LAM - Swarthmore College



LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 1

LABORATORY 3

THE COUPLED PENDULA SYSTEM

OBJECTIVES

1. To observe the behavior of a system of two loosely coupled pendula experimentally.

2. To model the behavior of a coupled physical system using simulation software SIMULINK in MATLAB.

3. To compare the data obtained experimentally and from the software simulation to analytical results.

APPARATUS

Coupled pendula with transducer to convert angular position to voltage, oscilloscope, scale, weights, pc with SIMULINK.

ANALYSIS

Consider the single pendulum (ignore the spring and second pendulum in Figure 1 below).

[pic]

Figure 1: Coupled Pendula

By summing torques (consider contributions from inertia, the weight of the bob, and the spring force; neglect friction and the weight of the rod), obtain the equation of motion (note that L = ℓ and r = ℓs).

[pic]

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 2

LABORATORY 3

THE COUPLED PENDULA SYSTEM

Here the moment of inertia J is given by

[pic]

Use the identity

[pic]

and assume small angles so that

[pic]

to obtain

[pic].

For two coupled pendula, the equations of motion are as follows:

[pic]

and

[pic]

Rewriting with the coefficients of the highest order derivatives set to unity, and grouping terms, gives

[pic]

[pic]

For compactness, let [pic]so that

[pic]

[pic]

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 3

LABORATORY 3

THE COUPLED PENDULA SYSTEM

In state variable form, let

[pic]

where the state equations are

[pic]

In matrix form, [pic] where [pic] is a column vector of the derivatives of the state variables, [pic] is the matrix of constants describing the motion of the pendula, and [pic] is a column vector of the state variables. Written out,

[pic]

The eigenvectors of the matrix M give the (characteristic) modes of the system, and the eigenvalues of M give the (characteristic) frequencies of the modes. (Recall that each eigenvalue has an associated eigenvector.)

To find the eigenvalues, set det (λI-M) = 0, and solve for λ.

[pic]

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 4

LABORATORY 3

THE COUPLED PENDULA SYSTEM

[pic]

The determinantal equation is thus

[pic]

with solutions for the eigenvalues (characteristic frequencies) of

[pic]

and eigenvectors V(λ), that satisfy [pic], of

[pic]

Solutions are formed by a superposition of contributions from each eigenvector to each state variable, with arbitrary multiplicative coefficients Aj to be adjusted to satisfy the initial conditions of the system. (A1 multiplies V1, A2 multiplies V2, etc.)

[pic]

where, recall, xi are the state variables representing [pic]

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 5

LABORATORY 3

THE COUPLED PENDULA SYSTEM

Written out, the equations of motion are as follows.

[pic]

Using Euler’s Identity, [pic] the equations of motion can be equivalently rewritten as follows.

[pic]

Determine the behavior of the system for three sets of initial conditions.

Case (i) [pic].

The coefficients A, B, C, D are evaluated as follows.

[pic] , and the equations of motion are

[pic]

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 6

LABORATORY 3

THE COUPLED PENDULA SYSTEM

Case (ii) [pic].

The coefficients A, B, C, D are evaluated as follows.

[pic] , and the equations of motion are

[pic]

Case (iii) [pic].

The coefficients A, B, C, D are evaluated as follows.

[pic] , and the equations of motion are

[pic]

Here [pic]

A plot of Position θ1(t) vs Time (t) is shown below in Fig. 2.

LAM ENGINEERING 12: PHYSICAL SYSTEMS ANALYSIS 2005

page 7

LABORATORY 3

THE COUPLED PENDULA SYSTEM

[pic]

Figure 2: Position of Pendulum 1 vs. Time for Case (iii)

LABORATORY REPORT

1. Annotate and fill in the missing steps from the derivations shown. As examples, explain how the terms in the equations for the sum of the torques is obtained, and verify the last two equations for Case (iii) above.

2. Include a printout of the program and graphical output from the Simulink simulation for Cases (i) – (iii) in the Results Section.

3. Include Tables in the Results Section showing results from the analysis, experiment, and simulation for the frequencies (or periods) found for Cases (i) - (iii). Include percentage errors.

4. Include a Table with the parameters measured from the experimental apparatus.

5. Include a comparison of the results and errors from the analysis, experiment, and simulation for Cases (i) – (iii) in the Discussion Section.

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