Free vibration

Free vibration occurs when a mass is displaced through a distance x and then let to vibrate freely. This displacement is due to an impulse excitation of the structure and no external forces are applied to the structure. The mass oscillates around the point of equilibrium.

When the structure is in static equilibrium, the weight of the mass (mg) is equal to the spring force (Δk), as shown in the following graphic. The spring force is defined as the product of a stiffness constant, k, and the elongation of the spring at rest, Δ.

Each structure has one or more natural frequencies that are calculated from free vibration. The natural frequency (also called the resonance frequency or resonant frequency) is the frequency at which stiffness and inertial forces of the structure cancel each other out as the amplitude peaks. In modal analysis, the peaks in a frequency response function are used to identify the natural frequencies and modes of a structure.

Undamped free vibration

In the case of undamped free vibration, the mass oscillates at the natural frequency around the point of equilibrium indefinitely as there is no energy dissipation.

When the mass is displaced through distance x, the expression for the spring force becomes

Fk = k(Δ + x)

and the resultant force acting on the mass becomes

F = mg - k(Δ + x) = - kx

Recalling Newton's fundamental equation F = ma and noting that acceleration is the second derivative of x, it follows that m = - kx which gives the following equation of motion for the undamped free vibration:

m + kx = 0

Plot of the system displacement as a function of time
	Static equilibrium
	Displacement through x

A plot of the resulting undamped vibration as a function of time generates a pure sine wave with a magnitude x. The natural frequency of this structure is the frequency of the sine wave.

Damped free vibration

In the case of damped free vibration, the mass oscillates at the natural frequency around the point of equilibrium with a magnitude that declines to zero due to dissipation of energy.

Because a friction force is directly proportional to the velocity of the mass, the term for damping is obtained by multiplying the damping constantc by velocity . Damping is introduced as a negative value into the resultant force. The expression for the resultant force is then set equal to mass times acceleration:

mg - k (Δ + x) - c = m

Recalling that at equilibrium the forces on the structure give mg = kΔ, the equation of motion for a damped free vibration can be expressed as a second order differential equation:

m + c + kx = 0

Note that all structural terms are present: mass, damping, stiffness, and displacement.

Plot of the system displacement as function of time
Frequency response function

A plot of the system displacement as a function of time generates a damped sine wave. The frequency response function shows the natural frequency of the structure.

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