Estimation of Rigid Body Modes for System Model Development

Rigid body modes are a necessary set of modes used in the development of component system models. Often these modes are difficult to obtain during modal testing due to instrumentation limitations or test difficulties. Using a combination of singular value decomposition, modal parameter estimation to purge higher order mode effects and structural dynamic modification, a set of appropriately scaled rigid body modes are derived. Several variations of this approach are presented for a simple structure to show the use of the technique. Advantages and limitations of the technique are discussed. INTRODUCTION System modeling applications are abundant in many industries and applications. The component modes can be obtained from either analytical or experimental sources. When using analytical models, the modes describing the system are generally available and contain the appropriate sets of scaled modes for the definition of the component. These modes consist of rigid body modes, flexible modes in the frequency range of interest and the higher modes to account for residual effects. All of these modes are necessary for the development of an accurate system model representation. Especially important are the rigid body modes which are critical to the success of the component mode synthesis; the rigid body modes define important mass inertia characteristics necessary for component coupling. However, when working with experimental representations of these components to define the system representation, there are several issues that cause difficulties. One is that the higher frequency residual modes may not be available and approximations may need to be derived. Another is that the rigid body modes are often difficult to obtain due to the low frequency nature of the modes and may be difficult to extract in the presence of higher frequency modes that dominate the dynamic range of the measurement system. As such, the rigid body modes are generally more difficult to extract and the measurements are generally not of the same quality level as the higher frequency modes in the same measurement frequency range. These rigid body modes are critical to the success of the overall system model representation and need to be approximated for a proper representation of the component modes. If an analytical model is available, then the analytical rigid body modes can be used as an approximation of the modes necessary for the component representation. These modes may need to be scaled to form a consistently related set of modes for the component. But if an analytical model is not available then these rigid body modes must be obtained from alternate approaches if possible. Several test-based approaches are available for the calculation of rigid body modes through the determination of the structure’s inertial characteristics. Mass line approximation techniques can be used to approximate these rigid body modes and are available in software packages such as Test.Lab [1]. The techniques require a wire-frame model along with the weight of the structure. These mass-line techniques rely on the concept that at frequencies above the rigid body modes the inertance FRF will be zero slope and is directly related to, or is, the inverse of the mass value [2]. Another method, the Unchanged FRF’s, involves a direct mass line determination technique and is possible when sufficient space between rigid body modes and flexible modes exists resulting in a horizontal ‘mass-line’ in the frequency response of the structure. When insufficient spacing exists between the rigid body modes and the flexible modes, the unchanged FRF will no longer be suitable for the mass-line approximation. A second mass-line method uses ‘Corrected FRF’s’ with flexible modes subtracted from the original FRF data. The technique uses modal parameter estimation to obtain the modes of the system from experimental data. The flexible modes are then used to synthesize FRF’s without a lower residual that are then subtracted from the original experimental traces. The resulting ‘Corrected FRF’s retain the rigid body information and yield a zero slope mass line from which the rigid body modes may be determined. A third technique is designed for cases when accurate measurements at lower frequencies are not available and is referred to as a ‘Lower Residual’ method. The technique involves the utilization of a lower residual term which is determined through modal parameter estimation and represents the influence of modes below the frequency band selected for estimation. The lower residual term is taken from modal parameter estimation when curve fitting the lowest flexible mode of the system. This residual represents the effects of the rigid body modes and can be used to approximate the actual rigid body modes of the structure. Another technique allows for the generation of rigid body modes using the rigid body calculator in CADA-X[3]. The geometry and inertial properties for the structure must be supplied and based on provided values an approximation of the rigid body modes is performed. This method can be useful yet is strictly an approximation due to the assumption that a free-free situation exists. Appending the shapes into a mode set from experimental data in which the test structure is not free-free introduces error. The required input for this technique is often difficult to produce because it is based on the ability to accurately calculate moments of inertia which is difficult for complex structures. Several alternate approaches are presented in this paper that provide other methods to obtain the rigid body modes for the component. The basic approach utilizes the singular value decomposition of a frequency band where no modes are present (neither the rigid body modes nor the flexible modes of the system) and then performing mass structural dynamic modifications to identify the scaling of the modes shapes estimated from the singular value decomposition. As a variant of this approach (where a frequency band without any modes is difficult to obtain), modal parameter estimation of the flexible modes of the system is performed and the synthesized flexible modes are used to purge the effects of these modes before utilizing the approach just mentioned. The new proposed approach is presented next followed by test cases used to show the methodology proposed. These results are presented and advantages/disadvantages are discussed. THEORY RELATED TO APPROACHES FOR RIGID BODY MODE ESTIMATION In order to present the results of the studies performed herein, there are several basic fundamental theoretical approaches that need to be summarized. These relate to singular value decomposition, modal parameter estimation and structural dynamic modification. Each of these methods is only summarized here; details of the techniques are contained in the respective references or are commonly used approaches and techniques. Singular Value Decomposition of the Frequency Response Matrix: Singular Value Decomposition (SVD) has been widely used in many structural dynamic modeling applications. In one application, the Test Reference Identification Procedure (TRIP) [4] uses SVD to find dominant mode information for selection of references for the modal test. In terms of modal parameter estimation, SVD has been employed in the Complex Mode Indicator Function technique [5,6] to decompose a matrix of frequency response functions to find the underlying modal parameters. Typically this has been used over a band of interest where several modes exist. For use in the approaches described herein, this technique will be applied over a frequency range that is remote from modes of the system. The SVD of a set of FRFs is written as: [ ] [ ][ ][ ]H V S U H = (1) where [H] is the FRF matrix [U] is the left singular matrix corresponding to the matrix of mode shapes [S] is the diagonal singular value matrix [V] is the right singular matrix corresponding to the matrix of modal participations The number of dominant modes is identified in the singular values matrix [S]. The estimation of the modal vectors is contained in the left singular matrix [U]. The SVD process applied to a matrix of FRFs is pictorially shown in Figure 1. SVD of Multiple Reference FRF Data [ ] { } { } { } [ ] { } { } { } ⎥ ⎥ ⎥ ⎥ ⎥