langrange_deriva

# langrange_deriva - INERTIAL MECHANICS Neville Hogan The...

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INERTIAL MECHANICS Neville Hogan The inertial behavior of a mechanism is substantially more complicated than that of a translating rigid body. Strictly speaking, the dynamics are simple; the underlying mechanical physics is still described by Newton’s laws. The complexity arises from the kinematic constraints between the motions of its members. One powerful method to describe inertial mechanics is Lagrange’s equation, which is traditionally introduced using the variational calculus with Hamilton’s principle of stationary action. Here’s a more direct approach that may provide more insight. Inertial Mechanics page 1 Neville Hogan

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L AGRANGE S EQUATION FOR MECHANISMS Begin with the uncoupled members of the mechanism. x uncoupled coordinates (orientations, locations of mass centers) with respect to a non-accelerating (inertial) reference frame v velocities p momenta f forces These four fundamental quantities are related as follows. d x /dt = v d p /dt = f Inertial Mechanics page 2 Neville Hogan
The constitutive equation for kinetic energy storage (inertia) is: p = Mv M diagonal matrix of inertial parameters (masses, moments of inertia, e.g. about mass centers) Kinetic co -energy is the dual of kinetic energy: E k * = p t d v = 1 2 v t Mv = E k * ( v ) Thus, by definition: p = E k * / v Inertial Mechanics page 3 Neville Hogan

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The underlying mechanical physics is fundamentally independent of choice of coordinates. Therefore, these may be regarded as tensor equations. By the usual conventions: v is a contravariant rank 1 tensor (vector) M is a twice co-variant rank 2 tensor p is covariant rank 1 tensor (vector) These observations become more useful when we consider transformations of variables. Inertial Mechanics
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## langrange_deriva - INERTIAL MECHANICS Neville Hogan The...

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