Center of rigidity

The roof diaphragm is designed to transfer wall loads to the side shear walls. The diaphragm is fixed at both ends by continuous attachment to the walls. The center of mass coincides with the center of rigidity indicating no incidental torsion... 

Torsionally Coupled (TC) Effect An effect that describes the coupling of translation and torsion of a building due to the inconsistency of the center of rigidity and the center of mass. 

We now consider the formulation of the equations of motion for a rigid body pinned at its center of mass and acted on by a (possibly nonlinear) potential field. The Lagrangian in this case is... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

Cleaning is frequently aided mechanically. Foam balls scour the center of tubes, and hoUow-filter systems can be back-flushed. HoUow fibers and membranes attached to rigid supports can be back-pressured, thereby eliminating the pressure drop that holds redispersed films on the membrane surface. 

Let us consider systems which consist of a mixture of spherical atoms and rigid rotators, i.e., linear N2 molecules and spherical Ar atoms. We denote the position (in D dimensions) and momentum of the (point) particles i with mass m (modeling an Ar atom) by r, and p, and the center-of-mass position and momentum of the linear molecule / with mass M and moment of inertia I (modeling the N2 molecule) by R/ and P/, the normalized director of the linear molecule by n/, and the angular momentum by L/. 

FIG. 13 Average center-of-mass position of flexible chains of length / with respect to the nearest solid surfaces for different /. Diamonds denote a system of semi-rigid chains in which the opposite effect is observed [28]. 

Figure 13-23 suggests recommended pulse level (peak-to-peak) pressure pulsations for acceptable pipe vibration. Figure 13-24 presents allowable machinery and pipe vibration at safe limits and damage levels, and Figure 13-25 presents allowable pressure pulsations for various pipe spans between rigid supports when a 5 mil peak-to-peak vibration is allowed at the center of the pipe spans for the pipe sizes noted. 

The foregoing are volume integrals evaluated over the entire volume of the rigid body and dw is an infinitesimal element of weight. If the body is of uniform density, then the center of gravity is also called the centroid. Centroids of common lines, areas, and volumes are shown in Tables 2-1, 2-2, and 2-3. For a composite body made up of elementary shapes with known centroids and known weights the center of gravity can be found from... 

When dealing with the motions of rigid bodies or systems of rigid bodies, it is sometimes quite difficult to directly write out the equations of motion of the point in question as was done in Examples 2-6 and 2-7. It is sometimes more practical to analyze such a problem by relative motion. That is, first find the motion with respect to a nonaccelerating reference frame of some point on the body, typically the center of mass or axis of rotation, and vectorally add to this the motion of the point in question with respect to the reference point. 

When looking for the velocities of points on a rigid body, the method of instantaneous centers can often be used. If the velocity of two points on the body are known, those points and all other points on the body can be considered to be rotating with the same angular velocity about some motionless central point. This central point is called the instantaneous center of zero velocity. The instantaneous center generally moves through space as a function of time and has acceleration. It does not represent a point about which acceleration may be determined. 

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

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