Determination of the stiffness matrix of flat springs for modeling of the boundary condition at a pipeline support

Abstract

Abstract Elastic mounting elements are often used in mechanical systems for reduction of undesired loads. In many engineering applications one-dimensional (1D) approximation of absorber reaction is an acceptable approach however in general, a six order stiffness matrix of an elastic element should be known. This can also be the case of water hammer with fluid-structure interaction in a pipeline fixed with elastic supports. For the standard model of this phenomenon the general boundary condition (BC) at the support should take into account all degrees of freedom of the pipe element. In this paper various methods of flexibility matrix determination of actual, self manufactured flat springs (FS) used as supports at a laboratory pipeline are discussed. Analytical calculations are based on modeling of elastic strain energy of a bar under external loads and formulas being developed for the FS flexibility coefficients are presented. Experimental verification of selected coefficients was performed and the results were consistent with theoretical findings. A general experimental method and a data processing scheme based on linear least squares (LS) method is also proposed and shortly discussed. Numerical investigations are based on FEM modeling of the FS structure and computations of their load-displacement characteristics with the ANSYS software. Flexibility matrix was determined with that same LS data processing scheme. Analytical, experimental and numerical results are compared and discussed. For non-singular flexibility matrix the stiffness matrix used in the BC can be easily determined as its reverse. Otherwise, for a singular case, a special treatment is required. A solution to this problem is developed and presented in the paper. It has been evidenced that singularity results in dimension reduction of the BC.