J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, Harcourt Brace Jovanovich, Publishers, third edition, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....local lattices by expressing (s l ; t l ; u l ) as a function of another level of lattices in a fashion similar to Equation (3) 12 mass points Figure 6: Mass distribution for a 2D table 2. 5 Equations of Motion To incorporate dynamics into our flexible models we use a Lagrangian formulation [19]. A mass distribution is associated with the object by discretizing the object in material coordinates using mass points. For example, a 2D table could be assigned a discrete mass distribution as shown in Fig. 6. In practice, approximating an object s distribution using 4 Gamma 10 mass points is ....

J. B. Marion and S. T. Thornton, Classical Dynamics of Particles and Systems, Harcourt Brace Jovanovich, Publishers, third edition, 1988.

....that we apply local deformations using (3) before global deformations using (2) 4 The dynamics formulation presented next will ensure that a force which produces a local deformation appropriately affects the global deformations. 4. 4 Equations of Motion We use a Lagrangian formulation [19] to make our flexible models dynamic. A mass distribution is associated with the object by discretizing it in material coordinates using mass points. For example, a 2D table could be assigned a discrete mass distribution, as shown in Fig. 5. In practice, approximating an object s distribution ....

J.B. Marion and S.T. Thornton, Classical Dynamics of Particles and Systems, third edition. Harcourt Brace Jovanovich, 1988.

....periods between colliding objects. The rest of this chapter presents an overview of existing solutions for simulating the motion of flexible models. The next section presents background information concerning Lagrangian dynamics. 2. 1 Lagrangian Dynamics The Lagrangian dynamics formulation [18, 28, 61] is a formulation of the physical laws of motion. It is a very general formulation and in many cases it results in compact equations. It generalizes the well known Newtonian law F = m x into Q = M q where Q are the generalized forces, M is the generalized mass matrix and q is the vector of the ....

....well known Newtonian law F = m x into Q = M q where Q are the generalized forces, M is the generalized mass matrix and q is the vector of the generalized accelerations. The Lagrangian formulation can be derived from Hamilton s Principle, a variational principle which can be summarized as follows [28]: Of all the possible paths along which a dynamical system may move from one point to another in configuration space within a specified time interval, the actual path followed is that which minimizes the time integral of the Lagrangian function of the system. Configuration space implies the ....

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Jerry B. Marion and Stephen T. Thornton. Classical Dynamics of Particles and Systems. Harcourt Brace Jovanovich, Publishers, third edition, 1988.

.... q(t) Delta p Delta q(t) v(t) Delta q(t) Delta p Delta q(t) 2v(t) Delta q(t) Delta p Delta q(t) 2v(t) Delta p(t) 2v(t) Theta p(t) 15 When we interpret (t) 2v(t) 2 R 3 as the angular velocity, the above is exactly the same as the formula given in classical dynamics [19, 28]: p 0 (t) t) Theta p(t) 16) 3.3 The Jacobian Matrix of the Transformation U Given fixed 3D points p i 2 R 3 (for i = 1; m) let p i (t) be the rotated point of p i by the 3D rotation R q(t) of the unit quaternion q(t) Then, we have p 0 i (t) t) Theta p i (t) ....

Marion, J., and Thornton, S., Classical Dynamics of Particles and Systems, 3rd Ed., Harcourt Brace Jovanovich, Pub., Orlando, Florida, 1988.

....The popularity of symplectic integrators stems from the fact that the time t flow of any Hamiltonian system is a symplectic or canonical mapping. One consequence of this is that the mapping is volume preserving in phase space. This is a consequence of Liouville s Theorem (see, for example, [56]) which states that the phase space density of trajectories inside an ensemble remains constant with time. A symplectic integrator also preserves phase space volume to within the machine precision. As a result, symplectic integrators tend to preserve qualitative properties of phase space ....

Jerry B. Marion and Stephen T. Thornton. Classical Dynamics of Particles and Systems. Harcourt Brace Jovanovich, third edition, 1988.

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