langrange_deriva

langrange_deriva - INERTIAL MECHANICS Neville Hogan The...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 14

langrange_deriva - INERTIAL MECHANICS Neville Hogan The...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online