For conservative systems (i.e. no friction or energy lose), the equation of motion can be found from the conservation of energy, which states that the sum of kinetic and potential energies is a constant,

     T + V = constant

Taking the time derivative gives

     d(T + V)/dt = 0

From this, the equation of motion for vibrations can be determined.


Development of Energy Equation

To understand how energy methods can be used, a simple mechanical system with linear and rotational kinetic, spring potential, and gravity potential energy will be analyzed. For the system shown, summing the kinetic and potential energies gives


Here, C is an arbitrary constant (total energy). Also, the displacement variable, yo, needs to be related to the rotation variable, θ since only one variable can be used with the energy method.

     yo = r θ

The moment of inertia for the pulley is

     I1 = 1/2 m1 r2

Substitute these equations in the total energy equation gives,


Take the first time derivative and rearrange,


Since the velocity is not zero at all times, the equation of motion is



At static equilibrium, the acceleration is zero, so the static equilibrium position is

     yo = m2 g/k

Rewriting the equation of motion in the new coordinate

     y = yo - m2 g/k

gives the familiar equation


where the natural frequency ωn is


The general solution is

     y = A sinωnt + B cosωnt

The constants A and B are determined from the initial conditions:


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