tailieunhanh - Applied Structural and Mechanical Vibrations 2009 Part 5

Tham khảo tài liệu 'applied structural and mechanical vibrations 2009 part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | where G1 s and G2 s are the transforms of g1 t and g2 t respectively. From a table of transforms we get so that the convolution integral is zero for t t1 leaving only the first term in eq and 1 r . 1 - sin - T dr - 1 - cos Lự í - Í1 J i for t t1. The inverse transformation of eq finally yields _ 1 x t 1 cos v t ti k . 1 r . . x i 7 cos ưn t - Í1 - cos cc t t ti which aside from the constant f0 are exactly eqs and . So far we have not yet considered the possibility of obtaining directly the final solution satisfying given initial conditions. This is one advantage of solving linear differential equations with constant coefficients by the Laplace transform method. By standard methods one finds a general solution containing arbitrary constants and further calculations for the values of the constants are needed to solve a particular problem. The Laplace transforms of derivatives given in Chapter 2 eqs and will now be used to clarify this point. Let us consider the general equation of motion for a damped SDOF system mx ex kx fit with initial conditions x 0 Xồ and The Laplace transformation of both sides gives wii2X - sxo - no c sX - Xo kx F s where as customary we are using lower-case letters for functions in the time domain and capital letters for functions in the transformed domain. Solving for X s and rearranging leads to F s x0 s 2 n v0 5 s2 ứị iQjJnS s2 u 2 2 .ưns 2 a 2 2Qx S Copyright 2003 Taylor Francis Group LLC The first term on the right-hand side product of two functions of s transforms back to the convolution integral 1 0 f r e T sin - r dr the second term transforms back to see any list of Laplace transforms _ -1 Xo I cosier sinWdt Wrf and we have already considered the third term whose inverse transform is Vo . e sinwji The sum of the three expressions b and c finally gives the general response IT. . . x t ----- f T e r sirtu t r dr muj Jo . -r . . . I . . v0 xo . e I Xo cos udt .