1. 10. Harmonic oscillator 10.1. Simple harmonic motion
Harmonic oscillator is an example of periodic motion, where the displacement of
a particle from the origin is a following function of time:
A – amplitude
- phase
φ – phase constant
ω – angular frequency
T – period (time for one cycle), unit: s
f – frequency (No of oscillations per second), unit: hertz (Hz)
Examples of harmonic oscillators:
spring-mass system,
simple pendulum,
physical pendulum,
torsion pendulum
electrical circuit with capacitance and inductance

2. Example Oscillations of a particle are given by equation:
where: [x] = m, [t] = s
What is:
a) the displacement of a particle at t = 2.0 s
b) the velocity at t = 2.0 s
c) the period of oscillations
d) the frequency of oscillations
Figure from HRW,2
(a) The displacemnt x(t),
(b) velocity v(t) = dx/dt and
(c) acceleration a(t) = dv/dt
in a simple harmonic motion.
Phase angle φ = 0.

3. 10.2. Spring – mass oscillator
The spring oscillator consists of a spring with spring constant k and an attached
mass m.
An alternative definition of a simple harmonic motion:
This is the motion of a particle under the influence of
a force proportional to the displacement but opposite
in sign.
The force can be written as
(10.1)
According to the second Newton’s law
(10.2)
and taking into account that one gets
(10.3)
stretched
spring
compressed
spring
Mass m attached to a spring
moves on a frictionless
horizontal floor.
In this way we obtained the differential equation (second order) of motion of a
simple harmonic oscillator: (10.4)

4. Spring oscillator, cont.
Solution of eq.(10.4) can be easily guessed if one takes into account that the second derivative of x(t) must be equal to x(t) with adequate coefficient and the reverse sign. This property is characteristic of the function:
(10.5)
In order to prove this we calculate
(10.5a)
and next
(10.5b)
Substituting (10.5) and (10.5b) into (10.4) we get
(10.6)
Therefore if then function (10.5) is a solution of eq.(10.4). Thus the magnitudes of m and k determine the oscillator frequency or the period:
(10.7)
The constants A and φ depend on the initial conditions.

5. 10.3. The simple pendulum
A simple pendulum consists of a particle of mass m suspended from one end of an unstretchable , masless string of length L.
There are two forces acting on m: gravitational Q and tension R forces. The net force is
(10.8)
which plays the role of a restoring force (acts opposite the displacement to bring the mass back toward its equilibrium position). For small angles θ we make an approximation
The equation of motion for the simple pendulum can then be written as
(10.9)
Comparing (10.9) with (10.4) we see that the constant is equivalent to k and according to (10.7) one obtains
The string makes an angle θ with the vertical. Q = mg is resolved into radial Qr and tangent Qs components.

6. 10.4. The physical pendulum
The physical pendulum is a rigid body suspended from a fixed
pivot point O and oscillating under the influence of gravity.
The analysis of motion is similar to that for a simple pendulum but instead of a mass point we have here the moment of inertia I of a body and instead of restoring force – restoring torque τ.
The restoring torque for small angles can be approximated as
(10.10)
The equation of motion for angle displacements (Chapter 7), taking into account (10.10), can be written as
(10.11)
After some ordering eq.(10.11) takes the form
(10.12)
Figure from HRW,2
The gravity force is applied to the center of mass C placed at a distance h from a pivot point O.
Taking into account the formal similarity of eqs(10.12) and (10.4) one can write
expression for the period of a physical pendulum at small amplitudes, in analogy
to eq.(10.7):

7. 10.5. Energy of an oscillator
The energy of the oscillating object transfers back and forth between kinetic and potential energies.
Kinetic energy
The kinetic energy of an oscillator is associated entirely with a mass of a moving body, then one obtains (for ):
(10.13)
Potential energy
The change in oscillator potential energy is associated with the work of a conservative force (a spring force) and as was discussed in Chapter 4 is
(10.14)

8. Energy of an oscillator, cont.
Total energy
The total (mechanical) energy of an oscillator is a sum of kinetic and potential energies
(10.15) As
one obtains from (10.15)
(10.16)
The total energy of an oscillator is constant,
independent of time and is proportional
to the square of an amplitude.
Every oscillating system contains an element
of springiness storing potential energy and an
element of inertia storing kinetic energy.
Potential, kinetic and total
energies of a linear oscillator
as functions of displacement x.

9. Energy of an oscillator, cont.
Average values of energy
The average value of time dependent function x(t) is defined as follows:
(10.17)
For the periodic function one gets
(10.18)
The average kinetic energy of an oscillator is then equal:
Taking into account that
one obtains

10. Average values of energy
Finally one obtains for the average kinetic energy
(10.19)
The average potential energy is
Taking into account that
one obtains
Accordingly the average potential energy
(10.20)
Thus, for the average total energy one gets
(10.21)
what was expected because the total energy is constant. The average potential and kinetic energies are equal. This is not true for the damped and anharmonic (nonlinear) oscillators.

11. 10.6. Damped harmonic oscillator
Real oscillators are always damped. The damped oscillator shown in the figure consists of a mass m, a spring of constant k and a vane submarged in a liquid. The liquid exerts a damping force which in many cases is proportional to the velocity (with opposite sign):
b – damping constant (10.22)
In this case the equation of motion can be written as
(10.23)
After rearrangement we have
(10.24)
Figure from HRW,2
Introducing the substitutions: one gets
(10.25)
The solution of (10.25) for a small damping is:
(10.26)
where

12. Damped harmonic oscillator, cont.
Solution (10.26) can be regarded as a cosine function with a time dependent amplitude Time t = τ, after which the amplitude decreases e1/2 times is called the average lifetime of oscillations or the time of relaxation.
The angular frequency ω of the damped oscillator is less than that of undamped oscillator ωo. For the small damping, i.e. for ωo>> β, solution (10.26) can be approximated by
(10.27)
Energy losses for the damped oscillator
The amplitude for the damped oscillator decreases exponentially with time.
For the oscillator with a small damping one obtains for the average energy:
(10.28)
The average power of losses is:
(10.29)
Therefore the average power of losses is related to the average energy as
(10.29a)

13. Damped harmonic oscillator, cont.
For the oscillating system with damping one introduces the dimentionless factor, called quality factor Q, defined as follows:
(10.30)
For low damping one obtains from the last equation substituting for power of losses from (10.29a)
(10.31)
Applying the last result to the electrical oscillator L,R,C, and introducing the analogous quantities
one obtains for the quality factor
(10.32)
Examples of quality factors Q
resonance radio circuit several hundreds
violin string 103
microwave resonator 104
excited atom 107

14. 10.7. The driven oscillator with damping
The damped oscillator responds to a periodic driving force. In this case on the right side of eq.(10.24) we introduce the driving force F(t)
(10.33)
Moving support
Substituting
and one obtains from (10.33)
(10.34)
Let the driving force be
Then the solution of eq.(10.34) takes the form
(10.35)
where the amplitude x0 is given by
(10.35a)
and the phase angle φ:
(10.35b)
The system is driven by a moving support that oscillates at an arbitrary angular frequency ω. The natural frequency of a freely oscillating system is ω0.

15. The driven oscillator with damping, cont.
Analysis of solution (10.35).
driving frequency much lower than the natural frequency ω0 (ω << ω0)
In this case:
resonance – the condition at which
the amplitude of a displacement (or velocity, or power) of oscillations is maximum.
For ω = ω0 the amplitude x0 is not generally maximum:
when is minimum.
Amplitude of the driven oscillations as a function of the driving force frequency for varying damping
(b1<b2<b3). Smaller damping gives taller and narrower resonance peak.
Thus the condition for minimum is:
what yields:

16. The driven oscillator with damping, cont.
The velocity of the driven damped oscillations can be calculated by differentiation
of eq.( 10.35):
(10.36)
The amplitude of a velocity is then given by:
From (10.37) it follows that the amplitude of
a velocity is maximum exactly for ω = ω0 .
From the analogy between oscillating mechanical
system and electrical circuit:
one obtains that the maximum
of electrical current amplitude
(resonance) is for ω = ω0.
Amplitude of a current in a resonance electrical circuit for a varying supply voltage frequency.