Mass on a Spring

Consider a mass m on a spring with spring constant k, as shown in Fig. 3.5. In addition to the spring force acting on the mass (-kx), we will also assume that dissipative forces act on the mass. These dissipative forces could be generated within the spring itself, or due to the fluid through which the mass moves. We will assume that the dissipative forces can be written as where v is the velocity of the mass.

   


Figure: Mass on a spring

We can write F=ma for the mass:

 

This is a second order differential equation whose general solution should have two functions with two arbitrary constants. We can solve by rewriting Eq. (3.2) as follows:

 

where the operator acts on all terms to the right of it. The general solution of Eq. (3.3) is the sum of solutions obtained from setting the result of each of each first order operation on x equal to zero.

We can set and giving:

which gives for x(t):

 

Clearly q can be complex (for example, take b=0, corresponding to the case where there is no damping). We can distinguish three different regimes for different possible values of q

1. Overdamped (q real > 0)
   Here x(t) is a decaying function of time with two different decay time constants: and . and are determined by the intial conditions (such as position and velocity). For example, if the initial velocity is zero, then (you should check this) and .

2. Critically Damped (q real = 0)
   Here q=0. Since the two first order equations are the same, they only give one function and constant. Since we need two independent functions and constants to satisfy possible boundary conditions we return to the original equation with q=0.

   

   We substitute

   

   giving

   

   Integrating both sides with respect to time gives

   

3. Underdamped (q imaginary)
   With q imaginary, we can write where

   

   where would be the oscillation frequency of the system in the absence of damping.

   We can rewrite the general solution [Eq. (3.8)] for x(t):

   

   but x is real (after all it is the position of an object), therefore (x equals its complex conjugate). This gives:

   

   Therefore:

   

   We can write

   

   giving:

   

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