Simple Elastic Systems, An Introduction Based on Geometry

Elastic systems like strings, beams, membranes or plates are special approximations of the linearized equations of simple elasticity. Based on the principles conservation of mass and balance of linear momentum the equations of simple elasticity are derived. Balance of momentum of momentum is taken into account by the strong constitutive assumption, the Cauchy stress tensor is symmetric. The equations of motion of simple elasticity can be rewritten as Lagrangian or Hamiltonian equations with distributed ports, which describe the energy exchange with the environment. Since these equations are often too complex, a reduction procedure is applied. In the case of holonomic constraints the reduction can be applied to the Lagrangian or the Hamiltonian model or to the equations such that the results coincide. This fact is demonstrated for the rigid body and the Euler Bernoulli beam exemplarily.