L 40 : Solution of vibration problems by Energy Methods
In the conservative system, the total energy is constant and the differential equation of motion can also be established by the principle of conservation of energy. For the free vibration of an undamped system, the energy is partly kinetic and partly potential. The kinetic energy T is stored in the mass by virtue of its velocity, whereas the potential energy is stored in the form of strain enrgy in elastic deformation or work done in a force field such as gravity. The total energy being constant, its rate of change is zero as illustrated by the following equation:
constant
where T is the kinetic energy and U is the potential energy.
If our interest is only in the natural frequency of the system, it can be determined by the following consideration. From the principle of conservation of energy, we can write
T1+U1 = T2+U2
where 1 and 2 represent two instances of time. Let 1 be the time when the mass is passing through its static equilibrium position and choose U1 = 0 as reference for the potential energy. Let 2 be the time corresponding to the maximum displacement of the mass. At this position the velocity of the mass is zero, and then T2 = 0. We then have,