Rotary inertia

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

Free Vibrations with Rotary Inertia and Shear Stress... 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

By assuming that the mode shapes are approximately sinusoidal, as occurs in the absence of shear and rotary inertia, the new solution proposed for Eq. (17.143) is... 

It should be noted that the rotary inertia is relevant only at high frequencies (f > lO Hz). In most cases,... 

Ffere, m equals the body mass, I3 is the 3x3 unity matrix and J is the 3x3 rotary inertia tensor. 

In section Nonlinear Flexural Dynamic Analysis of Beams with Shear Deformation Effect of this chapter, the geometrically nonlinear dynamic flexural analysis of homogeneous prismatic beam members taking into account shear deformation and rotary inertia effects (Timoshenko beam theory) is presented. The differential equations of... 

In this application the free vibrations of Timoshenko beams with very flexible boundary conditions is examined, since in this case the natural frequencies and the corresponding modeshapes are highly sensitive to the effects of shear deformation and rotary inertia (Aristizabal-Ochoa 2004). Thus, the linear free vibration analysis of piimed-free beams of length L = 5.0 m of various cross sections is examined (p = 2.40 tn/m, E = 25.42 GPa, G = 11.05 GPa). In Table 3 the geometric and inertia constants as well as the shear correction factors of the examined cross sections are presented (re = 1/a). The differential equations for the linear free vibrations of this special case can be obtained by Eq. 14 for zero axial stress resultant and external force as... 

Vmax, w max and the periods Ty, of the first cycle are presented for the aforementioned cases of analysis, including or ignoring shear deformation effect. In all of the aforementioned cases, rotary inertia has been taken into account. From the obtained results, the influence of... 

Brigham E (1988) Fast Fourier transform and its applications. Prentice Hall, Englewood Cliffs Foda MA (1999) Influence of shear deformation and rotary inertia on nonlinear free vibration of a beam with pinned ends. Comput Struct 71 663-670 Kausel E (2002) Nonclassical modes of unrestrained shear beams. ASCE J Eng Mech 128(6) 663-667 Mohri F, Azrar L, Potier-Ferry M (2004) Vibration analysis of buckled thin-walled beams with open sections. J Sound Vib 275 434-446... 

MEMS find wide applications in microsensors such as acoustic waves, biomedical, chemical, inertia, optical, pressure, radiation, and thermal microactuators like valves, pumps, and microfluidics electrical and optical relays and switches grippers tweezers and tongs as well as linear and rotary motors, etc., in various fields. They also find application in microdevice components such as palmtop reconnaissance aircrafts, minirobots and toys, microsurgical and mobile telecom equipment, read/ write heads in computer storage systems, as well as ink-jet printer heads [4]. 

A.xes, pi ineupal, of inertia, 284 of polarizability, 44 of symmetry, 78 alternating, 79, 80 improper, 79 jrroper, 78 rotary-reflection, 79... 

Section 8.2 described how different rotary rheometers are designed to control and to measure rotation rate, angular position, torque, temperature, and other variables. Equally important is the analysis of these measurements, conversion of the raw millivolts to material functions. Twenty years ago this was all done by hand, but today commercial rheometers spit out materials functions like G and G" in real time. Data analysis software is becoming a more and more important part of rheometer design. We have already seen that the inertia correction algorithms illustrated in Figure 8.2.11 can significantly extend the performance of controlled stress rheometers. 

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