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Forced equations and oscillations · Differential Equations | Math Infinitum

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Differential Equations
Forced equations and oscillations

Differential Equations · Chapter 05

Forced equations and oscillations

What a push does to a linear system. A forced equation is one particular solution plus every homogeneous one, found by guessing its shape or by letting the constants vary; then the mass on a spring, free, damped, driven, and driven at its own frequency.

What a push does to a linear system. A forced equation is one particular solution plus every homogeneous one, found by guessing its shape or by letting the constants vary; then the mass on a spring, free, damped, driven, and driven at its own frequency.

m y^{''} + c y^{'} + k y = F (t)

ω_{0} = \frac{k}{m}

y = y_{h} + y_{p}

F_{0} cos ω t

e^{- γ t}

ω = ω_{0}

L [y_{p}] = g

y_{p} = u_{1} y_{1} + u_{2} y_{2}

Lessons

Build the foundation

01Non-homogeneous equationsA forced equation has exactly one solution for each value and slope, just as the homogeneous one does — but the whole family is shifted. Every solution is one particular solution of the forced equation plus the general solution of the homogeneous one. So solving a forced equation is two jobs: the homogeneous job, which the last chapter finished for constant coefficients, and finding just one solution of the forced equation, which the next two lessons do by two different methods.Read lesson →
02Undetermined coefficientsFor constant coefficients and forcing built from polynomials, exponentials, sines and cosines, a particular solution has the same shape as the forcing, with unknown coefficients. Substitute the guess, match coefficients, and solve a small linear system. The one trap is a guess that already solves the homogeneous equation: it can then only produce zero, never the forcing, and multiplying the guess by x repairs it.Read lesson →
03Variation of parametersUndetermined coefficients needs luck: constant coefficients and a forcing of a few shapes. Variation of parameters needs none. Take the general solution of the homogeneous equation and let its two constants vary, as functions. One condition on them is free to choose, and choosing it well leaves two linear equations for their derivatives, whose determinant is the Wronskian.Read lesson →
04Mass–spring systemsHang a mass on a spring and pull it down. Hooke's law says the spring pulls back with a force proportional to the stretch; friction, if there is any, resists in proportion to the speed; and anything else pushing on the mass is the forcing. Newton's law adds these up into a second-order linear equation with constant coefficients — the mass, the friction and the stiffness — in which each of the three numbers means something you could point to.

Read lesson →

05Free oscillationsWith no friction and no forcing the characteristic roots are purely imaginary, and the motion is a pure oscillation that never dies: a cosine at the natural frequency, shifted in time. Its amplitude is set by how the motion was started, and its period does not depend on the start at all.Read lesson →

06DampingAdd friction and the energy leaks away. Whether the characteristic roots are real or complex now depends on the square of the friction against four times the mass times the stiffness, and that comparison sorts the motion into three kinds: underdamped, which still oscillates while it decays; critically damped, the quickest return without oscillating; and overdamped, a slower creep back. All three die away.Read lesson →

07Forced oscillationsPush a damped spring with a force that swings back and forth like a cosine, and it ends up moving at the frequency of the push, not its own. Every solution is the particular solution plus a homogeneous one, and with friction the homogeneous part — the transient — dies away. What remains is the steady state, a cosine at the driving frequency, whose amplitude depends on how the driving frequency compares with the natural one.Read lesson →

08ResonanceDrive an oscillator at its own natural frequency and each push arrives just when it helps most. Without friction the amplitude then grows without bound — the particular solution is t times a sine, the modification rule of undetermined coefficients in physical form. Close to the natural frequency the motion beats, swelling and fading; with friction the amplitude stays finite but peaks sharply near the natural frequency.Read lesson →

09Chapter practiceBring the chapter together: choose between undetermined coefficients and variation of parameters, solve a forced problem with its conditions, classify a damped spring, compute a steady amplitude, and recognise resonance. No worked example sits above the answers.Read lesson →