东南大学：《固体力学基础》课程教学课件（英文讲稿）01 Mathematical Preliminary

Sample Problem - Solution106119618(matrix)a,ajk=anaik+a2a2k+ai3ak=1076[10]a,b, =a,b, +a,2b, +a,b, =16(vector)1[8a,b,b, =abb, +ai2b,b +ar3b,b +a2ib,b, +...= 84 (scalar)bb,=b,b,+b,b,+b,b,=4+16+0=20(scalar)0848160(matrix)b.b10000202 111114442P21(matrix)1223211(00002-1104324-111(matrix)220320-121I16

    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                                                                          1 1 2 2 3 3 1 1 2 2 3 3 11 1 1 12 1 2 13 1 3 21 2 1 1 1 2 2 3 3 1 10 6 6 19 18 (matrix) 6 10 7 10 16 (vector) 8 84 (scalar) 4 16 0 20 (scalar) ij jk i k i k i k ij j i i i ij i j i i i j a a a a a a a a a b a b a b a b a b b a b b a b b a b b a b b b b b b b b b b b b                                           4 8 0 8 16 0 (matrix) 0 0 0           Sample Problem - Solution 16

Coordinate Transformation. Unit base vectors for each frameX3[e,] =(e1,e2,e3];[e'] = (e',e2,e')InTransformation matrixO, = cos(xf,x,)eeree' = Qne1 +Q12e2 +Q13e3X2ee, = Q21e1 +Q22e2 +Q23e3ee' = Q31e, +Q32e2 +Q33e3XRelations between base vectorse; = Q,ej;e; =Qje'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               123 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