Tulsa Community System Dynamics of A Pendulum Coil System Lab 3 Report

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Tulsa Community College

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Lab 3: System Dynamics of a Pendulum-Coil System Finding an expression for term 𝐹 𝑙 which depends on the 𝑚 constant magnetic field (Figure 2), Ɓ and the varying magnetic flux. Abstract— A copper wire coil is used as a pendulum bob which swings through a magnetic field. This scenario involves transduction in energy between both mechanical and electrical domains, each possessing its own respective dynamic properties. Both of these affect the performance of the pendulum. The purpose of this lab is to use s-domain modeling to describe an engineering system that contains both mechanical and electrical elements as well and obtain variables within the characteristic equation. I. Introduction The simplified linearized version of the pendulum that was used in the lab can be described by simple second order dynamics. Excluding the electrical portion of the system, the pendulum is only subject to friction in the pivot joint which dissipates small amounts of energy during each cycle. By adding the electrical component to the system, another factor arises which must be taken into account. The copper wire coil bob enters and leaves the magnetic field each time the pendulum swings, resulting in additional damping. Figure 2: Coil moving through magnetic field. The magnetic field is perpendicular to page. Dots are field coming out of page, crosses represent going into page The total magnetic flux Φ, through the coil is expressed in Equation 3. 𝜑=∫∫ ∙𝑑𝐴 𝑆 (3) The dA is a vector differential element that is pointing of the page and is aligned with B. For this lab, however, the residual magnetic field that is located outside of the magnet can be neglected. As the coil swings through the magnetic field, the back EMF generated across the coil is proportional to the change of flux per unit time. When applying this approximation, Equation 4 is obtained. 𝑉emf =≈ ∫∫ 𝑑𝐴 =𝐵 (4) Figure 1: Free Body Diagram of Pendulum −𝑚𝑔𝑙 sin 𝜃 − 𝐵 𝑑𝜃 𝑑𝑡 + 𝐹𝑎𝑙 − 𝐹𝑚𝑙 = 𝑚𝑙2 𝑑 𝑑2𝑡𝜃2 (1) This equation contains many parts such as l, which represents the length from the pivot to the center of mass, Fm which is the force that the magnetic field imposes on the coil. FA is the fictitious “applied force” of an external entity driving the pendulum. Lastly, , is the rotational damping coefficient. Using small angle approximation, the formula is rearranged, and Equation 2 is obtained 𝑚𝑙2 𝑑𝑑2𝑡𝜃2 + 𝐵 𝑑𝜃𝑑𝑡 + 𝑚𝑔𝑙𝜃 + 𝐹𝑚𝑙 = 𝐹𝐴𝑙 (2) A(t) however is a complicated nonlinear function that corresponds to the shaded region of overlapping circles which is represented in Figure 2. Taking into account the angle of the pendulum, the function can be linearized about the center of the swing for small displacements which in turn produces Equation 5. 𝑉𝑒𝑚𝑓 = 𝐾 𝑑𝜃 𝑑𝑡 (5) When a circuit is connected across the coil terminals, back EMF induces a current 𝑖. A feedback connection exists between the two domains where (𝑡)depends on the system dynamics of the circuit and 𝐹𝑚is the force the magnetic field exerts on the coil. Using Faraday’s law and taking the differential element of the coil, 𝑑𝑙𝑐 which is the force on that element, one can obtain Equation 6. 𝑑 𝐹 = 𝑖 𝑑 𝑙 𝑐 𝑣̂ × Ɓ 𝑑2𝜃2 = 𝑑𝑡 (6) the plane, and Ɓ is normal to the plane, Equation 7 can be reached. 𝑑𝐹 = 𝑖𝐵𝑑𝑙𝑐𝑢̂ (7) where 𝑢̂ is a unit vector pointing toward or away from the coil, depending on the sign of Ɓ. By applying the assumptions made thus far, taking the line integral of dF around the coil, and linearizing the result, Equation 8 is attained. 𝐹 =Kti (8) where 𝐾𝑡is the first derivative in the Taylor series expansion. Thus, if the coil is disconnected from the circuit, no current can flow, 𝐹 = 0, and the system dynamics represent a simple pendulum. If there are other elements attached to the circuit, the coupling between the two domains will affect the pendulum’s performance. Next, by taking the linearization approximations into account in equations 5 and 8, the response equation of the coupled system can be computed. Taking the force magnitude equation, Fm=Kti (9) Substituting in Ohm’s Law 𝑉 𝑅 And Equation 5 (11) 𝑉𝑒𝑚𝑓 𝐾 𝑑𝑡 and simplifying: 𝐾 𝐹𝑚 = 𝑅 𝑡 𝐾𝑒 𝑑𝜃 (12) 𝐵𝑑𝑡−𝑚𝑔𝑙𝜃𝑚𝑙−𝐹𝑚𝑙+𝐹𝐴𝑙 =− (14) Finally, in order to examine the decay of the functions as well as the damping coefficient, ζ, the logarithmic decrement can be used as seen in Equation 15. 𝜁 (15) Where x(t) is the angle at the first peak within the angular displacement vs time graph and xnT represents the nth peak within the angular displacement vs time graph. II. PROCEDURE For this lab, the pendulum swing was first evaluated with an open circuit; meaning that there was no current flow through the circuit since the wires on the pendulum were not connected. Next, the circuit of the pendulum was shorted (connected wires); meaning that only the 100 ohms of the internal resistance of the coil was accounted for within the system. After having an open and closed circuit, the system was then evaluated at three different resistance values: 2766 ohms, 5630 ohms, and 9860 ohms. The potentiometer was attached to a 5V voltage source while the oscilloscope was attached to the central terminal to measure the output of the system. The remaining last terminal of the potentiometer was connected to the ground. 𝑑𝜃 The pendulum was then released, which initiated the trigger to begin collecting the response data on the oscilloscope. The same initial angle was used for each run of the system using the different resistance values. Once the pendulum stopped oscillating, the data was captured and then transferred into Excel. To determine the value of zeta for each data set, a MATLAB program was created which detected the number of peaks which used logarithmic decrement (Equation 15). Example code for determining the value for zeta for the open-circuit case can be seen in the appendix. 𝑑𝜃 𝑑𝑡 𝑙 𝑑𝑡 Rearranging Equation 2, 𝑑2𝜃2 − 𝑑𝜃𝑑𝑡 (𝑚𝑙𝐵2 + 𝐾𝑚𝑙𝑅𝑡𝐾𝑒) − 𝑔𝜃 In measuring the response values of the system using the oscilloscope, the “Single” trigger mode was then enabled. The trigger time was set at 100 seconds in total. When running the open system first, the pendulum was pulled back so that the far edge of the bob was in line with the outermost edge of the apparatus frame (10) 𝐼= Now, substituting equation 9 into equation 2 and simplifying, the equation then becomes Equation 14: 2 (13) Next, the block diagram of the open circuit and short circuit, was generated within MATLAB Simulink The damping value, B, calculated from equation X is 0.000423. This value remains constant when changing the system resistance. The zeta value, 𝜁, was found to be similar for all resistance values, with a slight decrease upon increasing the resistance. Using equations X and X, it is possible to develop an equation for 𝐾𝑡𝐾𝑒, shown below: 𝐾𝑡𝐾𝑒 = [(2𝜁𝜔𝑛)(𝑚𝑙2) − 𝐵] ∗ 𝑅 Figure 3: Block Diagram of System Figure 3 shows the block diagram of pendulum coil system, with gain values shown below in Table 1. For the open coil (R = ∞) both input Fa and Fm are set to zero. Table 1: Gain Block Equations Gain # Equation 1 1 𝐿 1 2 (18) 𝐿 The zeta and 𝐾𝑡𝐾𝑒 values for all resistances tested is shown in Table X. Table 2: 𝐾𝑡𝐾𝑒 and 𝜁Values for Varying Resistance Resistance (Ω) Zeta (𝜁) ∞ (open) 0.0014 - 100 (short) 0.0016 0.0127 2766 0.0016 0.3611 5630 0.0014 0.0096 9860 0.0013 -0.5163 𝐾𝑡 𝐾𝑒 𝑚𝐿2 3 𝑑 4 𝑚𝑔𝑙 Because the zeta values are similar for all resistances, it can be seen that increasing resistance decreases 𝐾𝑡𝐾𝑒. The 𝐾𝑡𝐾𝑒 value for a resistance of 9860 Ohms is negative because as resistance increases, the system response becomes closer to the open circuit response (infinite resistance). The transfer function from this diagram is shown in Equation 16, where the damping value, B, is shown in Equation 17. Θ(𝑠) 𝑙 = 2𝑠2+𝐵𝑠+𝑚𝑔𝑙 𝐹𝐴(𝑠) (16) 𝑚𝑙 𝐵 = 2 𝜁𝜔𝑛𝑚𝑙2 (17) Note: the simulation returns a signal even if the input is identically 0 for all time because of noise within the system. III. RESULTS AND DISCUSSION Figure 4: Simulated and Oscilloscope Data for Open Coil Figure 4 compares the response of oscilloscope data to the simulated (Block Diagram Scope) for the open coil system. The responses were recorded for the same starting angle, 11.6°, and time interval, 100 seconds. There is a variation between the simulation and experimental data. This is because the peaks for the simulated data follow an exponential curve, while the peaks for the oscilloscope data follow a linear fit. If one more circuit, such as an RC, RLC, or RCC, was connected across the coil, the system would increase in order by one. IV. CONCLUSION A pendulum with a folded bob was displaced via a magnetic field by the force of gravity with changing resistances released from a consistent starting angle. The system can be represented by a simple second order transfer function and a block diagram. Experimental data for the pendulum’s motion is taken with an oscilloscope and compared to a simulation of the system’s transfer function (block diagram) on Simulink. How the log decrement equation can be used to approximate zeta as well as calculating the system’s damping value is discussed in this report. This lab found that 𝐾𝑡𝐾𝑒 decreases and zeta decreases for increasing resistance, proven by data collected for four varying resistance values.
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Lab 3: System Dynamics of a Pendulum-Coil
System
Finding an expression for term 𝐹 𝑙 which depends on the
𝑚 constant magnetic field (Figure 2), Ɓ and the varying
magnetic flux.

Abstract—

A copper wire coil is used as a pendulum bob which
swings through a magnetic field. This scenario involves
transduction in energy between both mechanical and
electrical domains, each possessing its own respective
dynamic properties. Both of these affect the
performance of the pendulum. The purpose of this lab
is to use s-domain modeling to describe an engineering
system that contains both mechanical and electrical
elements as well and obtain variables within the
characteristic equation.

I.

Introduction

The simplified linearized version of the pendulum that
was used in the lab can be described by simple second
order dynamics. Excluding the electrical portion of the
system, the pendulum is only subject to friction in the
pivot joint which dissipates small amounts of energy
during each cycle. By adding the electrical component to
the system, another factor arises which must be taken into
account. The copper wire coil bob enters and leaves the
magnetic field each time the pendulum swings, resulting
in additional damping.

Figure 2: Coil moving through magnetic field. The
magnetic field is perpendicular to page. Dots are field
coming out of page, crosses represent going into page
The total magnetic flux Φ, through the coil is expressed
in Equation 3.
𝜑=∫∫ ∙𝑑𝐴 𝑆

(3)

The dA is a vector differential element that is pointing of
the page and is aligned with B. For this lab, however,
the residual magnetic field that is located outside of the
magnet can be neglected.
As the coil swings through the magnetic field, the back
EMF generated across the coil is proportional to the
change of flux per unit time. When applying this
approximation, Equation 4 is obtained.

𝑉emf =≈ ∫∫

𝑑𝐴
=𝐵

(4)

Figure 1: Free Body Diagram of Pendulum
−𝑚𝑔𝑙 sin 𝜃 − 𝐵

𝑑𝜃

𝑑𝑡 +

𝐹𝑎𝑙...


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