tailieunhanh - MODELLING OF MECHANICAL SYSTEM VOLUME 2 Episode 2

Tham khảo tài liệu 'modelling of mechanical system volume 2 episode 2', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 18 Structural elements written as pX _ --------- GAX À G grad div X fe r t Vr e V -A r Ấ div XI n G grad X grad X I I n Ks X te r t Vre S V t V 0 V S t S 0 S Though vectors may be considered as being merely tensors of first rank it is preferred to mark the gradient of a scalar quantity by an upper arrow instead of a double bar in order to stress that the result is a vector. The first equation governs the local equilibrium at time t of a material particle located at r and the second equation stands for elastic boundary conditions. No prescribed motion has been assumed as it would bring nothing new to the formalism at least at this step. Finally the system taken as a whole is said to be homogeneous if no external loading of any kind is applied either to V or to S even as non-zero initial conditions. Otherwise it is said to be inhomogeneous. . Hamilton s principle Hamilton s principle has already been introduced and extensively used in AXI 04 for deriving the Lagrange equations of discrete systems. It is recalled that this variational principle is expressed analytically as 5 A t1 t2 5 L dt 0 where 5 denotes the operator of variation. A t1 t2 is the action between two arbitrary times t1 and t2 of the extended Lagrangian L defined as L Ek Ep Wq Ek q denotes the kinetic energy of the system Ep q the internal potential energy expressed in terms of the generalized displacements and velocity vectors q and q . Wq is the work function of extra external or and internal generalized force vectors Q applied to the system which are not necessarily conservative. The dimension of all the vectors just mentioned is equal to the number ND of the degrees of freedom DOF of the system. Here Hamilton s principle will be extended to continuous media providing us with a very efficient analytical tool for Solid mechanics 19 dealing with 1. the kinematical constraints 2. the boundary conditions 3. various numerical methods for obtaining approximate solutions of the differential .

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