3 Modelling forced mechanical oscillations

Suppose now that the mass is subject to a force f ( t ) after the initial disturbance. Then the equation of motion is

m d 2 x d t 2 + k d x d t + n x = f ( t )

Consider the case f ( t ) = F cos ω t , that is, an oscillatory force of magnitude F and angular frequency ω . Choosing specific values for the constants in the model: m = n = 1 , k = 0 , and ω = 2 we find

d 2 x d t 2 + x = F cos 2 t

Task!

Find the complementary function for the differential equation

d 2 x d t 2 + x = F cos 2 t

The homogeneous equation is

d 2 x d t 2 + x = 0

with auxiliary equation λ 2 + 1 = 0 . Hence the complementary function is

x cf = A cos t + B sin t .

Now find a particular integral for the differential equation:

Try   x p = C cos 2 t + D sin 2 t  so that  d 2 x p d t 2 = − 4 C cos 2 t − 4 D sin 2 t . Substituting into the differential equation gives

( − 4 C + C ) cos 2 t + ( − 4 D + D sin 2 t ) ≡ F cos 2 t .

Comparing coefficients gives − 3 C = F and − 3 D = 0    so that D = 0 , C = − 1 3 F and  x p = − 1 3 F cos 2 t .  The general solution of the differential equation is therefore

x = x p + x cf = − 1 3 F cos 2 t + A cos t + B sin t .

Finally, apply the initial conditions to find the solution for the displacement x :

We need to determine the derivative and apply the initial conditions:

d x d t = 2 3 F sin 2 t − A sin t + B cos t .

At t = 0 x = x 0 = − 1 3 F + A and d x d t = 0 = B

Hence B = 0 and A = x 0 + 1 3 F .

Then x = − 1 3 F cos 2 t + x 0 + 1 3 F cos t .

The graph of x against t is shown below.

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If the angular frequency ω of the applied force is nearly equal to that of the free oscillation the phenomenon of beats occurs. If the angular frequencies are equal we get the phenomenon of resonance . Note that we can eliminate resonance by introducing damping into the system.