A substructure synthesis formulation for vibration isolation

Abstract

The new modeling method presented here is classified as a substructure synthesis (SS) technique. The distinction between the new SS method and the component mode synthesis formulation is that no transformation between local coordinates and generalized coordinates occurs in the new SS method. The advantage of this is a retention of physical insight into the model and the ability to form equations of motions directly with generalized coordinates. The new formulation differs from other substructure synthesis formulation because it satisfies geometric, natural, displacement and force constraints between substructures into one mathematical process, instead of using both kinematic chains and boundary condition approximation methods. This has the advantage of reducing the complexity of the integrals that are required in the computation. The new formulation also results in global eigenfunction approximations and global generalized coordinates, which eventually satisfies the inclusion principle which means eigenvalue estimates converge from above their actual values. The analysis method also facilitates the examination of boundary conditions in a unique manner. The method is unique because constraints are explicitly examined and selectively satisfied. This allows the identification of extraneous constraints and provides guidance in the selection of admissible functions. The new SS formulation may be divided into two steps. The first step is to satisfy geometric boundary conditions of substructures with appropriate admissible functions. The second step is the modification of these admissible functions to minimally satisfy geometric constraints imposed by the interaction of substructures. Natural constraints can also be satisfied to improve convergence to the exact eigenvalues.

The MAF-SS formulation results in explicit knowledge of the constraints coupling substructures. Changing these constraints with active feedback results in a modified structure. The effect of active feedback of terms proportional to the coupling constraints is to lower the stiffness of the structure. This increases the isolation between substructures. The ability to improve isolation using this unique type of feedback is demonstrated. The concept of structural modification through substructure constraint alteration 1s applied to systems using a multivariable feedback method. This is accomplished by combing the MAF-SS method with a standard eigenstructure assignment technique. This method uses the MAF-SS formulation to define a system with substructure constraint eigenstructure properties, the active feedback gain that realizes these systems is calculated with an eigenstructure assignment method. The MAF-SS has application to active control formulation, the result of this control can be an improvement in substructure isolation.