Forced Vibrations
Flick a metal ruler clamped to a bench and it vibrates briefly at its own natural frequency before settling down. Bolt that same ruler to a machine with a spinning, slightly unbalanced shaft, and it will instead be dragged into vibrating at whatever frequency the shaft turns at. The difference between these two situations — vibrating freely versus vibrating because something keeps pushing you — is the difference between free and forced vibration.
Free vibration versus forced vibration
A free vibration is what happens when a system is displaced from equilibrium once, and then left alone with no ongoing external driving force acting on it. Released, it oscillates at its own natural frequency — the frequency set purely by the system's own physical properties, such as its stiffness and mass — and (in the idealised case with no damping) would continue doing so forever at constant amplitude. A struck tuning fork, a plucked guitar string, or a bracket flicked once and released are all, immediately afterwards, undergoing free vibration.
A forced vibration, by contrast, is oscillation driven by a continuous external periodic force that keeps acting on the system for as long as the vibration continues. The driving force has its own frequency — the driving frequency — which is entirely independent of the system's natural frequency, fixed instead by whatever is generating the force. Crucially, once a forced vibration has settled into its steady state, the system ends up oscillating at the driving frequency, not at its own natural frequency — the driving force essentially takes over and dictates the rhythm of the motion.
Amplitude away from resonance
The amplitude of a forced vibration depends strongly on how close the driving frequency is to the system's natural frequency. When the driving frequency is different from the natural frequency, the system still settles into oscillating at the driving frequency, but generally with a much smaller amplitude than it would show if driven right at resonance. This connects directly back to the previous lesson: resonance is simply the special case of forced vibration where the driving frequency matches the natural frequency, and it is precisely at that special case that the amplitude rises to its peak. Move the driving frequency away from the natural frequency in either direction, and the amplitude of the steady-state forced vibration falls away, following the shape of the resonance curve described in the previous lesson.
This is why forced vibration and resonance are best understood as two views of the same underlying phenomenon: forced vibration is the general behaviour of a driven oscillator at any driving frequency, and resonance is what that general behaviour does at one particular, critical driving frequency.
Real engineering sources of forced vibration
Forced vibration in engineering almost always originates from some component that repeats a motion or a load periodically. Common sources include:
- Rotating machinery with imbalance. Any rotating shaft, wheel or rotor with even a small amount of mass distributed off-centre delivers a periodic driving force to its mountings once per revolution, at a driving frequency equal to the rotation frequency of the shaft.
- Engines. The cyclic firing of pistons in an internal combustion engine, or blade-passing effects in a turbine or compressor, produce driving forces that repeat at frequencies tied directly to the engine's operating speed.
- Out-of-round wheels or worn bearings. A wheel that isn't perfectly circular, or a bearing with surface damage, delivers a small periodic "bump" to whatever it's mounted in, once per revolution.
- Periodic loads from moving parts. Reciprocating mechanisms, cams, and other components that repeat a motion cycle apply a driving force at the frequency of that repeating cycle.
In every one of these cases the driving frequency is tied to a rotation speed or a cycle rate that the engineer knows, or can calculate, in advance — which is exactly what makes it possible to check for a resonance risk before a machine is ever built.
Because the driving frequency of rotating machinery is fixed directly by its rotation speed, engineers have direct control over it simply by choosing what speed the machine runs at. This is exactly why designers deliberately select operating speeds — and, where a machine must pass through a wide speed range (such as during start-up), design mountings or add damping — to avoid dwelling for long at a speed that matches a supporting structure's natural frequency. Avoiding resonance is very often cheaper and simpler than trying to build a structure that can survive it.
Key takeaways
- Free vibration is oscillation at a system's own natural frequency after a single disturbance, with no ongoing external force; forced vibration is driven continuously by an external periodic force.
- A forced vibration settles into oscillating at the driving frequency, not the system's own natural frequency.
- Amplitude away from resonance is generally much smaller than at resonance; resonance is the special case where driving frequency equals natural frequency, producing the amplitude peak.
- Common engineering sources of forced vibration include rotating machinery with imbalance, engines, out-of-round wheels or worn bearings, and periodic loads from moving parts.
- For rotating machinery, the driving frequency in Hz equals the rotation speed in rev/min divided by 60.
- Engineers deliberately choose operating speeds — and keep a safety margin — to avoid driving supporting structures near their natural frequencies.