Equivalent moment of inertia

The terms (/q + n /m) and (Co + rP-Cm) are ealled the equivalent moment of inertia /g and equivalent damping eoeffieient Cg referred to the output shaft. 

Rib design An example of how ribbing will provide the necessary equivalent moment of inertia and section modulus will be given. A flat plastic bar of IV2 in. x 3/8 in. thick and 10 in. long, supported at both ends and loaded at the center, was calculated to provide a specified deflection and stress level under a given load. The favorable material thickness of this plastic is 0.150 in. Using... 

The second approach ignores the strength contribution of the core and assumes that the two outer skins provide all the required rigidity (Fig. 6-21). The equivalent moment of inertia is then equal to ... 

The thin-wall bellows element should be designed for membrane stresses to conform to code-allowable stresses. The sum of membrane and secondary bending stresses should not exceed 1.5 times the yield stress in order to prevent the collapse of the corrugations caused by pressure. Bellows subjected to external pressure can be analyzed in a manner similar to a cylinder, utilizing an equivalent moment of inertia. The fatigue life can be estimated based on the sum of deflections and pressure stresses as compared to S/N curves based on bellows test data or using the curves in B31.3 Appendix X, Metal Bellows Expansion Joints. Formulas for the stress analysis of bellows are available in the Expansion Joints Manufacturing Association (EJMA) Standards (37). 

Dodge comments on this method in WRC Bulletin 198 Although the equivalent moment of inertia procedure is simple and direct, it was not derived by any mathematical or logical reasoning which would allow the designer to rationalize the accuracy of the results. ... 

The second approach ignores the strength contribution of the core and assumes that the two outer skins provide all the rigidity (Fig. 5-61). The equivalent moment of inertia is then equal to Ix = b h - hi)/l2. This formula results in conservative accuracy, since the core does contribute to the stress-absorbing function. It also adds a built-in safety factor to a loaded beam or plate element when safety is a concern. 

An example of how ribbing can provide the necessary equivalent moment of inertia and section modulus follows. 

A better estimate of the shape of the polymer molecules, since they are highly anisotropic, is a representation of each molecule in terms of an equivalent spheroid with the same moment of inertia [45,46]. This is achieved by diagonalizing the moment of inertia tensor to obtain the eigenvectors a, b, and c and the principal moments 7, I/,/, and Icc. The moment of inertia tensor of molecule j is given by... 

Moment of inertia The rotational equivalent of mass required to overcome the rotation of a body, depending on the mass and distance of the mass from the axes of rotation. 

The MMI (mass moment of inertia), EXC (excitation factor), and ZPE (zero point energy) terms are defined on successive lines of Equation 4.145. For reactions involving heavier isotopes the effects are no longer concentrated in the ZPE term and it is convenient to apply the Teller-Redlich product rule (Section 3.5.1) and eliminate the moments of inertia by using Equations 4.79,4.79a, and 4.141, thus obtaining an equivalent relation... 

Ethylene was one of the first systems subjected to detailed vibrational analysis using HOCM modified to account for lattice anharmonicity. Agreement with experiment is excellent (Fig. 5.5). The differences in the VPIE s of the equivalent isotopomers cis- trans-, and gem-dideuteroethylene (Fig. 5.6) are of considerable interest since they neatly demonstrate the close connection between molecular structure and isotope chemistry. The IE s are mainly a consequence of hindered rotation in the liquid (moments of inertia for cis-, trans-, and gem-C2D2H2 are slightly... 

There are N(N-l)/2 distinct distances in a cluster of N atoms, disregarding symmetry-dictated equivalencies. This set of distances is of course redundant 3N-6 Cartesian coordinates are sufficient to determine molecular geometry, apart from the position of the center of mass and the orientation of the principle moments of inertia. 

Few details of the apparatus are given in ISO 4663, it is simply stated that means shall be provided to measure frequency to 1% ( 5% in a transition region), amplitude to 1% and, for method C, the supplied energy to 2%. It is suggested that a moment of inertia of about 0.03 gm is suitable for the inertia member which may be a disc or rod. For methods B and C the torsion wire should be of such dimensions that its restoring torque is not more than 25% of the total restoring torque due to the test piece and suspension. BS 903 (equivalent to method B of ISO 4663) suggests that moments of inertia between 50 and 500 g cm are suitable and states that the tensile strain on the test piece should be between 0 and 5%. The British... 

A qualitative idea of the surface viscosity is given by noting the ease with which talc can be blown about the surface. Most insoluble films have surface viscosities of c. 10 6 kg s l to 10 3 kg s-1 (for films 10-9 m thick this is equivalent to a bulk viscosity range of c. 103 kg m"1 s 1 to 106 kg itT1 s"1). These films can be studied by means of a damped oscillation method (Figure 4.20). For a vane of length / and a disc with a moment of inertia /,... 

The accepted mean density of Mars, based on its measured volume and determination of its mass from spacecraft orbits, is 3.9335 0.0004 g cm (Lodders and Fegley, 1998). The density of the elastic lithosphere (approximately equivalent to the crust), estimated from models of the relationship between gravity and topography from Mars Global Surveyor data, is 2.95-2.99 g cm (McKenzie et al., 2002), which is similar to the density of basalt. The planet s dimensionless moment of inertia (0.3662 0.0017), calculated from Mars Pathfinder measurement of the rate at which its spin pole precesses (Folkner et al., 1997), constrains the core radius to —1,300-1,500 km, depending on core composition. 

According to Rudolph [33], structures determined by fitting structural parameters directly to the experimentally determined rotational constants Bq, the moments of inertia Iq or the planar moments Pq of a series of isotopomers are called rg structures. If the covariances of the observables are transformed correctly, the structures resulting fi-om any of these fits are identical. However, if covariances are not transformed, these structures are not identical nor equivalent (for examples, see [34]). Therefore, one should always specify what kind of covariance matrix is used with a particular fit. Pseudo-Kraitchman (p-Kr) or pseudo-r fits are those in which the structures are fit to differences of rotational constants, moments of inertia or planar moments. The structures called and again are equivalent to each other if the covariances are transformed correctly. On the other hand, the structure is theoretically different fi om the and structures because of a lack of a linear relation between the AB and A/. In practice, structures are very close to other p-Kr structures. The rig structure is obtained when the isotopic moments Iq are fitted to structural parameters and three e parameters. It is... 

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