ORDINARY DIFFERENTIAL EQUATIONS POWERTOOL

Unit 28 -- Application: Unforced Harmonic Motion

Outline of Unit 28

Initialization

> restart;

> with( DEtools ):

> with( plots ):

> with( linalg ):

Warning, the name changecoords has been redefined

Warning, the name adjoint has been redefined

Warning, the protected names norm and trace have been redefined and unprotected

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The general form of the ODE for unforced, or free, harmonic motion is

> ode1 := a * diff( x(t), t$2 ) + b * diff( x(t), t ) + c * x(t) = 0;

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When < 0, the characteristic values are complex conjugates that can be written as

> lambda[1] := alpha + beta*I;

> lambda[2] := alpha - beta*I;

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where and .

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The general solution is a linear combination of exponentially damped trigonometric functions

> X[1] := exp( -alpha * t ) * cos( beta * t );

> X[2] := exp( -alpha * t ) * sin( beta * t );

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> X[h] := c[1] * X[1] + c[2] * X[2];

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or, equivalently,

> X[h2] := A * exp( -alpha*t ) * sin( beta*t + phi );

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where , , and . In this form, is the damped amplitude, is the quasi-frequency, is the quasi-period, and is the phase angle.

The decaying exponential term forces all solutions to decay to the zero function as increases.

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When = 0 there is a double root to the characteristic equation:

> lambda[1] := -b / 2;

> lambda[2] := -b / 2;

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Two linearly independent solutions to the ODE in this case are

> X[1] := exp( lambda[1]*t );

> X[2] := t * X[1];

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and the general solution is

> X[h] := c[1]*X[1] + c[2]*X[2];

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Subcase 1: > 0

So long as b > 0 the common exponential factor overwhelms the linear growth and forces all solutions to zero. The decay to zero is not necessarily monotonic. The general solution to the critically damped equation has a critical point at

> dX[h] := factor( diff( X[h], t ) );

> t_crit := solve( dX[h]=0, t );

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which is positive if and only if and < .

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Subcase 2: = 0

Note that if = 0, then = 0 and > 0 imply that . In this case the ODE simplifies to

> eval( ode1, [a=1,b=0,c=0] );

a fundamental set of solutions is

> eval( X[1], b=0 );

> eval( X[2], b=0 );

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and the general solution is a linear function. Hence, solutions do not tend to zero as increases. In fact, except in special cases where the solution is a constant function, the solutions become unbounded as increases.

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The third case is the most straightforward. If > 0, then there are two distinct real characteristic values.

> lambda[1] := eval( lambda, charvals[1] );

> lambda[2] := eval( lambda, charvals[2] );

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that produce a fundamental set of solutions

> X[1] := exp( lambda[1]*t );

> X[2] := exp( lambda[2]*t );

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and general solutions

> X[h] := c[1]*X[1] + c[2]*X[2];

Observe that, because 0 < <= , both characteristic values are negative. In fact, < < 0. All solutions to an overdamped equation ultimately decay to zero as t 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 );

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To determine when the critical point occurs with > 0, observe that the location of the critical point can be expressed as = - ln( ). This time is positive precisely when 0 < < 1.

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To conclude, note that except for the trivial case with = = 0, all solutions to the ODE for simple harmonic motion are transient solutions. This means that the steady-state behavior of a solution for a forced harmonic motion problem depends on the forcing function, i.e. , on a particular solution to the nonhomogeneous problem.

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