AP Physics C Mechanics: Driven Oscillations and Resonance

Driven oscillations explain why a bridge can collapse from marching soldiers, how a radio tunes to a specific station, and why pushing a child on a swing at the right time yields the biggest arc. Understanding the response of a system to an external driving force is critical in engineering, from designing earthquake-resistant buildings to preventing turbine failures. This analysis centers on solving the fundamental differential equation to find the steady-state response and uncovering the powerful, and sometimes destructive, phenomenon of resonance.

The Driven Harmonic Oscillator Equation

We begin with a damped harmonic oscillator—like a mass on a spring moving through a viscous fluid—that is subjected to an external, sinusoidal driving force. The force law is $F_d=F_0\cos (\omega t)$, where $F_0$ is the maximum magnitude of the driving force and $\omega$ is the driving frequency (not necessarily the system's natural frequency). Applying Newton's second law yields the governing differential equation:

$$m\frac{d^{2}x}{d{t}^2}+b\frac{dx}{dt}+kx=F_0\cos (\omega t)$$

It's standard to rewrite this equation by defining key parameters: the natural angular frequency ${\omega }_0=\sqrt{k/m}$, the damping constant $\gamma =b/(2m)$, and the maximum driving acceleration $A_0=F_0/m$. This gives the canonical form:

$$\frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_0^{2}x=A_0\cos (\omega t)$$

The total solution to this equation is the sum of two parts: a transient solution and a steady-state solution. The transient solution is the solution to the homogeneous equation (with $A_0=0$), which takes the form $x_t=e^{-\gamma t}[C_1\cos ({\omega }^{\prime }t)+C_2\sin ({\omega }^{\prime }t)]$, where ${\omega }^{\prime }=\sqrt{{\omega }_0^2-{\gamma }^2}$. This solution decays exponentially over time due to the $e^{-\gamma t}$ factor. After a sufficiently long time, only the long-term, steady-state solution remains, which oscillates at the driving frequency $\omega$, not the natural frequency ${\omega }_0$.

Finding the Steady-State Solution

The steady-state solution is a particular solution to the full non-homogeneous differential equation. Because the driving force is a cosine function, we assume a solution of the form $x_p(t)=A\cos (\omega t-\delta )$, where $A$ is the steady-state amplitude and $\delta$ is the phase constant (or phase lag) between the driving force and the system's displacement. The phase lag $\delta$ tells us how much the displacement response "lags behind" the applied force.

To find $A$ and $\delta$, we substitute $x_p(t)$ and its derivatives into the differential equation. Using trigonometric identities, this leads to a system of equations. Solving this system yields the amplitude and phase as functions of the driving frequency:

$$A(\omega )=\frac{A_0}{\sqrt{({\omega }_0^2-{\omega }^2{)}^2+(2\gamma \omega {)}^2}}$$

$$\tan \delta =\frac{2\gamma \omega }{{\omega }_0^2-{\omega }^2}$$

These two equations are the core results for driven oscillations. The amplitude function $A(\omega )$ is particularly important. Notice that the denominator has two terms: $({\omega }_0^2-{\omega }^2{)}^2$ and $(2\gamma \omega {)}^2$. When the driving frequency $\omega$ is far from the natural frequency ${\omega }_0$, the first term is large, making the amplitude $A$ small. The system simply can't follow a force that is oscillating too fast or too slowly relative to its preferred rhythm.

Resonance Curves and the Resonance Peak

A plot of the steady-state amplitude $A$ versus the driving frequency $\omega$ is called a resonance curve or frequency response curve. Its most striking feature is a peak. To find the precise frequency ${\omega }_r$ at which this peak (maximum amplitude) occurs, we take the derivative of $A(\omega )$ with respect to $\omega$, set it equal to zero, and solve. This gives the resonance frequency:

$${\omega }_r=\sqrt{{\omega }_0^2-2{\gamma }^2}$$

Notice that for light damping ($\gamma \ll {\omega }_0$), the resonance frequency is approximately equal to the natural frequency: ${\omega }_r\approx {\omega }_0$. However, for stronger damping, the resonance peak occurs at a frequency slightly less than ${\omega }_0$. This is a key distinction often tested on the AP exam.

Substituting ${\omega }_r$ back into the amplitude formula gives the maximum amplitude at resonance:

$$A_{max}=\frac{A_0}{2\gamma \sqrt{{\omega }_0^2-{\gamma }^2}}\approx \frac{A_0}{2\gamma {\omega }_0}\text{(for light damping)}$$

The resonance curves for different damping constants $\gamma$ reveal crucial physics:

• Light Damping ($\gamma$ small): The curve is tall and narrow. The amplitude at resonance $A_{max}$ is very large, and the system is extremely sensitive to frequencies near ${\omega }_r$.
• Heavy Damping ($\gamma$ large): The curve is short and broad. The maximum amplitude is much smaller, and the system responds to a wider band of driving frequencies.

This explains the swing analogy: light damping (low friction) requires very precise timing (frequency matching) for large swings, while a heavily damped swing (one moving through mud) would never achieve a large amplitude no matter how well you time your pushes.

Common Pitfalls

1. Confusing ${\omega }_0$, ${\omega }_r$, and ${\omega }^{\prime }$. These are three distinct frequencies.

• ${\omega }_0=\sqrt{k/m}$ is the natural frequency of the undamped, free oscillator.
• ${\omega }^{\prime }=\sqrt{{\omega }_0^2-{\gamma }^2}$ is the damped natural frequency of the free (un-driven) oscillator.
• ${\omega }_r=\sqrt{{\omega }_0^2-2{\gamma }^2}$ is the resonance frequency at which the driven oscillator's amplitude is maximized.

Remember: ${\omega }_r<{\omega }^{\prime }<{\omega }_0$ for any $\gamma >0$.

1. Forgetting that the steady-state solution oscillates at $\omega$, not ${\omega }_0$. A common conceptual error is thinking the mass oscillates at its natural frequency. In the steady-state, the driver is in complete control of the oscillation frequency. The system's natural frequency ${\omega }_0$ only determines how large the response will be to a given driving frequency $\omega$.

1. Misinterpreting the phase lag $\delta$. The equation $\tan \delta =\frac{2\gamma \omega }{{\omega }_0^2-{\omega }^2}$ dictates the phase relationship.

• When driven well below resonance ($\omega \ll {\omega }_0$), $\delta \approx 0$. The displacement is in phase with the driving force (force and displacement peak together).
• At resonance ($\omega ={\omega }_r\approx {\omega }_0$), $\delta =90^{\circ }$ or $\pi /2$ radians. The displacement lags the force by a quarter cycle; velocity is in phase with the force, maximizing power transfer.
• When driven well above resonance ($\omega \gg {\omega }_0$), $\delta \approx 180^{\circ }$ or $\pi$ radians. The displacement is out of phase with the driving force; when the force pushes right, the mass is displaced left.

Summary

• The driven harmonic oscillator is modeled by $\frac{d^{2}x}{d{t}^2}+2\gamma \frac{dx}{dt}+{\omega }_0^{2}x=A_0\cos (\omega t)$. Its long-term steady-state response is $x(t)=A\cos (\omega t-\delta )$, oscillating at the driving frequency $\omega$.
• The steady-state amplitude is $A(\omega )=\frac{A_0}{\sqrt{({\omega }_0^2-{\omega }^2{)}^2+(2\gamma \omega {)}^2}}$. It peaks at the resonance frequency ${\omega }_r=\sqrt{{\omega }_0^2-2{\gamma }^2}$, which is slightly less than ${\omega }_0$ for damped systems.
• Damping $\gamma$ critically shapes the resonance curve: light damping produces a tall, narrow peak; heavy damping produces a short, broad curve, reducing the maximum amplitude.
• Resonance occurs when the driving frequency matches the system's resonance frequency, leading to a maximal amplitude response. This is a cornerstone concept for understanding energy transfer in oscillatory systems, from mechanical engineering to electrical circuits.
• The phase lag $\delta$ between the driving force and displacement varies from $0^{\circ }$ (low frequency) through $90^{\circ }$ at resonance to $180^{\circ }$ (high frequency), indicating how the system's inertia and stiffness dominate in different frequency regimes.