Fast generation of stability charts for time-delay systems using continuation of characteristic roots

Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.Comment: 12 pages, 6 figure

Parametric instabilities in variable angle tow composite panels

Advancement in the automation of composite manufacturing techniques like automated fiber place- ment, embroidery fiber placement, and continuous tow shearing enables the three-dimensional elastic tailoring and the concept of tow placement. This technology allows the tows to be steered in prin- cipal load direction paths in composite structures. These composites with curvilinear fibers exhibit spatially varying stiffness, and are termed as variable angle tow (VAT) composite laminates. Elas- tic tailoring has been successfully performed to improve the modal, buckling, and post-buckling performance of the VAT structures. Aircraft structures like fuselage and wings are often subjected to dynamic loads in addition to the static loads. These structures, under periodic in-plane loads, exhibit parametric resonance for certain combinations of the excitation frequencies and loads. Therefore, the present thesis work focuses on investigating the dynamic stability behavior of VAT composite panels under in-plane periodic compression load. Since the fiber orientation in the VAT panel changes with the position, the stiffness properties vary continuously across the lamina. As a result, unlike straight-fiber composites, in VAT panels, the in-plane stress distributions over the laminate are non-uniform even under uniform compression load. Therefore, pre-buckling analysis has to be carried out first to determine the non- uniform in-plane stress distributions due to the applied load along the edges. Then the evaluated stress distributions are used to investigate the dynamic stability characteristics of the VAT panel. A flat VAT panel, a delaminated VAT panel with a cutout, and a curved VAT panel are tailored for enhancing the dynamic performance under periodic axial compression load. A linear fiber-angle variation with symmetric VAT layup has been considered, and the effect of fiber-angle variation on the dynamic stability behavior of the VAT composites is studied. The performance of the VAT panel is then compared with the straight-fiber laminates. In addition, the effect of various other parameters like boundary conditions, orthotropy ratio, aspect ratio, span-thickness ratio, delamination area, and radius of curvature on the dynamic stability behavior of the VAT panel is studied. Further, new evidence on the benefits of tow-steering over the straight-fiber composites in tailoring the dynamic stability and stiffness properties of the VAT panel simultaneously is reported. Subsequently, an implicit Floquet analysis is used to determine the dynamic stability character- istics of the VAT panel. In the literature dealing with the stability of structures with time-periodic loads, Bolotin’s method is widely used for determining the instability regions. However, Bolotin’s approach is an approximate method, and the evaluated instability regions are accurate only up to certain limit. To avoid this, Floquet theory can be used to determine the accurate dynamic stability characteristics of the systems. In Floquet theory, a Floquet transition matrix (FTM) is computed, and the dominant eigenvalue of the FTM determine the stability of the system. However, for large degree of freedom systems like finite element models of VAT panels, calculation of FTM becomes computationally expensive. Whereas, an implicit Floquet analysis can significantly reduce the com- putational load. In this technique, the dominant eigenvalue of the FTM is computed without the explicit computation of the full FTM matrix. Also, unlike Bolotin’s method, Floquet analysis pro- vides information about the effective damping present at different locations in the parametric space and the nature of bifurcation through which the stability is lost

Galerkin-Ivanov transformation for nonsmooth modeling of vibro-impacts in continuous structures

This work deals with the modeling of nonsmooth vibro-impact motion of a continuous structure against a rigid distributed obstacle. Galerkin's approach is used to approximate the solutions of the governing partial differential equations of the structure, which results in a system of ordinary differential equations (ODEs). When these ODEs are subjected to unilateral constraints and velocity jump conditions, one must use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a computationally demanding task. Ivanov proposed a nonsmooth transformation for a vibro-impacting multi-degree-of-freedom system subjected to a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions; however, he did not explore its application for simulating distributed collisions in spatially continuous structures. We adopt Ivanov's transformation to deal with multiple unilateral constraints in spatially continuous structures. Also, imposing the velocity jump conditions exactly in the modal coordinates is nontrivial and challenging. Therefore, in this work we use a modal-physical transformation to convert the system from modal to physical coordinates on a spatially discretized grid. We then apply Ivanov's transformation on the physical system to simulate the vibro-impact motion of the structure. The developed method is demonstrated by modeling the distributed collision of a nonlinear string against a rigid distributed surface. For validation, we compare our results with the well-known penalty approach

Nonsmooth modeling of distributed impacts in spatially discretized continuous structures using the Ivanov transformation

This work deals with the modeling of nonsmooth impacting motions of a structure against a rigid distributed obstacle. Finite element methods can be used to discretize the structure, and this results in a system of ordinary differential equations (ODEs). When these ODEs are subjected to unilateral constraints and velocity jump conditions, one has to use an event detection algorithm to calculate the time of impact accurately. Event detection in the presence of multiple simultaneous impacts is a nontrivial and computationally demanding task. Ivanov (Ivanov, A., 1993. Analytical methods in the theory of vibro-impact systems. Journal of Applied Mathematics and Mechanics, 57(2), pp. 221-236.) proposed a nonsmooth transformation for a vibro-impacting multidegree-of-freedom (MDOF) system subjected to only a single unilateral constraint. This transformation eliminates the unilateral constraints from the problem and, therefore, no event detection is required during numerical integration. This nonsmooth transformation leads to sign function nonlinearities in the equations of motion. However, they can be easily accounted during numerical integration. Ivanov used his transformation to make analytical calculations for the stability and bifurcations of vibro-impacting motions, but did not explore its application to simulating distributed collisions in discretized continuous structures. We adopt the Ivanov transformation to deal with multiple unilateral constraints in discretized continuous structures. The developed method is demonstrated by modeling the distributed collision of a string and a beam against a rigid surface. For validation, we compare our results with the penalty approac

Implicit Floquet analysis for parametric instabilities in a variable angle tow composite panel

In the literature dealing with the stability of time-periodic structures, Bolotin’s method is widely used for generating the approximate stability charts. Further, increasing the order of approximation in Bolotin’s approach requires solving several eigenvalue problems and may not lead to the rapid convergence of stability charts. Floquet theory can be used to calculate accurate stability charts for time-periodic systems. In Floquet theory, a Floquet transition matrix (FTM) is computed, the dominant eigenvalue of the FTM determine the stability of the time-periodic system. For a large degree of freedom systems like finite element models of VAT panels, the calculation of FTM becomes computationally expensive. However, an implicit Floquet analysis can significantly reduce the computational load. In this technique, the dominant eigenvalue of the FTM is computed without calculating the full FTM matrix. In this work, the stability regions obtained from the implicit Floquet analysis are compared with the results from Bolotin’s approach and verified using the time response of the panel obtained from numerical integration. Also, unlike Bolotin’s method, Floquet analysis provides information about the damping present in different areas of stability charts and the nature of bifurcation through which the stability is lost

Dynamic instability analysis of variable angle tow composite plate with delamination around a cutout

The dynamic instability analysis of a simply supported variable angle tow (VAT) composite with delamination around a cutout subjected to periodic axial compression load is investigated. The governing equations of motion for the VAT plate are derived based on first-order shear deformation theory and solved using finite element method. The equations of motion contain time periodic coefficients and their stability is determined using the Bolotin’s approach. A parametric study is carried out to analyze the effect of linearly varying fiber orientation angle on the dynamic instability of VAT laminate with delamination around a circular cutout

Dynamic instability of curved variable angle tow composite panel under axial compression

Variable angle tow (VAT) composites have demonstrated better performance in buckling and post-buckling over straight fiber composites based on the mechanics of load redistribution from critical regions to supported edges. In this work, the dynamic instability behavior of a curved VAT composite panel subjected to periodic axial compression load is investigated. The governing energy functional of a curved symmetric VAT panel under external loading is derived using Donnell's shallow shell theory. Later, the discretized equations of motion are derived using the Rayleigh–Ritz method combined with the generalized differential integral quadrature method (GDIQM). Initially, the pre-buckling problem is solved by applying a uniform compression load to compute the stress resultant distribution which is used to evaluate the buckling load of the curved VAT panel. Subsequently, the dynamic/parametric instability region of a curved VAT panel subjected to periodic axial compression load is determined using Bolotin's first-order approximation. Then, the dynamic instability performance is evaluated for a curved VAT panel with linear fiber angle distribution and compared with straight fiber laminates. Finally, the influence of fiber angle orientation, the radius of curvature, aspect ratio and plate boundary conditions on the dynamic instability of VAT panel is presented

Spurious roots of delay differential equations using Galerkin approximations

The dynamics of time-delay systems are governed by delay differential equations, which are infinite dimensional and can pose computational challenges. Several methods have been proposed for studying the stability characteristics of delay differential equations. One such method employs Galerkin approximations to convert delay differential equations into partial differential equations with boundary conditions; the partial differential equations are then converted into systems of ordinary differential equations, whereupon standard ordinary differential equation methods can be applied. The Galerkin approximation method can be applied to a second-order delay differential equation in two ways: either by converting into a second-order partial differential equation and then into a system of second-order ordinary differential equations (the “second-order Galerkin” method) or by first expressing as two first-order delay differential equations and converting into a system of first-order partial differential equations and then into a first-order ordinary differential equation system (the “first-order Galerkin” method). In this paper, we demonstrate that these subtly different formulation procedures lead to different roots of the characteristic polynomial. In particular, the second-order Galerkin method produces spurious roots near the origin, which must then be identified through substitution into the characteristic polynomial of the original delay differential equation. However, spurious roots do not arise if the first-order Galerkin method is used, which can reduce computation time and simplify stability analyses. We describe these two formulation strategies and present numerical examples to highlight their important differences