Damped Harmonic Motion

1. Damped Harmonic Motion 4.3.1 State what is meant by damping. • If a mass on the end of a spring is pulled down and released it will continue to oscillate until something retards it’s motion.

2. Damped Harmonic Motion 4.3.1 State what is meant by damping. • Damped harmonic motion is harmonic motion with a frictional or drag force. If the damping is small, we can treat it as an “envelope” that modifies the undamped oscillation.

3. Damped Harmonic Motion • However, if the damping is large, it no longer resembles SHM at all. • A: underdamping: there are a few small oscillations before the oscillator comes to rest.

4. Damped Harmonic Motion • B: critical damping: this is the fastest way to get to equilibrium. • C: overdamping: system comes to rest without oscillating, but does not achieve in the shortest possible time.

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6. Damped Harmonic Motion • There are systems where damping is unwanted, such as clocks and watches. • Then there are systems in which it is wanted, and often needs to be as close to critical damping as possible, such as automobile shock absorbers and earthquake protection for buildings.

7. Damped Harmonic Motion 4.3.2 Describe examples of damped oscillations. Reference should be made to the degree of damping and the importance of critical damping. • Examples • Under-damping – mass on a stiff spring • Critical damping – door closing damper

8. Damped Harmonic Motion 4.3.2 Describe examples of damped oscillations. Reference should be made to the degree of damping and the importance of critical damping. • Damping is caused my dissipative forces, such as air viscosity, and work is taken from the energy of oscillation. • Damping is the process whereby energy is taken from the oscillating system • For example a playground swing • If you push it will oscillate • It will eventually slow down as energy is lost to friction • Energy needs to be supplied to keep it oscillating, that comes from you!

9. Forced Vibrations; Resonance 4.3.3 State what is meant by natural frequency of vibration and forced oscillations. • Forced vibrations occur when there is a periodic driving force. This force may or may not have the same period as the natural frequency of the system. • If the frequency is the same as the natural frequency, the amplitude becomes quite large. This is called resonance.

10. Forced Vibrations; Resonance • The oscillations so far have been free oscillations, or natural oscillations, which the system has been given some energy and left alone. • The frequency of oscillation depends on the inertia and elasticity factors of the system • For example • Guitar string, it will always play the same notes regardless of how hard you pluck it • A child’s swing, it will always swing at the same rate regardless of how hardyou push it • This is called the natural frequency, f0

11. Forced Vibrations; Resonance • Previously the oscillations have been given a single push to start them moving • Often oscillations are subjected to a constant force, called the driving force, f • The effect that the driving for has depends on its frequency

12. Forced Vibrations; Resonance 4.3.4 Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency of vibration. Students should be able to describe qualitatively factors that affect the frequency response and sharpness of the curve. • The damping of the system has these effects: • Amplitude • The amplitude is decreases with damping (cuts down the sharp peak) • The maximum amplitude is at a frequency less than the natural frequency • Energy • The power of the driver is controlled by damping

13. Wine glass demo • The glass can be forced to vibrate at it’s natural resonant frequency. • Resonance occurs when these forced vibrations reach maximum amplitude. • Microwaves – natural resonant frequency of 2450 MHz. • Gallstones, Kidney stones • Tacoma Narrows Bridge, Troops Marching

14. Applications of Resonance 4.3.6 Describe examples of resonance where the effect is useful and where it should be avoided. • Quartz Oscillators -A quartz feels a force if placed in an electric field and will oscillate when removed. -Appropriate electronics are added to generate an oscillating voltage from the mechanical movements of the crystal and this is used to drive the crystal at its own natural frequency. -These devices provide accurate clocks for microprocessor systems. • Musical Instruments -Produce their sounds by arranging for column of air or a string to be driven at its natural frequency, which causes the amplitude of the oscillations to increase.

15. Forced Vibrations; Resonance 4.3.5 State what is meant by resonance. • Resonance occurs when the an oscillator is acted upon by a driving force that has the same frequency as the natural frequency • The driving force easily transfers its energy to the oscillator • From the picture the amplitude of oscillation will become very high • This can be a useful and sometimes very bad

16. Applications of Resonance • Electricity, tuning a radio • The natural frequency of the radio circuit is made equal to the incoming electromagnetic wave by changing its capacitance • The electrons in the circuit will oscillate with the incoming electromagnetic wave. • The electric current will oscillate and this can be turned into sound, through a speaker • Microwave ovens • Microwaves are produced at the same frequency as the natural frequency of water molecules • Water molecules absorb the energy from the microwaves and transfer their energy to the food in the form of thermal energy

17. Applications of Resonance • A Driving force at resonance increases the oscillations, sometimes this is unwanted • Structures • Tacoma Narrows bridge, this bridge was destroyed as the wind (driving force) was at the same as the natural frequency. The bridge vibrated and shook itself apart

18. Applications of Resonance An additional unwanted resonance would be • Tower blocks, the same effect as the bridge the wind, or earthquakes, can cause vibrations to destroy the buildings • Vibrations in machinery, if the driving force equals the natural frequency the amplitude may get dangerously high. Ex. At a particular speed in a truck’s rear view mirror can be seen to vibrate This can be stopped by designing the building with heavy damping • High stiffness • Large mass • Shape • Good at absorbing energy

19. Energy in the Simple Harmonic Oscillator 4.2.1 Describe the interchange between kinetic energy and potential energy during SHM. The simple harmonic motion of a mass on a spring is an example of an energy transformation between potential energy and kinetic energy.

20. Energy in the Simple Harmonic Oscillator Potential Energy 4.2.2 Apply the expressions Ek=1/2 mω2 (xo2-x2) for the kinetic energy of a particle undergoing SHM, ET=1/2 mω2xo2 for the total energy and EP=1/2 mω2x2 for the potential energy. At extension x:

21. Energy in the Simple Harmonic Oscillator Kinetic Energy At extension x:

23. Energy in the Simple Harmonic Oscillator 4.2.3 Solve problems, both graphically and by calculation, involving energy changes during SHM. Total energy Total energy