Stability of plane oscillations of a satellite in a circular orbit

We study motions of a rigid body (a satellite) about the center of mass in a central Newtonian gravitational field in a circular orbit. There is a known particular motion of the satellite in which one of its principal central axes of inertia is perpendicular to the orbit plane and the satellite itself exhibits plane pendulum-like oscillations about this axis. Under the assumption that the satellite principal central moments of inertia A, B, and C satisfy the relation B = A + C corresponding to the case of a thin plate, we perform rigorous nonlinear analysis of the orbital stability of this motion.In the plane of the problem parameters, namely, the oscillation amplitude ε and the inertial parameter, there exist countably many domains of orbital stability of the satellite oscillations in the linear approximation. Nonlinear orbital stability analysis was carried out in thirteen of these domains. Isoenergetic reduction of the system of equations of the perturbed motion is performed at the energy level corresponding to the unperturbed periodic motion. Further, using the algorithm developed in [1], we construct the symplectic mapping generated by the equations of the reduced system, normalize it, and analyze the stability. We consider resonance and nonresonance cases. For small values of the oscillation amplitude, we perform analytic investigations; for arbitrary values of ε, numerical analysis is used.Earlier, numerical analysis of stability of plane pendulum-like motions of a satellite in a circular orbit was performed in several special cases in [1–4].     

Analysis of the structure of generalized forces in the Lagrange equations

Stability of mechanical systems in the sense of Thomson and Tait [1] can be judged from the type of forces applied to them. The forces are usually divided into potential (conservative), circular, dissipative, accelerating, gyroscopic, etc. The decomposition itself of generalized positional forces into conservative and properly nonconservative forces is well known for the case in which these forces linearly depend on the generalized coordinates (e.g., see [1–5]). Such a decomposition is associated with the unique representation of an arbitrary matrix of these forces as the sum of symmetric and skew-symmetric parts. Generalized forces linearly depending on the velocities can in a similar way be divided into dissipative and gyroscopic parts. In the present paper, we show how the same decomposition can be performed in the general nonlinear case.     

Existence of Solutions for a Mathematical Model Related to Solid–Solid Phase Transitions in Shape Memory Alloys

We consider a strongly nonlinear PDE system describing solid–solid phase transitions in shape memory alloys. The system accounts for the evolution of an order parameter χ (related to different symmetries of the crystal lattice in the phase configurations), of the stress (and the displacement u), and of the absolute temperature ?. The resulting equations present several technical difficulties to be tackled; in particular, we emphasize the presence of nonlinear coupling terms, higher order dissipative contributions, possibly multivalued operators. As for the evolution of temperature, a highly nonlinear parabolic equation has to be solved for a right hand side that is controlled only in L ¹. We prove the existence of a solution for a regularized version by use of a time discretization technique. Then, we perform suitable a priori estimates which allow us pass to the limit and find a weak global-in-time solution to the system.     

A self-stabilized algorithm for enforcing constraints in multibody systems

A new algorithm is developed for the enforcement of constraints within the framework of nonlinear, flexible multibody system modeled with the finite element approach. The proposed algorithm exactly satisfies the constraints at the displacement and velocity levels, and furthermore, it achieves nonlinear unconditional stability by imposing the vanishing of the work done by the constraint forces when combined with specific discretizations of the inertial and elastic forces. Identical convergence rates are observed for the displacements, velocities, and Lagrange multipliers. The proposed algorithm is closely related to the stabilized index-2 or GGL method, although no additional multipliers are introduced in the present approach. These desirable characteristics are obtained without resorting to numerically dissipative algorithms. If high frequencies are present in the system, i.e. the system is physically stiff, dissipative schemes become necessary; the proposed algorithm is extended to deal with this situation.     

A simple criterion for finite time stability with application to impacted buckling of elastic columns

The existing theories of finite-time stability depend on a prescribed bound on initial disturbances and a prescribed threshold for allowable responses. It remains a challenge to identify the critical value of loading parameter for finite time instability observed in experiments without the need of specifying any prescribed threshold for allowable responses. Based on an energy balance analysis of a simple dynamic system, this paper proposes a general criterion for finite time stability which indicates that finite time stability of a linear dynamic system with constant coefficients during a given time interval [0, t _f ] is guaranteed provided the product of its maximum growth rate (determined by the maximum eigen-root p₁ >0) and the duration t _f does not exceed 2, i.e., p₁t _f <2. The proposed criterion (p₁t _f =2) is applied to several problems of impacted buckling of elastic columns: (i) an elastic column impacted by a striking mass, (ii) longitudinal impact of an elastic column on a rigid wall, and (iii) an elastic column compressed at a constant speed (“Hoff problem”), in which the time-varying axial force is replaced approximately by its average value over the time duration. Comparison of critical parameters predicted by the proposed criterion with available experimental and simulation data shows that the proposed criterion is in robust reasonable agreement with the known data, which suggests that the proposed simple criterion (p₁t _f =2) can be used to estimate critical parameters for finite time stability of dynamic systems governed by linear equations with constant coefficients.     

Dissipative dynamics of geometrically nonlinear Bernoulli-Euler beams

We consider nonlinear dissipative vibrations of the Bernoulli-Euler beam. We find that, under the action of a transverse alternating load, the vibrations may become chaotic. We study a scenario in which harmonic vibrations become chaotic, namely, the Feigenbaum scenario, and find the Feigenbaumconstant. In the present paper, we paymuch attention to the reliability of the results obtained. To this end, we use two methods, the finite difference O(h ²) method and the finite element method in the Bubnov-Galerkin form, and verify the convergence of these methods.     

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.     

Instabilities of a thin viscoelastic liquid film flowing down an inclined plane in the presence of a uniform electromagnetic field

The effect of a uniform electromagnetic field on the stability of a thin layer of an electrically conducting viscoelastic liquid flowing down on a nonconducting inclined plane is studied under the induction-free approximation. Long-wave expansion method is used to obtain the surface evolution equation. The stabilizing role of the magnetic parameter M and the destabilizing role of the viscoelastic parameter Γ as well as the electric parameter E on this flow field are established. A novel result which emerges from our analysis is that the stabilizing effect of M holds no longer true for both viscous and viscoelastic fluids in the presence of electromagnetic field. It is found that when E exceeds a certain critical value depending on Γ, magnetic field exhibits the destabilizing effect on this flow field. Indeed, this critical value decreases with the increase of the viscoelastic parameter Γ since it has a destabilizing effect inherently. Another noteworthy result which arises from the weakly nonlinear stability analysis is that both the subcritical unstable and supercritical stable zones are possible together with the unconditional stable and explosive zones for different values of Γ depending on the wave number k.     

Periodic solutions of <Emphasis Type="Italic">n</Emphasis>-dofs autonomous vibro-impact oscillators with one lasting contact phase

In the continuum structural mechanics framework, a unilateral contact condition between two flexible bodies does not generate impulsive contact forces. However, finite-dimensional systems, derived from a finite element semi-discretization in space for instance, and undergoing a unilateral contact condition, require an additional impact law: Unilateral contact occurrences then become impacts of zero duration unless (i) the impact law is purely inelastic, or (ii) the pre-impact velocity is zero. This contribution explores autonomous periodic solutions with one contact phase per period and zero pre-impact velocity [case (ii)], for any n-dof mechanical systems involving linear free-flight dynamics together with a linear unilateral contact constraint. A recent work has shown that such solutions seem to be limits of periodic trajectories with k impacts per period as k increases. Minimal analytic equations governing the existence of such solutions are proposed, and it is proven that, generically, they occur only for discrete values of the period. It is also shown that the graphs of such periodic solutions have two axes of symmetry in time. Results are illustrated on a spring–mass system and on a 4-dof two-dimensional system made of 1D finite elements. Animations of SPPs with up to 30 dofs are provided.     

Nonlinear stability of two-dimensional steady flows of an ideal fluid with constant vorticity and their axisymmetric analogs

Flows with constant vorticity are widely used as local models of more complicated flows [1]. In many cases, such flows are stable against finite two-dimensional perturbations. In particular, the inviscid plane-parallel Couette flow has the property of nonlinear stability. Similar treatment of a class of axisymmetric flows yields nonlinear stability of a spherical Hill vortex and inviscid Poiseuille flow in a circular tube with respect to axisymmetric perturbations.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 16–21, October–December, 1981.     

Global Attractor for a Nonlinear Oscillator Coupled to the Klein–Gordon Field

The long-time asymptotics is analyzed for all finite energy solutions to a model\(\mathbf{U}(1)\)-invariant nonlinear Klein–Gordon equation in one dimension, with the nonlinearity concentrated at a single point: each finite energy solution converges as t→ ± ∞ to the set of all “nonlinear eigenfunctions” of the form ψ(x)e^{?iω t}. The global attraction is caused by the nonlinear energy transfer from lower harmonics to the continuous spectrum and subsequent dispersive radiation.We justify this mechanism by the following novel strategy based on inflation of spectrum by the nonlinearity. We show that any omega-limit trajectory has the time spectrum in the spectral gap [ ? m,m] and satisfies the original equation. This equation implies the key spectral inclusion for spectrum of the nonlinear term. Then the application of the Titchmarsh convolution theorem reduces the spectrum of each omega-limit trajectory to a single harmonic \(\omega\in[-m,m]\).The research is inspired by Bohr’s postulate on quantum transitions and Schrödinger’s identification of the quantum stationary states to the nonlinear eigenfunctions of the coupled\(\mathbf{U}(1)\)-invariant Maxwell–Schrödinger and Maxwell–Dirac equations.     

Destabilization paradox due to breaking the Hamiltonian and reversible symmetry

Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. The present paper shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle” and “Whitney umbrella” that govern stabilization and destabilization. In case of two degrees of freedom, approximations of the stability boundary near the singularities are found in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping.     

Lyapunov-Kozlov method for singular cases

Lyapunov’s first method, extended by Kozlov to nonlinear mechanical systems, is applied to study the instability of the equilibrium position of a mechanical system moving in the field of conservative and dissipative forces. The cases with a tensor of inertia or a matrix of coefficients of the Rayleigh dissipative function are analyzed singularly in the equilibrium position. This fact renders the impossible application of Lyapunov’s approach in the analysis of the stability because, in the equilibrium position, the conditions of the existence and uniqueness of the solutions to the differential equations of motion are not fulfilled. It is shown that Kozlov’s generalization of Lyapunov’s first method can also be applied in the mentioned cases on the conditions that, besides the known algebraic expression, more are fulfilled. Three theorems on the instability of the equilibrium position are formulated. The results are illustrated by an example.     

Mechanobiology and morphogenesis in living matter: a survey

Morphogenesis in living tissues is the paramount example of a time- and space-dependent orchestration of living matter where shape and order emerge from undifferentiated initial conditions. The genes encode the protein expression that eventually drives the emergence of the phenotype, while energy supply and cell-to-cell communication mechanisms are necessary to such a process. The overall control of the system likely exploits the laws of chemistry and physics through robust and universal processes. Even if the identification of the communication mechanisms is a question of fundamental nature, a long-standing investigation settled in the realm of chemical factors only (also known as morphogens) faces a number of apparently unsolvable questions. In this paper, we investigate at what extent mechanical forces, alone or through their biological feedbacks, can direct some basic aspects of morphogenesis in development biology. In this branch of mechano-biology, we discuss the typical rheological regimes of soft living matter and the related forces, providing a survey on how local mechanical feedbacks can control global size or even gene expression. We finally highlight the pivotal role of nonlinear mechanics to explain the emergence of complex shapes in living matter.     

On solitary wave diffraction by multiple,in-line vertical cylinders

The interaction of solitary waves with multiple, in-line vertical cylinders is investigated. The fixed cylinders are of constant circular cross section and extend from the seafloor to the free surface. In general, there are N of them lined in a row parallel to the incoming wave direction. Both the nonlinear, generalized Boussinesq and the Green–Naghdi shallow-water wave equations are used. A boundary-fitted curvilinear coordinate system is employed to facilitate the use of the finite-difference method on curved boundaries. The governing equations and boundary conditions are transformed from the physical plane onto the computational plane. These equations are then solved in time on the computational plane that contains a uniform grid and by use of the successive over-relaxation method and a second-order finite-difference method to determine the horizontal force and overturning moment on the cylinders. Resulting solitary wave forces from the nonlinear Green–Naghdi and the Boussinesq equations are presented, and the forces are compared with the experimental data when available.     

Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading

Summary Axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems are investigated by means of a so-called deformation map. The deformation map was further used for stability considerations of geometrically nonlinear shells, see Shilkrut [1, 2]. The map reveals the complete picture of the axially symmetric deformations and the stability of the investigated structure. The equilibrium differential equations for the above mentioned circular plate were derived by Timoshenko [3]. The boundary value problem of the investigated structure is transformed to an initial value problem (Cauchy's problem). Then the Runge-Kutta (R. K.) method can be used to solve numerically the equilibrium equations. The geometrically nonlinear, simply supported circular plate subjected to uniform radial force and uniform radial bending moment acting along the supported edge is investigated as example, and some new qualitative and quantitative results are obtained. This approach can be used without essential difficulties for the investigation of axially symmetric deformations and stability of a geometrically nonlinear circular plate subjected to multiparametrical static loading systems in elastic and non-elastic fields.

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Axialsymmetrische Verformung und Stabilität geometrisch nichtlinearer Kreisplatten unter mehrparametrischer statischer Belastung

Übersicht Zur Untersuchung axialsymmetrischer, geometrisch nichtlinearer Verformung von Kreisplatten und ihrer Stabilität bei mehrparametrischer Belastung wird eine sog. Deformationskarte benutzt. Sie wurde auch für Stabilitätsbetrachtungen geometrisch nichtlinearer Schalen benutzt, s. Shilkrut [1,2]. Die Karte zeigt das vollständige Bild der axialsymmetrischen Verformung und die Stabilität der untersuchten Struktur. Das Randwertproblem zu den differentiellen Gleichgewichtsbedingungen, die für die betrachtete Platte von Timoshenko [3] hergeleitet wurden, wird in ein Anfangswertproblem (Caudy-Problem) überführt, welches numerisch nach der Methode von Runge-Kutta gelöst wird. Als Beispiel wird die nichtlineare Kreisplatte unter radialer Zug-und Biegemomentenbelastung am einfach gestützten Umfang untersucht, und man erhält einige neue qualitative und quantitative Ergebnisse. Die Methode läßt sich ohne wesentliche Schwierigkeiten auch auf axialsymmetrische, nichtlineare Verformungen und die Stabilität von Kreisplatten unter anderen mehrparametrischen statischen Belastungen im elastischen und nichtelastischen Bereich anwenden.

     

Nonlinear dynamics of flexible tethered satellite system subject to space environment

The paper studies the nonlinear dynamics of a flexible tethered satellite system subject to space environments, such as the J ₂ perturbation, the air drag force, the solar pressure, the heating effect, and the orbital eccentricity. The flexible tether is modeled as a series of lumped masses and viscoelastic dampers so that a finite multidegree-of-freedom nonlinear system is obtained. The stability of equilibrium positions of the nonlinear system is then analyzed via a simplified two-degree-freedom model in an orbital reference frame. In-plane motions of the tethered satellite system are studied numerically, taking the space environments into account. A large number of numerical simulations show that the flexible tethered satellite system displays nonlinear dynamic characteristics, such as bifurcations, quasi-periodic oscillations, and chaotic motions.     

Application of the angular superposition method to the contact problem on the compression of an elastic cylinder

The angular superposition method is used to construct an approximate solution of the contact problem on the compression of an elastic cylinder by two rigid plates. The solution thus obtained has a closed-form analytic expression and can be used in the entire domain of the cylinder cross-section. We analyze the absolute error, which takes the largest value near the points of contact between the plates and the cylinder, where the boundary conditions are discontinuous. According to the von Mises criterion, when moving into the depth of the cylinder from the contact site along the symmetry axis, the second invariant J ₂ of the stress deviator tensor first decreases and then, after attaining a minimum, increases and attains the largest value at a small depth, which agrees with Johnson’s photoelastic experiments and Dinnik’s computations. We present the graphs of the displacement and normal stress distributions over the contact site, the dependence of the compressing force on the displacements of rigid plates, and the dependence of the invariant J ₂ on the coordinate along the symmetry axis. If 640 computation points are chosen on the cylinder boundary and the Hertz law for the normal pressure on the contact site is used, then the error in the approximate solution near the endpoint of the contact site is approximately 55%, and if the proposed two-parameter normal law is used, then the error is of the order of 4%. On the free lateral surface of the cylinder boundary, we find the critical pointM*, which separates the cylinder contraction and extension parts.The contact problems are the most difficult problems, and their solution is complicated by the discontinuous boundary conditions [1–5]. In [6], the contact problem is solved by the Fourier method, which can be used only for bodies of classical shapes. In such cases, the problem can be reduced to solving coupled integral equations [7]. The interaction between the bandage and a cylindrical body is considered in [2, 6, 7]. In [8], the possibility of using the finite element method is investigated in the case of contact problems for a differential wheel with roughness of the contacting surfaces taken into account. In [9, 10], the method of homogeneous solutions is used to consider contact problems for a finite-dimensional elastic cylinder loaded on its end surfaces. Note that only error estimates are given in the literature cited above; the absolute error over the entire domain of the elastic body is not studied, although this is one of the important characteristics of the obtained approximate solution. A sufficiently complete survey of the literature in the field of contact interactions of elastic bodies is given in [3–5].In what follows, we propose to solve contact problems by the angular superposition method [11]. This method can be used for bodies of nonclassical shapes, which can be multiply connected, and the friction on the contact site can be taken into account. In the present paper, as a first example of applied character, we show how this method can be used in the simplest case. The multiple connectedness and the curvilinearity of the shape of the body, as well as taking into account the friction on the boundary, do not create new essential difficulties in this method.     

Effective characteristics of corrugated plates

Corrugated plates are widely used in modern constructions and structures, because they, in contrast to plane plates, possess greater rigidity. In many cases, such a plate can be modeled by a homogeneous anisotropic plate with certain effective flexural and tensional rigidities. Depending on the geometry of corrugations and their location, the equivalent homogeneous plate can also have rigidities of mutual influence. These rigidities allow one to take into account the influence of bending moments on the strain in the midplane and, conversely, the influence of longitudinal strains on the plate bending [1]. The behavior of the corrugated plate under the action of a load normal to the midsurface is described by equations of the theory of flexible plates with initial deflection. These equations form a coupled system of nonlinear partial differential equations with variable coefficients [2]. The dependence of the coefficients on the coordinates is determined by the corrugation geometry. In the case of a plate with periodic corrugation, the coefficients significantly vary within one typical element and depend on the values of local variables determined in each of the typical elements. There is a connection between the local and global variables, and therefore, the functions of local coordinates are simultaneously functions of global coordinates, which are sometimes called rapidly oscillating functions [3].One of the methods for solving the equations with rapidly oscillating coefficients is the asymptotic method of small geometric parameter. The standard procedure of this method usually includes preparatory stages. At the first stage, as a rule, a rectangular periodicity cell is distinguished to be a typical element. At the second stage, the scale of global coordinates is changed so that the rectangular structure periodicity cells became square cells of size l × l. The third stage consists in passing to the dimensionless global coordinates relative to the plate characteristic dimension L. As a result, the dependence between the new local variables and the new scaled dimensionless variables is such that the factor 1/α, where α=l/L ? 1 is a small geometric parameter, appears in differentiating any function of the local coordinate with respect to the global coordinate. After this, the solution of the problem in new coordinates is sought as an asymptotic expansion in the small geometric parameter [1], [4–10].We note that, in the small geometric parameter method, the asymptotic series simultaneously have the form of expansions in the gradients of functions depending only on the global coordinates. This averaging procedure can be applied to linear and nonlinear boundary value problems for differential equations with variable periodic coefficients for which the periodicity cell can be affinely transformed into the periodicity cube. In the case of an arbitrary dependence of the coefficients on the coordinates (including periodic dependence), another averaging technique can be used in linear problems. This technique is based on the possibility of the integral representation of the solution of the original problem for the linear equation with variables coefficients in terms of the solution of the same problem for an equation of the same type but with constant coefficients [11–13]. The integral representation implies that the solution of the original problem can be represented in the form of the series in the gradients of the solution of the problem for the equation with constant coefficients [13].The aim of the present paper is to develop methods for calculating effective characteristics of corrugated plates. To this end, we first write out the equilibrium equations for a flexible anisotropic plate, which is inhomogeneous in the thickness direction and in the horizontal projection, with an initial deflection. We write these equations in matrix form, which allows one to significantly reduce the length of the expressions and to simplify further calculations. After this, we average the initial matrix equations with variable coefficients. The averaging procedure implies the statement of problems such that, after solving them, we can calculate the desired effective characteristics. By way of example, we consider the case of a corrugated plate made of a homogeneous isotropic material whose corrugations are hexagonal in the horizontal projection. In this case, we obtain approximate expressions for the components of the effective tensors of flexural rigidity and longitudinal compliance and expressions for the effective plate thickness.     

Studies of nonlinear deformation and stability of oval and elliptic cylindrical shells under axial compression

The stability of noncircular shells, in contrast to that of circular ones, has not been studied sufficiently well yet. The publications about circular shells are counted by thousands, but there are only several dozens of papers dealing with noncircular shells. This can be explained on the one hand by the fact that such shells are less used in practice and on the other hand by the difficulties encountered when solving problems involving a nonconstant curvature radius, which results in the appearance of variable coefficients in the stability equations. The well-known solutions of stability problems were obtained by analytic methods and, as a rule, in the linear approximation without taking into account the moments and nonlinearity of the shell precritical state, i.e., in the classical approximation. Here we use the finite element method in displacements to solve the problem of geometrically nonlinear deformation and stability of cylindrical shells with noncircular contour of the transverse cross-section. We use quadrilateral finite elements of shells of natural curvature. In the approximations to the element displacements, we explicitly distinguish the displacements of elements as rigid bodies. We use the Lagrange variational principle to obtain a nonlinear system of algebraic equations for determining the unknown nodal finite elements. We solve the system by a step method with respect to the load using the Newton-Kantorovich linearization at each step. The linear systems are solved by the Kraut method. The critical loads are determined with the use of the Silvester stability criterion when solving the nonlinear problem. We develop an algorithm for solving the problem numerically on personal computers. We also study the nonlinear deformation and stability of shells with oval and elliptic transverse cross-section in a wide range of variations in the ovalization and ellipticity parameters. We find the critical loads and the shell buckling modes. We also examine how the critical loads are affected by the strain nonlinearity and the ovalization and ellipticity of shells.