Note on static and dynamic instability of a non-uniform beam

The instability of a non-uniform beam analyzed by Lee and Reissner from a static point of view is reanalyzed by a dynamic approach. The beam is loaded by a parallel or tangential compressive force respectively. As expected the load causing static instability, obtained by both approaches, is exactly the same. A peculiar behaviour of the frequency for both cases of loading is revealed by the dynamic method. The tangential load causing dynamic instability is obtained.

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Zusammenfassung Die Instabilität eines Stabes mit konstantem Querschnitt, welche durch Lee und Reissner vom statischen Standpunkt aus untersucht worden ist, wird einer dynamischen Analyse unterzogen. Der Stab steht unter einer axialen bzw. tangentialen Drucklast. Wie erwartet, ist die Belastung, welche statische Instabilität erzeugt, bei beiden Untersuchungsmethoden dieselbe. Ein besonderes Verhalten der Frequenz für beide Lastfälle wird durch die dynamische Methode aufgezeigt, und es wird die tangentiale Last erhalten, welche dynamische Instabilität erzeugt.

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Preparation of this note has been supported by Sherutei Teuffa, Ltd., Haifa, Israel.     

Vibration of a stamp partially adhering to an elastic medium : PMM, vol. 42, no. 6, 1978, pp. 1085–1092

There is examined the problem of vibration of a stamp of arbitrary planform occupying a space Ω and vibrating harmonically in an elastic medium with plane boundaries. It is assumed that the elastic medium is a packet of layers with parallel boundaries, at rest in the stiff or elastic half-space. Contact of three kinds is realized under the stamp: rigid adhesion in the domain Ω₁, friction-free contact in domain Ω₂, there are no tangential contact stresses, and “film” contact without normal force in domain Ω₃ (there are no normal contact stresses, only tangential stresses are present.). It is assumed that the boundaries of all the domains have twice continuously differentiable curvature and Ω = Ω₁ Ω₂ Ω₃.

The problem under consideration assumes the presence of a static load pressing the stamp to the layer and hindering the formation of a separation zone. Moreover, a dynamic load, harmonic in time, acts on the stamp causing dynamical stresses which are of the greatest interest since the solution of the static problem is obtained as a particular case of the dynamic problem for ω = 0 (ω is the frequency of vibration). The general solution is constructed in the form of a sum of static and dynamic solutions.

A uniqueness theorem is established for the integral equation of the problem mentioned and for the case of axisymmetric vibration of a circular stamp partially coupled rigidly to the layer, partially making friction-free contact, the problem is reduced to an effectively solvable system of integral equations of the second kind, which reduce easily to a Fredholm system.

These results are an extension of the method elucidated in [1], where by the approach in [1] must be altered qualitatively to obtain them.     

Formation of the true contact area between high-elastic materials and a hard smooth surface

The change in the true contact area between rubber and glass on passing from static to dynamic friction has been studied experimentally. The extent of this change depends on the applied normal load and the contact time. The change of friction force is determined by the conditions of formation of the true contact area under the action of normal and tangential forces.Mekhanika Polimerov, Vol. 2, No. 2, pp. 263–268, 1966     

Buckling and stability analysis of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces

A study on the buckling and dynamic stability of a piezoelectric viscoelastic nanobeam subjected to van der Waals forces is performed in this research. The static and dynamic governing equations of the nanobeam are established with Galerkin method and under Euler–Bernoulli hypothesis. The buckling, post-buckling and nonlinear dynamic stability character of the nanobeam is presented. The quasi-elastic method, Leibnitz’s rule, Runge–Kutta method and the incremental harmonic balanced method are employed for obtaining the buckling voltage, post-buckling characteristics and the boundaries of the principal instability region of the dynamic system. Effects of the electrostatic load, van der Waals force, creep quantity, inner damping, geometric nonlinearity and other factors on the post-buckling and the principal region of instability are investigated.     

Stability analysis of cantilever carbon nanotubes subjected to partially distributed tangential force and viscoelastic foundation

Dynamic instability of cantilever carbon nanotubes conveying fluid embedded in viscoelastic foundation under a partially distributed tangential force is investigated based on nonlocal elasticity theory and Euler–Bernouli beam theory. The present study has incorporated the effects of nonlocal parameter, Knudsen number, surface effects and magnetic field. And two main parameters have also considered, namely partially distributed tangential force and foundation. It is assumed that viscoelastic foundation has modeled as Kelvin–Voigt, Maxwell and Standard linear solid types. The size-dependent governing equation of transverse vibration is derived using Hamilton’s variational principle and discretized by the Galerkin truncation method. A detailed parameter study is carried out, indicating the stability behavior of the nanotubes. In the light of numerical results, it is shown that variables considered in nondimensional equations have significant effects on natural frequencies and flutter velocities, especially for the foundation distribution length and model as well as the partially distributed tangential force.     

Classical and Nonclassical Dynamic Problems of Sandwich Shells with a Transversely Soft Core

Based on the hypothesis of similarity of transverse displacements in thin-walled sandwich shells with a transversely soft core under dynamic and static loads, refined geometrically nonlinear dynamic equations of motion are constructed in the case of large variations in the parameters of the stress-strain state (SSS) in the tangential directions. For shells structurally symmetric across the thickness and loaded with initial static loads, linearized dynamic equations are derived, which, upon introducing the synphasic and antiphasic functions of displacements and forces, can be used to describe the synphasic and antiphasic buckling forms in the transverse and tangential directions. For nonshallow cylindrical and shallow spherical shells, the nonclassical problems on all possible vibration forms realized at zero indices of variability of the SSS parameters in the tangential directions are formulated and solved. For shallow shells of symmetric structure, the resolving equations are obtained by introducing, instead of tangential displacements and transverse tangential stresses in the core, the corresponding potential and vortex functions.     

Natural and parametric vibrations of a cylindrical shell-filler system with tangential interaction taken into account

Conclusions 1. The spectrum of the eigenfrequencies and dynamical instability regions of a shell-filler system breaks up into two infinite spectra for each mode of wave formation; the first of them is determined by the shear modulus of the filler material, and the second — by the volume deformation modulus. The second spectrum is absent for incompressible fillers.2. It has been shown that taking tangential interactions into account has a strong effect on the arrangement and width of the dynamical instability regions belonging to the first spectrum and hardly changes the arrangement and widths of the regions of the second spectrum.3. As a result of the limiting transition as the frequency of the driving force tends to zero, expressions are obtained from the formulas of this paper for calculating the static stability of a shell with an incompressible filler. The numerical results obtained for this case correspond to results given in the literature.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. P. Stuchki Latvian State University, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 503–509, May–June, 1977.     

Paradox of Nicolai and related effects

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form.     

The modelling of an electromagnetic energy harvesting architecture

In this paper a detailed mathematical model for an electromagnetic energy harvesting architecture based on a semi-analytical approach is introduced. This model estimates the generated energy of the architecture by computing the static and dynamic magnetic and electric fields that describe its dynamics. A comparison of the static fields with the results of a Finite Element Analysis simulation in COMSOL Multiphysics shows good agreement. The model also features increased accuracy and numerical stability. In the model the semi-analytical solutions for the electromagnetic damping force exerted by the induced current coil and the induced electromotive force on the coil provide additional insight into the interactions of electromagnetic induction and damping. Additionally, the energy estimation could be used as a figure of merit in an maximization process to identify the optimal dimensions of the energy harvester.     

Paradox of Nicolai and related effects

The paper presents a general approach to the paradox of Nicolai and related effects analyzed as a singularity of the stability boundary. We study potential systems with arbitrary degrees of freedom and two coincident eigenfrequencies disturbed by small non-conservative positional and damping forces. The instability region is obtained in the form of a cone having a finite discontinuous increase in the general case when arbitrarily small damping is introduced. This is a new destabilization phenomenon, which is similar to well-known Ziegler’s paradox or the effect of the discontinuous increase of the combination resonance region due to addition of infinitesimal damping. It is shown that only for specific ratios of damping coefficients, the system is stabilized due to presence of small damping. Then, we consider the paradox of Nicolai: the instability of a uniform axisymmetric elastic column loaded by axial force and a tangential torque of arbitrarily small magnitude. We extend the results of Nicolai showing that the column is stabilized by general small geometric imperfections and internal and external damping forces. It is shown that the paradox of Nicolai is related to the conical singularity of the stability boundary which transforms to a hyperboloid with the addition of small dissipation. As a specific example of imperfections, we study the case when cross-section of the column is changed from a circular to elliptic form.     

Multiple duffing problem in a folding structure with hill-top bifurcation

This paper reviews the theoretical basis and its application for a multiple type of Duffing oscillation. This paper uses a suitable theoretical model to examine the structural instability of a folding truss which is limited so that only vertical displacements are possible for each nodal point supported by both sides. The equilibrium path in this ideal model has been found to have a type of “hill-top bifurcation” from the theoretical work of bifurcation analysis. Dynamic analysis allows for geometrical non-linearity based upon static bifurcation theory. We have found that a simple folding structure based on Multi-Folding-Microstructures theory is more interesting when there is a strange trajectory in multiple homo/hetero-clinic orbits than a well-known ordinary homoclinic orbit, as a model of an extended multiple degrees-of-freedom Duffing oscillation. We found that there are both globally and locally dynamic behaviours for a folding multi-layered truss which corresponds to the structure of the multiple homo/hetero-clinic orbits. This means the numerical solution depends on the dynamic behaviour of the system subjected to the forced cyclic loading such as folding or expanding action. The author suggests simplified theoretical models for hill-top bifurcation that help us to understand globally and locally dynamic behaviours, which depends on the static bifurcation problem. Such models are very useful for forecasting simulations of the extended Duffing oscillation model as essential and invariant nonlinear phenomena.     

Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads

In the present paper, the dynamic stability of circular cylindrical shells is investigated; the combined effect of compressive static and periodic axial loads is considered. The Sanders–Koiter theory is applied to model the nonlinear dynamics of the system in the case of finite amplitude of vibration; Lagrange equations are used to reduce the nonlinear partial differential equations to a set of ordinary differential equations. The dynamic stability is investigated using direct numerical simulation and a dichotomic algorithm to find the instability boundaries as the excitation frequency is varied; the effect of geometric imperfections is investigated in detail. The accuracy of the approach is checked by means of comparisons with the literature.     

Departure from linear mechanical behaviour of a helical spring

We model small deviations from linear mechanical behaviour of a loaded vertically suspended helical spring. A Taylor expansion of the elasticity equations governing the axial extension of the spring is used to determine the relative magnitudes of linear and (quadratic and cubic) nonlinear terms in the force–extension relationship. This relationship is the basis for the derivation of a model for the static extension of a loaded spring, and a wave equation that models small amplitude oscillation. The models account for the natural decline in pitch angle down a suspended spring, and provide accurate fits to measurements of static extension and periods of oscillation that are not adequately represented by equations based on Hooke’s law. The static and dynamic data yield consistent estimates of the spring rate.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

Large deformation analysis of a gel using the complex variable element-free Galerkin method

Deformation of a gel is a complicated multiphysical process involving the diffusion of solvent and the mechanical stretching of polymeric networks. This process leads to a mechanical and diffusional equilibrium after a certain period of evolution. Here we present a large deformation analysis of a gel in the equilibrium state under environmental triggers using the complex variable element-free Galerkin (CVEFG) method. The work mainly addresses the numerical challenges encountered while a gel in contact with a solvent deforms only with the geometric constraints but without any force boundary conditions (implying that the force column in the final standard algebraic equations equals zero) and emphasizes the implementation of a different CVEFG approach. The material model is based on the work of Hong et al. (2008) and incorporates our previous efforts in the numerical implementation. The discretized equation system is derived from the complex variable moving least squares approximation and the Galerkin method. The essential boundary conditions are imposed through the penalty method. The proposed approach is verified by the simulation of the swelling-induced large deformation and surface instability of a confined single gel layer.     

弹性地基输流管道的耦合模态颤振分析

推导出了弹性地基输流管道的固-液耦合振动微分方程,用幂级数法计算了Winkler模型地基和双参数模型地基输流管道的临界流速和复频率,分析了弹性地基对输流管道静力稳定性与动力稳定性的影响.结果表明,与不考虑弹性地基的情况相比较,弹性地基的作用可使管道发生静力失稳和动力失稳的临界流速增大,并且增大弹性地基参数可提高静力失稳和动力失稳的临界流速,从而推迟发散与颤振的发生.当质量比β较大时,管道会在某个地基参数组合下,在发生静力失稳后,会在较高流速下出现再稳定和再发散现象,然后发生耦合模态颤振.     

Magneto hydrodynamic stability of self-gravitational fluid cylinder

The self-gravitating instability of a fluid cylinder pervaded by magnetic field and endowed with surface tension has been discussed. The dispersion relation is derived and some reported works are recovered as limiting cases from it. The capillary force is destabilizing only in the small axisymmetric domain and stabilizing otherwise. The magnetic field has a strong stabilizing effect in all modes of perturbation for all wavelengths. The self-gravitating force is destabilizing in the axisymmetric perturbation. However the magnetic field effect modified a lot the destabilizing character of the model and could overcome the capillary and self-gravitating instability of the model for all short and long wavelengths.     

A DAE formulation for geared rotor dynamics including frictional contact between the teeth

A DAE approach is presented for geared rotor dynamics simulations with rigid helical evolvent gears. It includes the normal contact force between the teeth as well as tangential components. Given the evolvent tooth flank geometry of gear 1 and gear 2 [1], the contact line and the velocity difference in the contact are found. The requirement of no penetration of the teeth yields a non-holonomic constraint and the contact normal force. The friction caused force and moment are obtained by applying Coulomb's friction model. This approach is used to investigate the dynamics of two ideal rotors with translational DoFs, which are connected by gears to one another. The driving rotor has a given angular speed, while the driven rotates unrestrainedly and is connected to a rotational damper. Because of the periodic friction terms, the solution is periodic. A direct time integration or a harmonic approach can be used for the numerical computation. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)