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Mechanics of Microelectromechanical Systems - N.Lobontiu and E.Garcia Part 9

Tham khảo tài liệu 'mechanics of microelectromechanical systems - n.lobontiu and e.garcia part 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 228 Chapter 4 where D is the dielectric displacement vector and ecơ is the electrical permittivity matrix the subscript Ơ indicates that the matrix is determined under constant-stress conditions . The vector D is defined as D - Dị D2 D Jr 4.108 and the symmetric permittivity matrix is ell g12 e J -2 22 22 4.109 31 2 2 633. When premultiplying Eq. 4.99 by the vector 1 2 ơ T the following equation is obtained which contains only specific energy energy per unit volume terms ơ T Em ưm UPZT 4.110 where ưm is the mechanical energy and is formulated as ưm ơf cE ơ 4.111 and Upz.T is the piezoelectric energy defined as UPZT ơ T d E 4.112 The following equation can be obtained from Eq. 4.107 through leftmultiplication by 1 2 E r E T D UPZT Ue 4.113 where the electric energy Je is Ee E T EJ 4.114 4. Microtransduction actuation and sensing 229 The energy formulation is useful as it allows introducing an amount the piezoelectric coupling factor which is defined as 4.115 and which gives the measure of the degree of energy conversion efficiency. Example 4.12 Determine the coupling factor kpe for the case defined in Example 4.11 knowing that the electrical permittivity Ễ33 is 15 X 10 9 F m. Solution For the particular problem of the previous example the stress vector reduces to the Ơ3 component. Similarly the compliance matrix is singletermed as it only contains C33 the permittivity matrix reduces to its Ễ33 component and the electric field vector reduces to E3. The piezoelectric energy will be in this case UpzT - 2 33 3Ơ3 4.116 The mechanical energy is rr 2 4.117 and the electrical energy simplifies to 4.118 By substituting Eqs. 4.116 4 117 and 4.118 into the definition equation - Eq. 4.115 the following equation is obtained for the coupling factor kpe - 33 ylcĩĩEĩĩ 4.119 and it can be shown by analyzing the strain-stress relationship of Eq. 4.105 - Example 4.11 - that C3 3 1 E where E is the Young s modulus about direction 3 or z . As a consequence Eq. 4.119 yields a coupling .