We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt -=y de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (7, O): ) dy = -9₁ sinx i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system
We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the undamped motion of the pendulum of length Lis: dx dt -=y de where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration due to gravity (see Figure (left)). (i) Determine all of the critical points for the system. (ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution. (iii) Consider the critical points at the origin and (7, O): ) dy = -9₁ sinx i. State the expected type and stability of each critical point, given the eigenvalues of the linearized system ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system
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Simple harmonic motion
Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.
Simple Pendulum
A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.
Oscillation
In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.
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Transcribed Image Text:We consider a simplified version of an oscillating system, namely, we focus on the undamped pendulum where there is no friction or
air resistance to slow down the motion of the pendulum. In this case, the non-linear system of differential equations governing the
undamped motion of the pendulum of length Lis:
do
dt
dy
)
=y
वह
de
where x = 0 is the angular position, y = at is the angular velocity, and g is the acceleration
due to gravity (see Figure (left)).
(i) Determine all of the critical points for the system.
(ii) Determine the linearised system for each critical point in part (i), and find the corresponding eigenvalues of the general solution.
(iii) Consider the critical points at the origin and (π, O):
-=-은
sinx
i. State the expected type and stability of each critical point, given the
eigenvalues of the linearized system
ii. Discuss whether we can expect the linearised system to approximate the behaviour of the non-linear system
Transcribed Image Text:L
Ꮎ
mg
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