EXPERIMENTAL DETERMINATION OF AN IRREGULAR OBJECT’S MOMENT OF INERTIA Y. Minami Koyama, H. G. Serrano Miranda, A. Monrroy Cano, B. Senzio-Savino Barzellato Universidad Nacional Autónoma de México, Facultad de Ingeniería
[email protected] ABSTRACT The relevance of obtaining the moment of inertia of a mechanical system’s mobile components is associated with the component’s capability to make slow or abrupt rotations. This concept idea is used for motor, robot’s moving parts, and mechanism’s dynamic design, among other applications. One way to measure this physical property for a certain object is by using a mechanical system made from a homogeneous circular platform, horizontally suspended by three vertical strings equidistantly placed on the platform’s border. The form to proceed consists of placing an object, whose moment of inertia will be determined, over the platform, in a way that both mass centers coincide in a vertical axis. This axis will be the reference to measure the desired moment of inertia. Then, the object–platform system is turned a certain angle from the axis and released in order to produce a helical oscillation. The object–platform system’s oscillatory movement’s mathematical model is obtained, and from the angle function’s differential equation’s solution it is verified that the centroidal moment of inertia depends of the oscillation period, the object-platform system’s weight, the strings’ length, and the distance between the centroidal axis and the strings. The oscillation period, the strings’ lengths and tensions were measured with a photogate, a motion sensor, and a force sensor, respectively. Data was acquired via USB, and then processed by a program that calculates the moment of inertia’s numeric value. The previous result is obtained with a 3% maximum relative error that takes into account a reasonable discrepancy between the mass centers, and the system’s off-center oscillation. The time variable’s relative error causes the greater uncertainty in this result. Keywords: Moment of inertia, irregular objects, experimental method, oscillating platform. RESUMEN La importancia de la determinación del momento de inercia en las componentes móviles de un sistema mecánico, está asociada a las capacidades que pueden tener estos elementos para poder realizar cambios lentos o bruscos en sus movimientos rotacionales. Esta idea conceptual se maneja en los diseños de motores, partes móviles de robots, maquinaria, entre otros. Una forma de cuantificar esta cantidad física para un determinado cuerpo, es mediante el empleo de un sistema mecánico formado por una plataforma circular homogénea, suspendida horizontalmente por medio de tres cuerdas verticales que la sujetan, en tres puntos de su periferia equidistantes entre sí. La forma de operar consiste en colocar sobre la plataforma, el cuerpo cuyo momento de inercia se desea determinar, de tal forma que los centros de masa de ambos cuerpos coincidan sobre una
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recta vertical. Este eje define la referencia para la medición del momento de inercia que se desea conocer. Posteriormente, al sistema cuerpo–plataforma, se le proporciona un giro en torno a este eje y se suelta con el fin de que tenga una oscilación helicoidal en torno a éste. Se determina el modelo matemático del movimiento oscilatorio del conjunto cuerpo-plataforma, y a partir de la solución de la ecuación diferencial en función del ángulo, se verifica que el momento de inercia centroidal del cuerpo queda en función del periodo de oscilación, los pesos de la plataforma y el cuerpo, la longitud de las cuerdas y la distancia del eje centroidal a las cuerdas. Para las mediciones del periodo de oscilación, longitud y tensión en las cuerdas, se emplearon una compuerta optoelectrónica, un sensor de movimiento y sensor de fuerza, cuyas señales son adquiridas por el puerto USB y procesadas por una aplicación de software, que proporciona directamente el valor numérico del momento de inercia. El resultado anterior se obtiene con un error relativo máximo del 3% el cual incluye la posible discrepancia entre los centros de masa, así como la oscilación descentrada del sistema. El error en la variable de tiempo provoca la mayor incertidumbre. Palabras clave: Momento de inercia, objetos irregulares, método experimental, plataforma oscilante. 1 INTRODUCTION The moment of inertia of an object around a given axis is one of the properties that determine the relationship between the applied mechanical torque and the angular acceleration produced. Few practical applications can theoretically quantify this property from the mathematical definition (given by an integrative function) since this resource is applicable only when the body is homogeneous and symmetric to one or more planes. Actually, most of the rotating elements for machinery present opposed characteristics, such as: a) Irregular forms, since most of the time these objects do not have symmetry for a given plane i.e. the cranes, crankshafts and cams of an internal combustion engine. b) They are made from different elements, making them heterogeneous like DC motors (since its winding and shaft are made from different materials). For the particular case of mobile robots, the entire set, which consists of the power supply, motors, circuit boards, sensors and mechanical components, it is not possible to conceive them as homogeneous. There are many different instruments used for measuring an object’s mechanical properties in the market, such as the mass center and the moment of inertia around a given axis, which use either sophisticated air suspension systems with actuators to produce rotational movement or special materials that emulate torsion springs (which produce the object’s oscillation). These instruments are very expensive, but can offer low error measurements, up to one in a thousand. In spite of this situation, there is a chance to quantify the moment of inertia with cheap and easy to build and operate physical model’s manipulation. A reasonable option consists of indirectly measure this physical property from experimental procedures. The proposed procedure [1] consists of applying Newton’s equation to a mechanical system that, under certain physical conditions, is nearly approximate to a homogeneous rigid body’s plane movement known as barycentric rotation. Once the equation related to the torques applied to the mechanical system with the moment of inertia and angular acceleration is obtained, a mathematical model is determined. This model takes the desired body’s moment of inertia in function of various quantities such as the system’s geometric dimensions, masses and mass center’s experimental location, as well as the oscillation period, so that it is possible to calculate this physical quantity.
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2 PHYSICAL MODEL’S ANALYSIS The proposed measurement system’s basic elements consist of two wooden circular platforms 0.40 m in diameter; one of them is fixed on top of a steel frame with 2.50 m in height and 0.80 m in width and the other one is mobile and horizontally suspended by three vertical strings of the same length that are equidistantly placed with the use of hooks and eyelets. The mobile platform’s mass is 0.250 kg. Additionally, three auxiliary electronic devices were adapted: an ultrasonic rangefinder that measures the string’s lengths, a force sensor that registers the string’s tensions, and a photogate that registers the oscillatory period measurements. The first two are placed on the top platform and the third one is fixed to the frame and near the mobile platform’s border. Every signal is acquired by an interface connected to a PC’s USB port that process data with a software application, which automatically calculates the desired object’s moment of inertia’s numeric value. Figure 1 shows a picture of the system’s lower part, in which a mobile robot is placed on the platform for measuring purposes.
Figure 1 The measurement system lower part. Prior to measure the moment of inertia, the mobile platform must be leveled; this can be achieved by placing a spirit level and adjusting the length of two of the three holding strings with a set of eyelets and setscrews located on the fixed platform (top platform). In order to correctly determine the desired parameter, the body’s mass center must be coincident as near as possible with the circular platform’s vertical centroidal axis; although the fact that a 5 mm deviation produces an error up to 0.125% in 1 kg objects when obtaining the final result (given the fact that the resting values are exact). To match the mass centers of the irregular body and the circular platform, it is only necessary to measure the tension of each of the strings, with which it is possible to determine the position on the wooden platform from the body’s mass center, and determine the relative displacement required to achieve the mentioned coincidence. After making the two adjustments described in the preceding paragraphs, it is necessary to rotate the circular platform at a slight angle (about 30° or more) from the mass center’s vertical axis and release, so that will produce oscillations whose period is necessary to measure. The fact that the oscillations of the body-platform system is not around its mass center results in a significant deviation of the system’s oscillation period.
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3 MEASURING ELEMENT’S ANALYSIS In order to determine the moment of inertia with the procedure proposed on this paper, it is necessary to measure the platform’s and the object of study’s weights, the strings’ vertical length, the strings’ holding positions (respect to the platform’s center), and finally, the object–platform system’s oscillation period around its mass center, when released from a given initial position different to its equilibrium position. The latter parameter is irrelevant for obtaining the moment of inertia. For automating the moment of inertia’s calculation, it was decided to use a force sensor to measure the weights, an ultrasonic rangefinder to measure the strings’ lengths and a photogate to measure the oscillation period. Since the strings’ holding positions had to be set when building the mobile platform, this parameter is not necessary to be obtained for each experiment because it is constant. For the platform used, it was decided that the strings’ holding distances would be 0.20 m from the platform’s center. The force sensor employed was a PASCO CI-6537, which has a measuring range of 50 N, and a tare calibration button. The sensor has strain gauges mounted on a specially designed “binocular beam.” The beam deflects less than 1 mm, and has built-in over-limit protection so it will not be damaged if a force greater than 50 N applied [2]. Also, the selected ultrasonic rangefinder was a PASCO CI–6742A, known as Motion Sensor II, that has a measuring range from 0.15 to 11.00 m, with a 1 mm resolution for distances less than 1.50 m. It produces a series of ultrasonic pulses and detects the sound reflecting back from an object in front of it. The interface used with this sensor measures the times between outgoing pulses and returning echoes. From these measurements is possible to determine the position, velocity, and acceleration of the object [3]. Figure 2 shows the devices mentioned above.
Figure 2 The force sensor and the ultrasonic rangefinder are fixed on the top platform. Finally, the selected photogate was PASCO’s model ME–9498A. The device, as shown on Figure 3, has a narrow infrared beam and a fast fall time that provide very accurate signals for timing. When the infrared beam between the source and detector is blocked, the output of the photogate is low, and the red LED on the photogate goes on. When the beam is not blocked, the output is high, and the LED is off [4].
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The three sensors were connected to the ScienceWorkshop 750 USB interface, which is able to process simultaneously four digital and three analog signals [5]. The force sensor sends an analog signal whereas the rangefinder and the photogate send digital signals.
Figure 3 The photogate is held to the frame. 4 OBTENTION OF THE MATHEMATICAL MODEL Figure 4 presents the object-platform system’s free body diagram, with a whole mass m.
Figure 4 Free body diagram of the object-platform system that was rotated a certain (angle. With this configuration, the strings’ lengths should be at least five times greater that the platform’s radius, and as previously shown, both mass centers must match with the platform’s vertical centroidal axis, since this condition is required to achieve three fundamental mechanical aspects: a) When on the equilibrium position, the tension on each of the strings must be equal to the third of the sum of the platform’s and the body’s weights.
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b) The vertical centroidal axis of both objects is the axis on which the moment of inertia of the bodies will be experimentally determined which corresponds to the algebraic sum of the objects’ moment of inertia. c) By providing the object–platform initial rotation angle around its centroidal axis, after being released it will have oscillations that can be considered as barycentrical in a horizontal plane, and this dynamic state is only achievable if the condition of the string’s length not being too long is met. If this condition is not fulfilled, the points on the platform’s border will have helical oscillations. d) If the generated oscillations are relatively small and the long string’s condition is taken into consideration, then the magnitude of the tension in both cases (equilibrium position and oscillatory state) can be considered invariable. Based on the mentioned free body diagram, if the air viscous friction force is neglected, and considering that each of the strings’ tension forces, T, have the same value and are equivalent to the third of the total weight, and given that the horizontal components of those forces are their magnitude times the cosine of the angle formed between the strings and the platform, whose value can be approximated to the quotient of s = rG and L, the sum of the static moments, MG, that are produced from the object–platform system’s mass center, G, is:
MG
m g rG2 L
(1)
where g is the standard gravity of the place’s gravitational camp, r G is the distance between the platform’s center and the strings’ holding point, whose length is L, and is the angular position of the table respect to its equilibrium position. Therefore, based on the Newton’s equation [6]: m grG2 IGG (2) L in which IGG is the moment of inertia of the object–platform system, is the angular acceleration of the oscillatory movement, and given that is the second derivative of the angular position , the equation’s eigenvalues are:
1, 2 rG
mg i IGG L
(3)
Note that i is the imaginary number, equal to the square root of -1. The differential equation’s (2) solution that models the system’s movement is: m g 0 cos rG t I L GG where 0 is the initial angular position. Therefore, it can be verified that the oscillation’s angular speed, , equals to: mg rG IGG L From (5), it is easy to demonstrate that the system’s centroidal moment of inertia is: r2 mg IGG G 2 L
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(4)
(5)
(6)
Finally, the irregular’s object moment of inertia will be, according to Steiner’s theorem: r2 mg (7) IGG,obj G 2 IGG,plat L where IGG,obj is the desired object’s moment of inertia, and IGG,plat is the circular platform’s moment of inertia, whose value is determined prior to realizing the experiment. 5 SENSITIVITY OF THE MODEL’S VARIABLES For obtaining the values of the parameters required to be measured, the sensors where used along with an interface as described on part 3, which is connected to a PC. In order to recognize the signals from the USB port, a program was developed with C language. This program is in charge of processing the whole data packages of the signals from the USB port. This could be achieved based on the reverse engineering process applied to the PC’s signal from the port connected to the interface. To calculate the irregular’s object moment of inertia with the experimental method proposed in this paper, it is necessary to perform the following steps: 1) The discrepancy of the position of the body’s mass center, whose moment of inertia is going to be calculated, is obtained from the mobile platform’s center. In order to do that, the strings’ tension is measured; prior to the measuring it is necessary to press the tare calibration button before attaching the platform to the vertical strings. The software then asks for the Enter key in order to save the force data. Then the platform is released, it is turned 120° clockwise and then reattached, so that the tension from other string is measured and its value can be saved in the program. Finally, this process is repeated, in order to obtain the third string’s tension data. As soon as the third value is input, the program calculates instantly the difference and it is displayed on the screen, so that the object can be moved to a position that matches the platform’s center’s vertical axis. The platform has two perpendicular reference axis labeled to facilitate the displacement of the irregular object to be measured. 2) Once the object is positioned, it is highly recommendable to repeat step 1 in order to verify the required position, and just in case that an error is presented, there is a chance to fix it. Now that the correct position is secured, it can be said that the tension of any of the string is the same, and therefore, the total weight, including the body and the circular platform, equals three times the tension measured by the force sensor. 3) The last step consists of turning the mobile platform around its center, at an initial angular position of 30° or more from its equilibrium position, and then release it so that it is possible to measure the system’s oscillation period, with the use of the photogate, which is turned on and off by a flexible tab attached to the platform’s perimeter. It is possible to observe the red colored tab on Figure 3. The interface used continuously measures the corresponding sensor’s force as well as the distance between the rangefinder and the mobile platform, which is equal to the strings’ length, with a sampling rate of 50 samples per second. Hence, as soon as the system calculates the time it takes for 10 complete oscillations to be produced, all the necessary values to calculate and display the desired irregular’s body moment of inertia on screen are obtained. From the expression (7) of the previous part, it can be verified that it is required to perform five multiplications, a division and a subtraction to obtain the result. Based on the error propagation analysis [7], the maximum relative error obtained on the calculation of the moment of inertia, is basically in function of the relative errors of each of the values involved. Given that rG = 0.20 m it can be considered exact, or, without relative error, and that the angular speed is = 2T, where T is the oscillation period, the maximum relative error will be approximately:
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er,max er m g er L 2 er T
(8)
where er (mg) is the relative error from the object–platform system’s weight measurement, er (L) is the relative error from measuring the strings’ length, and er (T) is the relative error from measuring the period. The measuring instruments used in the proposed experiment have the following relative errors: e r (mg) = 0.5% for a 10N (m = 1kg) weight, er (L) = 0.1% for a 1 m length, and er (T) = 1% for a 2 s period. Therefore, it can be seen easily that the irregular object’s moment of inertia’s calculation is very sensitive to the relative error generated when measuring each oscillation’s time. For a 1% error, the maximum relative error will be, at least, 2%. If a manual measure is to be performed, the errors produced from determining the weights and lengths would be similar, or even smaller, than the ones obtained from the automatic instruments. However, when measuring time with a stopwatch, error would be, at least, two or three times greater. Therefore, it can be observed that the use of a time measuring device precise enough and with a good resolution is necessary, to assure a reduced maximum relative error, in the object’s moment of inertia calculation method from this proposal. Another important aspect to take into account is that, as mentioned before on part 2, the object– platform system must oscillate around the platform’s center, so that the measured period does not deviate significantly and increase the period error in the already cited moment of inertia calculation. 6 RESULTS The proposed experimental development was applied to measure four objects whose geometry could be such that their moment of inertia could be theoretically calculated based on its physical dimensions and their mass. Those objects were a cylinder, a parallelepiped, a large steel cylindrical slim bar and a thin wall plastic pipe. Table 1 shows summarized the most representative values for measuring the moment of inertia, as well as the relative error committed respect to the theoretical value expected. As it can be seen from the table, the maximum relative error produced with the experimental measurements performed was 2.85%. It is possible that some bodies may present more air viscous friction, due to their greater area, such as the pipe, which had a greater oscillation period respect to the slim bodies (i. e. the large bar). Table 1 Results of the experimental method applied to the measurement of the moment of inertia of four bodies with “regular geometry.” dimensions, in mm
calculated moment of inertia, in kg m
experimental moment of inertia, in kg m
relative error, in %
mass, in kg
length
width
cylinder
0.250
410
410
6.5
0.005253
0.005150
1.96
parallele piped
1.930
300
95
57
0.02854
0.02805
1.72
large bar
0.910
915
15
15
0.06349
0.06411
0.98
pipe
0.720
520
110
110
0.01727
0.01777
2.85
object
Height
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7 CONCLUSIONS AND FURTHER WORK With the proposed experimental procedure, the moment of inertia of known bodies could be obtained with a maximum relative error less than 3%, provided that sufficient care is taken in order to comply with the minimum conditions to perform the experiment: the length of the platform holding strings must be at least five times greater than the distance between the platform’s center and the strings’ holding points, and both the object’s and circular platform’s mass centers must match the same vertical centroidal axis, and the barycentric oscillation of the object–platform system. The maximum relative error produced in the moment of inertia calculations with this method is very sensitive to the discrepancy generated from time measurement. Therefore, to assure a minimum relative error when applying this experimental process, it is necessary to use a measuring instrument that can provide a relative error less than 1%. This experiment might work as a base to create academic experiments for the Statics’, Kinematics and Dynamics’, and Physical system’s dynamic’s subjects, with the focus of experimentally determining irregular bodies’ mass centers with whimsical shapes, obtaining an irregular object’s moment of inertia, and verifying Steiner’s theorem for determining compose bodies’ moment of inertia, as well as an introduction for the study of mechanical vibrations. Finally, this experimental development is proposed to characterize the dynamic parameter of mobile robots, required for achieving their adequate navigation control, since their rotational dynamics are very linked to the mass center and centroidal moment of inertia properties. 8 REFERENCES [1] Huang, T. C., Mecánica para ingenieros, Tomo II, Fondo Educativo Interamericano, México, 1974. [2] PASCO Force Sensor Model No. CI–6537 Instruction Manual, 012–05804C, EUA. [3] PASCO Motion Sensor II Model No. CI–6742A, Instruction Sheet, 012–09624A, EUA. [4] PASCO Photogate Head Model No. ME–9498A, Instruction Sheet, 012–06372A, EUA. [5] PASCO ScientificWorkshop 750 Model No. CI–7599 Instruction Manual, 012–08101, EUA [6] Beer & Johnston, Mecánica vectorial para ingenieros, Dinámica, Octava edición, McGraw–Hill Interamericana, México, 2007. [7] McCracken, D., Métodos numéricos y programación Fortran, Limusa, México, 1986.
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