By Carl T. F. Ross

This ebook is aimed toward senior undergraduates, graduates and engineers. It fills the distance among the varied textbooks on conventional utilized Mechanics and postgraduate books on Finite point tools.

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4 Orthogonal matrices In engineering, these are usually met when certain physical properties which are orthogonal, are transformed from one set of cartesian coordinates to another. Typical examples of such cases are two-dimensional stress and strain systems and principal second moments of area. 63): /σ„ τ. \ re ,Ί/σ, τνψ where σ β = direct stress at any angle Θ σ<χ)+ο= direct stress at any angle 90 + Θ -,Ί Sec. 5] Matrix inverse 45 τ 0 = shear stress in 0 - (90 + Θ) plane σχ= direct stress in x direction Oy= direct stress in y direction xxy= shear stress in x-y plane 0= angle with x axis c= cosG s=sin0 .

1 — Chio's method for evaluation a determinant 19Θ 110 12Û 130 140 156 15Θ 170 ISO ISO JOO REN CHIO'S METHOD OF EXPANDING DETERMINANTS INPUT"TYPE IN ORDER OF DETERMINANT";N DIM FKN,N> PRI NT"TYPE IN THE MATRIX" FOP 1=1 TO H FOR J=l TO N PR I NT " TYPE IN FK " : I . " . T ; " :■ " INPUT fl(I,J> NEXT :,I DT=1 FOR I=1T0 N-l 210 ΡΙ=Μ·:Ι, ι:< 220 FOP J-I TO N 220 HÎI,,T>=A('I,J:J/PI 240 250 260 270 2S0 290 200 210 32© 330 240 350 NEXT J FOR K=I TO N-l CN=H FOR J=I TO N A=A'::K+1, J>-fla, J>#CN NEXT J NEXTK DT=DT*PI NEXT I DT=DT*FKN,N> PRI NT"DETERMINflNT=";DT END Equating coefficients, the following is obtained: 2 =/„x 1 + 0x0 - 2 = /21 x 1 + /22 X 0 /2i = - 2 6 = / n w 12 + 0 x l 6 4 = / 2 ι«ΐ2 + ' 2 2 χ 1 /22 = 4 - /21 u l2 = 4 - ( - 2) x 3 Sec.

48 Matrix algebra [Ch. 73). ^22*2 + /*23*3 + . . + A\„X„ = C2l A\2 x2 + A j-, x, + . . + A I, xn = C\ '. 73) An2X2 + ^ n 3 * 3 + · · · · + AnnX„ = C„ The above process can be repeated by eliminating xx, x3, etc. nxn = + B„nx„ = kn . 74) BLxn = kxn . 74), etc. the other unknowns can be determined. 71). 76). 76). 76). If this process is repeated to the nth equation, then the following set of simultaneous equations is obtained: Sec. 77). Multiply this equation by A\2, A\2, etc. and take these equations away from the third, fourth, etc.

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