Forced Oscillations

Push a spring at a steady rhythm and its motion splits into two parts: its own natural wobble, which dies out, and a steady response that copies your push. The particular solution $y_p$ is that steady response.

y

mass position

y_h

natural part

y_p

forced part

F(t)

push

How it works

Nonhomogeneous second-order ODEs

In one sentence

For a mass $m$ on a spring of stiffness $k$ with damping $c$ , driven by $F_0\cos\omega t$ , every solution is $y = y_h + y_p$ : $y_h$ solves the homogeneous equation and dies out when $c > 0$ , and $y_p$ is the steady response at the driving frequency $\omega$ .

General formula

{\htmlData{sym=0}{m\,y''}} + {\htmlData{sym=1}{c\,y'}} + {\htmlData{sym=2}{k\,y}} = {\htmlData{sym=3}{F_0\cos(\omega t)}}, \qquad {\htmlData{sym=4}{\omega_0}} = \sqrt{k/m}

Each term is a force. $my''$ is inertia, $cy'$ is drag from the damper, $ky$ is the spring pulling back, and the right side is the push. The equation says they balance at every instant.

What the symbols mean

$m\,y''$
Inertia (mass × acceleration)
The block resisting changes to its motion. A heavier block needs more force to speed up or slow down.
$c\,y'$
Damping force (internal)
Friction inside the system, like a shock absorber. It pushes against the motion, harder the faster the block moves; c sets its strength.
$k\,y$
Spring force (internal)
The spring pulling the block back toward rest, harder the further it is stretched. k is the spring's stiffness.
$F_0\cos(\omega t)$
External driving force
The outside push that keeps the block moving. F₀ is how hard it pushes; ω is how many times per second it repeats (in radians).
$\omega_0$
Natural frequency
How fast the spring swings on its own: √(k/m). Driving at ω close to ω₀ makes the swings grow huge. That is resonance.

Turn the push off and solve $my''+cy'+ky=0$ . That gives $y_h$ , the spring's natural motion.
Guess a response shaped like the push, $y_p = A\cos\omega t + B\sin\omega t$ , plug it in and solve for $A$ and $B$ . That is undetermined coefficients.
The full motion is $y = y_h + y_p$ . Damping shrinks $y_h$ toward zero, so after a while only $y_p$ is left.
When $\omega$ nears $\sqrt{k/m}$ the response spikes. That spike is resonance.

Learn the math

Derivatives of Trig FunctionsCalc I · Lesson 18, Derivatives Differential Equations: DefinitionsDiff Eq · Lesson 1, Basic Concepts Second Order Linear EquationsDiff Eq · Lesson 13, Second Order Differential Equations Repeated RootsDiff Eq · Lesson 16, Second Order Differential Equations Undetermined CoefficientsDiff Eq · Lesson 19, Second Order Differential Equations Mechanical VibrationsDiff Eq · Lesson 21, Second Order Differential Equations

Learn the math

Derivatives of Trig FunctionsCalc I · Lesson 18, Derivatives

Differential Equations: DefinitionsDiff Eq · Lesson 1, Basic Concepts

Second Order Linear EquationsDiff Eq · Lesson 13, Second Order Differential Equations

Repeated RootsDiff Eq · Lesson 16, Second Order Differential Equations

Undetermined CoefficientsDiff Eq · Lesson 19, Second Order Differential Equations

Mechanical VibrationsDiff Eq · Lesson 21, Second Order Differential Equations