Practical Random Vibration Analysis

Some commercial and proprietary structural dynamic finite element analysis codes have either a capability to perform only very elementary random vibration analyses, or none at all. In particular, some codes permit only the approximate analysis of mean square response, not fully accounting for cross-spectral effects. This paper shows that without regard for stated analysis capabilities, practically any linear structural dynamic finite element code can be used to obtain the information necessary to perform an exact random vibration analysis. The formulas to do so are presented, as well as numerical examples that demonstrate the operations. INTRODUCTION Structural dynamic finite element codes offer a relatively “complete” or limited random vibration analysis capability, or none at all. But even those codes that offer a complete analysis capability may not perform all the operations an analyst may desire. Even so, it is possible to express practically all the fundamental random vibration operations one may desire to perform, and use practically any structural dynamic finite element code to obtain the information necessary to perform the operations in a utility code like MATLAB ® (2004). This paper lists the fundamental formulas of stationary random vibrations and shows how the measures of structural behavior required to evaluate the formulas can be obtained using a structural dynamic finite element code. (In the process, the formulas used to perform structural dynamic transient analysis are developed.) The results of two practical examples involving large, complex finite element models are presented. The developments concerning deterministic structural dynamics are covered, for example, by Clough and Penzien (1975), and Meirovitch (1967). The developments concerning finite element analysis are covered by Bathe (1976). The developments concerning signal analysis are covered by Bendat and Piersol (2000). The developments concerning random vibration analysis are covered by Wirsching, Paez and Ortiz (1995). STRUCTURAL DYNAMICS WITH FINITE ELEMENT CODES A mechanical structure is a physical system constructed of parts that carry load and a dynamic environment is the aggregate of surrounding conditions that cause a structure to respond via motions through space. We seek to establish general relations that can be used to describe how mechanical structures respond to dynamic environments. There are several questions one might ask regarding the behavior of a structure exposed to dynamic environments: • What is the generic character of system response to low, medium, and high level environments? Is the system behavior substantially linear? • What is a system’s linear (or quasi-linear) response potential? What are its modal frequencies, mode shapes, and modal damping factors? • How does a system response to specific excitations? (Deterministic analysis) • How would a system respond to an ensemble of excitations? (Random vibration analysis) We seek to develop answers to the final three sets of questions to consider different approaches to random vibration analysis. The main problem we will solve is this: When a mechanical system is excited by stationary random inputs applied at discrete locations and jointly characterized by their matrix of spectral density functions, compute the autoand cross-spectral density functions of the responses at a set of locations of interest. A set of measures of the behavior of a structural dynamic system required to perform the analysis described above is the matrix of frequency response functions (FRF), ( ) 0 ≥ f , f xq H , of the system. (The meanings of the subscripts x and q will be given below.) These can be obtained with a finite element (FE) code using direct integration dynamic analysis or modal analysis. The typical, deterministic, structural dynamic finite element code is constructed to compute responses to specific excitations, so it can be used to establish characteristics (measures) of structural behavior. Consider the generic system shown in Figure 1. An FE computer code permits the modeling of the system with a collection of discrete, “finite” parts (elements), the aggregate of which is intended to collectively simulate the behavior of the actual system. For example, Figure 2 might represent the FE discretization of the system. The behavior of each element is described by its motions at exterior points on the element – the node points of the element. A structural dynamic finite element code automatically develops a collection of ordinary differential equations governing motions at element node points. The equations reflect the particular dynamics and geometry of the collection of elements. Boundary conditions can be imposed at any of the node points, in any direction or form (translation, or rotation if appropriate) desired. The motions of the nodes of a structure in specific directions constitute the degrees of freedom (DOF) of structural motion. Today, the behaviors of structural dynamic systems are being simulated with finite element models with up to a million or many millions of DOF. The ordinary differential equations governing motion of a structural dynamic system can be written in the form ( ) ( ) 0 0 0 0 v x x x q kx x c x m = = = + + & & & & , (1) where x is the 1 × N vector of system displacements at the DOF of the model, dots denote differentiation with respect to time, m is N N × mass matrix of the system, c is the N N × damping matrix of the system, k is the N N × stiffness matrix of the system, q is the 1 × N vector of forcing functions applied to the system at its DOF, and 0 0 v x , are the displacement and velocity initial conditions of the system, respectively, at the N system DOF. Referring to the governing equations and the problem solved by the finite element code we show, first, how the matrix of FRFs of a structure (and other measures of system behavior) can be obtained by direct integration dynamic analysis, and second, how they can be obtained through modal analysis. Figure 1. A generic structural dynamic system. Figure 2. Finite element model of the system. q1(t) q2(t) q3(t) x1(t) x2(t) q1(t) q2(t) q3(t)