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The Equations of Motion for a System
Linearization of the Equations of Motion
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absolute force unbalance accelerations Additionally algorithm application approximation becomes chain rule chapter Cipra computed configuration constant coordinate system coordinate vector damping defined degrees of freedom dependent developed differential equations displacement dynamic eigensystem eigenvalues and eigenvectors element equal to zero equations of motion error evaluate example problem expression figure force unbalance tolerance force vector formulation function geometric global coordinates gravitational identity matrix impact in-joint independent joint variables inertial input integration inverse iteration joint variable matrix joint variable velocities kinematic constraints kinetic energy Lagrange Lagrange's equations loop equation magnitude mass mathematical matrix products modal non-linear notation numerical oscillation partial derivative particular solution pendulum piston plow point-to-point springs potential energy predictor-corrector premultiplication R.dmR relative force unbalance respect revolute joint Runge-Kutta methods second term simulation spatial mechanisms specified step is reduced stiffness matrices suspension mechanism Taylor's series technique term of Lagrange's torque transformation matrix velocity matrix vibration XYZ coordinate system yields