Firdaus E. Udwadia

Abstract This paper presents recent advances and reviews several notions on constrained systems that have evolved over the years. It questions several of the ‘received (conventional) views’ on both holonomic and non-holonomic systems and shows their erroneous nature in light of recent developments. By using simple illustrative examples, it draws attention to several key ideas that differ from conventional views on the behavior of constrained systems, and especially their stability. Using the explicit equation of motion for constrained systems, we show that an degree of freedom unconstrained dynamical system upon which holonomic and/or nonholonomic constraints are imposed, remains an n degree of freedom system with the initial conditions allowed by the constraints. A central result on the stability of systems using the explicit equation is developed. It encompasses holonomic and/or nonholonomic constraints, including a wide class of nonlinear nonholonomic constraints. Several commonly held notions related to constrained systems are examined and found to require refinement, alteration, or rejection.

Abstract This article develops a fundamental result in linear algebra by providing the necessary and sufficient conditions for the simultaneous quasi-diagonalization of two symmetric matrices and two skew-symmetric matrices by a real orthogonal congruence. This result is used to study the uncoupling of general linear multi-degree-of-freedom (MDOF) structural and mechanical systems described by arbitrary damping and stiffness matrices through quasi-diagonalization, and real orthogonal coordinate transformations. The uncoupling leads to independent subsystems, each having at most two degrees-of-freedom with a specific structure. The results encompass the different physical categories of linear MDOF systems identified by engineers, mathematicians, and physicists and provide the necessary and sufficient conditions for their maximal uncoupling. A total of 16 conditions are shown to exist. However, the number of such conditions for physical systems that are commonly met in nature as well as in aerospace, civil, and mechanical engineering are shown to be considerably less, dwindling at times to two or three, thereby making the results applicable to numerous high-order real-life linear MDOF dynamical systems. Several new analytical results are obtained and corroborated through numerical examples.

The decomposition of linear multi-degree-of-freedom systems with damping, circulatory, and potential forces is considered through a real linear coordinate transformation generated by an orthogonal matrix. Criteria are derived that establish the conditions under which such a transformation exists, allowing these systems to be decomposed into independent, uncoupled subsystems, each with a maximum dimension of two. These criteria are expressed in terms of the properties of systems? coefficient matrices. Several numerical examples are provided to demonstrate the analytical results.

Abstract A new central result that gives the necessary and sufficient conditions for two n by n skew-symmetric matrices and one symmetric matrix to be simultaneously quasi-diagonalized by a real orthogonal congruence is proved. Based on this result, the decomposition of linear multi-degree-of-freedom dynamical systems with gyroscopic, circulatory, and potential forces is investigated through a real linear coordinate transformation generated by an orthogonal matrix. Several sets of conditions, applicable to real-life structural and mechanical systems arising in aerospace, civil, and mechanical engineering, under which such a coordinate transformation exists are found, thereby allowing these systems to be decomposed into independent, uncoupled subsystems, each with a maximum of two degrees of freedom. The conditions are expressed in terms of the coefficient matrices of the system. A specific form for the circulatory (gyroscopic) matrix is posited, and when the gyroscopic (circulatory) matrix is simple—a situation that commonly appears in real-life applications—it is shown that just a single necessary and sufficient condition is required for the decomposition of the multi-degree-of-freedom system. Numerical examples are provided throughout to demonstrate the analytical results.

Abstract This paper provides the necessary and sufficient conditions for a multi-degrees-of-freedom linear potential system with an arbitrary damping matrix to be uncoupled into independent subsystems of at most two degrees-of-freedom using a real orthogonal transformation. The incorporation of additional information about the matrices, which many structural and mechanical systems commonly possess, shows a reduction in the number of these conditions to three. Several new results are obtained and are illustrated through examples. A useful general form for the damping matrix is developed that guarantees the uncoupling of such systems when they satisfy just two conditions. The results provided herein lead to new physical insights into the dynamical behavior of potential systems with general damping matrices and robust computational procedures. It is shown that the dynamics of a damped potential system in which the damping matrix may be arbitrary is identical to that of a damped gyroscopic potential system with a symmetric damping matrix. This brings, for the first time, these two systems, which are seen today as belonging to different categories of dynamical systems, under a unified framework.

Abstract This paper deals with the uncoupling of linear damped multi-degree-of-freedom gyroscopic potential systems in which the damping is taken to have a specifically chosen form. Necessary and sufficient conditions are obtained that guarantee the uncoupling of such damped systems into independent subsystems with at most two degrees-of-freedom. Along with several other results, it is shown that when the potential (stiffness) matrix of the damped system has distinct eigenvalues—a situation commonly found in civil, mechanical, and aerospace engineering, as well as in nature—the damping matrix must have this specifically chosen form for any such multi-degree-of-freedom system to be capable of being uncoupled.

Abstract This paper explores the decomposition of linear, multi-degree-of-freedom, conservative gyroscopic dynamical systems into uncoupled subsystems through the use of real congruences. Two conditions, both of which are necessary and sufficient, are provided for the existence of a real linear coordinate transformation that uncouples the dynamical system into independent canonical subsystems, each subsystem having no more than two-degrees-of-freedom. New insights and conceptual simplifications of the behavior of such systems are provided when these conditions are satisfied, thereby improving our understanding of their complex dynamical behavior. Several analytical results useful in science and engineering are obtained as consequences of these twin conditions. Many of the analytical results are illustrated by several numerical examples to show their immediate applicability to naturally occurring and engineered systems.

Abstract This paper develops a general form of Gauss’s Principle of Least Constraint, which deals with the manner in which Nature appears to orchestrate the motion of constrained mechanical systems. The theory of constrained motion has been at the heart of classical mechanics since the days of Lagrange, and it is used in various areas of science and engineering like analytical dynamics, quantum mechanics, statistical physics, and nonequilibrium thermodynamics. The new principle permits the constraints on any mechanical system to be inconsistent and shows that Nature handles these inconsistent constraints in the least squares sense. This broadening of Gauss’s original principle leads to two forms of the General Gauss Principle obtained in this paper. They explain why the motion that Nature generates is robust with respect to inaccuracies with which constraints are often specified in modeling naturally occurring and engineered systems since their specification in dynamical systems are often only approximate, and many physical systems may not exactly satisfy them at every instant of time. An important byproduct of the new principle is a refinement of the notion of what constitutes a virtual displacement, a foundational concept in all classical mechanics.