Energy Methods

As we all know the simple harmonic motion of a body is due only to gravitational and elastic restoring forces acting on the body. Since these forces are conservative, it is also possible to use the conservation of energy equation in order to obtain the body’s natural frequency or period of vibration.

Now let’s consider a system with the block and spring mode. 
 

Figure 1- Undamped Free Vibrational System

When the block is displaced at distance x from the equilibrium position, the kinetic energy: 

$$T=\frac{1}{2}m{{v}^{2}}=\frac{1}{2}m{{\dot{x}}^{2}}$$

and the potential energy is:

$$V=\frac{1}{2}k{{x}^{2}}.$$

Since the energy is conserved, we can write: 

$$\begin{align} & T+V=\text{constant} \\ & \frac{1}{2}m{{{\dot{x}}}^{2}}+\frac{1}{2}k{{x}^{2}}=\text{constant} \\ \end{align}$$

To get the differential equation that describes the motion of the system we need to derivate the equation and than we get: 

$$\begin{align} & m\dot{x}\ddot{x}+kx\dot{x}=0 \\ & \dot{x}\left( m\ddot{x}+kx \right)=0 \\ \end{align}$$

Analyzing the previous equation we can conclude that the speed or x’ is not always equals zero so the general assumption would be:

$$m\ddot{x}+kx=0$$

As you can see we have got the differential equation which describes the vibrations of the system. If the conservation of energy equation is written for a system of connected bodies, the natural frequency or the equation of motion can also be determined by time differentiation. It is not necessary to dismember the system to account for the internal forces because they do no work.