东南大学：《弹性力学》课程教学课件（英文讲稿）02 Mathematical Preliminaries

Sample Problem - Solution106161918(matrix)a,ajk=a,aik+az2a2+aga3k=61071016(vector)a,b, =a,b +azb, +ab, =8a,bb, =a.,bb +ai2bb, +ajbb +a,bb +..=84 (scalar)b,b, = b,b, + b,b, +b,b, = 4+16+ 0 = 20 (scalar)4808b.b160(matrix)00020112011204142A1(matrix)smm20132222/027012042410--12(matrix)aant2321020-1116

    symm. anti. 1 2 0 1 0 2 1 1 1 1 1 1 0 4 3 2 4 1 1 4 2 (matrix) 2 2 2 2 1 2 0 3 2 1 2 2 1 2 0 1 0 2 0 1 1 1 1 1 0 4 3 2 4 1 1 0 1 (matrix) 2 2 2 2 1 2 0 3 2 1 1 0 ij ji ij ji a a a a a a                                                  Sample Problem - Solution 16

Coordinate Transformation: Unit base vectors for each frame4X3[e] = (ei,e2,es)[e.] = (er,e2,e3];TransformationmatrixQ, = cos(x,x,)Lee3e.e' = Que, +Q2e2 +Q13e3X2e2ee, = Q21e +Q22e2 +Q23e3ee', = Q3,e, +Q32e, +Q33e3XRelations between base vectorse, =Q,e'e; =Q,ej;17

Coordinate Transformation e e ,e ,e e e ,e ,e i 1 2 3 i 1 2 3     ;         • Unit base vectors for each frame 1 11 12 13 21 1 22 2 23 31 32 33 cos( , ) Q x x ij i j Q Q Q Q e Q e Q Q Q Q               1 2 3 2 3 3 1 2 3 e e e e e e e e e e ; Q Q ij ji e e e e   i j i j   • Transformation matrix • Relations between base vectors 17