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 $F_{d} = F_{0} cos (ω t)$ , where $F_{0}$ is the maximum magnitude of the driving force and $ω$ is the driving frequency (not necessarily the system's natural frequency). Applying Newton's second law yields the governing differential equation:

$m \frac{d ^{2} x}{d t ^{2}} + b \frac{d x}{d t} + k x = F_{0} cos (ω t)$

It's standard to rewrite this equation by defining key parameters: the natural angular frequency $ω_{0} = k / m$ , the damping constant $γ = b / (2 m)$ , and the maximum driving acceleration $A_{0} = F_{0} / m$ . This gives the canonical form:

$\frac{d ^{2} x}{d t ^{2}} + 2 γ \frac{d x}{d t} + ω_{0}^{2} x = A_{0} cos (ω t)$

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 $x_{t} = e^{- γ t} [C_{1} cos (ω^{'} t) + C_{2} sin (ω^{'} t)]$ , where $ω^{'} = ω_{0}^{2} - γ^{2}$ . This solution decays exponentially over time due to the $e^{- γ t}$ factor. After a sufficiently long time, only the long-term, steady-state solution remains, which oscillates at the driving frequency $ω$ , not the natural frequency $ω_{0}$ .

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 $x_{p} (t) = A cos (ω t - δ)$ , where $A$ is the steady-state amplitude and $δ$ is the phase constant (or phase lag) between the driving force and the system's displacement. The phase lag $δ$ tells us how much the displacement response "lags behind" the applied force.

To find $A$ and $δ$ , 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:

$A (ω) = \frac{A _{0}}{( ω _{0}^{2} - ω ^{2} ) ^{2} + ( 2 γ ω ) ^{2}}$

$tan δ = \frac{2 γ ω}{ω _{0}^{2} - ω ^{2}}$

These two equations are the core results for driven oscillations. The amplitude function $A (ω)$ is particularly important. Notice that the denominator has two terms: $(ω_{0}^{2} - ω^{2})^{2}$ and $(2 γ ω)^{2}$ . When the driving frequency $ω$ is far from the natural frequency $ω_{0}$ , 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 $ω$ is called a resonance curve or frequency response curve. Its most striking feature is a peak. To find the precise frequency $ω_{r}$ at which this peak (maximum amplitude) occurs, we take the derivative of $A (ω)$ with respect to $ω$ , set it equal to zero, and solve. This gives the resonance frequency:

$ω_{r} = ω_{0}^{2} - 2 γ^{2}$

Notice that for light damping ( $γ ≪ ω_{0}$ ), the resonance frequency is approximately equal to the natural frequency: $ω_{r} \approx ω_{0}$ . However, for stronger damping, the resonance peak occurs at a frequency slightly less than $ω_{0}$ . This is a key distinction often tested on the AP exam.

Substituting $ω_{r}$ back into the amplitude formula gives the maximum amplitude at resonance:

$A_{ma x} = \frac{A _{0}}{2 γ ω _{0}^{2} - γ ^{2}} \approx \frac{A _{0}}{2 γ ω _{0}} (for light damping)$

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

Light Damping ( $γ$ small): The curve is tall and narrow. The amplitude at resonance $A_{ma x}$ is very large, and the system is extremely sensitive to frequencies near $ω_{r}$ .
Heavy Damping ( $γ$ 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

Confusing $ω_{0}$ , $ω_{r}$ , and $ω^{'}$ . These are three distinct frequencies.

$ω_{0} = k / m$ is the natural frequency of the undamped, free oscillator.
$ω^{'} = ω_{0}^{2} - γ^{2}$ is the damped natural frequency of the free (un-driven) oscillator.
$ω_{r} = ω_{0}^{2} - 2 γ^{2}$ is the resonance frequency at which the driven oscillator's amplitude is maximized.

Remember: $ω_{r} < ω^{'} < ω_{0}$ for any $γ > 0$ .

Forgetting that the steady-state solution oscillates at $ω$ , not $ω_{0}$ . 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 $ω_{0}$ only determines how large the response will be to a given driving frequency $ω$ .

Misinterpreting the phase lag $δ$ . The equation $tan δ = \frac{2 γ ω}{ω _{0}^{2} - ω ^{2}}$ dictates the phase relationship.

When driven well below resonance ( $ω ≪ ω_{0}$ ), $δ \approx 0$ . The displacement is in phase with the driving force (force and displacement peak together).
At resonance ( $ω = ω_{r} \approx ω_{0}$ ), $δ = 9 0^{\circ}$ or $π /2$ 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 ( $ω ≫ ω_{0}$ ), $δ \approx 18 0^{\circ}$ or $π$ 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 $\frac{d ^{2} x}{d t ^{2}} + 2 γ \frac{d x}{d t} + ω_{0}^{2} x = A_{0} cos (ω t)$ . Its long-term steady-state response is $x (t) = A cos (ω t - δ)$ , oscillating at the driving frequency $ω$ .
The steady-state amplitude is $A (ω) = \frac{A _{0}}{( ω _{0}^{2} - ω ^{2} ) ^{2} + ( 2 γ ω ) ^{2}}$ . It peaks at the resonance frequency $ω_{r} = ω_{0}^{2} - 2 γ^{2}$ , which is slightly less than $ω_{0}$ for damped systems.
Damping $γ$ 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 $δ$ between the driving force and displacement varies from $0^{\circ}$ (low frequency) through $9 0^{\circ}$ at resonance to $18 0^{\circ}$ (high frequency), indicating how the system's inertia and stiffness dominate in different frequency regimes.

AP Physics C Mechanics: Driven Oscillations and 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:

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

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:

$$\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = A_0 \cos(\omega t)$$

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 $x_t = e^{-\gamma t}[C_1 \cos(\omega' t) + C_2 \sin(\omega' t)]$ , where $\omega' = \sqrt{\omega_0^2 - \gamma^2}$ . 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:

$$A_{max} = \frac{A_0}{2\gamma \sqrt{\omega_0^2 - \gamma^2}} \approx \frac{A_0}{2\gamma \omega_0} \quad \text{(for light damping)}$$

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.

Common Pitfalls

Confusing $\omega_0$ , $\omega_r$ , and $\omega'$ . These are three distinct frequencies.

$\omega_0 = \sqrt{k/m}$ is the natural frequency of the undamped, free oscillator.
$\omega' = \sqrt{\omega_0^2 - \gamma^2}$ 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: $\omega_r < \omega' < \omega_0$ for any $\gamma > 0$ .

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 $\omega$ .

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 $\frac{d^2x}{dt^2} + 2\gamma\frac{dx}{dt} + \omega_0^2 x = A_0 \cos(\omega t)$ . 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.

AP Physics C Mechanics: Driven Oscillations and Resonance

AP Physics C Mechanics: Driven Oscillations and Resonance

The Driven Harmonic Oscillator Equation

Finding the Steady-State Solution

Resonance Curves and the Resonance Peak

Common Pitfalls

Summary

AP Physics C Mechanics: Driven Oscillations and Resonance

The Driven Harmonic Oscillator Equation

Finding the Steady-State Solution

Resonance Curves and the Resonance Peak

Common Pitfalls

Summary

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AP Physics C Mechanics: Driven Oscillations and Resonance

The Driven Harmonic Oscillator Equation

Finding the Steady-State Solution

Resonance Curves and the Resonance Peak

Common Pitfalls

Summary