Vibration, Resonance & Isolation

Vibration is repeated exchange between kinetic and stored energy. For the one-degree-of-freedom model

the undamped natural frequency is \(\omega_n=\sqrt{k/m}\), and damping ratio is \(\zeta=c/(2\sqrt{km})\). Forcing near \(\omega_n\) can create a large response: resonance.

Worked frequency estimate

A 20 kg instrument sits on mounts with combined stiffness 80,000 N/m.

If a nearby machine excites the floor at 10 Hz, the mounting system is poorly placed. Changing mass or stiffness shifts the natural frequency; damping controls the peak and transient decay. Effective high-frequency isolation normally requires the forcing frequency to be well above the mounted natural frequency, but soft mounts also permit more static and transient travel.

Sources and paths

Do not jump directly to “add rubber.” Trace:

• source: imbalance, gear mesh, reciprocating mass, stick-slip, impacts, torque ripple, or control oscillation;
• path: frame, floor, fastener, cable, fluid line, or air;
• receiver: sensor, bearing, tool, person, or product.

Fixing the source—balancing a rotor or changing a motion profile—often beats isolating everything downstream. Stiffening can move a mode above the operating band; softening can move it below; neither is automatically correct.

Measurement discipline

Measure operating speed and vibration together. A frequency spectrum relates peaks to shaft orders, mesh frequencies, or structural modes. A coast-down test helps separate speed-related forcing from fixed structural resonance. Always check sampling rate, sensor mounting, orientation, and units before interpreting a plot.

Design exercise

Create a frequency inventory for one automated axis: motor speed range, pulley or gear tooth-passing frequency, cycle rate, impacts, and control update rate. Estimate its first structural natural frequency. Mark overlaps and propose one source, path, or receiver intervention for each. Your output is a frequency-separation plan, not a promise that one isolator cures all vibration.