An Updating Method for Finite Element Models of Flexible‐Link Mechanisms Based on an Equivalent Rigid‐Link System

Abstract

This paper proposes a comprehensive methodology to update dynamic models of flexible-link mechanisms (FLMs) modeled through ordinary differential equations. The aim is to correct mass, stiffness, and damping matrices of dynamic models, usually based on nominal and uncertain parameters, to accurately represent the main vibrational modes within the bandwidth of interest. Indeed, the availability of accurate models is a fundamental step for the synthesis of effective controllers, state observers, and optimized motion profiles, as those employed in modern control schemes. The method takes advantage of the system dynamic model formulated through finite elements and through the representation of the total motion as the sum of a large rigid-body motion and the elastic deformation. Model updating is not straightforward since the resulting model is nonlinear and its coordinates cannot be directly measured. Hence, the nonlinear model is linearized about an equilibrium point to compute the eigenstructure and to compare it with the results of experimental modal analysis. Once consistency between the model coordinates and the experimental data is obtained through a suitable transformation, model updating has been performed solving a constrained convex optimization problem. Constraints also include results from static tests. Some tools to improve the problem conditioning are also proposed in the formulation adopted, to handle large dimensional models and achieve reliable results. The method has been experimentally applied to a challenging system: a planar six-bar linkage manipulator. The results prove their capability to improve the model accuracy in terms of eigenfrequencies and mode shapes.

Flexible-Link Multibody System Eigenvalue Analysis Parameterized with Respect to Rigid-Body Motion

“…In Equation 13, symbol * denotes the conjugate transpose, as λ, φ, and ψ may be complex valued. The quadratic eigenvalue problem in Equations (12) and (13) has 2n do f eigenvalues, which are symmetric with respect to the real axis of the complex plane, being the system matrices real [35]. This means that the system eigenvalues can be either real or complex, but in the last case, they occur in complex conjugate pairs as well as the corresponding eigenvectors: if λ 1 = λ r + iλ i is a system eigenvalue, then λ 2 = λ r − iλ i is a system eigenvalue too, and the corresponding eigenvectors are φ 1 = φ r + iφ i and φ 2 = φ r − iφ i , respectively.…”

“…This means that the system eigenvalues can be either real or complex, but in the last case, they occur in complex conjugate pairs as well as the corresponding eigenvectors: if λ 1 = λ r + iλ i is a system eigenvalue, then λ 2 = λ r − iλ i is a system eigenvalue too, and the corresponding eigenvectors are φ 1 = φ r + iφ i and φ 2 = φ r − iφ i , respectively. Although the quadratic eigenvalue problem in Equations (12) and (13) could be directly solved, it is usually transformed in a generalized eigenvalue problem…”

“…where N ∈ R n do f ×n do f can be any nonsingular matrix; here it is considered N = M. The quadratic (Equations (12) and (13)) and the generalized (Equations (14) and (15)) eigenvalue problems have the same 2n do f eigenvalues, whereas the corresponding eigenvectors are correlated by the following relations,…”

“…The dynamic behavior of a mechanical system can be easily studied by means of modal analysis, which provides the system modal parameters or characteristics (i.e., natural frequencies, mode shapes, and damping ratios). The knowledge of the modal characteristics is a fundamental requirement for the implementation of several advanced model-based engineering techniques, such as motion planning [1], control design [2,3], stability analysis [4,5], model reduction [6][7][8][9][10], model updating [11][12][13], and structural modification [14][15][16].…”

“…A common practice used in the literature to apply the modal analysis to FLMBSs consists of linearizing the nonlinear dynamic model about a selected configuration so that the modal parameters can be computed. In particular, if the system model is formulated by means of ODEs, the eigenvalue analysis can be straightforwardly applied to the linearized system equations [12,22]. Conversely, if the system model is formulated by means of DAEs, two different strategies are possible for computing the eingensolutions: (a) transform the motion equations from DAEs to ODEs, linearize the resulting model, and then compute the eigenpairs [5,23,24]; (b) perform a direct eigenanalysis, i.e., the eigenanalysis for the system of equations resulting from the direct linearization of the DAEs [25,26].…”