Vibration of a single-degree system

Many structures can be pictured, at a first approximation, as a single mass on a spring with a damper: a water tank on a slender staging, a machine on its foundation, a footbridge swaying sideways, or one storey of a building. Displaced and released, the mass swings back and forth at its natural frequency, which depends only on its mass and the stiffness of its support. Heavier means slower; stiffer means faster.

Damping removes energy each cycle, so free vibration dies away. Structural damping is small, typically a few percent of critical for concrete and steel buildings, and it barely changes the natural frequency. It is measured from a free-vibration record: the ratio of successive peaks gives the logarithmic decrement, which for light damping is about two pi times the damping ratio.

When a harmonic force drives the system, the response depends on the frequency ratio, driving frequency over natural frequency. Well below resonance the structure follows the force almost statically. Near a ratio of one the dynamic amplification can be very large, limited only by damping; at 2 percent damping the peak is about 25 times the static deflection. Well above resonance the response falls below the static value, which is why machine foundations are often tuned to be either much stiffer or much softer than the running speed.

The same model underlies earthquake engineering: a response spectrum is simply the peak response of many single-degree systems with different periods to the same ground motion. Footbridge comfort checks, machine foundation design and wind-induced vibration of chimneys all start here before moving to multi-degree models.