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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