Resonance. The purpose of this experiment is to observe and evaluate the phenomenon of resonance.

Transcription

1 Resonance Objective: The purpose of this experiment is to observe and evaluate the phenomenon of resonance. Background: Resonance is a wave effect that occurs when an object has a natural frequency that corresponds to an external frequency. The resonance results from a periodic driving force with a frequency equal to one of the natural frequencies of the object. When a stretch string or an object is acted on by a periodic driving force with frequency equal to one of its natural frequencies, the oscillations have large amplitudes. This phenomenon is called resonance and in this case is the maximum energy transferred to the system. Frequencies at which the response amplitude is a relative maximum are known as the system's resonant frequencies, or resonance frequencies. At these frequencies, even small periodic driving forces can produce large amplitude oscillations. Resonance phenomena occur with all types of vibrations or waves: there is mechanical resonance, acoustic resonance, electromagnetic resonance, nuclear magnetic resonance (NMR), electron spin resonance (ESR) and resonance of quantum wave functions. Resonant systems can be used to generate vibrations of a specific frequency (e.g., musical instruments), or pick out specific frequencies from a complex vibration containing many frequencies (e.g., filters). Mechanical resonance is the feature of a mechanical system to absorb more energy when the frequency of its oscillations matches the system's natural frequency of vibration than it does at other frequencies. Each object has its own natural frequency. The resonance phenomena can cause violent swaying motions and even catastrophic failure in improperly constructed structures including bridges, buildings, trains, and aircraft. When designing objects, engineers must ensure the mechanical resonance frequencies of the component parts do 1

2 not match driving vibrational frequencies of motors or other oscillating parts, a phenomenon known as resonance disaster.(see Tacoma bridge event) For example, when an opera singer is emitting a pitch at the same natural frequency as a champagne glass, she can cause the glass to resonate and break. In everyday life we encounter systems that are oscillating. Only ideal systems oscillate indefinitely. In real systems, friction forces retard the motion of these systems. Friction reduces the total energy of the system and the oscillation is reduced or damped. For many oscillating systems, we can say that the force that causes damping is proportional to the speed of the oscillating object: F = bv (1) The negative sign means that the damping force is opposite to the motion of the object. The variable b is the damping coefficient. The damping force is the same as the restoring force of Hooke s Law. This causes the amplitude to decrease exponentially with time: A = A 0 e bt 2m (2) This exponential decrease is shown in the figure below: 2

3 The image shows a system that is underdamped it goes through multiple oscillations before coming to rest. A critically damped system is one that relaxes back to the equilibrium position without oscillating and in minimum time; An over damped system will also not oscillate but is damped so heavily that it takes longer to reach equilibrium. If the driving frequency is close to the natural frequency, the amplitude can become quite large, especially if the damping is small. This is called resonance. Here we are going to focus on resonance like the one shown in the figure above. However our graphs in capstone will not produce the same as the figure. 3

4 Any wave can be described by the following equation: Wave Equation: x(t) = A sin(ωt + φ) + C (3) A is the amplitude of the wave. ω is the angular frequency and t is the time. The last symbol is (φ) is the initial phase representing the value of x at t=0. From the previous lab we know that: angular frequency = ω = 2πf = 2π T Materials: (1) Pasco Scientific Cart Track (1) Pasco Scientific End-Stop (2) Pasco Scientific Dynamics Track Feet ME-9872 (1) Pasco Scientific Dynamics Cart ME-9430 (1) Pasco Scientific Mechanical Oscillator/Driver ME-8750 (1) Pasco Scientific Rotary Motion Sensor CI-6538 (1) Pasco Scientific Dynamics Track Spring Set ME-8999 (1) Pasco Scientific Mass and Hanger Set ME-9348 (1) Pasco Scientific Supper Pulley ME-9448B (1) Pasco Scientific Multi Clamp ME-9507 (1) Small Rod (1) 1.2 meter of string (1) Level 4

5 Procedure: 1) Gather above material. 2) Attach the Track feet to the track 3) Attach the Oscillator/Driver to the end of the track labeled 0cm. Make sure you leave enough room for the bar to rotate 4) Attach the Non-Adjustable End-Stop to same end of the track as the Oscillator/Driver 5) Attach the Multi Clamp to the opposite end of the track (120cm mark). 6) Slide the multi clamp as close to the 120 mark as possible and tighten into place Make sure the butterfly bolt is underneath the track or it will cause problems later in the lab. 7) Insert the rod into the multi clamp so as the rod is vertical. 8) Attach the rotary motion sensor to the rod. As seen in the pictures to the right. 9) Attach the supper pulley next to the multi clamp. 10) The Supper pulley and rotary motion sensor pulley should be in line and close to each other. This step is important to make sure the line does not leave the pulley before the end of the experiment. 11) Insert the yellow plug into digital input one and the black plug into digital input two. It may be best to run the wires from the rotary motion sensor under the track so the wires are not in the way. 12) Level the track (both long ways and sideways) 13) Cut a 120cm length of string. 14) Attach one end of the string to dynamics cart through the hole on top opposite the plunger. 15) Place the dynamics cart on the track with plunger end facing the oscillator driver. 16) Run the string down the center of the track to the motion sensor and pulley. 17) Guide the string under the bottom of the 3 position pulley on the sensor. 18) Guide the string over the top of the super pulley. Do not rush with the string, take your time as it will fall off the pulleys until you start the experiment. Do not get frustrated if it will not stay in place at first as this is normal. The super pulley can be adjusted up or down. Make sure the two pulleys do not touch! 19) Finally connect the string to the mass hanger. 20) Add a 50g mass to the hanger. 5

6 21) Connect the wire leader clip to the small hole on the plastic as seen in the picture to the right 22) Connect the spring clip to the plunger end of the dynamics cart by opening the clip and feeding the opening into the small hole next to the pin. 23) You can rotate the motion sensor on the rod so that the string is in line with the pulleys. You will not be able to perfectly align the string but the closer to alignment the better. See pic for example. 24) Plug in the Drive motor into the 850 interface. 25) Open the folder on your desktop labeled Capstone. 26) Open the Resonance file. 27) Click 28) Slowly increase the voltage in the signal generator by 0.1 volts to find at what voltage the spring starts to experience mechanical resonance. In this lab you will see the cart have a maximum displacement. You can play with the voltage up and down but do not exceed 5 volts. 29) Allow the cart to reach its max amplitude, if any part of the experiment fails or falls apart during this part then click. And reset the experiment. 30) Click after you have achieved resonance or step (29) 31) Click the button and select the last run. 32) Click the icon to scale every graph under all tabs. 33) Click the and select the Max, Min, options. 34) Click the and highlight the wave to just after the largest amplitude wave. 35) Click the 36) Select the Damped Sine Wave: Ae^(Bt)(sin(ωt+φ))+C 37) Repeat steps (31-36) for each tab. Do not select FIT for the last three tabs! 6

7 38) Print all Tabs using Landscape, Use recycled Paper if available. 39) Repeat step (27-38) for a total of three experimental trials. 40) Do steps (41-46) a separate piece of paper 41) Calculate the Amplitude by taking the max plus the min divided by 2 42) Calculate the period of the resonance wave (do not use capstone value) 43) Calculate the wavelength of the resonance wave(do not use capstone value) 44) Calculate the angular frequency of the resonance wave(do not use the capstone value) 45) Calculate the frequency. 46) Repeat steps (41-45) for each graph and each trial. Compare your calculated values with those given by capstone. Find the percent error between your calculation and capstone. Questions: 1) From the different trials that you conducted, how does the resonant frequency found compare with the natural frequency of oscillation? Justify your answer. 2) Can a singer with a perfect pitch really shatter glass? Explain your answer. 3) Soldiers normally march in unison. When approaching a bridge, the soldiers are commanded to break the step (stop marching in unison). Why? 4) Suppose, if you repeated the same experiment with another spring of different spring constant do you observe the same resonant frequency. 7