Analysis of transmitted vibration in discretely joined machine assemblies using component modes

In this study, a new analysis framework is presented for computing vibration transmitted among components joined at discrete locations in machine assemblies. The method is based on a modal synthesis approach, and predicts transmitted vibration on a narrow frequency band basis. Natural frequencies and modes of individual components are used to obtain assembly modal properties from a variational formulation, with Lagrange multipliers used to enforce constraints that motions of components at joint locations be identical. The modal properties of the assembly are used to compute its forced response. Since the Lagrange multipliers are equal to the interfacial forces at the joints, evaluating these from the assembly response provides an efficient means for computing the mechanical power transmitted among components. The new formulation is initially developed for proportionally damped assemblies, and an extended formulation for modal synthesis and vibration transmission in non-proportionally damped assemblies with complex-valued modes is developed subsequently. The extended formulation for modal synthesis in assemblies with general viscous damping is a new development, since it is more generally applicable than any developed previously. In addition, it permits the new formulation to be applied to examine the effects of concentrated damping at the joint locations. Example cases verify the accuracy of the original and extended formulations.;The proposed formulation is exact if the sets of component modes used are complete. Because most machine components have many more modes than may be practically included, incomplete sets of component modes often must be used. This admits the possibility of modal truncation errors. A new procedure for estimating modal truncation errors in assembly properties obtained from modal synthesis formulations is developed and applied to alternative methods. It is determined that synthesis methods which use component modes corresponding to free interface conditions are especially susceptible to truncation errors. Because the modal synthesis formulation used in the initial development of the new analysis framework is of this type, an alternative which is less susceptible to errors is developed. Example cases demonstrate that the alternative formulation yields accurate estimates of transmitted vibration even when small sets of component modes are used.