Engineering 12 Physical systems analysis Laboratory 3: The Coupled Pendulums system

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1 Engineering Physical systems analysis Laboratory 3: The Coupled Pendulums system Swarthmore College Department of Engineering Jeff Santer 09, Nick Villagra 09, Rio Akasaka 09 System behavior of two coupled pendulums when one is stationary and the other is released at an angle.

2 Abstract This experiment examines the behavior of a system consisting of two pendulums coupled by a spring. The motion of the pendulums was derived mathematically, after which the system was modeled in Matlab s Simulink and plotted. The discrepancy between experimental and theoretical values for the frequency were within 8% error, highlighting remarkable similarities despite significant sources of error. The angular frequency obtained for pendulum was 4.30 radians, and was for pendulum. The spring used for the experiment was found to have a spring constant K value of 4.. Introduction To deepen understanding of oscillatory motion, analysis was conducted of a coupled pendulum system under three different sets of initial conditions. After obtaining experimental data from the motion of the pendulums, equations of motion were analytically derived by summing the forces acting on each of the pendulums and applying state variables to the angular displacements and angular velocities of the respective pendulums. The program Simulink was also used to recreate the coupled pendulum system and thereby obtain theoretical representations of the oscillatory motion. The experimental results of motion served to verify the accuracy and thereby the utility of the analytical methods previously described, such as deriving state variable equations and using the program Simulink. This experiment introduces the tools and methodology necessary for broadly analyzing the oscillatory behavior of swinging masses, an important physical concept that is applicable to the fields of bridge design and skyscraper design.

3 Theory This experiment deals with a system of coupled pendulums, set up as shown in Figure. A free body diagram is shown below in Figure. Figure. Free Body Diagram of Pendulum Summing the torques caused by the above forces, using the small angle approximation that sinθ θ and cosθ, and setting the sum of the forces equal to Jθ && gives the equations of motion. We assume that the concentrated at a point, which means that rod is massless and the mass at the end is J = ml. The equations of motion for both pendulums are found to be: ml ml && θ && θ + mgl + mgl θ θ + kl + kl s s (θ (θ θ ) = 0 θ ) = 0 Making the substitutions g κ = l and s k l κ = m allows us to easily write the l system as below. ( ) ( ) && θ+κ +κ θ κθ = 0 && θ + κ +κ θ κ θ = 0.

4 The following substitutions are then made to rewrite the equations in state-variable form: x = θ x =θ x = θ & = x& x = θ & = x& 3 4 In state variable form, the equations of motion are thus: θ& θ θ θ& = ( ) 0 0 ω& κ +κ κ ω ω& κ κ +κ 0 0 ω ( ) Now, we must find the eigenvalues and eigenvectors of the matrix M that is multiplied by the state variables. To find this, we set det (λi-m) = 0, where the eigenvalue λ is a constant, and Ι is the identity matrix. We therefore have: λ 0 0 ( κ +κ) 0 λ 0 = 0 κ λ 0 ( κ +κ) Working this out: λ 0 Simplifying: ( κ) κ κ + λ κ λ 0 0+ ( ) 0 λ 0 λ κ +κ κ 0 0= 0 0 λ ( ) κ κ + κ λ 3 ( ( )) ( ( κ ) ( )) ( ) ( ) 4 4 λ + λ ( κ +κ) +κ + κκ = 0 λ λ +λ κ +κ +κ +κ λ κ +κ = 0 λ +λ κ +κ +κ +κ + κ κ κ +λ κ +κ = 0 By making the substitution q=λ, we can solve the equation using the quadratic formula.

5 ( ) ( ) q = κ +κ ± κ +κ κ κ κ q = κ κ ± 4 κ 4 q = κ, κ κ, λ =± q =± j κ, ± j κ + κ,,3,4, Now, we must find the eigenvectors by solving the matrix equation [ λ j I M] V j = 0 for each eigenvalue/eigenvector pair, where M is the matrix from the state-variable equation. jκ 0 0 V, 0 j 0 V κ, ( ) = κ +κ κ jκ 0 V,3 κ ( κ + κ) 0 jκ V,4 0 λ=κ j jκ V = V,,3 κ, =,4 j V V ( ),, (, ) V, ( ) V, j ( j V,) κ +κ V κ V + jκ jκ V = 0 κ + κ + κ + κ κ = 0 The last two equations show that V = V,, So V, V, = V,3 jκ V,4 jκ κ j 0 0 V, 0 j 0 V κ, ( κ +κ) κ jκ 0 V,3 ( ) 0 j V κ κ + κ κ,4 = 0 λ = jκ κ j V = V,,3 κ j V = V,,4 ( ) V, V, j ( j V, ) V, ( ) V, j ( j V,) κ +κ κ κ κ = 0 κ + κ + κ κ κ = 0 The last two equations show that V = V,, So V, V, = V,3 κ j V,4 κ j

6 j κ + κ 0 0 V3, 0 j κ + κ 0 V 3, = 0 κ +κ κ j κ + κ 0 V ( ) 3,3 V 3,4 κ κ + κ 0 j κ + κ ( ) λ = j κ + κ 3 j κ + κ V3, = V3,3 The last two equations V3, j κ + κ V3, = V V show that 3, 3,4 = V3, = V V 3, 3,3 j κ + κ ( κ +κ) V3, κ V3, + j κ + κ j κ + κv = 0 So V 3,4 j κ + κ κ V + κ + κ V + j κ + κ j κ + κ V = 0 ( 3, ) ( ) ( ) 3, 3, 3, j κ + κ 0 0 V4, 0 j κ + κ 0 V 4, = 0 κ +κ κ j κ + κ 0 V ( ) 4,3 V 4,4 κ κ + κ 0 j κ + κ ( ) λ = j κ + κ 3 j κ + κ V4, = V4,3 The last two equations V4, j κ + κ V4, = V V show that 4, 4,4 = V4, = V V 4, 4,3 j κ + κ ( κ +κ ) V4, κv4, j κ + κ j V = 0 So V 4,4 j κ + κ κ V + κ +κ V j κ + κ j κ + κ V = 0 ( ) 4, 4, ( κ + κ 4,) ( ) 4, There are two second order differential equations, so each solution has four linearly independent components. The solution for the i th state variable takes the following form: () N x t = A V λ i j= j i λ j ( ) t j e

7 Where the term i( j) V λ is the i th component of the eigenvector corresponding to the j th eigenvalue, each A is an arbitrary constant determined by boundary conditions, and x i (t) is the i th state variable. Renaming the variables ± jφ ±κ j =±ω j and ± j κ + κ =± jωand using Euler s Identity ( e = cosφ ± jsinφ ), the equations of motion can be written as x() t =θ () t = A cos( ω t) + Bsin ( ω t) + Ccos( ω t) + Dsin ( ωt) x( t) =θ ( t) = Acos( ω t) + Bsin( ωt) Ccos( ωt) Dsin( ωt) () =θ & () =ω ( ω ) ω ( ω ) +ω ( ω ) ω ( ω ) () =θ & () =ω ( ω ) ω ( ω ) ω ( ω ) +ω ( ω ) x t t Bcos t Asin t Dcos t Csin t 3 x t t Bcos t Asin t Dcos t Csin t 4 To determine the arbitrary constants, four boundary conditions must be known. In the experiment, the boundary conditions were the initial position and velocity of both pendulums. As an example, the following is the derivation for the Case (iii), using the initial conditions θ ( t = 0) = θ 0 = ; θ ( t = 0) = θ & ( t = 0) = θ & ( t = 0) = 0 x(0) = A + C = A = x(0) = A - C = 0 B = 0 x3(0) = ωb + ωd = 0 C = x4(0) = ωb - ω D = 0 D = 0 Substituting the coefficients A, B, C, D into our original equations of motion, x() t =θ () t = [cos( ω t) + cos( ωt )] x() t =θ () t = [cos( ωt) cos( ωt )]

8 Using the following trigonometric identities, substitutions can be made to the above: α+β α β cosα+ cosβ= cos cos α+β α β cosα cosβ= sin sin () θ t = cos t cos t () ω+ω ω ω ω+ω ω ω θ t = sin t sin t Which could further be reduced to () t cos( t) cos( t) () t sin( t) sin( t) θ = ω ω a b θ = ω ω a b ω+ω π T ω at = = a ω ω π ω bt = = Tb where the final substitution is changing from radians to frequencies.

9 Procedure Figure. Diagram showing the setup of the experiment, with two masses Picture obtained from Montana Uni versity The analysis involved the use of an apparatus setup as follows: two pendulums of identical material and dimensions were suspended with a metal stand from the same height and then connected together at approximately one-third length from their tops by a spring. An oscilloscope was then connected to sensors at the top of each pendulum to record their respective motion.

10 k r L θ θ m m Figure 3. Diagram showing the setup of the experiment, with two masses Image obtained from Prof. Molter With this apparatus, data was collected with the oscilloscope for the motion of the two pendulums under three different sets of initial conditions. Before each collection of data, it was verified that the pendulums were as stationary as possible. In the first case, both pendulums were lifted to a higher height and held still so that the values of θ and θ were equivalent. Mathematically, this satisfied the initial condition ( ) ( ) θ t = 0 = θ t = 0 = θ = ; 0 θ & ( t = 0) = θ & ( t = 0) = 0. Both pendulums were then released simultaneously, and data from the oscilloscope was recorded. This process was repeated for three trials. In the second case, one pendulum was lifted to a certain value of θ and the second pendulum was lifted to the same magnitude of θ but in the opposing direction, θ t = 0 = θ = ; θ & t = 0 = θ & t = 0 = 0. mathematically represented as θ ( t = 0) = ( ) ( ) ( ) 0 Again, data for three trials with this initial condition were collected after the pendulums were released.

11 In the third and final case, only one pendulum was lifted to a new height with valueθ while the second pendulum was kept in the neutral position, written as ( ) ( ) ( ) ( ) θ t = 0 = θ = ; θ t = 0 = θ & t = 0 = θ & t = 0 = 0. 0 Three trials of data were collected for this initial condition and the subsequent motion of the pendulums after the single pendulum was released. The spring constant, necessary for establishing the equations of motion, was also calculated by first measuring the weight and length of the spring and the force exerted on the spring by an attached mass, and then applying the equation F = k x. Results The following initial conditions were examined: Case : Both pendulums were pushed in the same direction with the same angle (as observed by the output on the oscilloscope) and then simultaneously released. Case : Both pendulums were pushed outward in the opposite direction, but at the same angle, and released simultaneously. Case 3: One pendulum was pushed out at and angle while the other remained stationary.

12 The following demonstrates graphs obtained by three different methods: ) Experimental: As the pendulum moved, the changing resistance of the potentiometer attached to the pivot caused variations in voltage, which were read by the two channels of the oscilloscope. The data were extracted using Excel s Agilent function, imported into Matlab, and plotted. ) Simulink: A model (shown under Discussion, Figure 4) was made using Matlab s Simulink platform, and the output was saved into two.mat data files which were then plotted for the appropriate time period to make comparisons easier. 3) Theoretical: The equations of motion developed in the theory section were plotted for each case. The following tabulates the parameters and measurements used in this experiment. Table. Measurements of mass and length Mass (kg).5435 Mass (kg).59 Average mass (kg).5360 Length of rod (m) Distance btw. Pivot and spring on rod (m) Distance btw. Pivot and spring on rod (m) Average length (m) 0.095

13 Table. Measurements for spring constant Actual weight of spring (kg) 0.03 Unstretched spring (m) Stretched spring (m) Measurement x (m) 0.03 Mass applied weight of spring (kg): = Corresponding force (N) Stretched spring (m) Measurement x (m) Mass applied weight of spring (kg): = Corresponding force (N).6359 Graph. Experimental output for case.

14 Note how the waves look very identical, but due to errors and slight differences in release time, as well as other external factors (see Error Dicussion), the lines do not match as well as they do in the Simulink or Theoretical graphs. Graph. Experimental output for case. The two waves appear to have the same amplitude and frequency, but are phase shifted from each other. Note how they begin to dampen after several seconds, due to non-ideal conditions, including friction and inequalities in weight (see Error Dicussion).

15 Graph 3. Experimental output for case 3. Here the discrepancies between the experimental output and the theoretical output are clear, in that the motion of Mass (which is initially stationary) must increase in proportion to the decrease in motion of Mass, which is released from some angle. In the theoretical data, the amplitude of Mass does not decrease as quickly, and the amplitude of Mass increases faster. This discrepancy is caused by factors such as a spring that is not initially taut, friction, and other details mentioned in Error Considerations.

16 Discussion The following tables describe the particular values obtained using calculations for the spring constant and the various substitutions made in the theory. Table 3. Spring constants from two trials calculated from table K (F/ x) Measurement (N/m).7 K (F/ x) Measurement (N/m) 6.06 Average (N/m) 4. Table 4. Calculations for variables used in substitution g κ = l κ, ω k ls κ = m l κ ( +κ ) κ + κ, ω 4.59

17 Graph 4. Theoretical output for case. Note once again that both waves are exactly identical, making them look as one. The following Matlab code was used for the theoretical form, obtained from the theory: omega = ; t = linspace(0,8) ; y = (cos(omega*t)) ; z = (cos(omega*t)) ; plot(t,y, m, t, z, b : ) title( Behavior of case according to equations of motion ) xlabel( Time (sec) ) ylabel( Angular displacement ) legend( Theta, Theta ) Code. Theoretical output for case.

18 Graph 5. Experimental data curve fitted in Kaleidagraph for case, x (t). We note how the curve fitting is extremely precise and almost super-positioned over the actual data points. Graph 6. Experimental data curve fitted in Kaleidagraph for case, x (t).

19 Graph 7. Simulink model for case. Note that since both waves are identical, they cannot be easily differentiated in print. We observe that the lines are rather jagged, due to the Simulink s discrete time intervals. Graph 8. Simulink output fitted in Kaleidagraph for case.

20 Here it can be easily seen that curve-fitting the Simulink results in almost exact fitting, demonstrating the accuracy of the modeling. Graph 9. Theoretical output for case. It is clear here that Theta is a cosine wave and Theta is an inverse cosine wave, as the theoretical output should be. The following details the Matlab code used for this specific case. Omega = 4.59 ; t = linspace(0,8) ; y = (cos(omega*t)) ; z = -(cos(omega*t)) ; plot(t,y, m, t, z, b : ) title( Behavior of case according to equations of motion ) xlabel( Time (sec) ) ylabel( Angular displacement ) legend( Theta, Theta ) Code. Theoretical output for case.

21 Graph 0. Experimental data curve fitted in Kaleidagraph for case (for x (t)- Note how m is negative). Curve-fitting the data is slightly more difficult as we see the damping of cosine curve as the graph progresses. The error, however, is still acceptable. Graph. Experimental data curve fitted in Kaleidagraph for case (for x ( t)- with m positive).

22 Graph. Simulink model output for case. Once again it can be seen that the waves are jagged in this Simulink-generated graph. Graph 3. Simulink model fitted in Kaleidagraph for case.

23 Once again, the curve fitting with the model provided by Simulink is very accurate B ehavior of cas e 3 according to equations of motion Theta Theta Angular dis placement Time (s ec) Graph 4. Theoretical output for case 3. The following Matlab code was employed to generate the above graph, the theoretical behavior of the system under initial conditions of case 3. Omega = ; omega = 4.59; t = [0:.:45]; y = ½*(cos(omega*t)+ cos(omega*t)); z = ½*(cos(omega*t)- cos(omega*t)); plot(t,y, m, t, z, b: ) title( Behavior of case 3 according to equations of motion ) xlabel( Time (sec) ) ylabel( Angular displacement ) legend( Theta, Theta ) Code 3. Theoretical output for case 3.

24 Graph 5. Kaleidagraph fitting for experimental data, case 3 (Graph for θ (t), note the positive sign). Graph 6. Kaleidagraph fitting for experimental data, case 3. (Graph for θ (t), note the negative sign).

25 It can be seen that due to the large number of data points Kaleidagraph has difficulty finding a good curve fit. With appropriate guess values for the curve-fit equation, however, the generated waveform fits the data points pretty nicely. Graph 7. Simulink model output for case 3. Due to the greater time period in which the data was recorded (45 seconds as opposed to 8), the jagged edges seen earlier are less obvious. We see evidence of phrase reversal just at around the 34-second mark.

26 Graph 8. Kaleidagraph fitting for Simulink data, case 3 (Graph for θ (t), note the positive sign). Graph 9. Kaleidagraph fitting for Simulink data, case 3 (Graph for θ (t), note the negative sign).

27 In the above cases, since Simulink s modeling is undoubtedly accurate, we retain the generated fitted curve and suggest Kaleidagraph s attempts at fitting are slightly inadequate. We do note however that the displayed equation shows a very minimal error, and ultimately the frequency values are almost identical to the theoretical values. Table 5. Results of values for ω for Experimental, Simulink and Theoretical curve fits Case, ω / ω Experimental Simulink Theoretical Case, ω 4.389/4.305 avg Case, ω 4.558/4.409 avg Case 3, ω 4.003/ /4.06 avg avg Case 3, ω 4.79/ /4.535 avg avg Average ω Average ω Table 6. Error values for each of the above cases Case, ω / ω Experimental Simulink Case, ω 6.5% 0.0% Case, ω 7.90% 0.0% Case 3, ω.% 0.0% Case 3, ω.04% 0.0% Average ω 4.8% 0.0% Average ω 4.97% 0.0%

28 Graph 0. Theoretical output for case 3 using MacOSX Grapher. The theoretical equations have been plotted using a different software, namely MacOS s Grapher. This shows a longer time span. Again, there is a phase reversal at 34 seconds. The following code reads the Excel file containing the data points read by the Agilent function, extracts the necessary columns, and plots it with a label, legend, and title. data=xlsread('h:\e Lab 3\CaseData.xls'); t=data(:,); c=data(:,); c=data(:,3); plot(t,c,'b:',t,c,'m-') title('oscilloscope output for case '); ylabel('scope Data (Volts)'); xlabel('time (Sec)'); legend('input','output') Code 4. Matlab code used to import Excel data sheet and graphing them using separate markers and a legend.

29 Simulink Block diagram The following diagram was constructed with Matlab s Simulink program based on the equations of motion as defined in theory. Figure 4. Simulink block diagram for coupled pendulums. Each case was defined by values placed with each of the gains, and where the summing block required a difference, the gain was made into a negative number. The diagram has been annotated with the variables from the equations of motion. The following Matlab code imports data from the outputs of the above block diagram (Theta.mat and Theta.mat), and plots it in the relevant time period (for easy comparison to our experimental data), adding labels for both the x- and y-axis and adds a legend.

30 load Theta load Theta plot(theta(,:44), Theta(,:44), 'm') hold on plot(theta(,:44),theta(,:44), 'b:'); title('simulink model with initial condition '); hold off legend('theta', 'Theta') ylabel('angular displacement'); xlabel('time (Sec)'); Code 5. Matlab code for importing data from Simulink and plotting. Error Discussion This lab required many assumptions to be made, and as a result undoubtedly contributed many sources of error, both systematic and random, only some of which are noted here. ) Our calculations for the pendulums assumed a massless rod. In reality, our rod had some amount of weight that contributed to the overall behavior of the pendulums. ) The pivot was considered to be frictionless. The ball bearings inside, though coated in a thin layer of oil, no doubt provide unavoidable amounts of friction, reducing the effect of the springs. Furthermore, the attachment of the transducer (in the form of a potentiometer, to measure the movement of the pendulum) added significant friction. 3) The properties of the spring. The spring was considerably longer than the span that separated both pendulums, and thus its effect was only limited to the cases where the pendulums initial positions forced the spring to be taut. For example, in case with both pendulums at a same angle in the same direction, the spring s effect was minimal in maintaining the distance between the two springs,

31 whereupon using a spring that was exactly the distance that spanned the two pendulums would have had a more significant effect. In addition, the spring was an extension spring, so it never exerted an outward force, although this force was assumed in the theoretical calculations. 4) The spring might have already suffered a strain on its elastic limit. As we were the last group to perform this experiment, it is more than likely that the spring was extended beyond its original state, lending itself to systematic error. 5) The distance between the pivot and the spring was not identical for both pendulums. As such, calculations made in this experiment employed an average of the two distances, although the difference ultimately means there was uneven application of the torque by the spring on the rod. 6) The masses were not identical. Once again, the assumptions made in the lab were that both masses were the same. The fact that they were not, and the approximation made in the calculations using the average mass, was likely to cause errors. 7) Random error in release of pendulums. No mechanical or systematic method was used to release the pendulums, except the precaution that one person should release both pendulums at the same time (since it would be nearly impossible for two people to let go of two different pendulums at the same time). The angle at which they were released was also only measured by the output shown on the oscilloscope due to the transducer connected to the pivot. 8) The measurement of the spring constant. It was decided upon that the spring constant would be measured using a spring scale, which unfortunately added more

32 sources of error, rather than minimize them, since the spring inside the spring scale had an effect on the overall measurement. 9) The small angle approximation. In the theoretical derivations, it was assumed that the pendulums would rotate through small enough angles that sinθ=θ and cosθ=. However, in some trials, especially for case 3, the pendulums swung through large enough angles that the small angle approximation would add considerable error. Conclusions In this experiment, a system of two pendulums attached with a spring was examined. It was analyzed theoretically using free body diagrams to find a set of differential equations. These equations were solved using the state variable method. The solutions for the differential equations agreed with the experimental results within 8% error. The average angular frequency obtained for pendulum was 4.30 radians, and was for pendulum. The spring used for the experiment was found to have a spring constant K value of 4.. Future Work Improvements that could have been made on this lab are numerous. An alternative system could be built, with the same exact distance from the pivots to the spring, and a better, newer spring with a span identical to the distance between the pivots could be used. Also, a spring that exerts a force when compressed should be used. The masses that are suspended beneath can also be modified so that they are the same. A mechanical

33 device could be employed to hold both masses at the same time and let go at a defined moment. Also, lower friction in the pivot would help greatly. Different forms of the same lab could also be performed, including using springs with different spring constants, rods of different length, masses of different weight, and a comparison of the system s behavior with different initial angular displacements. Another interesting comparison would be to conduct experiments with the spring further below the pivot. In the place of a spring, a cord or rope could be used, and the results could reveal their behavior under tension. With this experiment, however, it might have helped to pay a little more careful attention to minute details in order to minimize the errors encountered, even though such errors were present irregardless. For example, the measurement of the spring constant could have been done with a known (or measured) weight suspended from the spring itself (rather than a spring scale, which adds another factor of error). It did help however to conduct multiple trials of the same step, so that the best (or the average) measurement could be used for the experiment. Bibliography Moreshet, Tali. Laboratory 3: Coupled Pendula System < Acknowledgements Grateful acknowledgement is made to Prof. Moreshet for guidance in initial calculations and theory for the experiment.