The equations of motion of the pendulum were derived using the Lagrangian method. For the

-bob pendulum, there are

second-order nonlinear differential equations and

degrees of freedom. The equations are kept in their nonlinear form since

NDSolve was used for solving them.

*Mathematica* was used to do the analytical derivation due to the high complexity of algebra for the

case.

Initial position conditions can be changed by dragging the bob using the mouse. When you click in the display, there is a pause and then the mouse can be used to drag the bob to a new location. Logic inside the Demonstration detects which bob to drag based on the proximity of the current mouse location to known bob positions.

The top buttons labeled "play", "pause", "step", and "reset" are self-explanatory and used to control the simulation. The control labeled "duration" is used to set the maximum simulation time. When this time is reached, the simulation will restart from

automatically. The control labeled "

" is used to change the time step for the display. The smaller the time step, the more accurate the simulation, but it takes longer to complete.

The control labeled "number of bobs" is used to change the pendulum type to 1, 2, or 3 (simple, double, or triple). The controls below that are used to adjust the mass of the bobs, with units in

. All units in this simulation are in SI units. The controls below that are used to change the length of each pendulum bar.

The controls below that are used to set the initial conditions. Units are in radians for the angles and in radians per second for the angular velocities. The initial angle positions can also be set using the mouse as mentioned above; however, using the controls can provide more accurate settings if needed. For convenience, small buttons next to the control variables can be used to quickly set the value to zero.

The "gravity" control is used to select the gravitational constant

. The control labeled "show phase" is used to turn the phase portrait plot on and off. The red point in the phase portrait plot indicates the initial conditions, and the black point is the position at the end of the duration. The moving blue point is the position in phase space at the current time.

The control labeled "damping" is used to change the damping coefficient. At the bottom is a display of the energy plot, which shows the current kinetic energy (KE), potential energy (PE), and the total energy in Joules.

The angles

are all measured from the vertical line. When the pendulum is hanging vertically at rest, then all angles will have zero values at this position. The angles and velocities are taken to be positive in the counterclockwise direction.

[1] D. A. Wells,

*Schaum's Lagrangian Dynamics*, New York: McGraw-Hill, 1967.