On a stationarity principle for non-conservative dynamical systems

A stationarity principle for non-conservative, holonomic dynamical systems is formulated. It is based on the notion of Gâteaux directional derivative. Its relation to the classical and variational principle with non-commutative variational rules is discussed.     

On the nonlinear dynamic buckling mechanism of autonomous dissipative/nondissipative discrete structural systems

Summary Nonlinear dynamic buckling of nonlinearly elastic dissipative/nondissipative multi-mass systems, mainly under step load of infinite duration, is studied in detail. These systems, under the same loading applied statically, experience a limit point instability. The analysis can be readily extended to the case of dynamic buckling under impact loading. Energy, topological and geometrical aspects for the total potential energyV, which is constrained to lie in a region of phase-space whereV0, allow conclusions to be drawn directly regarding dynamic buckling. Criteria leading to very good, approximate and lower/upper bound dynamic buckling estimates are readily established without solving the highly nonlinear set of equations of motion. The theory is illustrated with several analyses of a two-degree-of-freedom model.     

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.     

Conservation laws in non-linear dissipative systems

In Hamiltonian theory, Noether's theorem commonly is used to show the conservation of linear momentum and energy as a consequence of symmetry properties. The possibility of enclosing Hamiltonian theory in a wider context by use of Gibbs-Falkian thermodynamical methods, offering the opportunity to cover mechanical and thermodynamical systems with the same mathematical tools, is considered. Consequently it is shown how Noether's identity can be extended for dissipative systems which are appropriate to describe real life phenomena. By use of the principle of least action an extended version of Noether's theorem is calculated, from which the conservation of linear momentum and total energy can be derived. Additionally, the condition of absolute invariance is shown to be too restrictive for physical applications.     

On damped non-linear dynamic systems with many degrees of freedom

Non-linear mass-spring-damper systems with many degrees of freedom are studied; all springs and/or all dampers may be strongly non-linear. It is shown that the ultimate state of completely damped systems is always rest, that of incompletely damped systems may be either rest or a periodic normal mode motion. Necessary and sufficient conditions are given for the existence of classical normal mode motion in completely or incompletely damped systems. When the system is linear, these reduce to known results found by Rayleigh and generalized by Caughey and O'Kelly.     

含高阶余项的非线性动力系统数值计算法

建立了一种求解非线性动力系统高精度数值计算的新方法,重构了等价的非线性动力系统方程,该方程考虑了非线性函数的任意高阶项,并给出了该方程的Duhamel积分表达式,在时间步长内用Newton-Raphson法进行数值迭代求解,该方法能连续满足微分方程而不只是在离散的步长端点满足方程,从而打破了传统的Euler型有限差分法。计算实例表明,该方法计算精度高于传统的Runge-Kutta,Newmark-β和Wilson-θ等方法。     

On the similarity solutions of wave propagation for a general class of non-linear dissipative materials

Deductive similarity analysis is employed to study one-dimensional wave propagation in rate dependent materials whose constitutive laws are special cases of Maxwellian materials (σ_t = φ(ε, σ)ε_t + ψ(ε, σ), ε = strain, σ = stress). The general problem is shown not to have a similar solution although many special cases have the independent similar variable (x ? c)/(t ? d)^e. These cases are studied and tabulated. Analytic similar solutions are presented for several cases and a discussion of permissable boundary conditions is given.     

On equilibrium instability for conservative and partially dissipative systems

The investigation brings some contributions to the classical problem of inverting the Lagrange-Dirichlet stability theorem. First, an example is given of a conservative holonomic mechanical system with a stable equilibrium at the origin, although the potential function is strictly negative along some rays issuing from the origin. Then, one establishes a new instability result in the conservative case. Last, by means of a vector auxiliary function, one proves an instability theorem for holonomic systems with partial dissipation.     

Broadband energy exchanges between a dissipative elastic rod and a multi-degree-of-freedom dissipative essentially non-linear attachment

We analyze complex, multi-frequency, non-linear modal interactions in the damped dynamics of a viscously damped dispersive finite rod coupled to a multi-degree-of-freedom essentially non-linear attachment. We perform a parametric study to show that the attachment can be an effective broadband energy absorber and dissipater of shock energy from the rod. It is shown that strong targeted energy transfer from the rod to the attachment occurs when there is strong stiffness asymmetry in the attachment. For weak viscous dissipation, a clear understanding of dynamical transitions in the integrated rod-non-linear attachment system can be gained by wavelet transforming the time series and superimposing the resulting wavelet spectra in the frequency-energy plot (FEP) of the periodic orbits of the underlying Hamiltonian system. Two distinct NES configurations are analyzed in detail, and their damped responses are analyzed by the Hilbert-Huang transform (HHT). We show that the HHT is capable of analyzing even complex non-linear damped transitions, by providing the dominant frequency components (or equivalently, time scales) at which the non-linear phenomena take place, and clarifying the series of non-linear resonance captures between the rod and attachment dynamics that are responsible for the broadband energy exchanges in this system.     

On a pure complementary energy principle and a force-based finite element formulation for non-linear elastic cables

A complementary-dual force-based finite element formulation is proposed for the geometrically exact quasi-static analysis of one-dimensional hyperelastic perfectly flexible cables lying in the two-dimensional space. This formulation employs as approximate functions the exact statically admissible force fields, i.e., those that satisfy the equilibrium differential equations in strong form, as well as the equilibrium boundary conditions. The formulation relies on a principle of total complementary energy only expressed in terms of force fields, being therefore called a pure principle. Under the assumption of stress-unilateral behavior, this principle can be regarded as being dual to the principle of minimum total potential energy, corresponding therefore to a maximum principle. Some numerical applications, including cables suspended from two and three points at the same level or at different levels, with both Hookean and Neo-Hookean material behaviors, are presented. As it will be shown, in contrast to the standard two-node displacement-based formulation derived from the principle of minimum total potential energy, the proposed dual force-based formulation is capable of providing the exact solution of a given problem only using a single finite element per cable. Both the proposed principle of pure complementary energy and its corresponding force-based finite element formulation can be easily extended to the case of cables lying in the three-dimensional space.     

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation $\text{dx}\text{dt}\text{= A(t)x + f(t,x)}$ by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.     

On the non-linear dynamic stability problem for thin elastic plates

In this paper the non-linear response of thin elastic plates under parametric excitation is investigated. A new analytical method is proposed. It gives the possibility to obtain all the characteristic features of the phenomenon considered, which are known from experiments—the existing of beats, their dependence on the excitation parameter, the influence of the initial conditions, the typical character of the vibrations in the different regions. Analog computer studies are carried out, and they show clearly the influence of different parameters on the output of the problem considered.     

On second-order non-linear systems with conservative limit-cycles

A class of non-linear systems whose limit cycles are continuously conservative is suggested and dealt with. Most of the discussion is carried out by considering the behaviour of certain members of the class. Some significant properties of such systems are demonstrated. These properties appear important in the development of electronic function generators of high precision. The analysis of certain pumps is also related to models of similar systems.     

An improved approach for random parametric response of dynamic systems with non-linear inertia

A non-Gaussian closure scheme is developed for determining the stationary response of dynamic systems including non-linear inertia and stochastic coefficients. Numerical solutions are obtained and examined for their validity based on the preservation of moments properties. The method predicts the jump phenomenon, for all response statistics at an excitation level very close to the threshold level of the condition of almost sure stability. In view of the increased degree of non-linearity, resulting from the non-Gaussian closure scheme, the mean square of the response displacement is found to be less than those values predicted by other methods such as the Gaussian closure or the first order stochastic averaging.     

On the normal modes of non-linear dual-mass systems

The motions of a class of non-linear dual-mass systems are investigated. A new characterization of non-linear normal modes in terms of solutions of initial value problems is developed that leads to interesting properties both on the existence of such non-linear normal modes and their continuous dependence upon the energies.