4. Show that the simple undamped pendulum, governed by the equation 0" + sin 0 =0 178 4 Stability of Linear and Alnmost Linear Systenis has an unstable (actually conditionally stable) equilibrium point at 0 = . [Hint: (a) Write as a system %D y =y2 - į sin y. L (b) Make the change of variable v, y1 - 7, V2 = y2 and show that the
4. Show that the simple undamped pendulum, governed by the equation 0" + sin 0 =0 178 4 Stability of Linear and Alnmost Linear Systenis has an unstable (actually conditionally stable) equilibrium point at 0 = . [Hint: (a) Write as a system %D y =y2 - į sin y. L (b) Make the change of variable v, y1 - 7, V2 = y2 and show that the
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Transcribed Image Text:4. Show that the simple undamped pendulum, governed by the equation
e" + sin 0 = 0
178
4 Stability of Linear and Alnmost Linear Systenıs
has an unstable (actually conditionally stable) equilibrium point at 0 = .
[Hint: (a) Write as a system
yi = y2
sin yı
|
(b) Make the change of variable v, = y1 – 7, V2 = y2 and show that the
corresponding linear unperturbed system has a saddle point at vi = 0, v2 = 0.
Expert Solution
Step 1
In case of simple pendulum, the forces acting on the pendulum are the tension along the string and gravitational force on the bob. on resolving the components of gravitational force, the component cancel out with the tension in the string. then the restoring force on the pendulum is provided by the gravitational force component .
The linear acceleration and angular acceleration are related as, , where l is the length of the pendulum and is the angular acceleration. So,
The equation can be written as,
This is the equation of the simple pendulum.
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