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© 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:
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:
q F Since:
© 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
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