Page 200 - Mechanics of Microelectromechanical Systems

4. Microtransduction: actuation and sensing 187
4.4 is the picture of a real thermal actuator which consists of five bent beams
that are placed in parallel‚ in order to enhance the output force capabilities.

Figure 4.4 Bent beam actuator realized by the MUMPs technology

In order to determine the two performance parameters that have been
defined for a straight free-fixed bar under thermal heating‚ and which are the
free displacement and the bloc force‚ it can be considered that the beam is
subject to a force which is the equivalent of the free expansion at end point 1‚
as given in Eq. (4.2)‚ and which has been denoted as (to highlight its
thermal nature) in Fig. 4.3 (b). As a result of boundary constraints‚ the
support reacts with the force and the moment which have to be
determined first. This can be done by using the conditions of zero
displacement about the x-axis and zero rotation about the z-axis at point 1‚
together with Castigliano’s displacement theorem. The deflection at point 1
about the y-direction can be found similarly after calculating and by
applying the same theorem and by considering that bending and axial
deformations produce the total strain energy. The equation for is:

where A and are the cross-sectional area and moment of inertia about the
z-axis‚ respectively. Equation (4.8) indicates that the deflection at point 1
depends linearly on the temperature increase and non-linearly on the
geometric parameters defining the half bent beam. An example will be
analyzed next that studies the free displacement in terms of the defining
geometry.