Constrained mechanical systems and gradient systems with strong Lyapunov functions

The characteristics of stationary and non-stationary gradient systems with strong Lyapunov functions are studied. The conditions under which holonomic mechanical systems, generalized Birkhoff systems and generalized Hamilton systems can be considered as gradient systems are given. The characteristics of the gradient systems can be used to study the stability of the mechanical systems. Some examples are given to illustrate the application of the results.     

Energy integrals for mechanical systems with nonlinear nonholonomic constraints

The work analyzes energy relations for nonholonomic systems, whose motion is restricted by nonlinear nonholonomic constraints. For the mechanical systems with linear constraints, the analysis of energy relations was carried out in [1], [2], [3], [4], [5], [6] …. On the basis of corresponding Lagrange’s equations, a general law of the change in energy dε/dt is formulated for mentioned systems by the help of which it is shown that there are two types of the laws of conservation of energy, depending on the structure of elementary work of the forces of constraint reactions. Also, the condition for existing the second type of the law of conservation of energy is formulated in the form of the system of partial differential equations. The obtained results are illustrated by a model of nonholonomic mechanical system.     

Gradient and vorticity banding

“Banded structures” of macroscopic dimensions can be induced by simple shear flow in many different types of soft matter systems. Depending on whether these bands extend along the gradient or vorticity direction, the banding transition is referred to as “gradient banding” or “vorticity banding,” respectively. The main features of gradient banding can be understood on the basis of a relatively simple constitutive equation. This minimal model for gradient banding will be discussed in some detail, and its predictions are shown to explain many of the experimentally observed features. The minimal model assumes a decrease of the shear stress of the homogeneously sheared system with increasing shear rate within a certain shear-rate interval. The possible microscopic origin of the severe shear-thinning behaviour that is necessary for the resulting nonmonotonic flow curves is discussed for a few particular systems. Deviations between experimental observations and predictions by the minimal model are due to obvious simplifications within the scope of the minimal model. The most serious simplifications are the neglect of concentration dependence of the shear stress (or on other degrees of freedom) and of the elastic contributions to the stress, normal stresses, and the possibility of shear-induced phase transitions. The consequences of coupling of stress and concentration will be analyzed in some detail. In contrast to predictions of the minimal model, when coupling to concentration is important, a flow instability can occur that does not require strong shear thinning. Gradient banding is sometimes also observed in glassy- and gel-like systems, as well as in shear-thickening systems. Possible mechanisms that could be at the origin of gradient-band formation in such systems are discussed. Gradient banding can also occur in strongly entangled polymeric systems. Banding in these systems is discussed on the basis of computer simulations. Vorticity banding is less well understood and less extensively investigated experimentally as compared to gradient banding. Possible scenarios that are at the origin of vorticity banding will be discussed. Among other systems, the observed vorticity-banding transition in rod-like colloids is discussed in some detail. It is argued, on the basis of experimental observations for these colloidal systems, that the vorticity-banding instability for such colloidal suspensions is probably related to an elastic instability, reminiscent of the Weissenberg effect in polymeric systems. This mechanism might explain vorticity banding in discontinuously shear-thickening systems and could be at work in other vorticity-banding systems as well. This overview does not include time-dependent phenomena like oscillations and chaotic behaviour.     

Convergence of Global and Bounded Solutions of a Second Order Gradient like System with Nonlinear Dissipation and Analytic Nonlinearity

In this paper we establish convergence to equilibrium of all global and bounded solutions of a gradient like system of second order with nonlinear dissipation and analytic nonlinearity. We estimate also the rate of convergence.     

Gradient elasticity and nonstandard boundary conditions

Gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality. The pertinent thermodynamic restrictions upon the gradient constitutive equations are derived, which are shown to include, besides the field (differential) stress–strain laws, a set of nonstandard boundary conditions. Consistently with the latter thermodynamic requirements, a surface layer with membrane stresses is envisioned in the strained body, which together with the above nonstandard boundary conditions make the body constitutively insulated (i.e. no long distance energy flows out of the boundary surface due to nonlocality). The total strain energy is shown to include a bulk and surface strain energy. A minimum total potential energy principle is provided for the related structural boundary-value problem. The Toupin–Mindlin polar-type strain gradient material model is also addressed and compared with the above one, their substantial differences are pointed out, particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations. A gradient one-dimensional bar sample in tension is considered for a few applications of the proposed theory.     

Solid damping in micro electro mechanical systems

This paper focuses on the problem of the numerical evaluation of dissipation induced by thermoelastic coupling in microelectromechanical systems. An ad hoc conceived, FE based, numerical procedure for the evaluation of the thermoelastic dissipation is proposed and the numerical results are compared with analytical solutions. In order to introduce in the numerical response a dependence on the size of the resonating devices, which is experimentally observed at very small dimensions, a new enhanced non-local coupled thermoelastic model is proposed and the first results are discussed. An erratum to this article can be found at     

Dynamics of mechanical systems involving impact and friction using an efficient contact detection algorithm

In this work, some new results are presented on the dynamics of a class of multibody mechanical systems, involving contact and friction. The main contribution refers to the development of a systematic, accurate and efficient method for detecting contact among the components of a system of solid bodies. For some simple geometries, this task is achieved by employing analytical means. For systems possessing components with complex geometric shapes a more involved numerical methodology is developed. In both cases, once a potential contact point is detected, the common tangent plane and normal vector are located and the penetration depth is calculated, leading to determination of the force arising between the contacting bodies. This information is then passed to a solver, providing the full dynamic response of the system. The validity and numerical efficiency of the methodology developed is first demonstrated by considering a number of examples with relatively small geometric complexity but large traditional value and interesting dynamic response. Some new results are obtained and presented on the dynamics of these systems. Finally, the same methodology is also tested in a more complicated and demanding mechanical application.     

Skew-gradient representations of constrained mechanical systems

The characteristics of stationary and non-stationary skew-gradient systems are studied. The skew-gradient representations of holonomic systems, Birkhoffian systems, generalized Birkhoffian systems, and generalized Hamiltonian systems are given. The characteristics of skew-gradient systems are used to study integration and stability of the solution of constrained mechanical systems. Examples are given to illustrate applications of the result.     

Stability analysis of relative equilibria of mechanical systems with cyclic coordinates: a direct approach

Nonlinear stability of relative equilibria of mechanical systems has been investigated during the past two decades by notable authors and has resulted in the so-called energy momentum method. Although it has numerous important engineering applications, this theory involves subtle mathematical methods such as group theory with which engineers usually are not familiar. This paper develops a simple and natural approach to the problem for the case of cyclic coordinates in the Lagrangian since many practical examples can be easily formulated in terms of cyclic coordinates. Referring to standard algebraic operations, a stability criterion for relative equilibria is derived. As a computational benefit the presented approach does not require knowledge of a system's complete kinetic energy, either for formulating steady-state equations or for checking stability. The application of the method, which is closely related to Routh's method, will be demonstrated using the example of a dumbell satellite.     

A controller for 2-DOF underactuated mechanical systems with discontinuous friction

We propose a controller for a class of 2-DOF underactuated mechanical systems with discontinuous friction in the unactuated joint. The control objective is the regulation of the unactuated variable while the position and speed of the actuated joint remain bounded. The unactuated joint is considered as a mechanical system with discontinuous friction but continuous, artificial control input given by a term depending on the actuated positions and velocities. The proposed controller guarantees the convergence of the position error of the unactuated joint to zero, and it is robust with respect to some uncertainty in the discontinuous friction coefficients. We illustrate the technique with its application to two systems.     

Lie Point Symmetries,Conservation and Balance Laws in Linear Gradient Elastodynamics

The aim of this work is the derivation of Lie point symmetries, conservation and balance laws in linear gradient elastodynamics of grade-2 (up to second gradients of the displacement vector and the first gradient of the velocity). The conservation and balance laws of translational, rotational, scaling variational symmetries and addition of solutions are derived using Noether’s theorem. It turns out that the scaling symmetry is not a strict variational symmetry in gradient elasticity.      

Stability of impulsive nonholonomic mechanical systems

Sufficient stability conditions for the manifold of equilibrium states of a nonholonomic mechanical system are established. The method of integral inequalities is used in combination with the comparison principle __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 135–143, March 2008.     

A generalized approach to tracking the nominals of mechanical systems

Tracking inputs and trajectories of mechanical systems as control objects is analyzed with an emphasis on the objects having the number of external inputs different from the number of their degrees of freedom (DOFs). The problems of the synthesis of the realizable nominals and control laws were considered especially for the case where the number of DOFs of the control object is larger than the number of its external inputs. The difference between control laws for nominals tracking and regulation laws is demonstrated.     

A critical review of microscale mechanical testing methods used in the design of microelectromechanical systems

Microelectromechanical systems (MEMS) technologies are evolving at a rapid rate with increasing activity in the design, fabrication, and commercialization of a wide variety of microscale systems and devices. The importance of accurate mechanical property measurement for successful design was realized early on in the development of this field. Consequently, there exist many different techniques to measure quantities such as the Young's modulus (E), yield strength (σ _Y ), fracture strength (σ _F ), residual stress (σ _F ), and residual stress gradient (∇σ _R ) of microscale structures and materials. We review and critically compare several of the important techniques including the microtension test, axisymmetric plate bend test, microbeam bend test, M-test, wafer curvature measurements, dynamic (resonant) tests, fabrication of passive strain sensors, and Raman spectroscopy. We discuss the characteristics of typical test structures, and the common sources of structure-related errors in measurement. A rational approach for the selection of test techniques for the design of microsystems is suggested.     

The analysis of arbitrarily-damped linear mechanical systems

Summary The analysis of arbitrarily-damped linear mechanical systems is the subject of this paper, the main issue being the analysis of nondecouplable systems. It is well known that decouplable systems occur when the damping matrix happens to be a linear combination of the mass and stiffness matrices. Systems with this type of damping are said to have proportional damping, which nevertheless seldom occurs in practice, but many a damped system is analyzed under the assumption that it is proportionally damped. In fact, this property allows the analyst to study these systems using the same approach as that applicable to their undamped counterparts. In this paper, we show that proportional damping need not be assumed in order to analyze the system at hand with the same approach as used to analyze undamped systems. Moreover, we propose an algorithm to determine the natural frequencies, the damped frequencies and the damping ratios of an n-degree-of-freedom damped system, that does not require the casting of the system into first-order form. In this way, the characteristic equation is derived naturally as a 2n-degree polynomial, computing its roots being then straightforward. Furthermore, we propose a semigraphical method to ease this calculation, which should be attractive to practicing engineers. Received 5 April 1999; accepted for publication 7 May 1999     

ROUTH’S EQUATIONS FOR GENERAL NONHOLONOMIC MECHANICAL SYSTEMS OF VARIABLE MASS

In this paper,Routh’s equations for the mechanical systems of the variable masswith nonlinear nonholonomic constraints of arbitrary orders in a noninertial referencesystem have been deduced not from any variational principles,but from the dynamicalequations of Newtonian mechanics.And then again the other forms of equations fornonholonomic systems of variable mass are obtained from Routh’s equations.     

Parametrization of sets of stabilizing controllers in mechanical systems

Procedures to parametrize a set of stabilizing controllers are reviewed. These procedures are the key ones in the frequency-domain synthesis of the optimal (minimum H ₂-and H _∞-norms) controller or filter for a linear stationary system. A relationship between the parametrization procedures proposed by different authors is shown. Examples of parametrization procedures in synthesis problems (delay problems, multichannel filtering problems, etc.) are given __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 6, pp. 3–27, June 2008.     

Principal parametric resonances of non-linear mechanical system with two-frequency and self-excitations

The principal parametric resonance of a single-degree-of-freedom system with non-linear two-frequency parametric and self-excitations is investigated. In particular, the case in which the parametric excitation terms with close frequencies is examined. The method of multiple scales is used to determine the equations that describe to first-order the modulation of the amplitude and phase. Qualitative analysis and asymptotic expansion techniques are employed to predict the existence of steady state responses. Stability is investigated. The effect of damping, magnitudes of non-linear excitation and self-excitation are analyzed.     

Gradient plasticity modeling of geomaterials in a meshfree environment. Part I: Theory and variational formulation

Deformation and strength behavior of geomaterials in the pre- and post-failure regimes are of significant interest in various geomechanics applications. To address the need for development of a realistic constitutive framework, which allows for an accurate simulation of pre-failure response as well as an objective and meaningful post-failure response, a strain gradient plasticity model is formulated by incorporating the spatial gradients of elastic strain in the evolution of stress and gradients of plastic strain in the evolution of the internal variables. In turn, gradients of only kinematic variables are included in the constitutive equations. The resulting constitutive equations along with the balance of linear momentum for the continuum are cast as a coupled system of equations, with displacements and plastic multiplier appearing as the primary unknowns in the final governing integral equations. To avoid singular stress fields along element boundaries, a finite element discretization of the governing equations would require C² continuous displacements and C¹ continuous plastic multiplier, which is undesirable from a numerical implementation point of view. This issue is naturally resolved when a meshfree discretization is used. Hence the developed model is formulated within the framework of a meshfree environment. The new constitutive model allows an analysis of grain size effects on strength and dilatancy of rocks. The role and effectiveness of the new gradient terms on regularizing the underlying boundary value problems of geomechanics beyond the initiation of strain localization will be assessed in a future paper.     

The issues of the uniqueness and the stability of the homogeneous response in uniaxial tests with gradient damage models

We consider a wide class of gradient damage models which are characterized by two constitutive functions after a normalization of the scalar damage parameter. The evolution problem is formulated following a variational approach based on the principles of irreversibility, stability and energy balance. Applied to a monotonically increasing traction test of a one-dimensional bar, we consider the homogeneous response where both the strain and the damage fields are uniform in space. In the case of a softening behavior, we show that the homogeneous state of the bar at a given time is stable provided that the length of the bar is less than a state dependent critical value and unstable otherwise. However, we also show that bifurcations can appear even if the homogeneous state is stable. All these results are obtained in a closed form. Finally, we propose a practical method to identify the two constitutive functions. This method is based on the measure of the homogeneous response in a situation where this response is stable without possibility of bifurcation, and on a procedure which gives the opportunity to detect its loss of stability. All the theoretical analyses are illustrated by examples.