Oscillations and Waves

Topic 4.3 Forced oscillations and resonance

Damped oscillations

• When a system executes true SHM then1. Its period is independent of its amplitude2. Its total energy remains constant in time

• In practice many bodies execute approximate SHM because1. F is not proportional to x2. The energy of the system decreases in time

Total energy E / J

Time / s

Energy of system

Total energy E / J

Time / s

Dissipated energy

State what is meant by damping.• Damping is caused by 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!• Damping involves a force that is always in the opposite

direction to the direction of motion of the oscillating particle and that the force is a dissipative force.

Describe examples of damped oscillations.

Types of damping 1

• Slight damping

• This results in a definite oscillation, but the amplitude decays exponentially

t/s

x/m

Types of damping 2

• Critical damping

• Critical damping is the damping required to make the oscillations stop in the quickest possible time without going past the equilibrium position.

t/s

x/m

T/4

Types of damping 3

• Heavy damping

• Damping force is much greater than the critical damping. The system returns to zero very slowly

t/s

x/m

Very slow return to zero displacement

State what is meant by natural frequency of vibration and forced oscillations

Natural oscillations

• The frequency at which an object tends to vibrate with when hit, struck, plucked, strummed or somehow disturbed is known as the natural frequency of the object.

• 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 fast you push it• This is called the natural frequency, f0

Forced oscillations

• Vibration that takes place under the excitation of external forces is called forced vibration.

• Often oscillations are subjected to a constant force, called the driving force, f

• The effect that the driving force has depends on its frequency

Describe graphically the variation with forced frequency of the amplitude of vibration of an object close to its natural frequency of vibration.

Damping of forced oscillations

Effects of forced oscillations

• The damping of the system has these effects:• Amplitude

– Decreases with damping (cuts down the sharp peak)

– Maximum amplitude is at a frequency less than the natural frequency

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 useful or disruptive

Describe examples of resonance where the effect is useful and where it should be avoided.

Examples of Useful resonance• 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

• Quartz oscillator found in watches, clocks - The crystal strains (expands or contracts) when an electrical voltage is applied. - Oscillation is sustained by taking a voltage signal from the resonator, amplifying it, and feeding it back to the resonator. - The rate of expansion and contraction is the resonance frequency, and is determined by the cut and size of the crystal. - The output frequency of a quartz oscillator is either the fundamental resonance or a multiple of the resonance frequency.

Examples of Useful resonance

• MRI (Magnetic Resonance Imaging)- physicians can make diagnoses without the patient having to undergo either surgery or x rays.- When a patient undergoes MRI, he or she is made to lie down inside a large tube-like chamber.- A technician then activates a powerful magnetic field that, depending on its position, resonates with the frequencies of specific body tissues. It is thus possible to isolate specific cells and analyze them independently, a process that would be virtually impossible otherwise without employing highly invasive procedures.

Examples of Useful resonance

When resonance should be avoided

• Structures like bridges and buildings– Tacoma Narrows bridge, this bridge was

destroyed as the wind (driving force) was at the same natural frequency of the bridge. The bridge vibrated and shook itself apart

– Tower blocks, the same effect as the bridge. Wind or earthquakes can cause vibrations to destroy the buildings

• Vibrations in machinery

FYI

Resonance occurring in buildings can be stopped by designing buildings with heavy damping

– High stiffness– Large mass– Shape– Good at absorbing energy

• Opera singer , why singing can break glass• Resonance is an important part in the design

of musical instruments. Pianos, acoustic guitars, violins use resonance to amplify and shape the sound produced