A dynamical model of remote-control model cars
AA dynamical model of remote-controlmodel cars
Álvaro Suárez , Daniel Baccino and Arturo C Martí Departamento de Física, Consejo de Formación en Educación, Montevideo, Uruguay Departamento de Física, Consejo de Formación en Educación, Montevideo, Uruguay Instituto de Física, Facultad de Ciencias, Universidad de la República, Montevideo, UruguayE-mail: [email protected] xxxxxxAccepted for publication xxxxxxPublished xxxxxx
Abstract
Simple experiments for which differential equations cannot be solved analytically can be addressed using an effective modelthat satisfactorily reproduces the experimental data. In this work, the one-dimensional kinematics of a remote-control model(toy) car was studied experimentally and its dynamical equation modelled. In the experiment, maximum power was applied tothe car, initially at rest, until it reached its terminal velocity. Digital video recording was used to obtain the relevant kinematicvariables that enabled to plot trajectories in the phase space. A dynamical equation of motion was proposed in which theoverall frictional force was modelled as an effective force proportional to the velocity raised to the power of a real number.Since such an equation could not be solved analytically, a dynamical model was developed and the system parameters werecalculated by non-linear fitting. Finally, the resulting values were substituted in the motion equation and the numerical resultsthus obtained were compared with the experimental data, corroborating the accuracy of the model.Keywords: dynamical model, remote-control cars, Tracker, phase space
1. Introduction
The kinematic and dynamic aspects associated with themotion of remote-control model (or toy) cars and theelectromagnetic aspects associated with the operation and theefficiency of their small built-in electric motor as well aswith the transmission and reception of electromagneticwaves for controlling their motion deserve the attention ofresearchers in and teachers of Physics.Wick and Ramsdell [1, 2] modelled the motion of toycars rolling down an arbitrarily defined track. In theirexperiment, turning points were expressed in terms of heightloss relative to a hypothetical frictionless situation based onthe static friction coefficient between the car and the track.The authors provided a detailed analysis of different trackshapes and the effects of air friction, but failed to account for the effects of rolling. In addition, unlike the remote controlcars of our study, the cars used by Wick and Ramsdell werenot driven by a built-in motor but rolled by the effect ofgravity. In a later work, Wick and Ramsdell [3] studied themotion of an electric toy train. The analysis focused onaspects of friction, electrically induced torque andelectromotive forces, and included other effective parametersneeded to develop a model that could be solved numerically.Unlike the case with remote-control cars, the power input ofa train can be changed arbitrarily by accurately adjusting thevoltage delivered by an external regulated source of directcurrent.Care must be taken by Physics teachers to avoidmisleading students into thinking that the behaviour of realsystems can always be described from a purely theoretical erspective and expressed in terms of simple equations thatcan be solved analytically. As an example, experiments withremote-control cars can be carried out easily but cannot beeasily modelled. The concepts and tools necessary fordeveloping a suitable model are described in this paper.The motion of a remote-control car can be modelled bydescribing trajectories in the phase space, with theacceleration and the velocity as variables [4]. The kinematicvariables can be obtained from the analysis of digital videorecording of the car in motion. In this work, use was made ofthe Tracker video analysis and modelling tool [5, 6] capableof determining the position of a moving object as a functionof time and then using it in numerical derivation schemes inorder to obtain other magnitudes, such as its velocity oracceleration. Tracker was also used to develop a dynamicalmodel describing the kinematic behaviour of the car. Non-linear fitting to the trajectory in the phase space was used todetermine approximate values of the parameters in themotion equation. Tracker also carried out numericalintegration of the motion equation, the results of which wereplotted and compared with the experimental plots.The experimental analysis of the evolution of differentphysical phenomena based on digital video recordings hasreceived particular attention in the literature in the past years[7, 8, 9, 10, 1 1 , among others]. In contrast, the same does nothold true for the use of dynamical models to verify thepredictive power of motion equations, the works by Wee [12,13] being notable exceptions. Clearly, where motionequations can be solved analytically—as is usually the casewith laboratory experiments—the numerical solution ofmodels appears to lack didactic value. However, real systemscan seldom be modelled from a purely theoreticalperspective.The dynamic behaviour of remote-control toy cars in thephase space cannot be described in terms of equations thatcan be solved analytically. In this paper, a model based on anon-linear motion equation was found to reproduce thesystem’s behaviour with a satisfactory degree of accuracy.The model was characterized based on the car’s trajectoriesin the phase space in terms of velocity and acceleration,created by Tracker. The predictive power of the model wasfinally verified by comparing the solution to the differentialequation with the car’s position at different times.
2. Experimental setup
The remote-control toy car was 17 cm in length, 7.3 cmin width, 3.8 cm in height, and 0.1736 kg in mass (Figure 1).The car was capable of moving along a straight line on aneven, level surface.In the experiment, applying maximum power by pulling the remote-control lever, the car started from rest andaccelerated until it reached its terminal velocity. The car inmotion was filmed with a Kodak PlaySport video cameramounted on a tripod. In order to obtain the sharpest imagepossible, light spots were used to improve the lightingconditions and reduce the shutter time of the camera. Therecording was analysed using the Tracker video analysis andmodelling tool. In order to determine the car’s position as afunction of time, the autotracker tool was enabled. This toolis capable of selecting a pattern within a frame and track itacross the rest of the recording. Figure 2 shows a screenshotof the autotracker interface in use.
Figure 1. Remote-control model car used in the experiments.Figure 2
Tracker screenshot showing car’s position, previousmarks (red romboids), axes (purple) and calibration tape (blue).Figure 3. Temporal evolution of the car’s position obtained with
Tracker.
3. Experimental results and analysis of kinematic variables
The temporal evolution of the car’s position was createdusing the autotracker tool of Tracker. An example is shownin Figure 3. The velocity and acceleration curves shown inFigures 4 and 5 were obtained by numerical derivationperformed by Tracker. These curves clearly show that the carapproached its terminal velocity asymptotically, consistentwith the expected behaviour. The position curve is noticeably mooth, the velocity curve is slightly noisy, and theacceleration curve is markedly noisier.Figure 4. Velocity of the model car as a function of time.Figure 5. Acceleration of the model car as a function of time. In the experiments, the car, starting from rest, acceleratedat maximum motor power until it reached its terminalvelocity. In the direction of motion, the car was acceleratedby the frictional force exerted on the car by the surface, andwas decelerated by resisting forces due to air friction. Theorder of magnitude of the forces acting on the car can beestimated from the analysis of the velocity and accelerationcurves as a function of time. Irrespective of the model usedto describe these forces, the system undergoes a transientstate in which the acting forces change with the car’svelocity, so that the car, initially at rest, eventually reaches aterminal velocity. At the initial time, as the velocity-dependent frictional force was zero, the net force acting onthe car was equal to the net driving force. Based on the initialacceleration and mass of the car, the net driving force wasestimated to be of the order of 1.4 m/s . 0.17 kg ~ 0.2 N.This value is similar to that of the effective dissipative forcedetermined when the car reached its terminal velocity.Based on the estimation of the effective dissipative forcein the stationary state, it is also possible to estimate therelative contribution of the drag forces acting on the car withrespect to other sources of dissipation. The air friction forceis a complex function even for objects with very simplegeometries like spheres or cylinders. Dimensional analysissuggests that the frictional force acting on an object of agiven geometry, expressed as a function of the dragcoefficient, can be related to the average velocity accordingto F d = ρ v C d A ( ) where ρ is the air density, A is the car’s frontal surface areaand C d is the drag coefficient [14], which in the case ofsports cars is about 0.3 [15].As the car’s terminal velocity was approximately 1.2 m/s,the maximum drag force, based on the car’s dimensions, wasof the order of 10 -3 N, amounting to less than 1% of themaximum effective dissipative force. Based on thesecalculations, it was demonstrated that the effectivedissipative force acting on a remote-control toy car originatesmainly in its internal mechanisms.
4. System dynamics
In order to determine the motion equation for the car. Onepossible approach would be to characterize the car’s motorand to model the dissipative effects associated with internalfriction forces within the car. Because of the numerousdetails that would need to be taken into account, thisapproach would be excessively time consuming and couldnot be addressed in basic university settings. Stemming fromthe notion of phase space, a simpler alternative relies on, thedetermination of an effective dynamical equation thatreproduces the main characteristics of the car’s behaviour. Inorder to develop the model, the force exerted by the surfacewas represented as the resultant of two forces: one associatedwith the motor drive, being constant in magnitude in thedirection of motion, and the other acting in the oppositedirection, being dependent on the velocity and encompassingall the dissipative effects associated with the motor andinternal friction. Therefore, the motion equation for the carcan be written as
M dvdt = F − k v n ( ) where M is the car’s mass, F is the driving force, assumed tobe constant, and parameters k and n are real numbers thatdefine the functional dependence of the dissipative force onthe velocity. Equation (2) shows that, as the dissipative forceincreases with the velocity, accelerating the car willeventually lead to a situation of dynamic equilibrium. Thecar’s terminal velocity is given by v limit = n √ Fk ( ) In order to characterize the dynamic behaviour, it isnecessary to determine
F, k and n , assuming the mass of thecar is known. However, Eq. (2) can be solved analyticallyonly when n =1 or n =2.The system dynamics was modelled based on the phase-space trajectory in terms of velocity and acceleration, created y Tracker. Assuming that the car’s motion can be suitablydescribed by Eq. (2), the phase space curve was fitted to thefollowing function a = C − D v n ( ) Figure 6 shows the fit of the data recorded in the a ( v )phase space. Knowing the mass of the car, it is possible tofully characterize its motion equation. Combining equations(2) and (4) gives F = C . m = N and k = D . m = N ( ms ) − n . Finally, the car’s motionequation can be written as follows a = − v ( ) Figure 6. Acceleration as a function of the velocity obtained with
Tracker: points (experimental results) and non-linear fitting (red line) . The dynamical model tool of Tracker was used to verifythe predictive power of the model. Figure 7 shows ascreenshot of the interface of the tool being used to create themodel by entering the value of each of the parameters in themotion equation and the initial conditions for the car’smotion. Detailed tutorials on how to create models can befound in the references [6, 1 3 ]. Figura 7. Screenshot of the
Tracker
Model builder.Tracker numerically solves the differential equation ofmotion using the fourth-order Runge-Kutta method. Thevalues obtained in this way were compared with thoseobtained with the autotracker tool. Figure 8 shows theposition as a function of time (top) and the velocity as afunction of time (bottom), with the experimental data shownin red and numerical results in blue. A high degree ofconcordance was found between the two data sets. This ishardly surprising in this case, in view of the closed-loopnature of the method— i.e. , the dynamical model was createdby fitting the experimental a ( v ) data to a non-linear function.Figure 8. Experimental data (red) and numerical results(blue) for the position (top) and the velocity (bottom) as afunction of time. sing Tracker, it is possible to view a plot of numericalresults in the same graphic interface as the experimentalplots, allowing to compare the position measuredexperimentally with that obtained numerically at differenttimes, as shown in Fig. 9.Figure 9. Tracker screenshot of the comparison between the experimental data and the numerical model.
5. Conclusions
The analysis of experimental systems whose differentialequations of motion have no explicit analytical solution—anaspect seldom discussed in introductory courses—has greatdidactic value, for it is usually the case with real systems. Insuch situations, the formulation of models is a very powerfultool. It is worth mentioning that our numerical/practicalapproach provides an equation that works fine withoutneeding to grasp in the details of the internal resistive forcesinvolved.Trajectories in the phase space are abstract constructswhose interpretation can prove conceptually very valuable asit allows students to qualitatively understand the temporalevolution of a system governed by a first-order differentialequation, as would be the case of a falling object subjected toa velocity-dependent drag force or a variable-mass system.Use of readily available computer tools like Trackerenables the analysis of experimental kinematic data in aphase space and the development of a dynamical modelbased on the numerical solution to the system’s motionequation.In a classroom setting, such tasks are found useful—asthey aid in engaging students more actively—and encouragethe learning process—as they allow students create their ownmodels and verify their predictions through experiment. Inaddition, the experiment is inexpensive and can be carriedout outdoors.
Acknowledgements
We acknowledge financial support from grant FSED_3_2016_1_134232 (ANII-CFE). Translated from theSpanish by Eduardo Speranza.
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