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15.1. Investigation of damped/driven oscillation including chaotic mo-tion of a damped/driven pendulum. 15.1.1. Physics. The dynamics of a non-linear pendulum can be modelled using the following differential equation (31) $\frac{d^2\theta}{dt^2} = -\frac{g}{l}sin\theta$ which should be split up into the two equations (32) $\frac{d\theta}{dt} = \omega$ (33) $\frac{d\omega}{dt} = -\frac{g}{l}sin\theta$ where $\omega$ is the angular velocity. Note that this splitting of a second order differential equation into two first order differential equation was also used for the simple harmonic oscillator above. To include damping and an oscillating driving force the equations are modified to (34) $\frac{d^2\theta}{dt^2} = -\frac{g}{l}sin\theta - q\omega + F_psin\Omega_D t$ which splits up into (35) $\frac{d\theta}{dt} = \omega$ (36) $\frac{d\omega}{dt} = -\frac{g}{l}sin\theta - q\omega + F_psin\Omega_D t$ where $q$ is a frictional damping coefficient, $F_D$ is the driving force and $\Omega_D$ the angular driving frequency in radians s?¹. Strictly, $F_D$ is the maximum angular acceleration due to the periodic driving force. 15.1.2. Structured work. (1) Write a program to model a non-linear pendulum. Use the value of (37) $l$ given by $l = 0.5 + \frac{5678}{10000} metres$ where 5678 are the last four digits of your student number. (2) Use your model to determine the period and frequency of oscillation of your pendulum with amplitudes of oscillation of 5°, 10°, 20° and 40°.

          15.1. Investigation of damped/driven oscillation including chaotic mo-tion of a damped/driven pendulum.
15.1.1. Physics. The dynamics of a non-linear pendulum can be modelled using the following differential equation
(31)
$\frac{d^2\theta}{dt^2} = -\frac{g}{l}sin\theta$
which should be split up into the two equations
(32)
$\frac{d\theta}{dt} = \omega$
(33)
$\frac{d\omega}{dt} = -\frac{g}{l}sin\theta$
where $\omega$ is the angular velocity.
Note that this splitting of a second order differential equation into two first order differential equation was also used for the simple harmonic oscillator above.
To include damping and an oscillating driving force the equations are modified to
(34)
$\frac{d^2\theta}{dt^2} = -\frac{g}{l}sin\theta - q\omega + F_psin\Omega_D t$
which splits up into
(35)
$\frac{d\theta}{dt} = \omega$
(36)
$\frac{d\omega}{dt} = -\frac{g}{l}sin\theta - q\omega + F_psin\Omega_D t$
where $q$ is a frictional damping coefficient, $F_D$ is the driving force and $\Omega_D$ the angular driving frequency in radians s?¹. Strictly, $F_D$ is the maximum angular acceleration due to the periodic driving force.
15.1.2. Structured work.
(1) Write a program to model a non-linear pendulum. Use the value of
(37)
$l$ given by
$l = 0.5 + \frac{5678}{10000} metres$
where 5678 are the last four digits of your student number.
(2) Use your model to determine the period and frequency of oscillation of your pendulum with amplitudes of oscillation of 5°, 10°, 20° and 40°.

15.1. Investigation of damped/driven oscillation including chaotic mo-tion of a damped/driven pendulum.
15.1.1. Physics. The dynamics of a non-linear pendulum can be modelled using the following differential equation
(31)
(d^2θ)/(dt^2) = -(g)/(l)sinθ
which should be split up into the two equations
(32)
(dθ)/(dt) = ω
(33)
(dω)/(dt) = -(g)/(l)sinθ
where ω is the angular velocity.
Note that this splitting of a second order differential equation into two first order differential equation was also used for the simple harmonic oscillator above.
To include damping and an oscillating driving force the equations are modified to
(34)
(d^2θ)/(dt^2) = -(g)/(l)sinθ - qω + Fpsin t
which splits up into
(35)
(dθ)/(dt) = ω
(36)
(dω)/(dt) = -(g)/(l)sinθ - qω + Fpsin t
where q is a frictional damping coefficient, FD is the driving force and the angular driving frequency in radians s?¹. Strictly, FD is the maximum angular acceleration due to the periodic driving force.
15.1.2. Structured work.
(1) Write a program to model a non-linear pendulum. Use the value of
(37)
l given by
l = 0.5 + (5678)/(10000) metres
where 5678 are the last four digits of your student number.
(2) Use your model to determine the period and frequency of oscillation of your pendulum with amplitudes of oscillation of 5°, 10°, 20° and 40°.

Added by Paula M.

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