1 10. Harmonic oscillator 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 A

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 (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. Figure from HRW,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 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) Mass m attached to a spring moves on a frictionless horizontal floor. In this way we obtained the differential equation of motion of a simple harmonic oscillator:(10.4) stretched spring compressed spring

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.

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 Q r and tangent Q s components.

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) 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): Figure from HRW,2

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 (10.13) where 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 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.

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 e 1/2 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 ω o. For the small damping, i.e. for ω o >> 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 circuitseveral hundreds violin string10 3 microwave resonator10 4 excited atom10 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 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 ω 0. 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 x 0 is given by (10.35a) and the phase angle φ: (10.35b)

15 The driven oscillator with damping, cont. Analysis of solution (10.35). a)driving frequency much lower than the natural frequency ω 0 (ω << ω 0 ) In this case: b)resonance – the condition at which the amplitude of a displacement (or velocity, or power) of oscillations is maximum. For ω = ω 0 the amplitude x 0 is not generally maximum: when is minimum. Amplitude of the driven oscillations as a function of the driving force frequency for varying damping (b 1 <b 2 <b 3 ). 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 = ω 0. From the analogy between oscillating mechanical system and electrical circuit: one obtains that the maximum of electrical current amplitude (resonance) is for ω = ω 0. Amplitude of a current in a resonance electrical circuit for a varying supply voltage frequency.