Complex Roots and Forced Oscillations

1Introduction

This lecture constructs free oscillations from complex characteristic roots and distinguishes transient from steady-state responses to periodic forcing. It then derives from the exact amplitude formula the different meanings of resonance in damped and undamped systems.

2Standard Form and Terminology

The free motion of a mass--spring system is governed by

where $$y\left(t\right)$$y(t) is displacement from equilibrium, $$m>0$$m>0 is mass, $$c\text{[PARSE ERROR: Undefined("Command("ge")")]}0$$c\ge0 is the viscous damping coefficient, and $$k>0$$k>0 is the spring constant. With periodic forcing, the equation becomes

where $$F_0>0$$F_0>0 is the forcing amplitude and $$\omega \text{[PARSE ERROR: Undefined("Command("ge")")]}0$$\omega\ge0 is the forcing angular frequency. The homogeneous solution determined by the initial conditions is the {homogeneous free response}. When this free response decays, it is also called the {transient response}. A particular solution that persists at the forcing frequency is the {steady-state response}.

3Complex Roots and Free Oscillation

If the characteristic roots are $$\alpha \pm i\beta$$\alpha\pm i\beta with $$\beta >0$$\beta>0, the real-valued homogeneous solution is

The amplitude decays exponentially when $$\alpha <0$$\alpha<0 and remains periodic without decay when $$\alpha =0$$\alpha=0. The angular frequency of the free oscillation is $$\beta$$\beta, and its period is $$2\pi /\beta$$2\pi/\beta.

For a mass--spring system with $$0<c<2\sqrt{mk}$$0<c<2\sqrt{mk}, the characteristic roots are

The quantity $${\omega }_n=\sqrt{k/m}$$\omega_n=\sqrt{k/m} is the {undamped natural angular frequency}, and $${\omega }_d$$\omega_d is the {damped natural angular frequency}. They are not equal in general.

4Amplitude of the Steady-State Response

Assume $$c>0$$c>0 and set

Comparison of coefficients gives

and hence

Writing $$y_s=M\left(\omega \right)\cos (\omega t-\delta )$$y_s=M(\omega)\cos(\omega t-\delta) gives the exact displacement amplitude

This is an equality containing the forcing amplitude, not merely a proportionality. A quadrant-sensitive expression for the phase lag is

The general solution is $$y=y_h+y_s$$y=y_h+y_s. When $$c>0$$c>0, the transient response $$y_h$$y_h decays, so the steady-state response dominates at long times.

5Resonance Frequency in a Damped System

Maximizing the displacement amplitude is equivalent to minimizing

For $$\omega >0$$\omega>0, the stationary condition is

Consequently, a positive displacement-resonance angular frequency exists only when $$c^2<2mk$$c^2<2mk, and it is

In general, this frequency equals neither $${\omega }_n=\sqrt{k/m}$$\omega_n=\sqrt{k/m} nor $${\omega }_d$$\omega_d. If $$c^2\text{[PARSE ERROR: Undefined("Command("ge")")]}2mk$$c^2\ge2mk, the displacement amplitude decreases monotonically from $$\omega =0$$\omega=0 and has no positive-frequency resonance peak. Frequencies maximizing velocity or acceleration amplitudes differ from the displacement-resonance frequency, so the response quantity must be specified whenever a resonance frequency is quoted.

6Boundary of Undamped Resonance

When $$c=0$$c=0 and $$\omega \ne {\omega }_n$$\omega\ne\omega_n, a steady-state particular solution is

At $$\omega ={\omega }_n$$\omega=\omega_n, the denominator vanishes because the forcing overlaps the homogeneous solution. No bounded periodic steady-state response exists at this boundary. Instead, one particular solution is

whose envelope grows linearly in time. This undamped resonance of the ideal linear model must be distinguished from the finite resonance peak obtained when $$c>0$$c>0.

7Example

For $$y^{\prime }\text{'}+4y=0$$y''+4y=0, the characteristic roots are $$\pm 2i$$\pm2i and

For the resonantly forced equation

we have $$m=1$$m=1, $$F_0=1$$F_0=1, and $${\omega }_n=2$$\omega_n=2. The undamped-resonance formula gives

and direct substitution confirms that $$y_{p^{\prime }}\text{'}+4y_p=\cos 2t$$y_p''+4y_p=\cos2t.

8Dimensional Consistency

If displacement and time have units $$\left[y\right]=\mathrm{m}$$[y]=\mathrm{m} and $$\left[t\right]=\mathrm{s}$$[t]=\mathrm{s}, then

Thus, $$m{y}^{\prime }\text{'}$$my'', $$c{y}^{\prime }$$cy', $$ky$$ky, and $$F_0\cos \omega t$$F_0\cos\omega t all have units of force, $$\mathrm{N}$$\mathrm{N}. The denominator in the amplitude formula has units $$\mathrm{N}/\mathrm{m}$$\mathrm{N/m} and the numerator has units $$\mathrm{N}$$\mathrm{N}, so $$\left[M\right]=\mathrm{m}$$[M]=\mathrm{m}. Moreover, both $$c^2/m^2$$c^2/m^2 and $$k/m$$k/m have units $$s^{-2}$$\mathrm{s}^{-2}, confirming the dimensional consistency of the formulas for $${\omega }_d$$\omega_d and $${\omega }_r$$\omega_r.

9Scope and Limitations

The amplitude formula and resonance conditions above assume a linear constant-coefficient model with sinusoidal forcing. In nonlinear oscillations, frequency may depend on amplitude and multiple stable periodic orbits may occur. Real systems also involve damping, nonlinearities, material limits, and forcing of finite duration. The unbounded linear growth of undamped resonance is therefore a conclusion of an idealized model.

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