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                                                 Resonant Micromechanical Systems

                              264   Chapter Five


                                                                           x
                              actuation      beam-spring                           k

                                                x                      Fcf           c
                                           m                                m


                               beam-spring       sensing
                                                                                   k



                                          (a)                                    (b)
                              Figure 5.41 Electrostatic actuation and  beam-spring  suspension: (a) schematic
                              microdevice; (b) lumped-parameter model.

                                                 mx ˙˙ + cx ˙ +2kx = F sin (Ȧt)           (5.92)
                                                                0
                                where it is assumed that the electrostatic force F cf  varies according to a sine
                                law. By using Eqs. (5.91) and (1.13), Eq. (5.92) is reformulated as
                                                    2
                                                  Ȧ z         F 0
                                                    r 0
                                                          2
                                               x ˙˙ +  + Ȧ x =   sin (Ȧt)                 (5.93)
                                                   ʌȝx ˙  r   m
                                Equation (5.93) is nonlinear as the velocity is in the denominator of the sec-
                                ond term of the left-hand side. This equation can be used to determine the
                                average damping ratio by applying a numerical integration scheme, such as
                                                             9
                                the Newmark procedure (see Wood,  for instance) which gives two equations
                                connecting the displacement, velocity, and acceleration of a single-DOF sys-
                                tem (such as the one studied here) at two consecutive moments in time in the
                                form:

                                                      (ǻt) 2         (ǻt) 2
                                       x   = x + ǻtx ˙ +  (1 íȕ ) x ˙˙ +  ȕ x ˙˙
                                        i+1   i     i   2      2  i   2   2 i+1           (5.94)
                                       x ˙  = x ˙ + ǻt(1 íȕ ) x ˙˙ + ǻt ȕ x ˙˙
                                        i+1   i        1  i    1 i+1
                                In Eqs. (5.94), ¨t is the sampling rate (the time interval between recording
                                two consecutive capacitive measurements), whereas ȕ 1  and ȕ 2  are numerical
                                control parameters, for values greater than 0.5, the Newmark scheme is un-
                                                                      9
                                conditionally stable, as mentioned by Wood.  Equation (5.93) can also be
                                sampled corresponding to the time stations i and i + 1, namely,









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