Mechanical and Electrical Vibrations and Resonance: every key term you need (+ practice quiz)
25 flashcard terms for Differential Equations Topic 6, written to match the course framework. Study them here, then drill them as interactive flashcards, or test yourself with the 15-question quiz — free, no account needed.
A second-order equation balancing inertia, a damping force proportional to velocity, and a restoring force proportional to displacement, with any external push as forcing. It is the canonical constant-coefficient application.
Hooke law restoring force
The assumption that a spring pulls back with a force proportional to its extension from the natural length. It supplies the displacement term of the vibration equation and is valid only for small extensions.
Damping coefficient
The constant multiplying velocity in the vibration equation, representing friction or a dashpot. Larger values remove energy faster and push the characteristic roots away from the imaginary axis.
Natural frequency
The angular frequency at which an undamped spring mass system oscillates freely, equal to the square root of the stiffness divided by the mass. It sets the reference frequency for every resonance discussion.
Simple harmonic motion
The undamped unforced solution, a pure sinusoid of constant amplitude at the natural frequency. Its amplitude and phase are set by the initial displacement and velocity and never change.
Amplitude and phase form
Rewriting a combination of cosine and sine at one frequency as a single cosine with an amplitude and a phase angle. It makes the peak size and timing of the motion immediately readable.
Underdamped motion
The regime in which damping is small enough that the characteristic roots are complex, producing oscillation inside a decaying exponential envelope. The system crosses equilibrium infinitely often as it settles.
Overdamped motion
The regime in which damping is large enough that the roots are real and negative, so the system returns to equilibrium without oscillating and can pass through equilibrium at most once.
Critically damped motion
The borderline case of a repeated real root, where the solution is an exponential multiplied by a linear factor. It is the fastest return to equilibrium without overshoot, which is why it is a common design target.
Quasi frequency
The oscillation frequency actually observed in underdamped motion, given by the imaginary part of the characteristic roots. It is always smaller than the natural frequency, and damping widens the gap.
Logarithmic decrement
The natural logarithm of the ratio of two successive peak displacements in underdamped motion. It is a laboratory-friendly way to measure the damping coefficient from a recorded decay trace.
Damping ratio
The dimensionless comparison of the actual damping with the critical value. Below one the system oscillates, at one it is critically damped, and above one it is overdamped, which is why it classifies behaviour cleanly.
Forced undamped resonance
The unbounded growth that occurs when an undamped system is driven exactly at its natural frequency. The trial solution duplicates the complementary solution, so the response acquires a factor growing with time.
Beats
The slow amplitude modulation produced when an undamped system is driven near but not at its natural frequency. Two close frequencies combine into a fast oscillation inside a slowly varying envelope.
Practical resonance
The peak in steady state amplitude of a damped forced system as the driving frequency is varied. The peak occurs slightly below the natural frequency and its height is finite and set by the damping.
Resonance peak sharpness
The narrowness of the amplitude curve near its maximum. Light damping produces a tall narrow peak that is very sensitive to driving frequency, while heavy damping flattens the curve until no peak remains.
Amplitude response function
The steady amplitude of a damped forced oscillator expressed as a function of driving frequency, forcing size, stiffness, mass and damping. Plotting it against frequency is the standard engineering diagnostic.
Phase lag at resonance
The steady output of a damped oscillator lags the drive by a quarter cycle exactly at the natural frequency, tending to no lag at very low frequencies and to half a cycle at very high ones.
Series RLC circuit equation
The loop voltage balance giving a second-order equation for charge, with inductance playing the role of mass, resistance the role of damping, and the reciprocal of capacitance the role of stiffness.
Mechanical electrical analogy
The correspondence letting one solved equation serve both domains, with displacement matching charge, velocity matching current, and applied force matching supplied voltage. Every vibration result transfers directly.
Impedance viewpoint
Treating a sinusoidally driven linear circuit or oscillator through a complex amplitude ratio whose magnitude gives the response size and whose argument gives the phase. It replaces differential algebra with complex arithmetic.
Quality factor
A dimensionless measure of how lightly damped a resonant system is, large when energy loss per cycle is small. It relates directly to the sharpness of the resonance peak and to how long free oscillations persist.
Energy in a vibrating system
The sum of kinetic energy from motion and potential energy stored in the spring. Damping strictly decreases it while forcing can add to it, which explains growth and decay without solving anything.
Equilibrium displacement under gravity
For a vertically hung mass, gravity shifts the rest position by the weight divided by the stiffness. Measuring displacement from that shifted position removes the constant term and recovers the standard equation.
Design implication of damping choice
Instruments and vehicle suspensions are usually tuned near critical damping to settle quickly without overshoot, whereas sensors meant to detect a narrow frequency band are deliberately left lightly damped.