Hamilton's Law Research Articles

Large-amplitude parametric response of fluid-conveying nanotubes due to flow pulsations

In this article, the nonlinear parametric response of viscoelastic nanotubes conveying pulsatile flow is investigated. A two-parameter scale-dependent elasticity-based model is developed within the framework of a nonlocal theory with strain gradient influences. To model the effects of fluid molecules, which slip on the internal nanotube wall, on the parametric response, Karniadakis–Beskok approach is used. Viscoelastic effects are also described via Kelvin–Voigt scheme. Hamilton law, Galerkin and continuation techniques are, respectively, utilized in this analysis for obtaining, discretising and solving nonlinear coupled equations. Both subcritical and supercritical nonlinear parametric responses are examined considering various parameters such as the speed variation amplitude and frequency. The viscoelastic nanotube conveying pulsatile flow exhibits a hardening nonlinearity in the subcritical regime while it displays a softening nonlinearity in the supercritical regime.

Hamilton's Law and Finance - Borrowing from the Brits (And the Dutch)

We live in an era when Modern Monetary Theory has gained purchase. Deficits do not seem to matter to nations or their finance ministers. States believe the can print their way to utopia without a reckoning; this is a scheme freighted with disaster. This article centers on the achievements of America’s greatest finance minister whose grasp of political economy was without equal. Hamilton charted the course for American economic power. His work is timely and worthy of study. Late-18th century America sought commercial growth and new manufacturers yet feared monopolistic economic power. Hamilton’s economic program, centered on a national bank and program for manufacturing, provided the framework for the finance-led transformation of America. In a short time, Hamilton’s elegant solutions (based upon the English financial model) transformed America from a defaulting debtor to a magnet that attracted immense amounts of capital in the 19th century. The burden of the Revolutionary War debt needed to be resolved. Hamilton’s program: the assumption of the states and Confederation debt, its monetization, the establishment of the Bank of the United States, and Report on Manufactures laid the foundation for a vibrant economy. The article demonstrates how Hamilton’s prudent program strengthened the new federal government while providing the blueprint for the commercial society that emerged in the 19th century.

SOLVING THE INVERSE PROBLEM OF AN AIRPLANE PLANNER MOTION VIA DIRECT APPLICATION OF HAMILTON'S LAW OF VARYING ACTION

The inverse problem for the airplane planner motion is analyzed through direct application of Hamilton's law of varying action. Using this technique one can approach the problem by solving a set of nonlinear algebraic equations only; i.e. no differential equations are needed. In this case the mass center trajectory and the required controls are approximated as a truncated power series. Direct application of Hamilton's law establishes the governing relations between those input and output coefficients. The problem is then solved in the coefficient domain to obtain the unknown control coefficients. The problem is formulated and solved for both time dependent and space dependent specified trajectories. Two illustrative numerical examples are provided to demonstrate the technique.

Structural response of porous FG nanobeams under hygro-thermo-mechanical loadings

Abstract We investigate the bending behavior of functionally graded (FG) nanobeams with internal porosity and subjected to a hygro-thermo-mechanical loading. The Eringen's Nonlocal Theory (ENT) is here applied for the numerical study, while considering a uniform porosity within the nanobeam. The mechanical and thermal properties of FG materials are assumed to vary throughout the thickness. The equations of motion are derived from the Hamilton's law and solved with the Navier's procedure. A key point of the work is to explore the effect of the material length-scale, power-law index, porosity volume fraction, temperature rise, and moisture concentration on the global deflection of nanobeams, as useful for practical applications.

Robot life: simulation and participation in the study of evolution and social behavior.

This paper explores the case of using robots to simulate evolution, in particular the case of Hamilton's Law. The uses of robots raises several questions that this paper seeks to address. The first concerns the role of the robots in biological research: do they simulate something (life, evolution, sociality) or do they participate in something? The second question concerns the physicality of the robots: what difference does embodiment make to the role of the robot in these experiments. Thirdly, how do life, embodiment and social behavior relate in contemporary biology and why is it possible for robots to illuminate this relation? These questions are provoked by a strange similarity that has not been noted before: between the problem of simulation in philosophy of science, and Deleuze's reading of Plato on the relationship of ideas, copies and simulacra.

C1‐continuous space‐time discretization based on Hamilton's law of varying action

We develop a class of C1‐continuous time integration methods that are applicable to conservative problems in elastodynamics. These methods are based on Hamilton's law of varying action. From the action of the continuous system we derive a spatially and temporally weak form of the governing equilibrium equations. This expression is first discretized in space, considering standard finite elements. The resulting system is then discretized in time, approximating the displacement by piecewise cubic Hermite shape functions. Within the time domain we thus achieve C1‐continuity for the displacement field and C0‐continuity for the velocity field. From the discrete virtual action we finally construct a class of one‐step schemes. These methods are examined both analytically and numerically. Here, we study both linear and nonlinear systems as well as inherently continuous and discrete structures. In the numerical examples we focus on one‐dimensional applications. The provided theory, however, is general and valid also for problems in 2D or 3D. We show that the most favorable candidate — denoted as p2‐scheme — converges with order four. Thus, especially if high accuracy of the numerical solution is required, this scheme can be more efficient than methods of lower order. It further exhibits, for linear simple problems, properties similar to variational integrators, such as symplecticity. While it remains to be investigated whether symplecticity holds for arbitrary systems, all our numerical results show an excellent long‐term energy behavior.

Hamilton's law of variable mass system and time finite element formulations for time‐varying structures based on the law

SUMMARYTraditional principles of mechanics are primarily conceived for constant mass systems, which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems, thus the numerical algorithms for time‐varying structures based on these principles are only suitable for this special case. In this paper, Hamilton's law of variable mass system is derived based on Meshchersky's fundamental equation, and two classes of novel time finite element formulations for linear systems with arbitrary continuous time‐varying parameters are developed based on the previous law. The formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated. Numerical examples demonstrate that compared with the algorithms for time‐varying structures that developed based on traditional principles of mechanics, the proposed algorithms provide extended capabilities in both time‐varying mass problems that mass is gained or lost at any velocity (such as rocket problem) and moving‐mass problems (such as vehicle‐bridge interaction problem) besides the time‐varying stiffness and damping problems, the proposed algorithms have a wider range of application. In particular, Hamilton's law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time‐varying structures. Copyright © 2014 John Wiley & Sons, Ltd.

NONLINEAR DYNAMICS ANALYSIS OF ALUMINUM HONEYCOMB SANDWICH PLATE WITH COMPLETED CLAMPED SUPPORTED

Study on the dynamics of aluminum honeycomb sandwich plates of composite structure material plays a key role for the special applications of the aerospace and automotive engineering. The nonlinear dynamics of honeycomb sandwich plate is explored.According to the classical plate theory and the large deformation,the governing equations of motion are established for the honeycomb sandwich plate subjected to the transversal excitation force by using the Hamilton's law of variation principle.The transversal damping is taken into consideration.The method of normalization is utilized to transform the nonlinear vibration equations to nonlinear system with double modes of freedom. Numerical simulation is used directly to investigate the nonlinear responses of the honeycomb sandwich plate.The results indicates that the change of transversal excitation force has a significant effect on the vibration of honeycomb sandwich plate because the hexagon cell of core and the amplitude of the first modal are bigger than the second modal.In the different ranges of transversal excitation force, the honeycomb sandwich plate exists different dynamical phenomena.Single periodic motion appears when the force value is small,and periodic motion,multi-period motion,chaotic motion appears with the increase of force.Experiments are conducted to validate the numerical simulation results.

On the Numerical Discretization in Space and Time: Part 1 - Hamilton's Law of Varying Action Involving Lagrangian/Hamiltonian/Total Energy Framework

The focus in Part 1 of this exposition is strictly restricted to holonomic-sceleronomic systems, and the applications of interest are elastodynamics, which are routinely encountered in a wide class of structural dynamics problems in engineering. Restricting attention to these considerations, new and different perspectives and equivalences are described which deal with finite element space discretization aspects employing three distinctly different frameworks with scalar formalism. It encompasses the Lagrangian mechanics, the Hamiltonian mechanics, and as an alternative, the framework with a built-in measurable quantity based on the Total Energy. Historically, traditional practices routinely employ the weighted residual method or equivalently the principle of virtual work in dynamics for developing finite element formulations. In contrast, the present developments stem from Hamilton's law of varying action (HLVA) as a starting point and involve distinctly different scalar descriptive functions (the Lagrangian [ℒ( q , [qdot] ) : TQ →ℝ], the Hamiltonian [ℋ( p , q ) : T* Q →ℝ], or the Total Energy [ℰ( q , [qdot] ):TQ →ℝ]). These developments naturally embody the weak form in space and the statement of the weighted residual in time. Complicated structural dynamical systems such as a rotating bar and the Timoshenko beam are particularly shown here simply for illustration. In Part 2, we describe the satisfaction of conservation properties of the fully discretized equations of motion in space and time with particular attention to the Total Energy framework (in contrast to the Lagrangian and the Hamiltonian), primarily because it is very natural and is computationally attractive and meaningful to conducting the time discretization [1] process. Instead of starting from the continuous form of representations, the particular focus is upon directly employing the discrete formulations for enabling algorithmic designs for the class of LMS methods for linear dynamic problems for illustration purposes. This is a necessary first step in the design of algorithms for integrating the equations of motion, and they serve as the parent algorithms for subsequent extensions to nonlinear dynamic problems.

Time modes and nonlinear systems

A vector-space implementation of the equation that has been known as Hamilton's Law of Varying Action is used with a non-virtual perspective to generate direct solutions to nonlinear initial value response problems in the time domain. This novel vector-space perspective unleashes the power of linear algebra to provide new capabilities with Hamilton's Law of Varying Action and new insights into nonlinear dynamics. In particular, this perspective permits the use of unconstrained temporal-basis-functions without the need to augment Hamilton's Law of Varying Action with Lagrange multipliers, and provides a linearly independent set of spatiotemporal functions spanning the solution space of the response. These spatiotemporal functions are descriptively denoted as fundamental-time-modes. By providing this basis of fundamental-time-modes, the concepts of linear algebra can be taken advantage of to accomplish model reduction and superposition for nonlinear systems. In particular, classical model reduction techniques are used to eliminate non-dominant fundamental-time-modes from the solution process, resulting in model reduction in the time domain. In addition, using the novel concept of eigen-direction iteration, linearly independent fundamental-time-mode response trajectories along eigen-directions are superposed to yield new response trajectories for nonlinear systems. These capabilities of the vector-space approach are demonstrated for nonlinear systems exhibiting nonlinear normal mode response. The article concludes with an Appendix A entitled “The Law of Evolutionary Energy, Hamilton's Law of Varying Action and the Principle of Virtual Work”, providing essential insight to the non-virtual working equation of the authors’ previous articles and the current article, referred to as Hamilton's Law of Varying Action, to distinguish it from virtual concepts and virtual variational principles of classical dynamics. It is shown that this particular equation referred to as Hamilton's Law of Varying Action is a special case of the Law of Evolutionary Energy which inherently encompasses the First Law of Thermodynamics involving quantifiable real changes (alterations) along the real dynamic path, the application of which to mechanical systems yields both the current non-virtual working form of Hamilton's Law of Varying Action and the original form of the only enunciated (non-virtual) “law of varying action” by Hamilton which are not the same expressions.

Numerical Investigation of Constrained Direct Solutions Using Hamilton's Law

This paper presents numerical results obtained for a dynamic system that is modeled using Hamilton's law of varying action and for which geometric constraints are enforced explicitly. These results are compared with numerical results for the same system that were obtained from ordinary differential equations, differential-algebraic equations, and a formulation based on Hamilton's law in which the constraints were enforced implicitly. This comparison had two objectives: first, to show that numerical results from the formulation based on Hamilton's law and explicit constraints were equal or superior to results from other formulations; second, to verify that explicit constraints must be enforced at the end of the time step. Both objectives were successfully accomplished.

Direct Optimal Vibration Control of a Piezoelastic Plate

An optimal control method for distributed parameter systems is developed and applied to active vibration control of a plate. Piezoelectric material is used as an actuator. The structural dynamics are modeled using the space—time assumed mode expansion technique (a Rayleigh—Ritz method applied to both time and space variables). In particular, the generalized coordinates (both mechanical and electrical) and all generalized forces are described in terms of expansion functions. Using Hamilton's law of varying action along with the space—time assumed mode expansion results in algebraic equations of motion. These are then used as the constraint equations in the optimal controller design. Using space—time expansion, the usual variational optimal control problem is transformed into an equivalent algebraic problem. Optimal solutions are then obtained in a closed form and the solution is global within the time period considered. The solution procedure does not lead to a Riccati equation as is the case in the conventional optimal control solutions. The direct optimal control problem of a vibrating plate is illustrated through simulations where a parametric study is undertaken to check the controller performance. This study shows that the developed optimal control method is simple and the structure—control interaction is an inherent property of the method.

Hamilton and the Law of Varying Action Revisited

According to history texts, philosophers searched for a unifying natural law whereby natural phenomena and numbers are related. More than 2300 years ago, Aristotle postulated that nature requires minimum energy. More than 220 years ago, Euler applied the minimum energy postulate. More than 200 years ago, Lagrange provided a mathematical “proof” of the postulate for conservative systems. The resulting Principle of Least Action served only to derive the differential equations of motion of a conservative system. Then, 170 years ago, Hamilton presented what he claimed to be a “general method in dynamics.” Hamilton's resulting “Law of Varying Action” was supposed to apply to both conservative and non-conservative systems and was supposed to yield either the differential equations of motion or the integrals of those differential equations. However, no direct evaluation of the integrals of motion ever resulted from Hamilton's law of varying action. In 1975, a scant 29 years ago, following five years of controversy with engineer mechanicians, Dr. Wolfgang Yourgrau, Editor, Foundations of Physics, published my first paper based on Aristotle's postulate, without mathematical proof. That and subsequent papers present, through applications, a true “general method in dynamics.” In this essay, I present the mathematical proof that is missing from my 1975 and subsequent papers. Six fundamental integrals of analytical mechanics are derived from Aristotle's postulate. First, however, Hamilton must be revisited to show why his H function and his “force function” prevents the law of varying action from being the general method in dynamics that he claimed it to be. I have found that Hamilton’s Law of Varying Action (HLVA), as Hamilton presented it, cannot be applied to systems for which the force function is non-integrable. In 1972, Dr. B.E. Gatewood and Dr. D.P. Beres (then a graduate student) discovered that the end-point term associated with the principle of least action does not vanish. I named the new equation, “the general energy equation.” In 1973, because I was doing with it what Hamilton claimed could be done with HLVA, I simply assumed that this new equation was HLVA. I gave the new equation the misnomer HLVA. In 2001, I learned that I had made a grave mistake. I found that HLVA is at most a special case of the general energy equation. My interpretation of Aristotle's postulate permits one to by-pass the differential equations of motion completely for both conservative and non-conservative systems (no calculus of variations).

The Unifying Laws of Classical Mechanics

It is shown that, at the time of Euler and Lagrange, a belief led to an assumption. The assumption is applied to derive the principle of least action from the vis viva. The assumption is also applied to derive Hamilton's principles from the vis viva. It is shown that Hamilton, in his 1834 paper, countered the assumption of the earlier mathematicians. Finally, Hamilton's law, completely independent of the principle of least action and Hamilton's principles, is obtained to verify the foregoing assertions. The unifying laws of classical mechanics, contained within Hamilton's 1834 paper, are identified.

TIME MODES AND LINEAR SYSTEMS

A vector space approach for generating the response of linear multi-degree-of-freedom time-invariant and time-varying dynamic systems using Hamilton's law of varying action (HLVA) is presented. The boundary (initial) condition constraints on the temporal-basis-function expansions of the time-dependent variables has been removed, while preserving HLVA in its original form. This provides for the broadest choice of basis functions. As a result of this new approach, it has been demonstrated that the response of dynamic systems are composed of temporal modes herein denoted as fundamental-time modes (FTM). Using these fundamental-time-modes, the general solution for the system response is obtained without reference to initial conditions or forcing functions. The unique response of the system is subsequently generated by using the initial conditions and forcing functions to scale the FTM. This new approach is demonstrated to provide the exact analytical response, as well as provide accurate numerical response solutions to a forced-damped-spring–mass system, using admissible temporal-basis functions (TBF). The numerical response solutions using Gaussian radial basis functions, are compared to those obtained by using power-series and third order Hermite polynomials. The new methodology, which will be referred to as the universal method, in conjunction with Gaussian TBF has permitted use of transition intervals (ordinarily referred to as time steps) of unprecedented length (larger than the period of the motion) while still maintaining an accurate response solution.

Nonlinear Control for Vibrations of a Load of Swing Crane.

The purpose of crane operation is to carry a load to a desired position quickly and safely. The oscillations of a suspended load occur in the operation. The sway motions in radius and circumference directions are coupled in a swing crane, so the oscillations in both directions are controlled at the same time. In this paper, the oscillations of a load are reduced by a nonlinear control theory. First, the algebraic equations of motion (AEM) is derived by Hamilton's law for transition interval of the system. Second, A direct optimal control is applied to the nonlinear system based on AEM. The direct optimal controller of the crane system based on AEM is obtained under the assumption that the crane boom is rigid and the wire rope and load system is a nonlinear pendulum. The numerical examples for crane models show the efficiency of the controller for the suppression of sway motions and the ability of positioning. The effectiveness of the controller is verified by the experimental results.

Parametrized formulations of Hamilton's law for numerical solutions of dynamic problems: Part I. Global approximation

Many dynamic problems can be solved numerically by using Hamilton's law. The solution is expressed as a series in the time domain with undetermined coefficients. The unknown coefficients are determined by satisfying the Hamilton's law when the solution is allowed to have certain types of variations. The advantage of the method is that it can directly generate a set of algebraic equations without considering the dynamic equilibrium or the governing differential equations. In this paper, the essential features of the Hamilton's law and its variations are re-examined from the numerical perspectives. A general version of variation is proposed and the parametrized formulations are presented. The parametrized formulations unify conventional formulations and also yield many new ones. Illustrative numerical examples in this paper demonstrate that the conventional formulations may not be optimal although they may be rational.

Parametrized formulations of Hamilton's law for numerical solutions of dynamic problems: Part II. Time finite element approximation

In part I of this paper, we presented a consistent mathematical perspective to the formulations of Hamilton's law and unified the formulations by parametrized form with global approximation. In part II of this paper, we extend the formulations to a proper form to develop high-performance time finite elements for numerical solutions of dynamic problems. The two-field mixed formulations are emphasized and the particular features of using lower order interpolation functions are discussed.

MODELLING OF STOCHASTIC DYNAMIC SYSTEMS USING HAMILTON'S LAW OF VARYING ACTION

This paper presents a model based on Hamilton's law of varying action for stochastic dynamic systems. In this model, the state variables are approximated as a linear sum of orthogonal polynomials. For deterministic systems, the coefficients of the polynomials are constant, but for stochastic systems, the coefficients are random variables. The initial conditions are treated as Gaussian random variables with specified mean values and covariance matrix, and the external forcing functions (input processes) are treated as stationary Gaussian processes with specified mean and correlation functions. Using the Karhunen-Loeve expansion, the random input processes are represented in terms of truncated linear sums of orthonormal eigenfunctions with uncorrelated random variables as coefficients. This method reduces the infinite dimensional input force vector to one with finite dimensions. A method to compute the mean and the variance functions for both the singular and the regular components of the state processes is developed. A single-degree-of-freedom spring-mass-damper systems subjected to random initial conditions and a random forcing function is considered to show the feasibility and effectiveness of the formulation. The response is obtained without any reference to the differential equations associated with the dynamics of the system. The study shows that the results of this formulation agree well with those obtained using other schemes.

A modeling via Hamilton's law with application to a model following resembling technique

As a result of direct application of Hamilton's law of varying action on the treatment of mechanical systems, solving both dynamic and control problems is done using algebraic equations only; no differential equations are needed. In these techniques both the dynamic response and control forces are assumed to be a truncated power series. The target was only to solve the concerned problem. Using the coefficients of the above-mentioned polynomials in the vector form, an input-output model for the mechanical system to follow a specified path, while it is perturbed out of that path, is introduced herein. The modeling uses only algebraic equations. In this work, a model following resembling technique is demonstrated as a useful application of this modeling way. That is to get the required input forcing function to obligate the mechanical system to behave like a reference system when it is shifted away from the specified path.