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Applied Structural and Mechanical Vibrations 2009 Part 7

Tham khảo tài liệu 'applied structural and mechanical vibrations 2009 part 7', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | premultiply both sides by PT and postmultiply by P to get L - in diag 2 jWj - W2I diag o 2 2iư i jjị u2 p R p which we can write as D PTR-1P where we define for brevity of notation D diag w2 2iúử ịu i lú1 From the above it follows that D 1 PTR-1P -1 P-1RP-T which after pre- and postmultiplication of both sides by P and PT respectively leads to 7.26 so that the solution 7.24 can be written as 7.27 and the jk th element of the receptance matrix can be explicitly written as 7.28 Now the term in brackets in eq 7.28 looks indeed familiar and a slightly different approach to the problem will clarify this point. For a proportionally damped system the equations of motion 7.22 can be uncoupled with the aid of the modal matrix and written in normal coordinates as ỷị IQưịỷị w 2y - pzTfoe iiJt 7.29 Each equation of 7.29 is a forced SDOF equation with sinusoidal excitation. We assume a solution in the form yi t -y e iưt where ỹj is the complex amplitude response. Following Chapter 4 we arrive at the steady-state solution the counterpart of eq 4.42 7.30 where fif ýi uị Copyright 2003 Taylor Francis Group LLC By definition the frequency response function FRF is the coefficient H a of the response of a linear physically realizable system to the input e iut with this in mind we recognize that 7.31 is the th modal because it refers to normal or modal coordinates FRF. If we define the nxi vector ỷ ỹi ỹ2 ỹn T of response amplitudes we can put together the n equations 7.29 in the matrix expression ỷ diag H M PTfo 7.32 and the passage to physical coordinates is accomplished by the transformation 7.2 which for sinusoidal solutions translates into the relationship between amplitudes z Py. Hence z Pdiag H Prf0 7.33 which must be compared to eq 7.27 to conclude that RM p diag f P7 7.34a Equation 7.34a establishes the relationship between the FRF matrix R of receptances in physical coordinates and the FRF matrix of receptances in modal coordinates. This latter matrix is diagonal because in .