ME 459 Dynamics of Machinery
Homework #15: Lagrange’s Equations with Configuration Constraints
1. The figure shows a pendulum made of a single slender
bar of mass m and length L.
a) Using Lagrange’s equations, formulate the equation of
motion of the pendulum using θ as the only
generalized coordinate.
b) Using Lagrange’s equations, formulate the equations of
motion of the pendulum using the set of constrained
generalized coordinates ( x, y,θ ). Then differentiate the
constraint equations to put them into the form of second
order ordinary differential equations. You should have
five equations in all. The equations will contain five
variables: x, y,θ and two Lagrange multipliers.
2. In Homework#13 you found the equation of motion of
the system shown using only the generalized coordinate
θ.
Using Lagrange’s equations, formulate the
equations of motion of the system using the set of
constrained generalized coordinates ( x, y,θ ). Then
differentiate the constraint equations to put them into
the form of second order ordinary differential
equations. You should have five equations in all. The
equations will contain five variables: x, y,θ and two
Lagrange multipliers.