Modal analysis of structures with holonomic constraints

Abstract

A formulation is presented to predict the modal parameters of constrained structural systems. The formula- tion employs the Lagrange multiplier technique to investigate structural vibration before and after holonomic constraints are imposed. Holonomic constraint can either be of the linear, scleronomic type that accounts for structure compatibility, or of the rheonomic, nonlinear type for structures with deployable or articulated components. The vibration and stability of an integrated, constrained structural system can be predicted from the modal parameters of the corresponding unconstrained system and the characteristics of holonomic con- straints. ECENT developments in aerospace technology have focused attention on the technical aspects of large-scale, complex, and constrained structural systems. Aircraft and spacecraft launch vehicles consist of a number of subsystems and components. Verification of structural design is typically performed using a limited series of tests on subsystem and component prototypes. To overcome the increasing size of the structural model and to ease the complexity of structural modification during the development phase, a method is needed wherein the vibration and stability of the integrated, con- strained structural systems can be evaluated. Large, complicated systems are often solved by either de- composition into components or subsystems and subsequent analysis, or system modeling with kinematic constraints. The former technique is similar to the so-called component mode synthesis method.5 Previous research applied these techniques to mechanism dynamic analysis,2 control system stability anal- ysis,3 and modal synthesis,4 in which modal parameters, such as natural frequency and mode shape of each subsystem or component, are first determined/Modal analysis of an inte- grated, constrained structural system can then be conducted based on the predetermined modal parameters of the subsys- tems and components. A number of subsystem and component coupling procedures have been proposed for solving undamped structural systems. Fuh and Chen5 developed a procedure to construct a transfor- mation matrix for formulating structure modal synthesis with constraints. Kuang and Tsuei6 presented a model reduction procedure for substructure mode synthesis. Although both pror cedures require an orthogonal complement of the constraint Jacobiano matrix, the matrix construction is neither efficient nor unique. Moreover, most of the structure analyses to date consider only linear, scleronomic constraints that account for structure compatibility at the interface of two adjacent com- ponents. Little attempt has been made to include holonomic constraint at the interface or to predict its effect on the struc- ture's natural frequencies when constraints are imposed. Nonlinear and/or rheonomic constraints have been applied to the area of mechanism dynamics where numerical integra- tion of system response is required. 7 Sohoni and Whitesell8 developed a procedure for calculating the natural frequencies of a machine dynamics system. However, their formulation demands numerical identification of the state variables before