Laboratory Exercise 1: Pendulum Acceleration Measurement and Prediction Laboratory Handout AME 20213: Fundamentals of Measurements and Data Analysis

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1 Laboratory Exercise 1: Penduum Acceeration Measurement and Prediction Laboratory Handout AME 20213: Fundamentas of Measurements and Data Anaysis Prepared by: Danie Van Ness Date exercises to be performed: 4 15 September, 2006 Due date: 21 September, 2006 Overview: In this experiment, the motion of a penduum wi be measured by a microcontroer based data acquisition system using an on-board acceerometer. The data acquisition card is mounted directy to the penduum, measuring the acceeration of the penduum system as we as a component of gravity. From the data, the period and the initia ange of the penduum reease can be determined. Data wi be coected for both a sma ange and a arge ange assigned to your ab group. Comparison of the sma ange case and the arge ange case wi be made. Differences between the theoretica and experimenta resuts wi be examined. Uncertainty in the resuts wi aso be computed. Deiverabes: There are six deiverabes for this ab: 1) Derive, in symboic form, the theoretica period for: a. Penduum 1 i. Point Mass, Sma Ange approximation ii. Distributed Mass, Large Ange approximation b. Penduum L i. Point Mass, Sma Ange approximation ii. Distributed Mass, Large Ange approximation Leave a soutions in terms of system variabes, i.e., ength, mass, etc., and do not substitute any numeric vaues. 2) Compute the period vaues for the experimenta cases you ran in ab, using the equations derived from Deiverabe (1): a. Penduum 1 i. Sma Ange ii. Large Ange b. Penduum L i. Large Ange c. Penduum S i. Large Ange Resuts shoud be given in seconds, with appropriate significant digits. 1

2 3) Determine the first and ast period for a the cases isted in Deiverabe (2). Compare these vaues with those you found in Deiverabe (2). How is the period affected by a arge or sma reease ange? What can you say about the decay of the period in time? Note, that for the sma ange approximation, the period is independent of the reease ange. 4) Determine the maximum apex ange for the first and ast period of data for a cases isted in Deiverabe (2). Compare these resuts with the reease anges recorded in ab. To find the maximum apex ange for the first and ast periods of the ten second data sets, use the maxima and minima of the data. (A data points for a given case make up a data set or data series.) How does the apex ange change from the beginning to the end of the ten second cyce? Is there a great effect on the arge ange case compared to the sma ange case? 5) Compare Experimenta Data with Theoretica Resuts. a. How we does theory predict the experimenta resuts? b. Determine the percent error between experimenta and theoretica resuts. c. How do the arge ange and sma ange data differ? d. Discuss your findings. 6) Compute the uncertainty in acceeration for the acceerometer and expain each uncertainty in detai, making sure to describe each type of uncertainty error in your own words. Introduction: Gaieo Gaiei 1 was the first person to discover that penduums keep good time. As a medica student in Pisa, he noticed that amps swinging in the cathedra osciated with a period that remained fairy constant. After Gaieo, Christiaan Huygens experimented with penduums and buit on the work the former scientist began. Huygens was abe to make penduum cocks that kept time accurate to better than one minute a day. The work that Gaieo initiay did ed to the creation of cocks and eventuay changed how the word thought about the order of the universe. For sma ange defections, Gaieo s idea that the period of a penduum is constant was and sti is true today. Understanding the centuries od experiment of an osciating penduum is the focus of this aboratory experiment. Thus, the equation of motion for a simpe penduum point mass system undergoing sma ange osciations is presented 2. One of the experimenta penduums used in this aboratory experiment is constructed of a thin meta rod connected to a thin disk on which the acceerometer is mounted. Figure (1) shows the forces that act on a penduum at some ange θ. The mass moment of inertia is counteracted by gravity, which tries to restore the penduum to an ange of zero. 2

3 Figure (1): Forces Acting on Penduum. The penduum is acted on by a component of gravity, the moment of which is defined as the cross product of the distance the mass is from the point of rotation with the gravitationa force acting on the mass: τ = mg τ = mg sin ( θ ) where is the ength of the rod from the axis of rotation to the center of the disk, m is the mass of the disk, g is the gravitationa acceeration due to gravity, and θ is the ange of the penduum with respect to the vertica axis. From Newton s Second Law, the ony other force acting on the penduum is that caused by the system s inertia. The moment created from the inertia force is defined as τ = I θ. Here, I is the mass moment of inertia of the disk andθ is the anguar acceeration of the penduum, or simpy the second derivative of acceeration with respect to time. For a simpe penduum in which the mass of the rod is negected, the moment becomes 2 τ = m θ. The sum of these moments represent a the forces that act on the system, so adding them together wi yied the foowing governing equation for this system 2 m θ + mg sin ( θ) = 0. This equation is noninear inθ, so the soution is not straightforward. Often, to simpify the soution, the foowing sma ange approximation is made sin ( θ ) θ. The resuting governing equation is a formua of simpe harmonic motion, 2 θ + p θ = 0, where p is the circuar frequency, defined by 3

4 g p =. There are four soutions for this differentia equation: g θ = Csin t g θ = Csin t, g θ = Ccos t g θ = Ccos t with C being a constant that corresponds to the initia reease ange. Based on the initia conditions of the penduum, the vaue of C can be computed by differentiating these soutions twice and pugging back into the origina differentia equation. During movement through one period, the argument of the sine or cosine moves from zero through 2π. So the period for this soution can be found by setting g T = 2π. Thus the period for a simpe penduum undergoing sma ange osciations is T = 2π. g This resut is the same that Gaieo found, that for sma ange osciations the period is not dependent on the initia reease ange of the penduum or on the mass of the penduum. For a distributed mass penduum or one which experiences osciations where the sma ange approximation is not vaid, the period wi differ from that stated above. For this aboratory experiment, you wi be required to derive the period of the penduum for both a point mass and a distributed mass using the sma ange approximation and arge ange perturbation soution. Acceerometer and Data Acquisition Equipment: The acceerometer 3 used in this experiment, an Anaog Devices mode ADXL105 singe axis unit, is a spring mass system that is sensitive to acceerations aong a specific axis of movement. This sensor is described in the accompanying iterature in the foowing manner: The sensor is a surface micromachined poysiicon structure buit on top of the siicon wafer. Poysiicon springs suspend the structure over the surface of the wafer and provide a resistance against acceeration-induced forces. Defection of the structure is measured with a differentia capacitor structure that consists of two independent fixed pates and a centra pate attached to the moving mass. A 180 degrees out-of-phase square wave drives the fixed pates. An acceeration 4

5 causing the beam to defect wi unbaance the differentia capacitor resuting in an output square wave whose ampitude is proportiona to acceeration. Phase sensitive demoduation techniques are then used to rectify the signa and determine the direction of the acceeration. The acceerometer outputs a votage that is proportiona to the force acting on the sensing eement. The acceerometer output votage is an anaog signa that is digitized by an anaog-to-digita (A/D) converter and stored in binary form. The output votage is samped and stored by the on-board data acquisition system, which in this ab is integrated into the microcontroer processing unit (MPU), aowing for remote storage of data. The MPU sampes the acceerometer votage for a finite number of points at a given samping frequency over a specified time interva. Here, the samping frequency is 100 Hz, and 1000 data points are acquired over a 10 second time period. This set of data points, commony caed a data set or data series, is stored on-board the MPU that can then be downoaded ater to a computer. The resuts in this aboratory experiment are converted to acceeration using LabVIEW with a caibration previousy conducted by the teaching assistant. The caibration consists of pacing the MPU with the acceerometer in a centrifuge and subjecting the unit to a range of known acceerations. The recorded votage of the acceerometer can be potted versus the known acceeration to obtain a inear caibration equation that reates output votage to acceeration vaues. Thus, the acceerations fet under unknown forcing can be reconstructed from acquired votages stored remotey on the MPU. In this experiment, the acceerometer is mounted so that it records acceerations perpendicuar to the ong axis, or radia component, of the penduum rod in the pane of motion. Therefore, when the penduum is moving, the acceerations recorded aways incude centripeta acceeration and a component of gravity, written as 2 a = r θ + gcos( θ ). To decipher the contributions of both acceerations to the resutant acceeration recorded by the data acquisition system, the acceeration of the penduum at its apex as we as at an ange of zero degrees can be used. At its apex ange where the inertia and gravitationa forces bring the penduum to a momentary stop, the centripeta acceeration is zero and the acceerometer measures ony the component of gravity. This appears in the set of data points as a minimum. As the penduum passes through zero degrees, both acceerations are incuded, which yieds a maximum in the data. As the penduum begins a period at its apex ange, it then swings through zero degrees, or a quarter cyce in the period (T/4) up to the other side, where it momentariy comes to rest (T/2). It then reverses direction and swings back through zero degrees (3T/4) up to another apex ange (T), competing one fu period. From the maximum acceeration, the apex ange is found by: amax g θm = arccos 1 [ rad] 2g where g is the gravitationa constant, and amax is the maximum acceeration being considered. From the minimum acceeration, the apex ange is: 5

6 where amin amin θm = arccos [ rad] g is the minimum acceeration being ooked at. Referring to Figure (2) beow, oca minima wi occur three times in one period. This can aso be appied to the maximum acceeration vaues as we. Figure (2): Period cacuation To determine the first and ast periods of a given data series, take the difference in time between the first three maxima apart, or first three minima apart as in Figure (2), and the difference in time between the ast three maxima or minima, respectivey. Experimenta Setup Physica Properties: The properties of the three penduum apparatus are isted beow. These wi be needed to compete the deiverabes required for the Technica Memo. Remember to incude the acceerometer/mpu assemby and digita protractor when computing the moment of inertia for penduum L and penduum S. For penduum 1, the acceerometer assemby is aready incuded as part of the thin disk. Aso be sure to use the parae axis theorem for each component. Penduum 1: Thin Disk properties: (This incudes a supports, mounts, data acquisition equipment, and battery.) Mass: kg Diameter: 0.30 m Note: Assume the center of mass of the disk is ocated at the center point of the disk. Sender Rod properties: Mass: kg Length: kg Diameter: m 6

7 Penduum S: I-beam support properties: Cross section: I COM = BH bh b: m h: m B: m H: m Overa ength: 1.22 m Bearing ocation offset from center of penduum: 0.29 m Mass: 1.03 kg Note: Penduum S has no added brass masses. Penduum L: I-beam support properties: Cross section: I COM = BH bh b: m h: m B: m H: m Overa ength: 1.22 m Bearing ocation offset from center of penduum: 0.57 m Mass: 1.03 kg Note: Penduum L has the foowing added brass masses. 7

8 Brass masses: Hoow Cyinder properties: Mass: kg each Outer diameter: 0.05 m Inner diameter: mm Thickness: mm Note: Each position hods 2 washers. Mass ocations from penduum L bearing ocation: W2: 0.17 m W3: 0.23 m W5: 0.51 m W m W7: 0.88 m Digita Protractor: Rectanguar properties: Mass: 0.23 kg Width: m Depth: m Offset from bearing ocation on Penduum L: 0.76 m Offset from bearing ocation on Penduum S: 0.48 m Acceerometer/MPU assemby: Mass: 0.20 kg Offset from bearing ocation on Penduum L: 1.15 m Offset from bearing ocation on Penduum S: 0.87 m Uncertainty: Notre: Assume assemby is a point mass for computing the moment of inertia. To understand the resuts obtained from an experiment, the error inherent in the resuts must be obtained. The uncertainty in a measured resut must be determined by performing an uncertainty anaysis 4 before the vaidity of the resut can be quantified. Since uncertainty anaysis has not yet been taught in this cass, the major errors associated with this experiment wi be expained and given to you. However, an uncertainty anaysis must be competed individuay in the future. The overa error, or uncertainty, is a combination of systematic (bias) and random (precision) errors. Systematic errors are errors between true and measured quantities, whereas precision errors are due to statistica fuctuations in a measured vaue. Systematic errors incude errors associated with the data acquisition equipment and errors that propagate through equations. 8

9 In this aboratory experiment, the errors which are of most importance are Random error, Quantization error, Misaignment error, and Temperature dependence error. The ast three errors isted here are bias errors. Random error (e R ) is the statistica fuctuation in a measured quantity under fixed conditions over a repeated number of acquisitions. Quantization error (e Q ) is the resut of digitizing an anaog signa. It occurs in measuring physica quantities, such as the initia reease ange, the ength of the penduum, and the masses of the disk, and the ength of the rod. It is aso found in the data recorded by the acceerometer, since the data acquisition system converts continuous data into a set of discrete points, which is imited by the resoution of the anaog to digita converter. The resoution of the device used to measure the signa causes to error in the fina measured resut. The resoution of an instrument is the smaest physicay indicated division that the instrument dispays or is marked. Normay, the quantization error is defined as one-haf the resoution of an instrument. For exampe, if a thermometer is ony marked in one-degree increments, the resoution is one degree and quantization error in reading this is one-haf of a degree. Misaignment error (e M ) is due to the acceerometer sensitivity direction being misaigned with the radia direction of the penduum. Due to manufacturing toerances, the ange at which the sensitive direction of the acceerometer is mounted may deviate by a sma ange. This wi resut in an incorrect acceeration being measured. Temperature Dependence error (e T ) is a resut of the acceerometer response being a function of ambient temperature. As the room temperature fuctuates during an experiment, the response wi be affected by this fuctuation. For this experiment, which contains a 12-bit anaog to digita converter with a 2.43 V fu scae range, 1 degree maximum misaignment, and a temperature dependence of 0.50% of fu scae, the errors are computed to be: e = m/s 2 R e = m/s 2 Q e M = m/s 2 e T = m/s 2 A eementa errors such as those above are combined to form the tota uncertainty, e, using the method of Kine and McCintock 5 : e= er + eq + em + et. Thus for this experiment the uncertainty in the acceeration is m/s 2. Other eementa errors that are important but which are not incuded in this simpe error anaysis incude: Hysteresis, Sensitivity, Zero-shift, Repeatabiity, and Stabiity. For a more detaied error anaysis in the future, you coud refer to the specifications sheets for the acceerometer 6 and data acquisition card 7 that ist a errors associated with these devices. For more information see Reference [4]. 9

10 References: [1] Gaieo, Huygens, Onine, Museum Victoria Austraia, January 18, 2005 [2] Hibbeer, R. C., Engineering Mechanics: Dynamics, Eighth edition, Prentice-Ha, Inc., Upper Sadde River, NJ, 1998 [3] Szarek, T., AME250 Laboratory Exercise 3A: Microprocessor-Based Data Acquisition on a Penduum, University of Notre Dame, 2004 [4] Dunn, P. F., Measurement and Data Anaysis for Engineering and Science, First Edition, McGraw-Hi, New York, NY, 2005 [5] Kine, S.J. and McCintock, F.A., Describing Uncertainties in Singe-Sampe Experiments, Mechanica Engineering, pp. 3-9, January [6] Anaog Devices Singe Axis Acceerometer, Onine, Anaog Devices, January 18, 2005 [7] Siicon Laboratories Microprocessor Chip Unit, Onine, Siicon Laboratories, January 18,