Dynamics and Vibration Workbench

Interactive workspace initializes in your browser. Engineering method, assumptions, validation, and references are available below.

Engineering reference

Dynamics & Vibration: theory, method, and sources

This mechanical workspace publishes 7 governing equations, 5 stated assumptions, 6 documented boundaries, and 1 source so the numbers it returns can be checked rather than taken on trust.

Calculations run locallyCalculation & source methodology

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.

Calculators and topics covered

dynamics
vibration
resonance
balancing
linkages
projectile motion
rotational inertia parallel axis theorem
mass spring damper simulator
natural frequency calculator
damping ratio logarithmic decrement
shaft critical speed whirl
two plane rotor balancing
four bar linkage Grashof condition
transmission angle linkage

Core equations

I = \sum \left(\bar{I} + md^{2}\right),\; k = \sqrt{\frac{I}{m}}

\omega_n = \sqrt{\frac{k}{m}},\; \zeta = \frac{c}{2 \sqrt{\mathrm{km}}},\; \omega_d = \omega_n \sqrt{1 - \zeta ^{2}}

Magnification = \frac{1}{\sqrt{\left(1 - r^{2}\right)^{2} + \left(2 \zeta r\right)^{2}}},\; \text{peak}\; at\; r = \sqrt{1 - 2 \zeta ^{2}}

\delta = \left(\frac{1}{n}\right) \ln \left(\frac{x_{0}}{x_{n}}\right),\; \zeta = \frac{\delta }{\sqrt{4 \pi ^{2} + \delta ^{2}}}

\begin{gathered}Static\; \text{balance}\; \sum mr = 0\\\text{dynamic}\; \text{balance}\; \text{also}\; \sum mra = 0\end{gathered}

\text{Grashof:}\quad s + l \le p + q

\text{Linear drag:}\quad \ddot{x} = - b \dot{x},\; \ddot{y} = - b \dot{y} - g

Method and assumptions

Assumptions

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 and design boundaries

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.

Sources and references

Primary sources are preferred for ratings, standards, manufacturer data, and externally defined constants.

Source policy

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.