Standing Waves Resonance Natural Frequency LT S6-8

Pendulum Demo... Natural Frequency – the frequency that the pendulum “naturally” swings

All objects have their own different natural frequency.

DAMPING – reduce amplitude Real Life... FRICTION HAPPENS What has happened to the amplitude of our pendulums? amplitude decreased Why? friction with the air This is called damped oscillations or damping.

Tuned mass dampers   Such as the one on the Taipei 101 is a landmark skyscraper located in Xinyi District, Taipei, Taiwan.   Stockbridge damper is a tuned mass damper used to suppress wind-induced vibrations on taut cables, such as overhead power lines

Millennium Bridge in London This pedestrian bridge happened to have a lateral resonant frequency on the order of 1 Hz. So when it started to sway (for whatever reason), people began to walk in phase with it (which is the natural thing to do). This had the effect of driving it more and further increasing the amplitude. Dampers were installed to reduce the amplitude of the oscillations.

Examples of Damping   shock-absorber assembly of a motor vehicle   critical damping in a door closer is achieved by viscous damping inside the piston cylinder actuator of the door.   Dampers on high-tension wires to prevent the wind from swining them to the breaking point.

Advanced Earthquake Resistant Design Techniques   One technique is to isolate the movement   If the building does move, then the motion is damped

Consider a person on a swing... If the frequency of the applied force is equal to the natural frequency. then... Increase in amplitude called... resonance RESONANCE – increase amplitude by forced vibrations

10+ min.

1 min.

Examples of Resonance  Milk Jug & Tuning Fork  Wine Glass The glass was forced to vibrate By the finger moving around the rim Making a loud sound In other words – a large amplitude wave. The air was forced to vibrate By the tuning fork Making a loud sound In other words – a large amplitude wave.

What are the three requirements for resonance? If the frequency of the applied force is equal to the natural frequency. then... Increase in amplitude called... resonance

STANDING WAVES – vibrating naturally Guitar String When a guitar string is plucked, the energy travels down the string. When it reaches the end it reflects back out of phase. If the period of a wave is equal to the amount of time it takes for the wave to travel to a fixed point and back, a standing wave is produced.

Standing Wave

If you apply a force that is a multiple of the natural frequency the standing wave pattern changes.

Wave Machine Demo

Node & Antinodes   Define node. - point where a disturbance caused by two or more waves destructively interfere and result in no displacement   Define antinodes. - point where a disturbance caused by two or more waves constructively interfere and result in maximum displacement   Note: Two antinodes (or two nodes) are separated by one-half wavelength.

2 FIXED ENDS Standing Waves in Strings When both ends are fixed These ends have to be nodes Therefore only certain frequencies will produce standing waves. These are called open.

Examples of Open Standing Waves   Guitar String In the CPO Wave Demo

1 FIXED END & 1 OPEN END Standing Waves in Air Columns When one end is fixed and one end is open The closed ends have to be nodes and the open ends have to be antinodes Therefore only certain frequencies will produce standing waves. These are called closed.

Examples of ClosedStanding Waves   Vibrating Wire (laserdisk) In the Long Air Column

Equations for Resonance in an Air Column This is not a random phenomenon. There is a relationship between the frequency, number of antinodes, wave speed, and the length of the resonator.

Open-pipe resonator Equation: f = nv 2L frequency = (number of antinodes)(wave speed) 2 Length Closed-pipe resonator Equation: f = nv 4L frequency = (number of antinodes)(wave speed) 4 Length open Air Column open closed Air Column fixed String

Keep the same wave speed  and frequency  L change the length  n increasing the number of antinodes In the CPO Wave Demo In the Long Air Column Keep the same length and wave speed  f increasing the frequency is required to make this pattern  n change the number of nodes

6+ min.

The Rubens' Flame Tube: Seeing Sound Through Fire 3+ min.

Fundamental Frequency To count antinodes we start with n = 1 this is called the fundamental frequency or first harmonic. Then we go to n = 2 this is called the first overtone or second harmonic. Then we go to n = 3 this is called the second overtone or third harmonic. Then we go to n = 4 this is called the third overtone or fourth harmonic.

Another way to define natural frequency is frequency at which a standing wave occurs. Fundamental Frequency

Beats Beats are... the slow oscillation in amplitude of a complex wave created when two waves or notes are played together. Equation: f beats = | f 2 – f 1 | Look at Example Problem on page 321

Homework: Standing Waves WS

Extra Info. on Damping   Overdamped (ζ > 1): The system returns (exponentially decays) to equilibrium without oscillating. Larger values of the damping ratio ζ return to equilibrium more slowly.exponentially decays   Critically damped (ζ = 1): The system returns to equilibrium as quickly as possible without oscillating. This is often desired for the damping of systems such as doors.   Underdamped (0 < ζ < 1): The system oscillates (at reduced frequency compared to the undamped case) with the amplitude gradually decreasing to zero.   Undamped (ζ = 0): The system oscillates at its natural resonant frequency (ω o ).