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Dechev university of victoria 12 examples of moment

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Unformatted text preview: s theorem’ © N. Dechev, University of Victoria 12 Examples of Moment of Inertia See Class Notes © N. Dechev, University of Victoria 13 Beam Deflection A unique analytical solution exists for beam deflection, given by: (a) beam geometry (b) loading conditions (c) boundary conditions Generally, beam curvature ! (rho), can be defined as: Additionally, we can also define ! as: where ‘v’ is the beam deflection from the initial position. This approximation is valid when v < 5% of beam length. © N. Dechev, University of Victoria 14 Beam Deflection Therefore, we can develop the following differential equation, which can be solved for any beam, given the specific beam (a) geometry, (b) loading condition and (c) boundary condition: It is beyond the scope of this course to solve these equations. For this course, we can use Appendix B in the textbook, which provides the deflections associated with this equation, for common ‘general cases’ found with MEMS beams. Example of beam deflection for cantilever beam: d q F © N. Dechev, University of Victoria 15 Beam Stiffness A useful concept in predicting the forces and deflections within MEMS beams is the concept of ‘stiffness’. The stiffness model normally associated with springs can be expressed as: Where K is a constant of proportionality that defines the relation between applied force, F, and the resulting spring deflection, x. © N. Dechev, University of Victoria 16 Beam Stiffness Given the equation for the tip deflection of a beam, we can define that beam’s stiffness as: Example of beam stiffness: Consider the cantilever beam in the previous example: d q F Since: Therefore: © N. Dechev, University of Victoria 17 Calculation of Combined Mechanical Stiffnesses Computation of Stiffness for Springs in Series. Computation of Stiffness for Springs in Parallel See Class Notes © N. Dechev, University of Victoria 18 Beam Torsion For some MEMS applications, the beams that allow the sensor or actuator to move undergo a twisting/torsional action. In these cases, it is useful to review the basic formulas governing the torsion of beams, to determine: (a) Maximum stress and it’s location (b) Beam Stiffness (c) Beam Deflection © N. Dechev, University of Victoria 19 Beam Torsion For Circular Beams The basic assumptions for the torsion of circular beams (a) All sections initially plane and perpendicular to the lengthwise axis, remain plane after torsion. (b) Following twisting, all cross-sections remain ‘undistorted’ and have a linear variation of stress from the center of twist (where !xy=0) to the outer surface (where !xy= !max). (c) Material is homogeneous and obeys Hooke’s law. © N. Dechev, University of Victoria 20 Beam Torsion For Circular Beams The governing equations for circular beam torsion are presented below, without derivation: where: " - shear stress T - applied torque r - radius from center to point of interest J - polar moment of inertia For circular x-section For deformation, we have: where: - angle of twist per unit length G - Modulus of shear and since where: - angle of twist 21 Beam Torsion For Non-Circular Beams The governing equations for non-circular beam tors...
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