Resonance Phenomenon

Calculus 2 · Axiom Academy

LESSON Resonance: When Frequency Matters Push a swing in step with its natural rhythm and tiny pushes build into enormous motion. Here is the differential equation that explains why. A mass on a spring left alone swings at one special rate, its natural frequency . Now attach a motor that pushes it back and forth at a driving frequency . The position y(t) obeys restoring pull vs. periodic push 2. Amplitude and the Resonance Curve Look for a steady response in step with the push, . Substituting into the equation and matching the terms (verified with sympy) gives the amplitude exactly: Plot A against the driving frequency and a sharp resonance curve appears, spiking where : 3. Inside Resonance: Growth Without Bound Set and the trial breaks down — the formula divides by zero. Solving the equation honestly (sympy, starting from rest) gives a response that is not a fixed-size wave but one whose envelope climbs in a straight line: Why it grows: each push arrives exactly when the mass is already moving the way the push points, so every cycle the push does positive work and feeds in more energy. The factor of t out front is that energy accumulating. The reality check: real systems have damping, which caps the peak at a large-but-finite value. The pure t -growth above is the idealized, frictionless extreme — and it is exactly why engineers design bridges to keep far from any wind or traffic frequency.

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