Damped Forced Oscillations Coupled Oscillations 1

The solution for x in the equation of motion of a damped simple harmonic oscillator driven by an external force consists of two terms: a transient term (‘=temporary’) and a steady-state term 2

The transient term dies away with time and is the solution to the equation discussed earlier: Transient Term This contributes the term: x = Ce pt 3 The steady state term describes the behaviour of the oscillator after the transient term has died away. Steady-state Term which decays with time as e -βt

Solutions Complementary Functions are transients Steady State behaviour is decided by the Particular Integral Both terms contribute to the solution initially, but the ultimate behaviour of the oscillator is described by the ‘steady-state term’. It always dies out if there is damping. As a practical matter, it often suffices to know the particular solution. 4

Driven Damped Oscillations: Transient and Steady-state behaviours 5

Forced oscillator: Damped case Time dependent function 6

7 Companion equation: General equation

Try steady-state solution (Particular solution) Complementary function:Transients 8

Consider a general sinusoidal drive force: Equation of motion becomes: The above equation is the real part of simplest complex differential equation: with x = Re(z) Try steady-state solution (Particular solution) (look at the flow of thought…) 9

For now, we write the angular frequency of motion in steady state as  s. A general guess would be: Guess a solution based on physical and mathematical intuition. It is not obvious whether the angular frequency of this motion would be that of: - Oscillation without damping or driving (  o ) - Oscillation with damping, but no driving (  ’) - Drive frequency (  ) - Or some combination of these? Plug into the equation: And obtain: 10

The left side oscillates at  s, while the right side oscillated at . So, if they are to be equal, we must have  s = . i.e., In the steady-state, the oscillator moves with the same angular frequency as the drive force. So, our guess now becomes: Now, solving the equation gives: REAL: IMAGINARY: 11

To isolate |z o |, square these 2 equations and add them to give: To isolate , divide the above 2 equations: Our guess is the solution, if |z o | and  are given as above. Then, the real part of the solution is: 12 REAL: IMAGINARY:

It may appear that the max amplitude appears at  =  o. However, the term is multiplied by the factor 1/ , which increase as  decreases, shifting the peak to a slightly lower value of  o. This point of maximum amplitude is the resonance. So that Writing the amplitude in a different form We also see that the phase  by which the oscillator’s response lags behind the drive force also depends on . When  =  o, then We see that the response amplitude at high frequencies approaches zero. 13 In the opposite limit

Amplitude: 14Forced Damped Osc.-2

Show amplitude resonance at: 15

Amplitude Resonance At ω = ω 0 16

The low and high frequency behaviour are the same as the situation without damping. The changes due to damping are in the vicinity of  =  o. Amplitude is finite throughout. The amplitude is maximum at: Amplitude:

Phase: 18 So that The phase  by which the oscillator’s response lags behind the drive force also depends on . When  =  o, then

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20 For mild damping (β <<  o ),  =  o ( approximately ).

21 - -

Forced oscillations for different resistances 22 In the case of larger resistance, the transients die faster.

Amplitude and phase 23

Different driving amplitudes 24 In the case of larger forcing amplitude, steady state is reached quicker.

Compare with F(t) = F 0 cos ωt The motion of the oscillator is  independent of mass or damping.  stiffness controlled.  in phase with external force. 25

The motion of the oscillator is  independent of stiffness or damping.  mass controlled.  out of phase with external force. 26

The motion of the oscillator is  independent of mass or stiffness.  damping controlled.  of phase lag with external force. 27

Amplitude and Phase - Maximum High frequency and low frequency behaviour unchanged by damping Amplitude is finite throughout Stiffness- Controlled Mass-controlled Resonance :Resonance 28 Summary

Velocity Resonance Show that velocity resonance frequency ω vres = ω 0 and velocity resonance amplitude

Instantaneous power absorbed by the system: Average power over one cycle: 30 (Power drawn by the oscillator from the external force)

We know:

Peak Power: Width at half peak power (FWHM) = 8 2 Maximum average power is absorbed at resonance Using this it can be shown that: From here one can measure the damping parameter: r 2,1 32

Average power absorbed = / 2 = 3 ω0ω0 ω2ω2 ω1ω1 Lorentzian Profile - Seen in situations where we have a resonance - Has a peak at  =  o 33

Resonant quality factor: Central frequency FWHM = Quantifies the sharpness of resonance curves. 34

Stored energy : Problem: What is the peak value and at what driving frequency? 35

Stored energy: = 3 36

Resonance (Interesting Examples) 37

If you push a person on a swing, you must put the energy in at just the natural frequency of the swing or some multiple of it. - Using Resonance to shatter a Kidney stone: By tuning ultra sound waves to the natural frequency of a kidney stone, we can rely on resonance to pulverize the stone. - MRI The motion will build up in amplitude to the point where it is limited by the damping forces on the system. If the damping forces are small, a resonant system can build up to amplitudes large enough to be destructive to the system. 38

Angers Bridge over river Maine in France Angers Bridge was a suspension bridge over the Maine River in Angers, France. The bridge collapsed when 478 French soldiers marched across it in lockstep. Since the soldiers were marching together, they caused the bridge to vibrate and twist from side to side, dislodging an anchoring cable from its concrete mooring. Though a storm also raged during the collapse, engineering experts indicate the collapse was due to the soldiers instead of the storm. 39 Collapsed under weight of marching troops, April 16, 1850, killing more than 200 soldiers.

40 When the Millennium Bridge was opened in 2000, the motion of pedestrians caused it to vibrate, and they fell into step with the vibrations, increasing them. The problem at the Millennium Bridge was corrected during the next two years. Soldiers break step while crossing bridges. Millennium Bridge, London

Tacoma Narrows Bridge The Tacoma Narrows Bridge is a mile-long (1600 meter) suspension bridge with a main span of 850 m (the third-largest in the world when it was first built) that carried Washington State Route 16 across the Tacoma Narrows of Puget Sound from Tacoma to Gig Harbor, Washington (collapsed in wind, 1940) 41

42 Coupled Oscillations

43 Coupled oscillations Waves

Coupled oscillators 44

45 θoθo xoxo k x1x1 θ1θ1 Mass-0 Mass-1 l Coupled pendula

Forces acting on a Simple oscillators 46

Equation of motion SHM Coupling term Considering small-angle approximation Total force on Mass-1 Total force on Mass-0 47 θoθo xoxo k x1x1 θ1θ1 Mass-0 Mass-1 l

Let Natural freq. of each pendulum Adding Subtracting 48

These are coupled differential equations. We have to identify linear combinations of x 0 and x 1 for which the equations become decoupled. In this case, we can easily identify 2 such variables (q 1 and q 2 ): 49

Normal Co-ordinates Normal modes Normal coordinates are coordinates in which the equations of motion take the form of a set of linear differential equations with constant coefficients in which each equation contains only one dependent variable. (Here, simple harmonic equations are in q 1 and q 2 only) A vibration involving only one dependent variable is called a normal mode of vibration and has its own normal frequency. The importance of the normal modes of vibration is that they are entirely independent of each other. The energy associated with a normal mode is never exchanged with another mode; this is why we can add the energies of the separate modes to give the total energy.

C-H Bending modeBreathing mode 51 Normal mode: A way in which the system can move in a steady state, in which all parts of the system move with the same frequency. The parts may have different (zero or negative) amplitudes. C-C Stretching mode

52 Normal modes of Vibration of molecule

Normal frequencies 53 Slow mode Fast mode

Normal mode amplitudes : q 10 and q 20 54

In-phase vibration (Pendulum mode) 55

Out-of-phase vibration (Breathing mode) 56

Initial conditions Coupled Oscillations To simplify discussion, let us choose: 57

Pendulum displacements 58

Behavior with time for individual pendulum 59

Condition for complete energy exchange For x o =0 The masses, M 0 and M 1, have to be equal. And, Else, neither of the two pendulums will ever be stationary 60

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Resonance 62 q1q1 q2q2 +

Normal mode frequencies 63

64

Stiff coupling Slow oscillation will be missing 65

Coupled oscillators 66

Equations of motion 67

Normal modes The two normal modes execute SHO with respective angular frequencies 68

q o represents Centre of mass Normal modes: Physical Interpretation Slow mode 69

q 1 represents Relative coordinate (relative motion of the two masses with the centre of mass unchanged) Fast mode 70

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The normal modes have solutions 73

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i.e., |A o | sin ψ = 0 which gives initial phase ψ = 0 75 Remember

Resonance 76 +

.. 77

x 0 (t) x 1 (t) Coupled oscillations (Resonance) t t TB/2TB/2 TCTC T B = Beats Period 78

79 k’<< k

Connecting two masses with rigid rod 80 k’ >> k