What is the Differential Equation for a Non-Linear Torsional Spring Pendulum?

[hancock.yang@]

Homework Statement

Consider a pendulum oscillation problem, where pendulum oscillates around the vertical in the downward configuration.
Assume that there is no friction at the pivot point around which the pendulum rotates, and assume that there exists a torsional spring that counter acts the pendulum motion. Let the spring force Fs be a non-linear function of the displacement of the pendulum θ from the vertical configuration, that is,
F=K\theta^{3}
Considering the presence of gravitational forces, ignoring external torques on the pendulum

To find a the differential equation governing the pendulum dynamics.

Kind regards

Homework Equations

The Attempt at a Solution

J\ddot{\theta}=T-m*g*sin(\theta)
I am tying to find T as a function of F

 

[Andrew Mason]

Homework Statement

Consider a pendulum oscillation problem, where pendulum oscillates around the vertical in the downward configuration.
Assume that there is no friction at the pivot point around which the pendulum rotates, and assume that there exists a torsional spring that counter acts the pendulum motion. Let the spring force Fs be a non-linear function of the displacement of the pendulum θ from the vertical configuration, that is,
F=K\theta^{3}
Considering the presence of gravitational forces, ignoring external torques on the pendulum

To find a the differential equation governing the pendulum dynamics.
...

The Attempt at a Solution

J\ddot{\theta}=T-m*g*sin(\theta)
I am tying to find T as a function of F

There are two restoring forces here: gravity and the spring force. Write out the expression for the restoring force of gravity as a function of angle \theta. Write out the expression for the restoring force of the spring as a function of \theta. Since the two forces are always the same direction, add them together to find the total force.

How is the total force on the pendulum bob related to its acceleration? How is this acceleration related to the angular acceleration (the rate of change of angular speed)?

Answer those questions and you will be able to set up the differential equation of motion.(hint: it is in the form:

\ddot{\theta} = -f(\theta)

AM

 

[hancock.yang@]

There are two restoring forces here: gravity and the spring force. Write out the expression for the restoring force of gravity as a function of angle \theta. Write out the expression for the restoring force of the spring as a function of \theta. Since the two forces are always the same direction, add them together to find the total force.

How is the total force on the pendulum bob related to its acceleration? How is this acceleration related to the angular acceleration (the rate of change of angular speed)?

Answer those questions and you will be able to set up the differential equation of motion.(hint: it is in the form:

\ddot{\theta} = -f(\theta)

AM

Thank you for your help.
I have already worked it out .