The Chemistry Maths Book, Second Edition

12.9 Forced oscillations 363

12.9 Forced oscillations

An important equation in the theory of forced oscillations in mechanical and electrical

systems is the inhomogeneous differential equation

(12.80)

This is the equation of motion for a body moving under the influence of the force

(12.81)

The term −kxtells us that we have a harmonic oscillator. The term−c 1 dx 2 dtis a

‘damping force’ proportional to the velocity representing, for example, the drag

experienced by the body moving in a fluid. The termF

0

1 cos 1 ωtis an external periodic

force that interacts with the motion of the oscillator. For example, a charge qin the

presence of an electric field experiences a forceqEin the direction of the field. In the

case of an alternating field,E 1 = 1 E

0

cos 1 ωt, the force acting on an electron, with charge

q 1 = 1 −e, is −eE

0

1 cos 1 ωt. If the electron is undergoing simple harmonic motion (in the

absence of the field) the total force acting on it is

3

F 1 = 1 −kx 1 − 1 eE

0

1 cos 1 ωt (12.82)

This is an example of (12.81) with no damping, and it is this case that we consider

here. Whenc 1 = 10 , equation (12.80) has the form

(12.83)

where is the angular frequency of the oscillator in the absence of the

field; the quantityν

0

1 = 1 ω

0

22 πis called the natural frequencyof the oscillator.

The general solution of the homogeneous equation corresponding to (12.83),

is (see Section 12.4)

x

h

(t) 1 = 1 d

1

1 cos 1 ω

0

t 1 + 1 d

2

1 sin 1 ω

0

t (12.84)

For the inhomogeneityr(t) 1 = 1 −A 1 cos 1 ωtin (12.83), we have the particular integral

(case 3 in Table 12.1)

x

p

(t) 1 = 1 c 1 cos 1 ωt 1 + 1 d 1 sin 1 ωt

dx

dt

x

2

2

0

2

+=ω 0

ω

0

= km

dx

dt

xA t

2

2

0

2

+=ωωcos

Fm

dx

dt

kx c

dx

dt

==−−+Ft

2

2

0

cosω

m

dx

dt

c

dx

dt

kx F t

2

2

0

++=cosω

3

Euler submitted his solution to the problem of the forced harmonic oscillator to the St Petersburg Academy

of Sciences on March 30, 1739.