Free rigid body

It has been observed by [27, 24] that the equations of motion of a free rigid body are subject to reduction. (For a detailed discussion of this interesting topic, see [23].) This leads to an unconstrained Lie-Poisson system which is directly solvable by splitting, i.e. the Euler equations in the angular momenta ... 

Favro, L.D. Theory of the rotational Brownian motion of a free rigid body. Phys. Rev. 1960,119, 53. 

Degree of freedom Independent variable defining the position. A free rigid body has six degrees of freedom, a ball joint three, and a hinge joint one. 

An elegant form of the equations of motion for the free rigid body is expressed in terms of the angular momentum vector referred to the body frame. 

These are the so-called Euler equations for the free rigid body. Let us observe that (4.41) is in the form... 

Each of these is, it turns out, easily integrated as a planar rotation (see Exercise 7). This yields a simple, efficient algorithm for the free rigid body which is symplectic in a generalized sense. 

In this section, a standard bond graph will be developed for a free rigid body with three-dimensional movement in which multi-bond graphs will be used. To define... 

In order to examine the effect of using a rubber shock absorber on the computed responses after impact, a simple numerical example of two free rigid bodies of equal masses that collide with a constant relative velocity has been performed. Two different circumstances have been considered regarding the area of ctMitact. In the first case, concrete-to-concrete impact was considered and the modified linear viscoelastic impact model was used (Eq. 1). fri the second case, a rubber bumper 5 cm thick was assumed to be incorporated at the area of contact, which is simulated using the nonlinear impact model with hysteretic... 

For the model of free point particles the Newtonian equations present by far the simplest and most efficient analytical fonnalism. In contrast, for chains of rigid bodies, there are several different, but equally applicable, analytical methods in mechanics, with their spe-... 

It is often necessary to compute the forces in structures made up of connected rigid bodies. A free-body diagram of the entire structure is used to develop an equation or equations of equilibrium based on the body weight of the structure and the external forces. Then the structure is decomposed into its elements and equilibrium equations are written for each element, taking advantage of the fact that by Newton s third law the forces between two members at a common frictionless joint are equal and opposite. 

Moment of inertia It is the ratio of torque applied to a rigid body free to rotate about a given axis to the angular acceleration thus produced about that axis. 

The chitobiose unit has been treated as a rigid body, and by using the full-matrix, least-squares, rigid-body, refinement procedure, the structure was refined to an R factor of 40.7%. Visually estimated intensities were used. The structure was found to be free from short contacts, and to be stabilized by an intrachain OH-3—0-5 hydrogen-bond and one interchain N-H—O hydrogen-bond. 

We have a collection of n point masses (the nuclei) held fixed relative to one another, but free to rotate as a whole. The rotational energy expression involves the moments of inertia of the body. The moment of inertia Ia of a rigid body about axis a is defined as... 

Two further energy functions based on Poisson-Boltzmann electrostatics and atomic solvation parameter (ASP)-based parameterizations of the solvation free energy changes43 were evaluated by Weng et al.57 in the context of side-chain optimization after rigid-body docking. 

The determination of accurate molecular structure from molecular rotational resonance (MRR) spectra has always been a great challenge to this branch of spectroscopy [/]. There are three basic facts which make this task feasible (1) the free rotation of a rigid body is described in classical as well as in quantum mechanics by only three parameters, the principal inertial moments of the body, Ig, g = x, v, z ... 

The rotational motion of the rigid set of mass points about any axis through its center of mass in the absence of exterior forces is known as the free rotation of the rigid body. The planar moment tensor for this motion, with the position vectors ra referred to an arbitrary basis system, can be compactly written as a dyadic (T denotes transposition) [8,32],... 

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