Vibrations
Comprehensive guide to mechanical vibrations, analysis methods, and control techniques.
Single Degree of Freedom Systems
A single degree of freedom (SDOF) system is the simplest vibrating system, consisting of a mass, spring, and damper. The equation of motion for a SDOF system is:
m(d²x/dt²) + c(dx/dt) + kx = F(t)
Where:
- m = mass (kg)
- c = damping coefficient (N·s/m)
- k = spring stiffness (N/m)
- x = displacement (m)
- F(t) = external force as a function of time (N)
The natural frequency of an undamped SDOF system is:
ω_n = √(k/m)
Damping Ratio
The damping ratio is a dimensionless measure that describes how oscillations in a system decay after a disturbance:
ζ = c / (2√(km))
Where:
- ζ = damping ratio (dimensionless)
- c = damping coefficient (N·s/m)
- k = spring stiffness (N/m)
- m = mass (kg)
Based on the damping ratio, systems can be classified as:
- ζ = 0: Undamped
- 0 < ζ < 1: Underdamped (oscillatory)
- ζ = 1: Critically damped
- ζ > 1: Overdamped
Resonance
Resonance occurs when a system is excited at its natural frequency, resulting in large amplitude vibrations. The amplitude magnification factor for a forced vibration is:
M = 1/√[(1-(ω/ω_n)²)² + (2ζω/ω_n)²]
Where:
- M = amplitude magnification factor
- ω = forcing frequency (rad/s)
- ω_n = natural frequency (rad/s)
- ζ = damping ratio
At resonance (ω = ω_n), the magnification factor becomes:
M = 1/(2ζ)
This shows that higher damping reduces the amplitude at resonance.