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.

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.

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.

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

1 min.

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.

10+ min.

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

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

Guitar String 1.) Just sitting there (Newton’s First Law of Motion) 2.) Pluck (applied force) it will vibrate at natural frequency 3.) Energy travels up and down the string 4.) Energy traveling up string will constructively interfere with the energy traveling down the string 5.) Therefore producing a wave pattern with a stationary outline called a standing wave

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 disturbance caused by two or more waves result in no displacement   Define antinodes. - point of maximum displacement of two superimposed waves   Note: Two antinodes (or two nodes) are separated by one-half wavelength.

Standing Waves in STRINGS When both ends are fixed these ends have to be nodes therefore only certain frequencies will produce standing waves.

Standing Waves in AIR COLUMNS Instruments open on BOTH ends Flute Just like strings Instruments open on ONE ends Clarinet Pipe Organ (laserdisk of wire vibrating) These are musical instruments create standing waves in a column of air. Some are closed on both ends some are closed on one end. Closed ends = nodeOpen ends = antinodes

6+ min.

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

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

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:

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 ).