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In the previous sections we looked at the properties of simple harmonic motion, in the context of a block attached to a spring.  A feature of that analysis is that the only force acting on the block was the linear restoring force of the spring.  In this section, we will consider a slightly more realistic situation in which an additional force is present.  This additional force is a resisting force that is proportional to the velocity of the block:

Resisting forces proportional to velocity can be used to model situations such as viscous drag in fluids at low velocities.  However, when velocities are high, a quadratic relationship is more appropriate, so the linear relationship between drag and velocity needs to be used with care.

With this resisting force and the restoring force of the spring, the forces acting on our block are as shown in the following diagram:

Newton’s Second Law for this system can be written as the following differential equation:

This can be rewritten into a form that will be more convenient later on:

where we have defined two new constants

In order to solve the governing differential equation, we try a solution of the form:

Substituting into the differential equation and cancelling the common exponential factor in each term leads to the following:

This is easily solved to give:

                                       (A)

Now, there are three cases to consider, depending on whether the quantity under the square root sign in (A) is positive, negative or zero.

Consider first the case in which the quantity under the square root sign in (A) is positive.  In this case we have

where p₁ and p₂ are the two real values of p in (A), and A and B are constants.  Thus, in this case, there is no oscillation at all in the system – the damping provided by the velocity dependent force is sufficiently strong that the block is unable to oscillate.  This situation is called “over-damping”.

Next consider the case in which the quantity under the square root sign in (A) is negative.  Expression (A) then evaluates to

where i is the square root of -1 and

The general solution can then be written in the form:

Therefore, in this case, we do have oscillations, but the oscillations are damped by the exponential factor at the beginning of the expression, and the frequency of the oscillations is lower than in the case of simple harmonic motion.  Nevertheless, the degree of damping is not sufficient to prevent oscillations, and this situation is called “under-damping”.

Next consider the case in which the quantity under the square root sign in (A) is zero.  In this case, expression (A) yields only a single solution, namely

Since the governing differential equation is of second order, we require a second solution.  It can easily be verified by direct substitution into the equation that the second solution is

The full solution is therefore

This is also a function that decays with time, and when this situation applies it is referred to as “critical damping”.  It marks the transition between purely decaying solutions and oscillatory solutions.  It also represents the most rapid return to the equilibrium (x = 0) position given an initial displacement of our block.

The following diagram illustrates the three solutions that we have just considered, including the case of undamped (c = 0) motion.  In all cases the constants A and B are chosen so that the initial displacement is unity and the initial velocity is zero.

The green curve shows the case of under-damping, and it can be seen that the time period of oscillation is slightly longer than in the undamped case (the blue curve).  Note also that the critical damping case (the black curve) shows a more rapid return to the equilibrium position than the over-damping case (the red curve).

Finally, it should be noted that the energy E of the block does not remain constant in time, since, using the governing differential equation,

The energy therefore decreases with time, as would be expected.

Next:  Forced Oscillations