Design Project #2 – Musical Instrument: Mechanical and Electrical Oscillations ENGR-111 Main Goal of the Project To build a musical instrument and to play a simple tune of your choice. Summary Sounds we hear are due to acoustic waves propagating in air, where the frequency of the acoustic wave determines the sound’s pitch. Humans can hear sounds in the frequency range from about 20 Hz to about 20,000 Hz. Thus, any object (such as a loudspeaker membrane or a beam) vibrating within this frequency range will create an audible sound, and the dominating frequency of vibration will determine the sound’s pitch. Low frequencies correspond to low pitch (e.g., bass guitar) and high frequencies correspond to high pitch (e.g., flute). In this project, mechanical vibration and electrical oscillation will be used in tandem to build an instrument capable of playing a tune with at least six musical notes (the frequencies will be chosen based on the selected tune). Each team will create two notes using electronic circuits and the remaining notes using selected mechanical components (one cantilevered beam, three tubular chimes and three palm pipes). Note that not all components need to be used. In the end, all the teams will perform their tune in front of other students in their section. Schedule Week 1 Specific Goals Project Introduction Learn the project goals, equipment, and underlying physics. Build a simple circuit and demonstrate Ohm’s Law using electronic breadboard. Electrical components Select tunes and identify the required frequencies to be generated. 2 Electronic Oscillation Generate electronic note(s): Build and test electronic oscillators to generate two notes. Observe the output signal on the oscilloscope and measure its frequency. Start working on Week 3. 3&4 Mechanical Vibration Explore mechanical vibration: Set up vibrating components (beams, chimes, palm pipes) and measure the frequency of selected components using piezofilm sensors. Determine the frequency dependence on length. Select the lengths of all components to get the right frequencies. Build the instrument: Combine the electronic and mechanical notes into one instrument. Music Machine 4 Finishing up and Performance Finishing up: Finish building the instrument. Practice playing the tune. Perform in front of other students. 1 Note: This document has two parts: “Schedule and Instructions” below and “Reference” at the end. Read entire document before working on detailed tasks. SCHEDULE AND INSTRUCTIONS Week 1: Project Introduction The aims of this week are to:         Form teams Understand the project goals thoroughly Understand the fundamentals of mechanical vibration, oscillating electrical circuits, natural frequency, and the relation between sound frequency and notes in a musical scale Introduce the available electronic and mechanical devices that can make musical notes Become familiar with all electronics and instrumentation used in the project such as power supply, multi-meter, and oscilloscope. This also includes brief demonstration of electronic components: resistors (including variable resistors, or potentiometers), capacitors, integrated circuits, etc. Introduce breadboard. Build a simple circuit on a breadboard to demonstrate Ohm’s Law. Identify all electrical components available to the teams in their project bins. Select a tune and identify the required frequencies of oscillation Introductory Activity: Building a simple circuit and demonstrate the Ohm’s Law. 1. Become familiar with the power supply (Figure 15 in the Reference), multi-meter, resistors, variable resistors (potentiometers) and electronic breadboard (Figure 18 in the Reference). 2. Using a breadboard, assemble one resistor and one potentiometer in series, as shown in Figure 1. Do not connect the power supply or the meters yet. 3. Using a multi-meter, measure the actual resistance R1 (it should be close to the nominal 1 k value). 4. Adjust the resistance R2 to the lowest possible (practically zero). Also measure resistance between nodes 1 and 3, which should be R1-3 = R1 + R2 ≈ R1 5. Connect the power supply and set it to 5V DC. 6. Measure current I, voltage V1-2 across resistor R1 and voltage V2-3 across resistor R2 7. Repeat steps 4, 5 and 6 for four more values of resistance R2. The second value should be R2 = R1 (measured in step 3). The last value should be the maximum resistance (about 10 k). 8. For each measurement, calculate and record the product R1-3 * I Figure 1 Circuit to investigate Ohm’s Law: Measuring current (top) and voltage (bottom) 2 Constant Values R1 = V1-3 = 5 (V) (k) Table 1: Ohms Law Variable Meas. 1 Meas. 2 Meas. 3 Meas. 4 Meas. 5 Values 0 R2 (k) R1-3 (k) I (mA) V1-2 (V) V2-3 (V) Core Activity: Selecting tunes and note frequencies Your instrument must generate between 6 and 8 notes. Play a tune at the end of the project. 1. Determine all frequencies needed to play the tune. 2. Decide which frequency will be generated using what method and record the frequencies in Table 2. Assign lower frequencies to mechanical components and higher frequencies to electronics. Table 2 Selected frequencies and methods Note # Note Frequency Method Name (Hz) (electronic or mechanical) 1 2 3 4 5 6 7 8 Table 3 lists the frequencies of musical notes from middle C to C’. The frequencies are based on the so called A 440 scale (440 Hz corresponding to the A note), which is the most widely used1 in music. Table 3 Frequencies of musical notes Note: If your tune contains notes outside the one octave shown, you Musical Frequency will need to research your frequencies. Note (Hz) C 261.6 C# 277.2 HW Assignment 1 – Graph Ohm’s Law results. Calculate theoretical D 293.7 value of resistance R2 for Week 2. – See Project 2 HW 1 in My D# 311.1 Assignments for details. E 329.6 F 349.2 F# 370.0 G 392.0 G# 415.3 A 440.0 A# 466.2 B 493.9 C 523.3 1 For physics experiments a “scientific” scale is often used, in which all C tones have power of 2 frequencies (for example 256 Hz or 512 Hz) 3 Week 2: Electronic Oscillation The goal of this week is for each team to build and test two electronic oscillators to generate the two assigned notes. The instruction below is for a single note. Core Activity: Generation one musical note with square wave oscillations Each note will require building an oscillator module based on the 555 timer (Integrated Circuit, IC) from individual components and using an electronics ‘breadboard’. The diagram of the oscillator module is shown in Figure 2. Theoretical value of resistance R2 was calculated in Week 1 HW Assignment. Building the Oscillator: 1. Collect all the components needed based on Figure 2. Note that R2 is a 50 kpotentiometer (different from the 10 k potentiometer used in the previous activity). 2. Set multimeter to resistance mode and adjust the variable resistor R2 to the value calculated in the Week 1 HW Assignment. Make sure to document whether the required Figure 2 Oscillator based on the 555 resistance is set from Pin 1 to Pin 2 or Pin 2 to Pin 3 on the timer (produces a square wave) variable resistor. It may be easier to place the variable resistor in the breadboard to complete this task. 3. Build the oscillator circuit (Figure 2) by placing all components on the left half of a breadboard. 4. Set up the DC power supply. Follow the dc power supply guide in the Reference section. 5. Connect the Ground (GND) and +5V leads from power supply to the breadboard. 6. Turn on the power supply and set the voltage on both voltage dials to +5V Confirming the output signal 7. Connect the oscilloscope ground clip to ground and the test clip to Pin 3 of the 555 timer. 8. Verify that a square wave similar to Figure 12 can be seen on the oscilloscope. Debug the connections, if needed. 9. Set the oscilloscope to frequency measurement and measure the signal frequency 10. If needed, adjust the potentiometer R2 until the frequency matches your target frequency Core Activity: Generating two notes from the tune and connecting to a speaker 1. Add second (virtually identical) oscillator to the breadboard 2. Add one headphone jack 3. Add two switches 4. Connect the speaker 5. Turn power supply on 6. Play two notes 7. Tune each of the notes using oscilloscope 8. Verify each frequency using electronic tuning device. Figure 3 Electronic module for generating two notes 4 HW Assignment 2 – Create theoretical curves f1(L) for Week 3. Calculate target lengths of the vibrating components based on theoretical calculations.—See Project 2 HW 2 in My Assignments. Optional Extension Activity: Cleaner sounds – converting square waves into sine waves While the square wave can deliver a sound with a dominating pitch corresponding to the square wave frequency, the signal also contains higher frequency harmonics, resulting in undesired ‘noise.’ In order to obtain cleaner sounds, these higher frequency harmonics need to be filtered out. Figure 4 shows one relatively simple system which can be used to accomplish this goal. It is a combination of a buffer amplifier (right side of Figure 4) and low pass filter module (left side). Note that a single Integrated Circuit (IC), a TL 082, provides two Op Amps needed for the two modules. The resistance and capacitance values were calculated in Week 2 HW Assignment. Figure 4 Combined LPF and Buffer Amplifier. Converts square wave to sine wave. Building the Oscillator with Filtering: 1. Collect all the components needed based on the values determined in the Week 2 HW Assignment. a. Standard resistors closest to the calculated values of R1 and R2 should be used b. C1 = 2C2 can be formed by using two capacitors C2 in parallel 2. Build the circuit. Place all components on right half of a breadboard 3. Connect with the output signal from the Oscillator 4. Connect the Ground (GND), +5V and -5 V leads from power supply 5. Turn on the power supply and set the voltage to +5V and -5V Confirming the output signal 6. 7. 8. 9. Connect the oscilloscope ground clip to GND and the test lead to Out 1 pin of the TL082 IC (Pin 1) Verify that the square wave has now become a sine wave. Debug the connections, if needed Set the oscilloscope to frequency measurement and confirm the frequency of the output signal. If needed, adjust the potentiometer R2 of the Oscillator (Figure 2) until the frequency matches your target frequency 5 Weeks 3 and 4: Mechanical Vibration Mechanical tones will be generated using any combination of the three methods: 1. Striking a steel cantilever beam and making it vibrate 2. Striking a tubular chime (made of either steel or copper) and making it vibrate 3. Striking a PVC palm pipe against the palm of your hand and making the air inside vibrate The vibrations of cantilevered beams or tubular chimes will be studied by using piezoelectric sensors. By placing double-stick tape on the underside of a sensor, the sensor can be attached (and easily removed) anywhere on the surface of the vibrating element. The main goals for this week are to observe free vibration of cantilevered beam and/or tubular chimes and to determine the dependence of the fundamental frequency of vibration on the lengths of vibrating element. Core Activity: Measurement of the fundamental frequency for various lengths of vibrating elements Select either a cantilevered beam or tubular chimes for this investigation Instructions for cantilevered beam: 1. Clamp the beam with the cross plates to the table edge using a large C-clamp such that the long axis extends past the edge of the table at a 90○ angle out from the table (see Figure 5). Start with a length of about 4.0 inches. Make sure that the clamp is tightened well. 2. Attach the sensor to the beam using double sided tape. Orient the sensor so that is mid-width of the beam and close to the point where the beam is clamped to the table. The long direction of the sensor should follow the length of the beam (see Figure 5). a. Be sure that the leads to the sensor are not clamped or are in the way of the vibrating beam. Also make sure any exposed metal on the sensor leads does not short out by touching the beam or clamp. 3. Turn on the oscilloscope, being sure that the scale factor used by the oscilloscope channels and the scale factor Figure 5 Clamped beam with sensor imprinted on the probes are the same, i.e. “X1” or “X10” or “X100” for the channel you wish to use. If this step is skipped the oscilloscope scale (as shown on the screen) will be off from the actual value by a factor of 10 or 100. 4. Pluck the free end of the beam and view the response on the oscilloscope. You may need to adjust the horizontal (time/div) and vertical (volts/div) to ensure a full signal is on the screen. Approximate settings for these are voltage: 100mV/div and time: 10ms/div. Determine the fundamental frequency f1 by “freezing” the oscilloscope trace and measuring the distance between signal peaks using vertical cursors on the oscilloscope. a. “Freezing” is done by pressing the Stop button shortly after plucking the beam. i. Cursors are turned on by pressing the Cursors button and choosing vertical bars. You can move the cursors using the knob to the left of the cursor button. You can choose which cursor to move by using the select button to the left of this knob. Horizontal distance between the two cursors is displayed automatically, and is defined as the period T of oscillation. b. Calculate the fundamental frequency as: 1 i. 𝑓1 = 𝑇 6 c. Sketch the “frozen” trace and neatly record, t1, t2, and T (see Figure 7 in the Reference section). 5. Determine the fundamental frequencies f1 three times for each length investigated. Select at least four lengths between 2.5 and 4 inches. Record all results in Table 4 and calculate the averages. Table 4 Vibration of cantilevered beam – quantities to be measured. Beam Length L ____inch ____inch ____inch ____inch ____inch Fundamental Freq. f1 (Hz) Measurement 1 Fundamental Freq. f1 (Hz) Measurement 2 Fundamental Freq. f1 (Hz) Measurement 3 Average Freq. f1 (Hz) 6. Using averaged results in Table 4 determine the beam length which will result in vibration at your target frequency (as listed in Table ). You can determine L by either linear interpolation or careful graphing of the data in Table 4. 7. Set the target beam length L and confirm the vibrating frequency with an oscilloscope. Instructions for tubular chimes: One set of tubular chimes of various lengths will be provided per section (you will need to share them with other teams). Perform the following steps for at least four chimes: 1. Measure its length 2. Attach the piezo sensor using double sided tape. Position the sensor near the half-length point with the long direction of the sensor following the length of the chime 3. Hang the chime using a zip tie positioned about 20% of the length from the top 4. Perform a similar procedure as described above for cantilever beams (steps 3 through 5). 5. Record all results in Table 5 below. Table 5 Vibration of tubular chime – quantities to be measured. Chime Length L ____inch ____inch ____inch ____inch ____inch Fundamental Freq. f1 (Hz) Measurement 1 Fundamental Freq. f1 (Hz) Measurement 2 Fundamental Freq. f1 (Hz) Measurement 3 Average Freq. f1 (Hz) 6. Using averaged results in Table 5 determine the chime length which will result in vibration at your target frequency (as listed in Table 2). You can determine L by either linear interpolation or careful graphing of the data in Table 5. 7 Core Activity: Building the instrument 1. 2. 3. 4. Make all mechanical tones listed in Table 2 using selected mechanical elements. Combine the electronic oscillators (from Week 2) and the mechanical tones into one ‘instrument’. Fine tune the frequencies, if needed. Electronic tuner will be available. Practice playing tunes for the performance. Week 3 HW Assignment -- Graph f(L) for both theoretical and experimental values for either cantilevered beams or tubular chimes. Add a trendline to experimental data. Compare with theoretical values. Determine the target value of L. See Project 1 HW 3 in My Assignments. Week 4: Performance – playing the tune Each team will perform their tune in front of other students in their section. You are allowed to bring additional instruments as long at least 6 notes are from the instrument built in this project. Competition Rules:   The tune must have at least 25 notes, total You can bring additional instruments, but the built instrument must dominate Optional (decided by your instructor)   Instant scoring by all people in the audience Prizes for the winning band HW Assignment 3 – Each student will write an Executive Summary of the project. See Project 2 Executive Report assignment in My Assignments. 8 REFERENCE Vibration of Cantilevered Beams A cantilevered beam is a beam fixed (cemented, clamped) at one end with the other end remaining free to move. This type of beam is very common in structures and machines, e.g. airplane wings, helicopter blades, beams supporting traffic lights. Even sky scrapers can be viewed as cantilevered beams. When a free end of a cantilevered beam is displaced and then released, the beam will start to oscillate back and forth. From the energy perspective, this oscillation manifests cyclic conversion from elastic (spring) energy to kinetic energy and back. Because of various energy losses in the system, the amplitude of the oscillation becomes smaller with time, and eventually stops. Cantilevered beam can vibrate in many modes, each with the associated so called natural frequency of vibration. The modes of vibration are commonly ordered Figure 6 First mode (top) and second mode of vibration of from the lowest frequency to the cantilevered beam. Right end free to move. highest, and the frequency associated with the first mode is called the fundamental frequency. Figure 6 present the first two modal shapes. The fundamental frequency of the cantilevered beam can be calculated by using (Eq. 1) below. (Eq. 1) 𝑓1 = 𝐾1 2𝜋 𝐿 2 𝐸𝐼 √ 𝜌𝐴 Where f1 represents the fundamental frequency; E and  are Young’s Modulus and density for the beam material, respectively; A and I are cross-sectional area and its moment of inertia, respectively; K1 is a constant associated with the fundamental frequency; and L is the length of the beam. (One can think of Young’s modulus as a ‘spring constant’ of the beam material F = kx where F is the spring force, k is the spring constant and x is the distance stretched.) The cantilevered beam used in this project is made of steel known as ANSI 1095 blue tempered spring steel (carbon steel with about 1% of Carbon and about 0.5% of Manganese). The beam dimensions are 6.6” x 1.0” x 0.062” (L x b x d). Table 6 lists the values for the constants needed in (Eq. 1) for the vibrating beams used in this project (1095 spring steel). Table 6 Physical Constants Used for Vibrating Beam Quantity Symbol Value Units 1 f1 Natural Frequency (Eq. 1) Hz K1 Constant 3.5156 unitless E Young’s Modulus 2.0 × 105 MPa d Beam Thickness 0.062 in b Beam Width 1.00 in I Moment of Inertia I= bd3/12 m4 A Cross-sectional area A= bd m2 Density of 1095 Steel 7,900 kg/m3  2 L Beam Length Adjust in 1. f1 represents the fundamental frequency 2. Beam length is measured from the free end to where the clamp is placed. 9 Figure 7 shows a typical oscilloscope trace generated by a vibrating beam. As one can see, the amplitude of the voltage signal decreases as time elapses after the cantilevered beam has been set in motion. Figure 7 Decaying amplitude of vibrations This is because the energy introduced into the system is absorbed by the surrounding hardware, table surface, table supports, and like structures. The waveform decreases in a logarithmic manner. The parameter that governs how fast this occurs is the time constant τ. This time constant can be found using (Eq. 2) given below using the values measured from the oscilloscope trace. (Eq. 2)   T / ln(V ' / V ) Where T is the period taken from one peak to the next, V is the amplitude of the first peak, and V’ is the amplitude of the succeeding peak. The “ln” stands for the natural logarithm. The negative sign appears since the ratio V’/V is less than unity which means that ln(V’/V) < 0. The time constant, τ is an important parameter in that in practice approximately 4 time constants (4τ) are required for the system to come to rest, i.e. settle down. Vibration of Tubular Chimes Vibration of a tubular chime is more complex than the one of a simple cantilevered beam. However, the dominating mode can be studied with vibrating beam theory. In this case, none of the beam ends is supported, (free-free configuration) resulting in different modes of vibration than for cantilevered beam. The first two modes are presented in Figure 8. Figure 8 First two modes of vibration of tubular chimes 10 Interestingly, the fundamental frequency of this free-free oscillation can be determined by the same equation as before (Eq. 1): 𝑓1 = (Eq. 1) 𝐾1 2𝜋 𝐿 2 𝐸𝐼 √ 𝜌𝐴 The only differences are that the constant K1 has a different value for this configuration, and that the material properties (E, ) and the properties of the cross sectional area (I, A) are different (see Table 7). There are two kinds of tubing available for tubular chimes in this project: steel and copper. The steel tube is referred to as ½ inch nominal size electrical metallic tubing, or EMT. It has the inside diameter (ID) of 0.622” and the outside diameter (OD) of 0.706". The copper tube is referred to as ½ inch nominal size Type M copper tubing , with 0.569" ID and 0.625" OD. Table 7 lists the values for the constants needed in (Eq. 1) for the tubular chimes used in this project. Table 7 Physical Constants Used for Tubular Chimes Quantity Symbol Quantity f1 Natural Frequency1 (Eq. 1) K1 Constant 22.373 E Young’s Modulus, Steel 2.0 × 105 E Young’s Modulus, Copper 1.1 × 105 ID Tube ID See text above OD Tube OD See text above I Moment of Inertia I= (OD4- ID4)/64 A Cross-sectional area A= (OD2- ID2)/4 Density, Steel 7,900  Density, Copper 8,960  L Beam Length Variable Units Hz unitless MPa MPa in in m4 m2 kg/m3 kg/m3 in 1. f1 represents the fundamental frequency in bending mode Vibration of Air in Palm Pipes A palm pipe is simply a short rigid tube. When one of the ends is stroke against the palm of your hand, the air in the tube gets excited and oscillates longitudinally. An excellent applet showing the longitudinal waves in a pipe (written by Walter Fendt) can be find here: http://www.walterfendt.de/html5/phen/standinglongitudinalwaves_en.htm Study the applet and try different lengths and modes of vibration. You can observe that when one of the sides is Figure 9 Longitudinal waves applet closed, the wavelength of the fundamental mode equals four times the tube length:  = 4 L. The period of this oscillation is T =  /c , where c is the speed at which the disturbance travels, also known as the speed of sound, which at room temperature is about 344 m/s. Recalling that f = 1/T, the oscillation frequency can be found as: f = c / = c /(4L) = 0.25 c /L (Eq. 3) You can observe that (Eq. 3) does not depend on the tube diameter. A more precise empirical equation (based on experiments with ½” Schedule 40 PVC tubes and accounting for pipe end effects) is: (Eq. 4) f = 0.23 c /L + 24 (Hz) 11 Complete Musical Instrument (electronic tones) Figure 10 presents block diagram of the electronic part of the musical instrument generating two tones. Examining the block diagram, one can see that there will be two oscillators, one for each electronic note selected by your team. Figure 10 Block Diagram of the electronic circuit generating two tones. Each component is described in more detail below. Note that an optional low pass filter (not shown) can be added between the output pin 3 and the switches to remove unwanted harmonics and convert square wave signal into a sine wave. Oscillator Figure 11 shows the connections needed for the NE 555 Timer to produce a series of output pulses as was shown in Figure 12. The NE 555 integrated circuit (abbreviated as an “IC”) produces a sequence of pulses with an output frequency set by a capacitor and two resistors. The frequency of pulses produced by the NE 555 Timer is given as: (Eq. 5) f= 1.46 (R1 +2R2 )C The 10nF capacitor shown in Figure 11 is optional, i.e. may be left open (no connection). The pin out of the NE555 Timer is shown in Figure 17. The timer, as the name suggests, is used as a low frequency clock, for instance in orchestrating the processing of a synchronous logic circuit. 12 Figure 11 Oscillator Using a 555 Timer. The 10nF capacitor is optional. To obtain 50% duty cycle (same time low signal as high) R1 should be much less than R2. It is recommended that R1 = 1kΩ and C = 0.1 F. Then R2 can be calculated and accurately set by using a potentiometer. The pulse train generated by the oscillator is shown in Figure 12 where the x-axis represents time and the y-axis represents voltage. Figure 12 A Pulse Train (In our case τ = T/2.) One can see that the ‘square wave’ signal differs from a single-frequency sinusoidal signal, and therefore the generated sound would not be clear. Some additional signal processing is needed to remove the unwanted frequencies Low Pass Filter (optional): The filter we can use to remove much of the unwanted frequencies (but not all) is a Sallen-Key Low Pass Filter (LPF). The LPF filter allows low frequencies to pass through and higher frequencies to be attenuated. Figure 13 illustrates the basic Sallen-Key LPF. We note that this form of a LPF is a very simple design requiring only two resistors and two capacitors along with an operational amplifier (or op amp, represented by the triangle in Figure 13). To design a Sallen-Key LPF, one must first determine what maximum frequency will be allowed to pass through. This is called the break frequency. Although the DC supplies are not shown, they must be included. Refer to Figure 4 for more detail. The resistors R1 and R2 are the same and equal to R. The break frequency, fb (for this project the fundamental frequency for the musical note) of the low pass filter can be calculated using the relationship below. 13 fb  (Eq. 6) 1 2R C1C2 Where C1 = 2C2 and R1 = R2 =R. As an example, by using C2 = 0.1μF (a standard size), then C1 = 0.2μF (which can be formed by placing two 0.1μF in parallel). With fb = 256 Hz, a musical C note, we calculate that R = 4398 Ω. Since this is not a standard size, choose the closest available value. Figure 13 Sallen-Key Low Pass Filter Buffer Amplifier: The function of a buffer amplifier (in this case referred to as a unity gain amplifier) is to minimize the problem of loading. Loading degrades the operation of the circuit due to the electrical characteristics of subsequent stages. The buffer acts to isolate one stage from its downstream neighbors. Figure 14 illustrates the circuit connections of the buffer amplifier. The pin out for the Op Amp used in this circuit is found in Figure 16. Remember that the DC voltages (+VCC and –VCC) are not shown in Figure 14 for simplicity, but must be included in practice. Refer to Figure 4 for more detail. Figure 14 Buffer Amplifier Using an Op Amp 14 Major Components and Equipment Piezofilm: The piezoelectric sensor that we will use is made of a special polymer, named polyvinylidene fluoride or PVDF for short. The material is heated and then rolled to form thin sheets. The material is allowed to cool in the presence of a strong electric field. The cooling “freezes” the alignment of the molecules in the crystalline structure in such a way that the net charge of the polymer is zero. The upper and lower surfaces are then metalized by a deposition process using silver ink. Cutting the metalized sheets into appropriate sizes, attaching electrical terminals with twisted-pair leads and covering the surfaces with a clear plastic coating completes the fabrication process of the sensor. Although the resulting piezofilm can be employed in various ways, for instance to measure temperature (pyro-electronic property) or to drive a speaker (electro-mechanical property), the mechano-electric property will be used in this experiment. A mechano-electric transducer transforms the energy of mechanical deformation into electrical energy. In its natural state, the symmetry of the charge distribution within the crystal leads to a cancellation of the net charge. However, when the piezofilm is mechanically deformed, the molecular arrangement is no longer symmetric in that local redistribution of alignments has occurred. A net charge is the result of the deformation. This gives rise to an electric field. This in turn is detected by an oscilloscope as a voltage. Multimeter: A multimeter is used to measure different electrical values of parts. For our purposes, it will be used to measure resistors (ohms) and capacitors (farads). To measure a component you must first have the probes in the correct holes. The black probe is always used in the COM. The red probe however is moved depending on whether you are measuring resistance or capacitance. For resistance use Ω and for capacitance use Cx. You then must set the dial to the correct position and value for resistance or capacitance. DC Power Supply: The DC Power Supply provides us with a constant -5volt +5volt and ground connections for our circuit. To set up the supply in this manner, place a wire from the + of the first set of probes to the – of the second set of probes (see Figure 15). Then make sure the tracking buttons are both in the out position to be set to independent. Lastly set the current knobs to 12 o’clock and the voltage knobs so both displays read 5v. The connections can be seen in Figure 15 below. Oscilloscope: This electrical measurement instrument can be thought of as a dynamic voltmeter, which can measure voltages as they evolve in time. The input to the oscilloscope is obtained by attaching probes that extend from the oscilloscope to the circuit. When the oscilloscope is turned on, one or two traces (depending on what option is selected) appear on the screen. By using the switches and control features of the oscilloscope, one can determine the frequency and amplitude of a waveform displayed on the screen. The “Run/Stop” buttons on the oscilloscope can be used to save the waveform on the screen at any given instant of time. One can determine the frequency and amplitude of an electrical system by using the cursors located on the oscilloscope. In the Electrical Engineering laboratories there are examples of both analog and digital oscilloscopes. The manner in which an oscilloscope is used is described in Wikipedia under the name “Oscilloscope”. It provides an overview of the console controls and the manner in which the oscilloscope probes are used. Additionally, videos are available through YouTube which demonstrates how the oscilloscope is set up to take measurements. Specifically, the video entitled, “The Oscilloscope: A Beginners Guide to the Oscilloscope” which was produced by Berkeley Engineering is especially helpful since it uses a similar scope to that available for this project 15 Figure 15 DC Power Supply Setup Op Amp: The major component in the buffer amplifier and LPF is an operational amplifier (op amp for short). It is an example of an integrated circuit (IC). The plastic package (in this case a “mini” DIP 8 pin, dual-in-line package) has eight pins. The “chip” is encased inside the black plastic package and connected to the surrounding circuit via metal pins. Looking down on the pins from above, the pins are numbered sequentially counterclockwise around the periphery of the IC from 1 to 8. A notch or imprint is provided on the top of the IC which indicates the location of pin 1. Each pin has a specific purpose. For example, pins 4 and 8 are reserved for the DC (constant voltage) inputs that serve to power the IC. The remaining pins are used according to the application. We will be using the Texas Instruments TL082 Op Amp. Refer to the Figure 16 below. Notice that there are two identical but separate op amps. The DIP 8 package is also shown. The result of connecting the proper pins together is an amplifier. An amplifier simply takes a voltage input and produces a larger version of this waveform at the output. A helpful YouTube short entitled 741 Op Amp Demo (the uA741 is similar to the TL082 but contains only one Op Amp.) by All American Five Radio provides additional technical details. Figure 16 TL082 Op Amp Pin Out and DIP-8 Plastic Package 16 Timer: The numbering scheme as described for the op amp is the same (but not the labels for each pin) that is used for the Timer. Although the packaging is identical, the timer does not amplify, but rather provide a series of pulses at the output that are 5.0 V in amplitude with a frequency that has been selected by including an appropriate capacitor and several resistors. We will be using the NE 555 Timer. Refer to Figure 17 below for the pin out and a view of the plastic package. The names for the terminals are ground (GND), trigger (TRIG), output (OUT), reset (RESET), discharge (DIS), threshold (THR), and a positive 5.0 V DC voltage source goes to pin (VCC). Figure 17 NE 555 Timer Op Amp Pin Out and DIP-8 Plastic Package Breadboard: Figure 18 shows a photo and a schematic of electrical connections in an electronics breadboard. The breadboard allows building electrical circuits without soldering. Figure 18 Photo and schematic of a breadboard. Lines in the schematic indicate electrical connections inside the breadboard. Note the placement of NE555 integrated circuit. Speakers: The speaker is an electro-mechanical device (a transducer) that converts electrical signals into mechanical vibrations that are in the audible range. A coil of wire is wrapped around the pointed end of a flexible, fabric cone. As the coil of wire receives current, a magnetic field is produced in proportion to the direction and the magnitude of the current. This field interacts with the magnetic field of a permanent magnet that is housed with the coil. This interaction creates forces on the cone which generate vibrations that are eventually perceived as sounds by the human ear. The speakers that we will be using are the XBOOM Mini, 3 W output portable speakers. 17 Musical machine Cantilevered beams Theoretical values are as per the A 440 scale. Note C Representation 1 in the graph C# 2 D 3 D# 4 E 5 F 6 F# 7 G 8 G# 9 A 10 B 11 C 12 The graph shows the frequencies corresponding to the Musical notes. Theoritical values Series 1 Linear (Series 1) 600 500 Frequency 400 300 200 100 0 0 2 4 6 8 10 12 14 Notes We can see in the graph that the frequencies go higher as we progress in the scale. Beam length (L) Fundamental freq. F1(Hz) Measurement 1 Fundamental freq. F2(Hz) Measurement 2 Fundamental freq. F3(Hz) Measurement 3 Average Freq. (Hz) 5.5 inch 62.60 4 inch 69.00 3 inch 185 2.5 inch 287.00 2inch 909.00 63.66 86.10 183 250.00 163.00 67.92 90.10 185 251.00 157.00 65.00 80.00 184 262.60 409.66 The graph bellow shows the frequencies produced with certain length of the beam. Beam Lenth L vs Avarage Frequency(Hz) 6 Beam Lenght (L) 5 4 3 2 1 0 0 50 100 150 200 250 300 350 400 Frequency (Hz) Figure 1 Experimental Gradient of the graph = (65, 4.5 and 400, 1.6) (4.5-1.6) / (65-400) = 2.9/-335 M=-0.0086567 Y = mx + c Y = beam length M= -0.0086567 X= frequency C = 5.1 Equation Y= -0.0086567X + 5.1 The Beam length can therefore be determined by extrapolating in the graph of using the equation to find the value of length for certain frequencies. Group A: Team A and Team B Happy 6 Notes (6 Frequencies) Needed: Musical Note E A D Frequency (Hz) 329.0 440.0 587.4 Beam length L (inch) 2.3 1.29 0.02 450 Tubular chimes Theoritical values Series 1 Linear (Series 1) 600 500 Frequency 400 300 200 100 0 0 2 4 6 8 10 12 14 Notes Chime length (L) Fundamental freq. F1(Hz) Measurement 1 Fundamental freq. F1(Hz) Measurement 2 Fundamental freq. F1(Hz) Measurement 3 Average Freq. (Hz) 16 inch 59.74 24 inch 71.93 31 inch 58.96 29 inch 60.34 20 inch 32.91 19.20 73.45 90.80 63.13 32.68 32.68 57.37 65.46 63.97 30.53 37.20 67.58 71.74 62.48 32.04 02 35 30 Chime Lenth (L) 25 20 15 10 5 0 0 10 20 30 40 50 60 70 Frequency (Hz) Figure 2 Experimental Coordinates (60, 30) and (30, 22) The gradient of the graph can be found as: 30-22/60-30 = 0.267 M=0.267 Y = 0.267X + 14 Group A: Team A and Team B Happy 6 Notes (6 Frequencies) Needed: Musical Note E A D Frequency (Hz) 329.0 440.0 587.4 Chime length L (inch) 97.84 127.48 166.83 80 0 1.000 4,94E-03 5 0 1 5 6 5 V1-2 (V) R2 (Ω) R1-3 (Ω) I (A) V1-2 (V) V2-3 (V) R1 (kΩ) V1-3 (V) Meas, 2 Meas, 3 Meas, 4 Meas, 5 1000 4000 8000 10000 2080 5270 8850 11400 2,55E-03 9,60E-04 5,60E-04 4,40E-04 2,54 0,94 0,99 0,99 2,43 4,03 0,88 10,15 1 1 1 1 5 5 5 5 4 3 2 1 0 0,E+00 1,E-03 V1-2 vs I graph removing out 6 5 Power (V) Meas, 1 4 3 2 1 0 0,E+00 1,E-03 V1-2 vs I graph I vs R graph y = 918,86x + 0,3553 R² = 0,9845 6,E-03 Intensity (A) 5,E-03 4,E-03 3,E-03 2,E-03 1,E-03 0,E+00 1,E-03 2,E-03 3,E-03 4,E-03 5,E-03 6,E-03 y = 1020,8x - 0,0487 R² = 1 2 vs I graph removing outliers 2,E-03 3,E-03 Intensity (A) 2000 4000 6000 Resistance (Ohm) I (A) 1,E-03 0 4,E-03 5,E-03 6,E-03 I vs R graph R2 R1-3 6000 8000 Resistance (Ohm) 10000 12000 R1 (kΩ) 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 Realistic oscillators R2 (kΩ) C (μF) 1 1 1,5 1 2 1 2,5 1 3 1 3,5 1 4 1 4,5 1 5 1 1 1 1,5 1 2 1 2,5 1 3 1 3,5 1 4 1 4,5 1 5 1 1 1 1,5 1 2 1 2,5 1 3 1 3,5 1 4 1 4,5 1 5 1 1 1 1,5 1 2 1 2,5 1 3 1 3,5 1 4 1 4,5 1 5 1 1 1 1,5 1 2 1 2,5 1 3 1 3,5 1 4 1 4,5 1 f (Hz) 486,6667 365 292 243,3333 208,5714 182,5 162,2222 146 132,7273 417,1429 324,4444 265,4545 224,6154 194,6667 171,7647 153,6842 139,0476 126,9565 365 292 243,3333 208,5714 182,5 162,2222 146 132,7273 121,6667 324,4444 265,4545 224,6154 194,6667 171,7647 153,6842 139,0476 126,9565 116,8 292 243,3333 208,5714 182,5 162,2222 146 132,7273 121,6667 Estimation of a frequency of an unknown oscillator Introduce the value for the resistance R1 (kΩ) Introduce the value for the resistance R2 (kΩ) Introduce the value for the capacitance (μF) Resulting frequency of the oscillator (Hz) 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 112,3077 265,4545 224,6154 194,6667 171,7647 153,6842 139,0476 126,9565 116,8 108,1481 243,3333 208,5714 182,5 162,2222 146 132,7273 121,6667 112,3077 104,2857 224,6154 194,6667 171,7647 153,6842 139,0476 126,9565 116,8 108,1481 100,6897 208,5714 182,5 162,2222 146 132,7273 121,6667 112,3077 104,2857 97,33333 243,3333 182,5 146 121,6667 104,2857 91,25 81,11111 73 66,36364 208,5714 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 162,2222 132,7273 112,3077 97,33333 85,88235 76,84211 69,52381 63,47826 182,5 146 121,6667 104,2857 91,25 81,11111 73 66,36364 60,83333 162,2222 132,7273 112,3077 97,33333 85,88235 76,84211 69,52381 63,47826 58,4 146 121,6667 104,2857 91,25 81,11111 73 66,36364 60,83333 56,15385 132,7273 112,3077 97,33333 85,88235 76,84211 69,52381 63,47826 58,4 54,07407 121,6667 104,2857 91,25 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 81,11111 73 66,36364 60,83333 56,15385 52,14286 112,3077 97,33333 85,88235 76,84211 69,52381 63,47826 58,4 54,07407 50,34483 104,2857 91,25 81,11111 73 66,36364 60,83333 56,15385 52,14286 48,66667 162,2222 121,6667 97,33333 81,11111 69,52381 60,83333 54,07407 48,66667 44,24242 139,0476 108,1481 88,48485 74,87179 64,88889 57,2549 51,22807 46,34921 42,31884 121,6667 97,33333 81,11111 69,52381 60,83333 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 54,07407 48,66667 44,24242 40,55556 108,1481 88,48485 74,87179 64,88889 57,2549 51,22807 46,34921 42,31884 38,93333 97,33333 81,11111 69,52381 60,83333 54,07407 48,66667 44,24242 40,55556 37,4359 88,48485 74,87179 64,88889 57,2549 51,22807 46,34921 42,31884 38,93333 36,04938 81,11111 69,52381 60,83333 54,07407 48,66667 44,24242 40,55556 37,4359 34,7619 74,87179 64,88889 57,2549 51,22807 46,34921 42,31884 38,93333 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 36,04938 33,56322 69,52381 60,83333 54,07407 48,66667 44,24242 40,55556 37,4359 34,7619 32,44444 121,6667 91,25 73 60,83333 52,14286 45,625 40,55556 36,5 33,18182 104,2857 81,11111 66,36364 56,15385 48,66667 42,94118 38,42105 34,7619 31,73913 91,25 73 60,83333 52,14286 45,625 40,55556 36,5 33,18182 30,41667 81,11111 66,36364 56,15385 48,66667 42,94118 38,42105 34,7619 31,73913 29,2 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 73 60,83333 52,14286 45,625 40,55556 36,5 33,18182 30,41667 28,07692 66,36364 56,15385 48,66667 42,94118 38,42105 34,7619 31,73913 29,2 27,03704 60,83333 52,14286 45,625 40,55556 36,5 33,18182 30,41667 28,07692 26,07143 56,15385 48,66667 42,94118 38,42105 34,7619 31,73913 29,2 27,03704 25,17241 52,14286 45,625 40,55556 36,5 33,18182 30,41667 28,07692 26,07143 24,33333 97,33333 73 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 58,4 48,66667 41,71429 36,5 32,44444 29,2 26,54545 83,42857 64,88889 53,09091 44,92308 38,93333 34,35294 30,73684 27,80952 25,3913 73 58,4 48,66667 41,71429 36,5 32,44444 29,2 26,54545 24,33333 64,88889 53,09091 44,92308 38,93333 34,35294 30,73684 27,80952 25,3913 23,36 58,4 48,66667 41,71429 36,5 32,44444 29,2 26,54545 24,33333 22,46154 53,09091 44,92308 38,93333 34,35294 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 30,73684 27,80952 25,3913 23,36 21,62963 48,66667 41,71429 36,5 32,44444 29,2 26,54545 24,33333 22,46154 20,85714 44,92308 38,93333 34,35294 30,73684 27,80952 25,3913 23,36 21,62963 20,13793 41,71429 36,5 32,44444 29,2 26,54545 24,33333 22,46154 20,85714 19,46667 81,11111 60,83333 48,66667 40,55556 34,7619 30,41667 27,03704 24,33333 22,12121 69,52381 54,07407 44,24242 37,4359 32,44444 28,62745 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 25,61404 23,1746 21,15942 60,83333 48,66667 40,55556 34,7619 30,41667 27,03704 24,33333 22,12121 20,27778 54,07407 44,24242 37,4359 32,44444 28,62745 25,61404 23,1746 21,15942 19,46667 48,66667 40,55556 34,7619 30,41667 27,03704 24,33333 22,12121 20,27778 18,71795 44,24242 37,4359 32,44444 28,62745 25,61404 23,1746 21,15942 19,46667 18,02469 40,55556 34,7619 30,41667 27,03704 24,33333 22,12121 20,27778 18,71795 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 17,38095 37,4359 32,44444 28,62745 25,61404 23,1746 21,15942 19,46667 18,02469 16,78161 34,7619 30,41667 27,03704 24,33333 22,12121 20,27778 18,71795 17,38095 16,22222 69,52381 52,14286 41,71429 34,7619 29,79592 26,07143 23,1746 20,85714 18,96104 59,59184 46,34921 37,92208 32,08791 27,80952 24,53782 21,95489 19,86395 18,13665 52,14286 41,71429 34,7619 29,79592 26,07143 23,1746 20,85714 18,96104 17,38095 46,34921 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 37,92208 32,08791 27,80952 24,53782 21,95489 19,86395 18,13665 16,68571 41,71429 34,7619 29,79592 26,07143 23,1746 20,85714 18,96104 17,38095 16,04396 37,92208 32,08791 27,80952 24,53782 21,95489 19,86395 18,13665 16,68571 15,44974 34,7619 29,79592 26,07143 23,1746 20,85714 18,96104 17,38095 16,04396 14,89796 32,08791 27,80952 24,53782 21,95489 19,86395 18,13665 16,68571 15,44974 14,38424 29,79592 26,07143 23,1746 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 7 7 7 7 7 7 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 20,85714 18,96104 17,38095 16,04396 14,89796 13,90476 60,83333 45,625 36,5 30,41667 26,07143 22,8125 20,27778 18,25 16,59091 52,14286 40,55556 33,18182 28,07692 24,33333 21,47059 19,21053 17,38095 15,86957 45,625 36,5 30,41667 26,07143 22,8125 20,27778 18,25 16,59091 15,20833 40,55556 33,18182 28,07692 24,33333 21,47059 19,21053 17,38095 15,86957 14,6 36,5 30,41667 26,07143 22,8125 20,27778 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 9 9 9 9 9 9 18,25 16,59091 15,20833 14,03846 33,18182 28,07692 24,33333 21,47059 19,21053 17,38095 15,86957 14,6 13,51852 30,41667 26,07143 22,8125 20,27778 18,25 16,59091 15,20833 14,03846 13,03571 28,07692 24,33333 21,47059 19,21053 17,38095 15,86957 14,6 13,51852 12,58621 26,07143 22,8125 20,27778 18,25 16,59091 15,20833 14,03846 13,03571 12,16667 54,07407 40,55556 32,44444 27,03704 23,1746 20,27778 18,02469 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 16,22222 14,74747 46,34921 36,04938 29,49495 24,95726 21,62963 19,08497 17,07602 15,44974 14,10628 40,55556 32,44444 27,03704 23,1746 20,27778 18,02469 16,22222 14,74747 13,51852 36,04938 29,49495 24,95726 21,62963 19,08497 17,07602 15,44974 14,10628 12,97778 32,44444 27,03704 23,1746 20,27778 18,02469 16,22222 14,74747 13,51852 12,47863 29,49495 24,95726 21,62963 19,08497 17,07602 15,44974 14,10628 12,97778 12,01646 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 27,03704 23,1746 20,27778 18,02469 16,22222 14,74747 13,51852 12,47863 11,5873 24,95726 21,62963 19,08497 17,07602 15,44974 14,10628 12,97778 12,01646 11,18774 23,1746 20,27778 18,02469 16,22222 14,74747 13,51852 12,47863 11,5873 10,81481 48,66667 36,5 29,2 24,33333 20,85714 18,25 16,22222 14,6 13,27273 41,71429 32,44444 26,54545 22,46154 19,46667 17,17647 15,36842 13,90476 12,69565 36,5 29,2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 24,33333 20,85714 18,25 16,22222 14,6 13,27273 12,16667 32,44444 26,54545 22,46154 19,46667 17,17647 15,36842 13,90476 12,69565 11,68 29,2 24,33333 20,85714 18,25 16,22222 14,6 13,27273 12,16667 11,23077 26,54545 22,46154 19,46667 17,17647 15,36842 13,90476 12,69565 11,68 10,81481 24,33333 20,85714 18,25 16,22222 14,6 13,27273 12,16667 11,23077 10,42857 22,46154 19,46667 17,17647 15,36842 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 13,90476 12,69565 11,68 10,81481 10,06897 20,85714 18,25 16,22222 14,6 13,27273 12,16667 11,23077 10,42857 9,733333 44,24242 33,18182 26,54545 22,12121 18,96104 16,59091 14,74747 13,27273 12,06612 37,92208 29,49495 24,13223 20,41958 17,69697 15,61497 13,97129 12,64069 11,5415 33,18182 26,54545 22,12121 18,96104 16,59091 14,74747 13,27273 12,06612 11,06061 29,49495 24,13223 20,41958 17,69697 15,61497 13,97129 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 11 12,64069 11,5415 10,61818 26,54545 22,12121 18,96104 16,59091 14,74747 13,27273 12,06612 11,06061 10,20979 24,13223 20,41958 17,69697 15,61497 13,97129 12,64069 11,5415 10,61818 9,83165 22,12121 18,96104 16,59091 14,74747 13,27273 12,06612 11,06061 10,20979 9,480519 20,41958 17,69697 15,61497 13,97129 12,64069 11,5415 10,61818 9,83165 9,153605 18,96104 16,59091 14,74747 13,27273 12,06612 11,06061 10,20979 9,480519 5 1 1 1 1 1 1 1 1 1 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 1,5 2 2 2 2 2 2 2 2 2 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 3 3 3 3 3 3 3 3 3 3,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 11 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 8,848485 40,55556 30,41667 24,33333 20,27778 17,38095 15,20833 13,51852 12,16667 11,06061 34,7619 27,03704 22,12121 18,71795 16,22222 14,31373 12,80702 11,5873 10,57971 30,41667 24,33333 20,27778 17,38095 15,20833 13,51852 12,16667 11,06061 10,13889 27,03704 22,12121 18,71795 16,22222 14,31373 12,80702 11,5873 10,57971 9,733333 24,33333 20,27778 17,38095 15,20833 13,51852 12,16667 11,06061 10,13889 9,358974 22,12121 3,5 3,5 3,5 3,5 3,5 3,5 3,5 3,5 4 4 4 4 4 4 4 4 4 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 4,5 5 5 5 5 5 5 5 5 5 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 1 1,5 2 2,5 3 3,5 4 4,5 5 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 18,71795 16,22222 14,31373 12,80702 11,5873 10,57971 9,733333 9,012346 20,27778 17,38095 15,20833 13,51852 12,16667 11,06061 10,13889 9,358974 8,690476 18,71795 16,22222 14,31373 12,80702 11,5873 10,57971 9,733333 9,012346 8,390805 17,38095 15,20833 13,51852 12,16667 11,06061 10,13889 9,358974 8,690476 8,111111 ency of an unknown oscillator r the resistance R1 (kΩ) r the resistance R2 (kΩ) r the capacitance (μF) the oscillator (Hz) 10 50 1 13,3 R2 (kΩ) 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 2,5 C (μF) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 f (Hz) 324,4444 224,6154 171,7647 139,0476 116,8 100,6897 88,48485 78,91892 71,21951 64,88889 59,59184 55,09434 51,22807 47,86885 44,92308 42,31884 40 37,92208 36,04938 34,35294 32,80899 31,39785 30,10309 28,91089 27,80952 26,78899 25,84071 24,95726 24,13223 23,36 22,63566 21,95489 21,31387 20,70922 20,13793 19,59732 19,08497 18,59873 18,13665 17,69697 17,27811 16,87861 16,49718 16,1326 15,78378 15,44974 350 300 250 f (Hz) R1 (kΩ) 200 150 100 50 0 0 5 10 2,5 2,5 2,5 2,5 47 48 49 50 1 15,12953 1 14,82234 1 14,52736 1 14,2439 f vs R2 10 15 20 25 R2 (kOhm) 30 35 40 45 50 Musical note C C# D D# E F F# G G# A A# B C Frequency (Hz) 261,6 277,2 293,7 311,1 329,6 349,2 370 392 415,3 440 466,2 493,9 523,3 Rfb (kΩ) R (kΩ) 0,249 0,235 0,221 0,209 0,197 0,186 0,176 0,166 0,157 0,148 0,139 0,132 0,124 7,21 6,80 6,42 6,06 5,72 5,40 5,10 4,81 4,54 4,28 4,04 3,82 3,60
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