弹篮波 注2:物理方程 Er=o-H(o, +o yya 2(1+) = E E 8y E [o,-(+01 2(1+p y E x 6:=l:-以(O+0, 2(1+) E E 16
16 注2：物理方程 [ ( )] 1 x x y z E = − + [ ( )] 1 y y z x E = − + [ ( )] 1 z z x y E = − + yz yz E 2(1+ ) = zx zx E 2(1+ ) = xy xy E 2(1+ ) =
Clastics wave Because the displacement component is difficult to be expressed by stress and its derivative, so movement equations of elasticity mechanics are usually solved according to the displacement Substitute the elasticity equations where stress components are expressed by displacement component into movement differential equations, and we let ou ov aw e=—+ Then we get Ox Yaz E de 2(1+)1-21Ox +V)+X-P2=0 at E 1 de +Vv)+Y-p-2=0 2(1+)1-2 Oy at E 1 ae 02w 2(1+)1-2Oz +V21)+Z-p =0 at 7
17 Because the displacement component is difficult to be expressed by stress and its derivative, so movement equations of elasticity mechanics are usually solved according to the displacement. Substitute the elasticity equations where stress components are expressed by displacement component into movement differential equations, and we let: Then we get: z w x y u e + + = ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t u u X x E e ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t Y y E e ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t w w Z z E e
弹篮波 由于位移分量很难用应力及其导数来表示,所以弹 性力学动力问题通常要按位移求解。将应力分量用位移」 分量表示的弹性方程代入运动微分方程,并令: Cu ov ow 十 Ox Oy az 得: E de 2(1+1)1-24 +V)+X-P2=0 at 2(1+)1-29,V2v)+-a20 E l de e1 de +V2)+Z-p 0 2(1+)1-2Oz at 18
18 由于位移分量很难用应力及其导数来表示，所以弹 性力学动力问题通常要按位移求解。将应力分量用位移 分量表示的弹性方程代入运动微分方程，并令： 得： z w x y u e + + = ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t u u X x E e ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t Y y E e ) 0 1 2 1 ( 2(1 ) 2 2 2 = + + − + − t w w Z z E e
Plastics wave These are the basic differential equations of movement equations solved by displacement. They are also called lame equations We need boundary conditions to solve Lame equations. Besides these we still need original conditions, because displacement components are the function of time variable In order to simplify calculation, usually we neglect body force. Now the movement differential equations of elastic objects can be simplified as E i de 0t22(1+)p1-2ax +V2) E de +V2v) 0n22(1+)p1-2Oy E 1 de 022(1+)1-2a×v1) 19
19 These are the basic differential equations of movement equations solved by displacement. They are also called Lame equations. We need boundary conditions to solve Lame equations. Besides these we still need original conditions , because displacement components are the function of time variable . In order to simplify calculation , usually we neglect body force. Now the movement differential equations of elastic objects can be simplified as : ) 1 2 1 ( 2(1 ) 2 2 2 u x E e t u + + − = ) 1 2 1 ( 2(1 ) 2 2 2 + + − = y E e t ) 1 2 1 ( 2(1 ) 2 2 2 w z E e t + + − =
弹篮波 这就是按位移求解动力问题的基本微分方程,也称 为拉密(Lame)方程。 要求解拉密方程,显然需要边界条件。除此之外, 由于位移分量还是时间变量的函数,因此求解动力问题 还要给出初始条件。 为求解上的简便,通常不计体力,此时弹性体的运 动微分方程简化为: E 1 de 0t22(1+p)p1-2uax +V2) E de +V2v) 0n22(1+)p1-2Oy E 1 de r22(1+)1-2a×v21p) 20
20 这就是按位移求解动力问题的基本微分方程，也称 为拉密（Lame)方程。 要求解拉密方程，显然需要边界条件。除此之外， 由于位移分量还是时间变量的函数，因此求解动力问题 还要给出初始条件。 为求解上的简便，通常不计体力，此时弹性体的运 动微分方程简化为： ) 1 2 1 ( 2(1 ) 2 2 2 u x E e t u + + − = ) 1 2 1 ( 2(1 ) 2 2 2 + + − = y E e t ) 1 2 1 ( 2(1 ) 2 2 2 w z E e t + + − =