Nasser M. Abbasi, updated Sept 10, 2012

This report is a description of the model used and mathematical derivation of equations of motion of the following dynamic system. It is a spring pendulum which carries as its “bob” another spring pendulum

The top spring represents the pendulum. It is a heavy spring with mass
and attached to the end of it is massless spring which holds a bob at the
end of it with mass . Both springs are assumed inﬁnitely rigid against
rotation and can only move along their length. If it helps visualize this, we can
imagine the lower spring to be wrapped around an inﬁnitely rigid thin massless
rod with the bob being a bead that slides along the rod attached to this
spring.

The bob has three degrees of freedom: the angle of rotation of the pendulum, the
distance of the bob along the lower rod and the current length of the top pendulum.
The top spring is assumed to have mass, while the lower spring is massless.

To derive the equations of motion (3 equations of motions are needed since there are
3 degrees of freedom) Lagrangian method is used. First the kinetic energy and then
the potential energy are derived.

Before deriving the Lagrangian, we need to discuss the top spring more. This spring will have relaxed length of which is the length when the spring is laying on a horizontal surface, hence its weight has no eﬀect. When this spring is held upright, it will now extend a little more due to its own weight. We use Rayleigh factor of to ﬁnd this additional extension. When we attach the bob, there will be an additional extension, called the static extension. It is at this state that the simulation starts. Any additional pulling of the spring at this time is what will cause the vibration.

Displacement of the top spring is measured from the static deﬂection position as shown in this diagram

To model the top spring, since it has mass and a length, its kinetic energy comes from rotational motion and motion due to translation of its center of mass in the top-down direction as it vibrates. For rotation it is modeled as rigid rod of moment of inertia where is the total current length of the top spring measured from the origin where the pendulum is attached. For kinetic energy due to translational vibration motion up and down, it is modeled as point mass located at the bottom of the spring of mass

For the lower spring, it is assumed to have zero relaxed length and the extension
of the lower spring is measured as starting from where the lower spring is
attached to the top spring. In the above diagram, all terms are assumed to be in the
positive sense of motion.

Given the above diagram, we can now ﬁnd the kinetic and potential energy of the
system and derive the equations of motion.

This is the position vector of the bob mass as measured in the inertial frame of reference. The origin of this frame of reference is where the top pendulum is attached. The position direction is towards the right, and positive direction is downwards.

This is speed of the bob after squaring it. It comes from the above expression.

Assume PE is zero at the bottom of the pendulum when it hangs in the vertical position.