tailieunhanh - Crc Press Mechatronics Handbook 2002 By Laxxuss Episode 1 Part 4

Tham khảo tài liệu 'crc press mechatronics handbook 2002 by laxxuss episode 1 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | FIGURE Coupled lumped parameter electromechanical system with single degree of freedom mechanical motion x t . circuits by defining the charge on the capacitor Q as another generalized coordinate along with x . in Lagrange s formulation q1 x q2 Q. Then we add to the kinetic energy function a magnetic energy function Wm Q x and add to the potential energy an electric field energy function W Q x . The equations of both the mass and the circuit can then be derived from dd T Wm dt dqk d T Wm 1 d W We _ ãĩT d k Q The generalized force must also be modified to account for the energy dissipation in the resistor and the energy input of the applied voltage V t . Q1 -cx Q2 -RQ V t . In this example the magnetic energy is proportional to the inductance L x and the electric energy function is inversely proportional to the capacitance C x . Applying Lagrange s equations automatically results in expressions for the magnetic and electric forces as derivatives of the magnetic and electric energy functions respectively . Wm 1L x Q2 1 LI2 We -1 Q2 2 2 2C x _ dWm x Q 12dL x V _ dWe x Q _ 1 2 d 1 Fm dx 2 dx Fe dx 2Q dx C x These remarkable formulii are very useful in that one can calculate the electromagnetic forces by just knowing the dependence of the inductance and capacitance on the displacement x. These functions can often be found from electrical measurements of L and C. Example Electric Force on a Comb-Drive MEMS Actuator Consider the motion of an elastically constrained plate between two grounded fixed plates as in a MEMS comb-drive actuator in Fig. . When the moveable plate has a voltage V applied there is stored electric field energy in the two gaps given by W e V x 1 0v2a - kk-2 2 d0 - x In this expression the electric energy function is written in terms of the voltage V instead of the charge on the plates Q as in Eqs. and . Also the initial gap is d0 and the area of the plate is A. 2002 CRC Press LLC FIGURE Coupled lumped .