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Unformatted text preview: Electromechanical Coupling A-PDF Split3.1.2 DEMO : Purchase from www.A-PDF.com to remove the watermark The variables 2 and x are independent. Thus dA and dx can have arbitrary values, and the equation must be satisfied by requiring the coefficients of dA and dx to be zero: i =w,,(2, x) = x) ,
fe x) (3.1.22)
(3.1.23) ax If the stored energy is known, the electrical and mechanical terminal relations can now be calculated. Equations 3.1.22 and 3.1.23 can be generalized to describe a system with arbitrary numbers of electrical and mechanical terminal pairs (see Table 3.1). To illustrate this generalization we consider again the electric field system of Nelectrical terminal pairs and Mrotational terminal pairs which was described mathematically by (3.1.14) to (3.1.18). We now take the total differential of (3.1.18),
dW, = _- " aw i=1 aq, dq, + _1 + Maw, dO,. i=1 a
T (3.1.24) Subtraction of (3.1.24) from (3.1.15) yields 0 =1 v-dq, -1 e + -] dO,. (3.1.25) All N of the q,'s and M of the B,'s are independent. Thus each coefficient of dq, and dOe must be equal to zero: v =
aqi i = 1,2,..., N, (3.1.26) (3.1.27) Tie= o, i =1, 2,...M. These expressions are generalizations of (3.1.22) and (3.1.23) to describe systems with arbitrary numbers of terminal pairs. They indicate that when the stored energy W, is known as a function of the independent variables all terminal relations can be calculated (see Table 3.1). It is usually easier in practice to determine the electrical terminal relations by calculation or measurement than it is to determine the mechanical terminal relations or the stored energy. We have seen that the electrical terminal relations are sufficient to evaluate the stored energy if we choose a path of integration in variable space that keeps electrical excitations zero while mechanical variables are brought to their final values. Once the stored energy is known, the forcef e can be calculated as a derivative of the stored _ __ __ ...
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- Fall '11