Free Body Diagram Method

The system consisting of a body with mass m and the spring of stiffens k is shown in the next figure. The body is displaced form equilibrium position and released.

Figure 2.2 a) Mass-spring system b) Free-body diagrams at an arbitrary instant. Directions of external and effective forces are consistent with positive direction of generalized coordinate x.

So now we will derive the differential equation of motion for this system by applying the FBD method. First we need to draw this body without connections. On this body we will add the weight force in downward directions and the force of the spring in upward direction.

Now we need to choose the generalized coordinate. Let’s say that x(t) is a generalized coordinate and the positive values of that coordinate is in the downward direction. Since the body is pulled down from equilibrium position and then released we need to add the distance from the point of release and the equilibrium position. 
  $$\begin{align} & \sum{{{F}_{ext}}=\sum{{{F}_{eff}}}} \\ & mg-k\left( x+{{\Delta }_{st}} \right)=m\ddot{x}, \\ & mg-kx-k{{\Delta }_{st}}=m\ddot{x}, \\ & mg=k{{\Delta }_{st}}, \\ & mg-kx-mg=m\ddot{x}, \\ & m\ddot{x}+kx=0. \\ \end{align}$$

After we add these forces to the body we can sum forces and derive the differential equation.

In general the procedure of FBD method is: 1)      Choose the generalized coordinate. This variable should represent the displacement of a particle in the system. If we analyze the rotational motion then the generalized coordinate could represent an angular displacement. 2)      FBD diagrams are drawn showing the system at an arbitrary instant of time. For any system you need to make two Free Body Diagrams one with external forces and one with effective forces. 3)      The appropriate form of Newton’s law is applied to free-body diagrams. 4)      Applicable assumptions are used along with algebraic manipulation. The result is the governing differential equation which describes the motion of  the analyzed system.