Vibrations Expert

Before Starting

1. Single DOF or multi DOF system?
2. Free, forced, or self-excited vibration?
3. Time domain or frequency domain analysis?

Core Expertise Areas

Single DOF Systems

Equation of motion: m x-double-dot + c x-dot + k x = F(t). Natural frequency: omega_n = sqrt of k over m. Damping ratio: zeta = c over 2 sqrt of km. Underdamped: oscillatory decay, frequency omega_d = omega_n sqrt of 1 minus zeta squared. Critically damped: fastest return to equilibrium without oscillation.

Forced Vibration

Steady-state amplitude: X = F0 over k times magnification factor. Resonance: maximum amplitude when excitation frequency equals natural frequency. Phase angle: response lags excitation, 90 degrees at resonance. Frequency response function: complex ratio of output to input versus frequency.

Multi DOF Systems

Mass and stiffness matrices: M x-double-dot + K x = F. Natural frequencies: eigenvalues of K inverse M. Mode shapes: eigenvectors, orthogonal with respect to mass matrix. Modal superposition: response is sum of individual mode responses.

Rotating Machinery

Unbalance: mass offset from rotation axis causes rotating force excitation. Critical speed: rotation speed equals natural frequency, resonance in rotor. Balancing: single plane and two plane balancing to reduce unbalance forces. Campbell diagram: plot natural frequencies and excitation orders versus speed.

Vibration Isolation

Transmissibility: ratio of transmitted to applied force. Isolation: below natural frequency amplifies, above attenuates vibration. Design rule: isolator natural frequency below one third excitation frequency. Soft mounts: low stiffness isolators, good isolation but large static deflection.

Best Practices

• Identify all resonances and ensure operating range avoids them
• Add damping if resonance cannot be avoided through design
• Measure actual natural frequencies to validate analytical models
• Design vibration isolation based on actual equipment operating frequencies

Common Pitfalls

Pitfall	Fix
Ignoring damping in resonance analysis	Even small damping limits response at resonance
Wrong isolation region	Ensure operating frequency is well above isolator natural frequency
Missing higher modes in truncated modal analysis	Include sufficient modes for accurate response
Unbalance in rotating equipment	Perform field balancing after assembly

Related Skills

• dynamics-expert
• statics-expert
• control-systems-expert