Mechanical Vibrations - Click here for a discussion on Forced Vibrations

In a Nut Shell: Vibrations of a spring, mass, and damper mechanical system with an

applied forcing function (forced vibration) is an important application involving second

order, linear, ordinary differential equations with constant coefficients. If there is no

forcing function, f(t), then the mechanical system is said to undergo a free vibration.

Let x(t) be the displacement of the mechanical system (figure below) with mass m

spring constant K (lb/ft, N/m), and with damping constant C (lb sec/ft or N sec/m).

For free, undamped vibration: m d²x/dt² + k x = 0

For free, damped vibration: m d²x/dt² + C dx/dt + k x = 0

For forced, undamped vibration: m d²x/dt² + k x = f(t)

For forced, damped vibration: m d²x/dt² + C dx/dt + k x = f(t)

where m = mass of the system (slugs or kg)
k = spring constant (lb/ft or N/m)

C = damping constant (lb sec /ft or N sec / m)

d²x/dt² represents physically the acceleration of the mass (magnitude)

dx/dt represents physically the velocity of the mass (magnitude)

x(t) = displacement of the mass (ft or m)

f(t) = applied forcing function ( lb or N )

Natural frequency: ω = √ (K/m) rad/sec

Critical damping: C_c = 2 √ (Km) (lb sec/ft or N sec/m)

C < C_c underdamped C > C_c overdamped

Two initial conditions are needed to find the two constants of integration.

x(0) = x_o and dx(0)/dt = v_o Click here to continue.

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