Analyzing SHO of a double pendulum - Wabash College · WJP, PHY381 (2010) Wabash Journal of Physics...

WJP, PHY381 (2010) Wabash Journal of Physics v4.2, p.1

Analyzing SHO of a double pendulum

W.C. Beard ∗, R. Paudel∗, B. Vest, T. Warn, and M.J. Madsen

Department of Physics, Wabash College, Crawfordsville, IN 47933

(Dated: May 4, 2010)

We measured the angle versus time by video capture for a physical double pen-

dulum for small angles, and found that the period of the second normal mode did

not agree with the Lagrangian model. We hope to improve our model to better

approximate our setup in the future.

Interests in non-linear motion have motivated a number of experiments. In an undergrad-

uate laboratory, a double pendulum serves as a good apparatus to model a chaotic device

[1]. We can anaylze the motion of the double pendulum to calculate the Lypanov exponent

which is a key indicator of the chaotic motion[1, 2].

A double pendulum can be easily built in an undergraduate laboratory by attaching one

metal rod to another [3]. The dynamics of such pendulum have been topic of interest for

a number of years[1]. In the past, polarized photos of the pendulum were used to analyze

the dynamics. In this project, we used a high speed camera[4] to analyze the motion of a

double pendulum. [5].

The double pendulum built at Wabash Advanced Laboratory was used to study the

motion. We modeled the upper arm and lower arm of the pendulum as solid blocks, and

developed a Lagrangian model of the dynamics of the double pendulum. Following the

analysis by Thornton and Marion[5], we find that the kinetic energy T of the two arms is

composed of the motion of their cener of mass, (x2i + y2

i ) for the two arms i = 1, 2, and the

rotational kinetic energy about the rotation axis as shown in Figure 1, 1/2Iiθ2i . The kinetic

energy of the entire system is:

T =1

2m1(x1

2 + y12) +

1

2I1θ1

2+

1

2m2(x2

2 + y22) +

1

2I2θ2

2(1)

where m1 and m1 are the masses of the upper and lower arms of the pendulum respectively,

I1 and I2 are the moments of inertia of the upper and lower arms (as shown in Figure 1).

∗ WCB and RP equally contributed to this paper.


We can find the moment of inertia, I1 and I2, modeling the two arms as solid blocks

which is rotating about the pivot which is (lf/2 −D/2) away from the center of mass y10.

The moment of inertia of the upper arm is I1 = 1/6(m1(y210 +w2

1)) +m1(lf1/2−D1/2)2 and

that for the lower arm is I2 = 1/6(m2(y210 + w2

2)) +m(lf2/2−D2/2)2

The potential energy of the system is defined as the height of the center of mass of each

arm above the equilibrium position. The offset potential V0 is a constant that defines the

potential energy in this frame.

V = m1gy1 +m2gy2 + V0 (2)

We now shift the coordinate system from Cartesian coordinates x, y to a generalized

coordinate system based on the two angles θi defined as the angle of deviation of the center

of mass of each arm from its equilibrium position about the rotation axis of the respective

arm. From the geometry, and noting that the sign in each of these equations is negative as

we are putting the origin at the top of the upper arm, we can see that,

y1 = −y10 cos θ1 (3)

x1 = −y10 sin θ1 (4)

y2 = −y20 cos θ2 + l1(cos θ1 − cos θ2) (5)

x2 = −y20 sin θ2 + l1(sin θ1 − sin θ2) (6)

The Lagrangian of the system is given by L = T − V . We then use the Euler-Lagrange

equation [5] to get the two coupled differential equations,

∂L

∂θi

− d

dt

∂L

∂θi

= 0 (7)

We can write the two equations for the two coordinates θ1 and θ2 as:

− g sin(θ1(t))(l1m2 + m1y10)− θ1′′(t)(i1 + l12m2 + m1y102

)+ l1m2(l1− y20) (8)

× θ2′′(t) cos(θ1(t)− θ2(t)) + l1m2(l1− y20)θ2′(t)2 sin(θ1(t)− θ2(t)) = 0 (9)

m2 (l1− y20)(g sin(θ2(t)) + l1θ1′′(t) cos(θ1(t)− θ2(t))− l1θ1′(t)2 sin(θ1(t)− θ2(t))

)(10)

− θ2′′(t)(i2 + m2(l1− y20)2

)= 0 (11)

We then look at the small-angle approximation to get,

(I2 + (y20 − l1)2m2θ2 −m2(l1 − y20)l1θ1 −m2(l1 − y20)gθ2 = 0. (12)


Arm 1 Arm 2

m (g) 471.4± .1 305.2± .1

lfi (cm) 27.40± .07 27.40± .07

D (cm) 3.79± .01 3.81± .01

w (cm) 3.18± .01 1.57± .01

TABLE I: This table shows the measured mass m, length lfi and width w of both arms i of the

double pendulum (see Figure 1). Though l1, the length between the centers of the two tracking

points on Arm 1, is used to calculate several k parameters, this quantity as well as the moments

of inertia I1 and I2 were calculated from direct measurements, including lf and D.

Following Rafat, we introduce new scaled variables to match those of the paper [2],

k1 = g(m2l1 +m1y10) (13)

k2 = I1 +m1l21 +m1y

210 (14)

k4 = m2(y20 − l1)2 + I2 (15)

k5 = −m2g(l1 − y20) (16)

k6 = −l1m2(l1 − y20). (17)

We can then find the two normal modes of oscillation,

ω2± =

k1k4 + k2k5 ±√

(k1k4 − k2k5)2 + 2k1k5k26

2(k2k4 − k26)

(18)

We can thus find the time period of these modes, using pendulum measurements that are

listed in Table I.

Using Equations (12-16), we get the values for the k parameters and are then able to use

Equation (17) to find the periods of the normal modes of oscillation, T− and T+, all of which

are shown in Table II. The predicted values of the periods of the first and second normal

mode were T− = 1.514± .007 and T+ = 0.983± .002, respectively.

We measured the normal mode frequency of the physical double-pendulum described

above as a preliminary step towards verifying our data collection and analysis procedures.

The double pendulum apparatus was constructed by Vest and Warn in the fall of 2009 [3],

whose dimensions are shown in Figure 1.

We captured the motion of the double pendulum with a high-speed camera in order to

get θi data as a function of time. We covered the background and the arms of the pendulum


y

x

(x1, y1)

y2-l1cosθ2

First normal mode Second normal mode

θ2

l1cosθ1

θ1

(x2, y2)

y

x

(x10, y10)x10=0

D2

l1

w1

lf1

lf2

y20

y20-l1

x20=0(x20, y20)

w0

D1

FIG. 1: This figure shows the two arms of the double pendulum, and the dimensions that were used

for calculating the normal mode periods of oscillation. These measured values are listed in Table I.

Below the dimensional diagrams are representations of the two normal modes of oscillation. The

periods T− and T+ of the first and second normal mode, respectively were calculated and compared

to the experimental periods.


FIG. 2: This figure compares a recorded video frame before (above) and after (below) being

processed with Photoshop. The contrast was increased, the image was converted to grayscale and

a Gaussian blur was applied to the image. Before applying the changes, the tracking software had

occasionally detected background noise as particles and sometimes counted a single spot as two, as

the brightness withing the dots were nonuniform. The Gaussian blur made the dots more uniform,

correcting this problem, and washed out most of the background noise.

Quantity Calculated Value

k1 9.29007× 106

k2 164569

k4 23564.1

k5 1.13656× 106

k6 11481.6

T− 1.514± .007

T+ 0.983± .002

TABLE II: This table shows the calculated k-parameters from Equations (12-16), as well as the

calculated periods of the first (T−) and second (T+) normal modes of oscillation.


with black felt-cloth, and then put three white reflective stickers at the end of each arm

and at the point of rotation in order to make the movement more visible. The lights were

turned off in the room, and the black background was used to minimize background light

onto the camera, and felt was used for this background to prevent glare. A projector was

used to emit white light onto the pendulum to maximize the amount of light reflected from

the white stickers, and was placed 2.58 m away from the apparatus. A high speed camera

was then set up next to the projector, 3 m away from the pendulum, so that the light from

the projector would reflect off of the reflective tape nearly directly back to the camera, and

not be scattered. It was our goal to move the projector and camera far enough back from

the pendulum that the changes in the angle of oscillation would not put the light from the

reflective dots at an excessively difficult angle for the camera to detect. We oscillated the

pendulum at angles of less than 24◦ and took high speed video at 300 fps, as high speed

movement of the pendulum has been known to produce video recording errors for frame

rates under about 100 fps [3].

High Speed Camera

Projector

Double Pendulum

3.5 m

2.58 m

0.5 m

FIG. 3: We set up the double pendulum and placed a high speed camera 3 m away. We placed a

projector to emit white light onto the pendulum and made the room dark while taking the high

speed video.

We converted each frame of the video to a tiff file, and ran these tiffs through Photo-

shop batch operations to increase the contrast between the dots and the background and to

Gaussian blur all of the dots, eliminating virtually all background light noise and distinctly

separating each of the dots (see Fig. 2). We used the Autocontrast command in Photoshop,


and set the Gaussian blur radius to 2.1 pixels. We input the series into ImageJ, a free-

ware image processing software package, about 900 frames at time, due to system memory

constraints [6]. ImageJ would track the trajectories and store the coordinate data of the

trajectories of the three reflected points’ pixels with respect to time, and output them into

a text file. These raw coordinates were then extracted from the text files into a spreadsheet,

which was then used to convert the relative position of the three reflective dots through time

into angles θi. Since the angle θi can be given as θi = tan(yi/xi), we were able to get the

angle data versus time by taking the inverse tangent of yi/xi for both of the angles. Once

we had the θi(t) data (shown in Figure 4), we applied a five-point smoothing function to

eliminate any small errors in the digitization of the image files, interpolated the data, and

then took the first time derivative of the interpolated data to plot the motion in phase space.

We found the period of the motion by measuring the time difference between subsequent

zero crossings of the interpolated data.

While we did not analyze chaotic systems, we did measure the periods of smaller angle

simple harmonic oscillation of the two normal modes, to compare the experimental results

with our model. As can be seen in Figure 5, the periods of oscillation varied periodically,

and the average period was measured to be T− = 1.089 ± .001 s for the first normal mode,

and the second normal mode period was measured to be T+ = .995 ± .062 s (95% CI).

(The varying oscillations seen in Fig. 5 is likely due to damping that was stronger on one

side of the pendulum, because of the large amount of felt material used to eliminate the

glare.) This compares to the predicted values according to the model given in Table II of

T+ = .983± .002 and T− = 1.514± .007 s (95% CI).

We were thus able to use video analysis to capture the motion of a double pendulum

and measure the periods of the first and second normal modes to compare to a model based

on the system’s Lagrangian. We found that our data do not agree with our model. One

of the reasons for this is because we used small angle approximation for our Lagrangian.

The maximum amplitude in our data is 24◦ which is slightly off from the small angle ap-

proximation. In the future, we can take video with amplitude less than 15◦ which will then

make it easier to use the small angle approximation. The other possibility is to extend the

model for larger angle. Future possibilities for experimentation along this line would involve

taking massive amounts of data of the double pendulum’s chaotic motion at larger angles,

and calculating Lyupanov exponents and constructing Poincare sections from the data to


5 10 15 20 25 30Time HsL

-0.006

-0.004

-0.002

0.002

0.004

0.006

Θ1, Θ2 HradiansL

2 4 6 8 10Time HsL

-0.4

-0.2

0.2

0.4

Θ1, Θ2 HradiansL

FIG. 4: These figures depict data of θ1 and θ2 versus time for the first and second normal modes,

top to bottom respectively, with θ1 shown in gray points and θ2 shown in black points.

quantify the system’s chaos.

[1] DeSerio, Robert. American Journal of Physics. 71, 250-257 (2003).

[2] Rafat, M. Z., Whealand, M. S., Bedding,T. R. American Journal of Physics. 77, 3 (2009).

[3] Vest, Brad C, Warn, Thomas and Madsen, Martin. Wabash Journal of Physics. 381, 1-10


5 10 15 20 25Integer Oscillation

1.088

1.089

1.090

1.091

1.092

1.093

1.094Period T HsL

FIG. 5: This graph shows the period data for θ1 for the small angle first normal mode simple

harmonic oscillations of the double pendulum. The y axis shows the time of the periods, measured

in seconds, and the x axis shows the numbering of which oscillation is represented.

(2009).

[4] L. Labous, A.D. Rosato, and R.N. Dave. Phys. Rev. E 56, 5717?5725 (1997)

[5] Thornton, Stephen T., Marion, Jerry B. Classical Dynamics of particles and systems, Brook

Cole.

[6] ImageJ. http://rsbweb.nih.gov/ij/

[7] Poincare, Henri. Sur le problme des trois corps et les equations de la dynamique. Divergence

des series de M. Lindstedt. Acta Mathematica 13, 1-270 (1890).

[8] Levien, R. B., Tan, S. M.American Journal of Physics. 61, 1038-1044 (193).

[9] Shinbrot, Troy; Grebogi, Celso; Wisdom, Jack; Yorke, James A. American Journal of Physics.

60, 491-499 (1992) 60, 6 (1992).

[10] Hilborn, Robert C. American Journal of Physics. 72, 425-427 (2004).

Analyzing SHO of a double pendulum - Wabash College · WJP, PHY381 (2010) Wabash Journal of Physics...

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