Page 323 - Mechanics of Microelectromechanical Systems
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310 Chapter 5
beam on its original longitudinal direction has the same length with the
unbent beam, the horizontal projection of the deformed beam is shorter by
the quantity Another major difference consists in the fact that the three
parameters defining the ultimate position of the microcantilever free tip,
namely: and are dependent. In bending, the essential difference
between the large- and the small-displacement theories consists in the way
the basic differential equation is formulated. While the large-displacement
theory takes the exact form:
the small-deflection theory ignores the slope second power in the
denominator of Eq. (5.128), and therefore studies the approximate
differential equation:
Another form of the exact Eq. (5.128) is:
as shown, for instance, by Timoshenko [4] or Gere and Timoshenko [5]. As
suggested in Fig. 5.44, the relationships between the curvilinear variable s
and its projections on the x and y axes are:
By taking into account that the bending moment at a generic position x is
simply equal to F times x, Eq. (5.130) can serve to take its derivative in terms
of the curvilinear coordinate s by also using Eq. (5.131), which results in:
The solution to this differential equation, by taking into consideration that the
curvature at the free end is zero, becomes:
where:
