describes the motion of the particle. It is, of course, the real part which actually describes the motion of the particle.

1.3 Phase Relationships

Equation (1.5) gives the x coordinate of the mass m at any instant. The velocity and acceleration can be obtained by successive differentiations:

(1.11)        dx/dt = -A w_o sin(w_o t + f )

(1.12)        d²x/dt² = - A w_o² cos(w_o t + f )

Now x, dx/dt and d²x/dt² are all periodic functions with precisely the same period. However, no two of the three quantities attain their largest positive values at exactly the same time. For example, if A>0, x attains its largest positive value at times t' for which

        w_o t' + f = 0, 2p , 4p , ...

At such times, dx/dt is zero and d²x/dt² has its largest negative value. When two sinusoidally varying quantities with the same period, attain their largest positive values at different
times, they are said to differ in phase, the phase difference being defined as

        d = 2p (t₁ - t₂)/T_o

where t₁ is a time at which one of the quantities attains its largest positive value, t₂ is the time nearest to t₁ at which the other quantity attains its largest positive value, and T_o is the period. The phase difference thus defined is in radians. Since dx/dt attains its largest positive value at times t'' for which

        w_o t" + f = 3p /2, 7p /2, 11p /2, ...

and d²x/dt² its largest positive value at times t''' for which

        w_o t''' + f = p , 3p , 5p , ...
It is evident that x differs in phase from dx/dt by p /2 radians or 90 degrees and from d²x/dt² by p radians or 180 degrees. If one uses the complex exponential form of the solution, one has {Note that differentiating or integrating A cos (w_o t + f ) +i A sin (w_o t + f ) is exactly equivalent to differentiating or integrating A exp (iw_o t ) as if it were a real function and A and i were real constants.}

        x = A exp(iw_o t)

        dx/dt = iw_oA exp (iw_o t)

        d²x/dt² = -iw_o² A exp(iw_o t )

At any given instant of time, x, dx/dt and d²x/dt² are complex numbers and may be represented in the complex plane as shown in Figure1.3.

Figure 1.3 Representation of x, dx/dt and d²x/dt² in the complex plane.

Note, that although the position of x is arbitrary, since it depends upon the particular instant of time chosen, once the position of x is selected, the positions of dx/dt and d²x/dt² are fixed from the relations dx/dt = -iw_ox and d²x/dt² = - w_o²x . Note further that the angle between x and dx/dt is p /2 or 90 degrees, precisely the phase difference between x and dx/dt while that between x and d²x/dt² is is 180 degrees, exactly the phase difference between x and d²x/dt². It should be apparent that the phase relations between the various quantities are more readily deduced from the complex exponential form of the solution than from the real form. In Figure 1.3, the projections of the vectors x, dx/dt, and d²x/dt² on the real axis are the real parts of these quantities and represent, respectively, the values of x, dx/dt, d²x/dt² at this particular instant.

As time increases, all three vectors rotate counterclockwise with an angular velocity w_o. Because dx/dt is 90 degrees counterclockwise from x it is said to lead dx/dt by p /2 or 90 degrees. The quantity d²xd/t² may be said to lead or lag x by p radians or 180 degrees, since one ordinarily speaks of quantities leading or lagging by angles of p radians or less.

Example 1.2
The real parts of x₁ = 5+12i exp(ip t/3) and x₂ = 4exp(ip t/3) represent simple harmonic motions.
(a) What are the real parts of these two expressions?
(b) What is the amplitude of each motion?
(c) Represent x₁ and x₂ as vectors in the complex plane at time t=0.
(d) What is the phase difference between these simple harmonic motions?
(e) Which leads?

Solution
(a) A complex number a+ib can be written in the form Ö (a² + b²) exp(ia ) where a = arctan (b/a). Thus x₁ = Ö (5² + 12²) exp(ip t/3 + 1.18). From Euler's theorem it is apparent that the real part is x₁=13 cos(p t/3 + 1.18). Similarly, x₂ = 4 cos(p t/3).

(b) From the real parts, one sees that the amplitudes are 13 and 4, respectively.

(c) At t=0       x₁ = 13 exp(1.18i)                      x₂ = 4

(d)(e) x₁ and x₂ are plotted in the complex plane at t=0 as shown in the figure.

From this figure it is seen that x₁ leads x₂ by by 1.18 radians or 67.4 degrees.

1.4 Energy

The total mechanical energy E of a harmonic oscillator is defined as the sum of its kinetic and potential energies. At an instant of timewhen the particle has a velocity dx/dt, its kinetic energy is defined as 1/2 m (dx/dt)² . The potential energy depends on the position of the particle, being equal to the work done by the spring force as the particle moves from the given position to a reference position. Taking the reference position as the equilibrium position, it follows that

when the particle is a distance x from the origin its potential energy is 1/2Kx². Thus, the potential energy is largest when the particle is furthest from its equilibrium position. The total energy at any instant may be written

        E = 1/2 m (dx/dt)² + 1/2 Kx²

Since the spring force is considered to be a conservative force, one would expect the total energy to be the same at every instant. One can easily verify that this is the case by substituting for x and dx/dt the values given by Equations (1.5) and (1.11) respectively. One obtains

        E = 1/2 m A² w_o² sin²(w_ot+ f ) + 1/2 KA² cos²(w_ot+ f )

which on substituting K/m for w_o² reduces to   E = 1/2 KA²

This can also be written as

        E = 1/2 w_o² mA² = 1/2 m v_max²

where v_max is the largest instantaneous value of the velocity of the particle, i.e., the velocity of the particle at the mid-point of its oscillation.
 

1.5 Damped Harmonic Motion

All of the oscillating systems with which one encounters in practice are subject to dissipative forces, and if left to themselves, i. e., if energy is not supplied from an outside source, the oscillations eventually cease. A model which provides a suitable description of many such systems, is a particle of mass m which is subjected both to a spring force and to a damping force, the latter being assumed proportional to and opposite in direction to the velocity of the particle. Although in real systems, the dissipation of energy may arise from various mechanisms, in the model, the dissipation of energy is usually thought of as arising from a frictional force exerted by the fluid through which the particle is moving. That such a force is present is often indicated schematically as depicted in Figure1.4

Figure 1.4
 

where a ``dashpot'' has been added to the spring and mass combination. A dashpot consists of a light piston moving in a suitable fluid, the piston being attached rigidly to the mass.

If the system depicted in Figure 1.4 is set in motion and the mass m isolated at a general time t, Newton's second law yields the equation

        m d²x/dt² + R dx/dt + K x = 0

where R is a constant whose value depends on the rate the real system is dissipating energy, and K is the spring constant. For convenience, let w_o = Ö (K/m) and a = R/(2m), so that the equation
of motion may be written

(1.13)        d²x/dt² + 2 a dx/dt + w_o² x = 0

It is possible to find a series solution of this differential equation using the same approach that was used for the undamped oscillator in Section 1.1. Although converting the series solution into a form which is easily interpreted is more involved, one can show that the general solution of (1.13) can be expressed in the form

(1.14)        x = exp(-a t) [ A cos (w_d t + f )]
where

        w_d = Ö (w_o² - a²) (1.15)

and A and f are arbitrary constants.

{There are three types of solutions of equation (1.13) depending on whether w_o is greater than, equal to, or less than a . The solution of most interest in the present discussion is Equation (1.14) which is the solution when w_o > a . For a discussion of all three cases, see Symon, K. R., Mechanics, 2nd edition, p 47 (Addison Wesley 1960)}
 
 

In Equation (1.14), the term in brackets is exactly the same form as (1.5) the solution of the undamped oscillator.

Figure 1.5 Motion of a damped harmonic oscillator

In Figure 1.5, this cosine term and the term exp(-a t) are plotted separately and then multiplied at each point to obtain x. It is seen that the motion is oscillatory with a gradually decaying amplitude. This decay in amplitude, which implies a corresponding loss in the total mechanical energy of the oscillator, arises from the presence of the dissipative force which converts the
mechanical energy of the oscillator into thermal energy in the fluid.

Although, strictly speaking, the motion of a damped oscillator is not periodic, one customarily defines the frequency of the oscillation as the number of times per second the particle passes through its equilibrium position in the positive direction. Since the cosine term in the brackets of Equation (1.14) determines the times at which x=0, it follows that the frequency, f_d of the damped oscillation is given by

        f_d = w_d/(2p ) = [Ö (w_o² - a²)]/(2p ) = [Ö (K/m - (R/(2m))²)]/(2p )

If (R/(2m))² is small compared to K/m, the frequency of a damped oscillator is only slightly smaller than the frequency of an undamped oscillator of the same mass and spring constant. When dealing with damped oscillating systems, it is customary to refer to a maximum occurring at some instant as the "amplitude" of the motion at that time.

Figure 1.6

Thus at time t₁ in Figure 1.6, the amplitude is A₁, at time t₂ the amplitude is A₂, and so on. Now the maxima (and minima) occur at times for which dx/dt=0 , and it is not difficult to show (Problem 1.12) that

        A_m = A_n exp(a (t_n - t_m))

where A_n is the amplitude at time t_n and A_m is the amplitude at time t_m. Thus, one can determine the damping term a =R/(2m) by measuring the amplitude at two different times.

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