"Begin with data, end with understanding: a real and a modelled double pendulum"

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1 "Bein with data, end with understandin: a real and a odelled double pendulu" Introduction Neil Challis, Harry Gretton and Dick Pitt n.challis@shu.ac.uk h.retton@shu.ac.uk r.pitt@shu.ac.uk School of Science and Matheatics Sheffield Halla University Sheffield S WB UK This paper is concerned with addressin two probles. The first is how to otivate students of enineerin to ive sufficient tie to atheatical studies. The second concerns the search for ways to encourae enineerin students to enae fully with atheatical ideas rather than treatin the "doin" of atheatics as a rote process. This eans lookin for ways to put physical flesh on the bones of abstract atheatical concepts. Motivation is crucial in ivin students the will to learn, and if any enineerin students are to enae in this process, then they need to see the relevance of each atheatical topic they are studyin, at the earliest stae. This paper discusses odellin a double pendulu, atherin real data usin a cheap hand-held data loer, and usin that data to support the establishent of the atheatical concepts and techniques of differential equations, eienvalues and eienvectors. This is another of our experiences with atheatical odellin of reality within the onoin T-TIME project [] Decidin how to teach topics such as these raises three ajor dileas. Should the initial approach be theory-based or context-based? Should we use new technoloy (screen or old (paper? Should the technoloy be hand-held or desktop? The answer in each case, as in so any others, is neither exclusively one nor the other, but an infored best of both. The theoretical atheatics ust be developed in the noral way as a basis, but real data alonside can provide a powerful otivational context. These topics are certainly not trivial, involvin for instance state variables and coplex eienvalues, with this latter topic bein one which frequently enerates the question "why are we studyin this?" However once the theory is developed, the echanics can nevertheless be handled on cheap hand-held achines such as a TI-85 or TI-86 raphic calculators, and these achines also are an interal part of the real data atherin equipent. Finally, the data processin to connect the athered data to the theoretical odel is perhaps ost conveniently done in a packae such as a spreadsheet (in this case Excel, and indeed the data are easily transferred fro raphic calculator to PC usin a link. The ethodoloy is to develop the theory, ather the data, fit the odel, copare the theoretical and easured outcoes, repeat the cycle as necessary, and et hoe in tie for lunch. Soewhere in there we hope that elusive thin called understandin will develop.

2 The current exaple: a double pendulu The double pendulu syste the students are investiatin consists of two weihts linked toether by strins and swinin fro a fixed point at the top as in Fiure. The physical structure is a little Heath-Robinson, but is nevertheless surprisinly robust, and certainly is fit for its purpose. L T q x T TI-85 CBL Ultrasonic otion sensor L q y Fiure Theoretical analysis Assuin "sall" oscillations, and nelectin air resistance we resolve the forces: Vertically: and Horizontally: T cos( θ = T cos( θ = + T cos( θ = + T sin( θ = y'' and T sin( θ + T sin( θ = x' ' For sall anles we ay linearise these equations in the usual way to ive: ( y x y'' = L and x'' = ( + x + ( y x L L Now definin state variables x = x, x = x', x 3 = y and x 4 = y' in the usual way, the equations of otion ay be written: c d x' = x e e

3 where c = ( + + L d = L and e = L L In this case the easureents are approxiately L = L = 3 feet inches and =. (We work here in feet and inches so that we can counicate with our colleaues at NASA! This ives c 5.56, d. and e. 5. The eienvalues of the syste ay now be found usin, for instance, the raphical calculator (TI-85/6 have an inbuilt facility with screens as below. Set up the atrix: Find the eienvalues: Interpretin these screens, the coplex eienvalues are ± j7. 4 and ± j The coplex eienvectors (and odal atrix P can also be found as on the screen below, which shows just the first colun of P. These screens betray one inevitable disadvantae of raphic calculators, that the screen is sall! The solution to the syste then is iven by λ t x e x = P* y y' P λt ' e e λ t 3 e λ t 4 x( x'( * y( y' (

4 In aiin to copare this odel with the physical syste, the coplex eienvalues λ ( i =...4 and the odal atrix P are known, but the vector of initial conditions i ( x( x'( y( y'( T B = is not. There is soe practical difficulty in easurin these initial values, for instance in akin t = coincide with releasin the pendulu, so they are calculated by fittin the above odel to the athered data on the otion of the pendulu. This is relatively siple to do by ipleentin a nonlinear least squares fit usin for exaple the Solver facility in Excel to iniise the su of squares. (It is well worth investin a little tie in findin out about this facility Gatherin the oveent data Data on the pendulu otion is athered siply as in Fiure, usin the inexpensive CBL TM Syste and a Vernier otion detector in consort with in this case a TI-85 []. x x', where y y'. In other words we observe the displaceent of the lower ass. We thus observe Y, which is in state space for is Y = Cx = ( C = ( A saple of the discrete data athered is shown in Fiure, as plotted after transfer to a spreadsheet: Double pendulu Displaceent Tie Matchin the odel to the data Fiure The eienvalues of the syste reveal the natural frequencies; the eienvectors are already known. The initial conditions have been calculated by fittin the odel usin least squares. Thus the odel is in principle atched to the data which has been athered. If it had been possible to find the initial conditions by easureent rather than by fittin, then the odel could be used predictively. As it is the closeness of the fit can still be verified raphically, but before doin so it is worth akin a few points fro the student's point of view. We wish to provide otivation by for instance ephasisin that coplex eienvalues of the for λ = ± jω ive rise to ters in the solution like e t cos( ωt and e t sin( ωt, and in order to relate to students' prior knowlede, we iht ask what are the coefficients of these ters. One iht think that a raphic calculator such as a TI-85 could not handle the extraction of such coefficients as it does not have an inbuilt CAS syste (althouh soe raphic calculators do!. However there are ways of extractin such values, for instance as described below.

5 Suppose the eienvalues are λ = + jω, λ = jω, λ3 = + jω and i λ = j. Suppose further that A i are the coefficients of t (i = 4. Then 4 ω ( A e t for instance A + is the coefficient of e t cos( ω t and j( A A is the coefficient of sin( ω t. This suests that if we let e λ L = L = L3 = L4 = then the coefficient of e t cos( ω t is C*P*(L+L*P - *x( the coefficient of e t sin( ω t is C*P*(L-L*P - *j*x( the coefficient of e t cos( ω t is C*P*(L3+L4*P - *x( the coefficient of e t sin( ω t is C*P*(L3-L4*P - *j*x( These are easily found in the raphic calculator as on the followin screens, as the odal atrix P is already held, and other atrices are siply created. Thus the final result ay be expressed as a real function which can then be plotted and copared to the easured data. The final fitted odel in this case is: y = {.7 cos(7.4t +.sin( 7.4t.5cos(.359t +.84sin(.359t} and to the sound of elation, Fiure 3 reveals that this is indeed a ood fit to the data. Double pendulu Displaceent Tie Discussion Fiure 3 The technoloy bein used here is inexpensive and easily available, with raphic calculators owned by ost of our students. It is easy to use, but still powerful enouh to deal with an advanced topic. The technoloy is personal - there is a culture of use, and ownership of the set-up and the data collected.

6 The echanical rind of findin eienvalues and eienvectors becoes trivial, and the focus can reain on extractin a eanin in ters of the enineerin syste. This application is real with real essy nubers, whereas exaples in books tend to be artificial, so that the nubers are uncoplicated. Visualisation is a stron feature - who needs odel answers when you can see it is valid? As for how this kind of activity actually works, one can either ather data toether in front of a roup, or let students do so theselves in roups. The data can either be shared directly then or shared via the web. We have used both approaches with siilar activities. Either way a roup diension enerates uch discussion. It is true that carryin out this kind of activity takes tie - but perhaps it is better to do less in the curriculu and to understand it ore? We have presented data and a odel which has worked: what if it hadn't? Such an experience is a frequent part of a student's experience (and ours!, and sortin out the probles is a valuable part of the learnin process. Is it an error in the theory, or is it a data proble, or This enuine proble solvin activity can only enhance a student's copetence. What do students say? Aonst the coents we would rather foret is "Is it on the exa?". We ust confess we feel depressed if this is the only thin that will otivate a student, but. Conclusion This is a two-levelled approach. Enineers start fro reality and they collect data of which they then need to ake sense. The theory ives a conceptual fraework for the understandin, with the technoloy allowin less ephasis on purely echanical use of techniques, as well as providin a powerful visualisation tool. To this extent the title of this paper is perhaps a slihtly isleadin, but the point we are akin is that teachin atheatics to enineers ust take their praatic approach into account. This practical activity is a useful contribution to their developent, consolidatin and clarifyin atheatical ideas. As for the future, we are interested in buildin up a library of data and inforation fro a rane of situations for siilar analysis, and invite interested people tryin this kind of approach to contact us and share experiences. With the advent of cheap video caeras and diitisin technoloy, we have another rich source of data to provide otivation for the classical theoretical approach. We feel that the skills involved in atchin reality to theory are and will continue to be uch in deand. Web references [] []

Vector Spaces in Physics 8/6/2015. Chapter 4. Practical Examples.

Vector Spaces in Physics 8/6/2015. Chapter 4. Practical Examples. Vector Spaces in Physics 8/6/15 Chapter 4. Practical Exaples. In this chapter we will discuss solutions to two physics probles where we ae use of techniques discussed in this boo. In both cases there are