Advanced Computer Graphics Spring 2008 K. H. Ko Department of Mechatronics Gwangju Institute of Science and Technology

Today’s Topics Rigid Body Motion Newtonian Dynamics Lagrangian Dynamics Euler’s Equations of Motion

Introduction Lagrangian Dynamics A framework for setting up the equations of motion for objects when constraints are present. The equations of motion are derived from the kinetic energy function and naturally incorporate the constraints. We could reduce the computational time of the simulation compared to a general-purpose system using Newtonian dynamics.

Newtonian Dynamics Dynamics describes how the particle must move when external forces are acting on it. We specify the acceleration and integrate to obtain the velocity and position. This is not always possible in a closed form. So many problems require numerical methods to approximate the solution.

Lagrangian Dynamics Inertial Frame Consider the motion of a mass over time F = ma = mv’ = mx’’. a: acceleration, v: velocity, x: position These quantities are measured with respect to some coordinate system, which is referred to as the inertial frame. It can be fixed, can have a constant velocity and no rotation. Any other frame of reference is referred to as a noninertial frame.

Lagrangian Dynamics The simplicity of Newton’s second law can disguise the complexity of the problem. F = ma We must represent all relevant forces for F. External forces and constraining forces that apply to the mass. This motivates what is called Lagrangian dynamics.

Lagrangian Dynamics Equations of Motion for a Particle on a Curve Given a curve x(q), q: parameter The Lagrangian equation of motion for a single particle constrained to a curve x(q) Fq is referred to as a generalized force. The force term in the equation eliminates the constraining forces

Lagrangian Dynamics Equations of Motion for a Particle on a Surface Given a curve x(q1,q2) The Lagrangian equation of motion for a single particle constrained to a curve x(q) I = 1,2

Lagrangian Dynamics Determining Constraint Forces The degrees of freedom may very well be greater than three. When constrained to a curve, you have two degrees of freedom and two equations governing the motion. A Lagrangian equation occurs for each degree of freedom. The construction that led to the Lagrangian equations applies equally well to additional parameters, even if those parameters are not freely varying. The generalized forces in these equations must include terms from the forces of constraint. These additional equations allow us to determine the actual constraint forces.

Lagrangian Dynamics Time-Varying Frames or Constraints If the frame of reference varies over time or if the constraining curve or surface varies over time, the Lagrangian equations of motion still apply. The Lagrangian formulation is the natural extension of Newton’s second law when the motion is constrained to a manifold (curve or surface).

Lagrangian Dynamics Equations of Motion for a System of Particles For each particle, we find a Largrangian equation of motion under each constraint of interest. The Lagrangian equations of motion are obtained by summing those for the individual particles, leading to

Lagrangian Dynamics Equations of Motion for a Continuum of Mass The Lagrangian equations of motion are also valid for a continuum of mass.

Lagrangian Dynamics Equations of Motion for a Continuum of Mass Using a transformation to local coordinates, V = vcen + wⅹr The first term is the energy due to the linear velocity of the center of mass. The last terms are the energies due to the angular velocity about principal direction lines through the center of mass.

Euler’s Equations of Motion Sometimes a physical application is more naturally modeled in terms of rotation s about axes in a coordinate system. A spinning top. The top rotates about its axis of symmetry. Simultaneously the entire top is rotating about a vertical axis.

Euler’s Equations of Motion Coordinate Systems

Euler’s Equations of Motion The angular velocity in world coordinates is

Euler’s Equations of Motion The angular velocity in body coordinates is

Euler’s Equations of Motion μi : the principal moments,