Dynamics and Vibration Workbench
Eight modules covering rotational inertia, the mass-spring-damper in free and forced response, natural frequency, damping ratio from decay, shaft critical speed, two-plane rotor balancing, four-bar linkages, and projectile motion with drag.
How this tool works
Every vibration module here is the same second-order system mẍ + cẋ + kx = F(t) seen from a different angle: natural frequency is its undamped root, damping ratio is measured from its decay, critical speed is where a rotor meets it, and the simulator shows the whole response.
The remaining modules cover how mass is distributed about an axis, how a rotor is corrected in two planes, and how a linkage or a projectile traces a path through space.
Each module opens with the question it answers, the governing relationship, and the specific thing worth noticing, rather than presenting a bare calculator.
Core equations
I = Σ(Ī + md²), k = √(I/m)ωn = √(k/m), ζ = c/(2√(km)), ωd = ωn√(1 − ζ²)Magnification = 1/√((1 − r²)² + (2ζr)²), peak at r = √(1 − 2ζ²)δ = (1/n)ln(x₀/xₙ), ζ = δ/√(4π² + δ²)Static balance Σmr = 0; dynamic balance also Σmra = 0Grashof: s + l ≤ p + qLinear drag: ẍ = −bẋ, ÿ = −bẏ − gAssumptions
- Vibrating systems are linear, with viscous damping proportional to velocity and constant stiffness.
- Each system is idealised to a single degree of freedom, lumping mass at one point and neglecting the mass of the beam or shaft itself.
- Rotor balancing treats the shaft as rigid at the operating speed.
- The four-bar linkage has rigid links, ideal pin joints, and no clearance or friction.
- Projectile drag is linear in velocity, and there is no wind, spin, or lift.
Limitations
- Single-degree-of-freedom idealisations give only the first mode. A real distributed structure has an infinite set, and higher modes matter once the excitation is broadband.
- Coulomb, hysteretic, and squeeze-film damping are all non-viscous and do not fit the ζ model used here.
- Rigid-rotor balancing fails above roughly half the first critical speed, where the shaft itself deforms and flexible-rotor methods are needed.
- Linear drag suits small slow objects. A ball or a shell is closer to quadratic drag, where the force goes as v² and the asymmetry is stronger.
- The linkage solves position and velocity only; it does not compute accelerations, inertia forces, or the shaking force transmitted to the frame.
- Nothing here is a substitute for measurement, a design code, or review by a licensed engineer.
References and verification
- Standard dynamics and vibration formulationsThe relationships used are the conventional ones from an introductory dynamics and vibration sequence: the parallel-axis theorem, the damped second-order response, logarithmic decrement, Rayleigh and Dunkerley critical speed estimates, two-plane balancing, Grashof classification with the vector loop closure, and projectile motion under linear drag.