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Mechanical Engineers Reference Book 12E Episode 17

Tham khảo tài liệu 'mechanical engineers reference book 12e episode 17', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Series and transforms 17 13 Figure 17.15 Sawtooth wave Basic formulae J t exp j2ir z d u j 5 u z exp -ịĨTTfl dt Change of sign and complex conjugates -z 5 - u z - Time and frequency shifts r and constant n z - t 5 t exp -j2Tr i- exp i2m z u z U f - d Scaling T constant u z 7 TU jT Products and convolutions zz Z v z U f V f uự v t Differentiation 0 j2rr G -j27rZzz z U f du t oi lda 5 d U f a da Integration 1 0 0 a and b real constants Figure 17.16 Pulse wave i r dr U f l lTTf 17.3.11 Square wave 4 X . sin 2n 1 Z 17.3.12 Triangular wave 8 . . 4. sin 2n - l wz . A s -J - TT 2n - 1 17.3.13 Sawtooth wave 2 X t .sin n ửt 77 n n 17.3.14 Pulse wave T 2t x- sin rz UT 7 2mrt M cos t i I rĩTĩT i i 17.3.15 Fourier transforms Among other applications these are used for converting from the time domain to the frequency domain. Interchange of functions t Z u -f Dirac delta functions 5 z 5 1 exp j2-rr 0 5 8Ự - fo Rect z unit length unit amplitude pulse centred on z 0 rect z i sin irf irf Gaussian distribution exp -7TZ2 exp -7r 2 Repeated and impulse delta function sampled waveforms u t -nT Í T UỢ ị ÔỰ - n T U f y 5 z - nT 5 1 7 y - n T -7- - X Parseval s lemma j u z v z dz j u z 2dz J Wf 2df 17.3.16 Laplace transforms vt v z exp -5Z Jo 17 14 Engineering mathematics Function Transform Remarks e 1 X a sin ù t i s2 bl2 cos bit s s2 bl2 sinh bit i 2 . .2 s Ù cosh bit s s2 I 2 t n lsn Hự i s H t - r - exp st s Heaviside step function x - f H t - r exp -sr x s Shift in t S t - r expt ST Dirac delta function exp atf x t x s a Shift in s exp -at sin bit Ừ s a 2 bl2 exp at cos bit 5 a x a 2 bl2 tx f dx s ds d 0 _ A dz .vx .v - x 0 dr s2x s - sx 0 - 0 d x v X dx x x s - s ix 0 - s -2x 0 . . . -sx -2 0 - x -1 Convolution integral Xi ơ- x2 r - a da- Xi s x2 .t 17.4 Matrices and determinants 17.4.1 Linear simultaneous equations The set of equations fliHi a12x2 . . . alnx b 21 1 a22x2 . . . 2n n bl a iXi a 2x2 a ztx b may be written symbolically Ax b in which A is the matrix of the coefficients