Third-order partial differential equations arising in the impulsive motion of a flat plate

We obtain numerical solutions to a class of third-order partial differential equations arising in the impulsive motion of a flat plate for various boundary data. In particular, we study the case of constant acceleration of the plate, the case of oscillation of the plate, and a case in which velocity is increasing yet acceleration is decreasing. We compare the numerical solutions with the known exact solutions in order to establish the validity of the method. Several figures illustrating both solution forms and the relative strength of the second and third-order terms are presented. The results obtained in this study reveal many interesting behaviors that warrant further study on the non-Newtonian fluid flow phenomena.     

Graphische Untersuchungen von erzwungenen Schwingungen mit unstetiger Geschwindigkeitsfunktion

Summary An oscillator which is being struck exhibits non-linear characteristics on account of the instabilities in speed induced. Through a wide range of parameters the processes of motion brought about by a stimulating force lead to a stationary condition via a transient phase (limit cycle). The law of motion for each periodic movement between two successive stimulations is basically known. There is both a free and a forced oscillation, and these are superimposed. Each can be illustrated as a vector rotating in the phase plane ( q), the point of which describes a phase curve. The resulting motion is described by a vector which is the sum of the two partial vectors. The author gives here a graphic method which he has developed from this. By establishing the connection between the angles described by the partial vectors and the time taken, the size and direction of the resulting vector at any moment is determined. The amount of the vector representing the forced oscillation remains constant during the whole process, while the vector describing the free oscillation changes its size and phase in each cycle.The author then extends his method to non-linear periodic movement by means of the so-called delta method.     

Chaotic and bifurcation dynamics of a simply-supported thermo-elastic circular plate with variable thickness in large deflection

This paper presents the conditions that can possibly lead to chaotic motion and bifurcation behavior for a simply-supported large deflection thermo-elastic circular plate with variable thickness by utilizing the criteria of fractal dimensions, maximum Lyapunov exponents and bifurcation diagrams. The governing partial differential equation of the simply supported thermo-elastic circular plate with variable thickness is first derived by means of Galerkin method. Several different features including Fourier spectra, phase plot, Poincar’e map and bifurcation diagrams are numerically computed. These features are used to characterize the dynamic behavior of the plate subjected to various excitations of lateral loads and thermal loads. Numerical examples are presented to verify the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. Numerical modeling results indicate that large deflection motion of a thermo-elastic circular plate with variable thickness possesses chaotic motions and bifurcation motion under different lateral loads and thermal loads. The simulation results also indicate that the periodic motion of a circular plate can be obtained for the convex or the concave circular plate. The dynamic motion of the circular plate is periodic for the cases including (1) the lateral loading frequency is within a specific range, (2) thermal and lateral loadings are operated in a specific range and (3) the thickness parameter is less than a specific critical value for the convex circular plate or greater than a specific critical value for the concave circular plate. The modeling results show that the proposed method can be employed to predict the non-linear dynamics of any large deflection circular plate with variable thickness.     

Nonlinear dynamic analysis of 2-DOF nonlinear vibration isolation floating raft systems with feedback control

In this paper, the average method is adopted to analysis dynamic characteristics of nonlinear vibration isolation floating raft system with feedback control. The analytic results show that the purposes of reducing amplitude of oscillation and complicating the motion can be achieved by adjusting properly the system parameters, exciting frequency and control gain. The conclusions can provide some available evidences for the design and improvement of both the passive and active control of the vibration isolation systems. By altering the exciting frequency and control gain, complex motion of the system can be obtained. Numerical simulations show the system exhibits period vibration, double period vibration and quasi-period motion.     

扁锥面网壳非线性动力分岔与混沌运动

对曲面为正三角形网格的3向扁锥面单层网壳,用拟壳法建立了轴对称非线性动力学方程．在几何非线性范围内给出了协调方程．网壳在周边固定条件下,通过Galerkin作用得到一个含2次、3次的非线性微分方程,通过求Floquet指数讨论了分岔问题．为了研究混沌运动,对一类非线性动力系统的自由振动方程进行了求解,继之给出了单层扁锥面网壳非线性自由振动微分方程的准确解,通过求Melnikov函数,给出了发生混沌的临界条件,通过数值仿真也证实了混沌运动的存在．     

非线性热弹耦合椭圆板的混沌运动

计及几何非线性大挠度效应和温度效应的影响,导出了椭圆板周期激励作用下热弹耦合的非线性动力方程,利用Melnikov函数法给出了系统发生混沌运动的临界条件,结合Poincaré映射、相平面轨迹和时程曲线进行数值分析,并对系统通向混沌的道路进行了讨论,从中得到了一些有益的结论.     

Refined geometrically nonlinear equations of motion for elongated rod-type plate

We derive new refined geometrically nonlinear equations of motion for elongated rod-type plates. They are based on the proposed earlier relationships of geometrically nonlinear theory of elasticity in the case of small deformations and refined S. P. Timoshenko’s shear model. These equations allow to describe the high-frequency torsional oscillation of elongated rod-type plate formed in them when plate performs low-frequency flexural vibrations. By limit transition to the classical model of rod theory we carry out transformation of derived equations to simplified system of equations of lower degree.     

Chaotic and bifurcation for a simply-supported thermo-mechanical coupling circular plate with variable thickness

This paper presents an approach to characterize the conditions that can possibly lead to chaotic motion for a simply supported large deflection circular plate of thermo-mechanical coupling by utilizing the criterion of the maximum Lyapunov exponent. The governing partial differential equation of the simply supported large deflection circular plate of thermo-mechanical coupling is first derived and simplified to a set of three ordinary differential equations by the Galerkin method. Several different features including time history, Power spectra, phase plot, Poincare map and bifurcation diagram are then numerically computed. These features are used to characterize the dynamic behavior of the plate subjected various geometric and excitation conditions. Numerical examples are presented to verify the validity of the conditions that lead to chaotic motion and the effectiveness of the proposed modeling approach. The modeling results of numerical simulation indicate that the chaotic motion may occurs in the lateral loads , η₁=1.1, β=0.5, and _∞=0.0007. As the thermo-elastic damping is great than a critical value, the dynamic motion of the thermal-couple plate is periodic. As the thickness parameter β of the concave circular plate is great than a critical value, the motion of the plate is periodic. The modeling result thus obtained by using the method proposed in this paper can be employed to predict the instability induced by the dynamics of the thermo-mechanical coupling circular plate in large deflection.     

Chaos control for the plates subjected to subsonic flow

The suppression of chaotic motion in viscoelastic plates driven by external subsonic air flow is studied. Nonlinear oscillation of the plate is modeled by the von-Kármán plate theory. The fluid-solid interaction is taken into account. Galerkin’s approach is employed to transform the partial differential equations of the system into the time domain. The corresponding homoclinic orbits of the unperturbed Hamiltonian system are obtained. In order to study the chaotic behavior of the plate, Melnikov’s integral is analytically applied and the threshold of the excitation amplitude and frequency for the occurrence of chaos is presented. It is found that adding a parametric perturbation to the system in terms of an excitation with the same frequency of the external force can lead to eliminate chaos. Variations of the Lyapunov exponent and bifurcation diagrams are provided to analyze the chaotic and periodic responses. Two perturbation-based control strategies are proposed. In the first scenario, the amplitude of control forces reads a constant value that should be precisely determined. In the second strategy, this amplitude can be proportional to the deflection of the plate. The performance of each controller is investigated and it is found that the second scenario would be more efficient.     

A cylindrical plate with a weakly fixed curvilinear edge made of a transversally isotropic material

Free oscillations and stability under an axial compression of a thin cylindrical plate with a weakly fixed rectilinear edge made of a transversally isotropic material with low stiffness with respect to transverse displacements are considered. The curvilinear edges of the plate are assumed to be hingedly supported. The oscillation frequencies and the critical load for a plate with a free or weakly fixed edge are smaller than those for a shell closed in the circumferential direction. The shapes of oscillations and the forms of stability loss localized near the weakly fixed edge and damped at a distance from it are considered. The Timoshenko-Reissner model is used. Localized forms are analyzed by using a system of equations for Timoshenko-Reissner shallow shells, which is derived for this purpose. The main special feature of this system is that it contains a separate equation describing a solution with large variability. For the example of the stability problem under consideration, the error involved in the system of equations for Timoshenko-Reissner shallow shells is studied. The critical load values obtained with the use of the Kirchhoff-Love and Timoshenko-Reissner models are compared.     

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.     

夹层椭圆形板的1/3亚谐解

研究了夹层椭圆形板的非线性强迫振动问题。在以5个位移分量表示的夹层椭圆板的运动方程的基础上,导出了相应的非线性动力方程。提出一类强非线性动力系统的叠加-叠代谐波平衡法。将描述动力系统的二阶常微分方程,化为基本解为未知函数的基本微分方程和派生解为未知函数的增量微分方程。通过叠加-叠代谐波平衡法得出了椭圆板的1/3亚谐解。同时,对叠加-叠代谐波平衡法和数值积分法的精度进行了比较。并且讨论了1/3亚谐解的渐近稳定性。     

Nonlinear dynamics and dynamic instability of smart structural cross-ply laminated cantilever plates with MFC layer using zigzag theory

In this paper, a novel dynamic model for smart structural systems cross-ply laminated cantilever plate with smart material Macro fiber composites (MFC) layer is presented by using zigzag function theory. The nonlinear dynamic response and dynamic instability of the smart structural systems are studied for the first time. The plate is subjected to the uniformed static and in-plane harmonic excitation conjunction with electrically loaded under different electric boundary conditions. The partial layer-wise theory which the first shear deformation theory is expanded by introducing the zigzag function in the in-plane displacement components is adopted. The carbon fiber reinforced composite material T800/M21and macro fiber composites (MFC-d31) M8528-P3 are implemented. By Lagrangian equation and Chebyshev polynomial, the equations of motion are derived for the laminated plate. The validation and convergence are studied by comparing results with literatures. The dynamic instability regions and the critical buckling load characteristics can be obtained for different layer sequences, geometric dimensions and also the electromechanical effects are considered. Nonlinear dynamic responses of the laminated plate are studied by using numerical calculation. It can be seen that in certain state the plate will loses stability and the periodic, multiple period as well as chaotic motions of the plate are found.     

具有缺陷的不可压缩neo-Hookean球壳的动力学行为的定性分析

研究了一类具有缺陷的不可压缩超弹性材料球壳的径向对称运动问题,该类材料可以看作是带有径向摄动的均匀各向同性不可压缩的neo-Hookean材料．得到了描述球壳内表面运动的二阶非线性常微分方程,并给出了方程的首次积分．通过对微分方程的解的动力学行为的分析,讨论了材料的缺陷参数和球壳变形前的内外半径的比值对解的定性性质的影响,并给出了相应的数值算例．特别地,对于一些给定的参数,证明了存在一个正的临界值,当内压与外压之差小于临界值时,球壳内表面随时间的演化是非线性周期振动;当内压与外压之差大于临界值时,球壳的内表面半径随时间的演化将无限增大,即球壳最终将被破坏．     

The problem of forward motion of a hovercraft: asymptotic analysis of a solution

The linear unsteady problem describing the forward motion of a hovercraft with an oscillating forward velocity is considered. A two-term asymptotic representation of a solution is derived provided that the oscillation period is small. Estimates for the remainder in Sobolev spaces and some hydrodynamical consequences are also obtained. Bibliography: 17 titles. Illustrations: 1 figure. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 40, May 2009, pp. 91–104.     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

Novel three-piston pump design for a slipper test rig

Slipper's micro motions including the squeezing motion, spinning motion, and tilting motion have a significant impact on its lubricating condition and dynamic behavior. However, few experimental studies are on these micro motions within a real axial piston pump, especially the slipper's spinning motion. The experimental investigations on the slipper in the past mainly focused on the parameters of the oil film such as pressure, thickness, and temperature. The sensors were often installed in the fixed swash plate when the cylinder block was chosen to rotate. Alternatively, the sensors were mounted in the fixed modified slipper when the swash plate rotated. The biggest challenge of the direct measurements of these micro motions is the space limitation for the sensor installation due to the compact structure of axial piston pumps as well as the slipper's macro motion. This paper presents a new three-piston pump for the slipper test rig which can provide enough installation space for the sensor. To realize the cylinder block balance, a hold-down plate is first introduced into this three-piston pump. In addition, a detailed set of relevant equations is derived to evaluate the functionality of the hold-down plate. Finally, the slipper's spinning motion was measured directly and continuously using this three-piston pump, which confirmed the capability of the slipper test rig.     

Exact and Approximate Analytical Solutions to a Problem on Plane Modes of Free Oscillations of a Rectangular Orthotropic Plate with Free Edges,with the Use of Triginometric Basis Functions

The linear problem on plane modes of free oscillations of a rectangular orthotropic plate with free unloaded edges is considered. A procedure for constructing displacement functions exactly satisfying the boundary conditions, with the use of double-trigonometric basis functions, is offered. Exact and approximated analytical solutions to the problem formulated are found, which presumably describe all plane modes of free oscillations of the plate in the class of the functions indicated. It is established that, in the use of variational principles, the variations of required functions must be considered not only arbitrary, but also mutually independent. Therefore, the solutions constructed give physically reliable results for the frequencies and modes of free oscillations only if the problem is stated in the form of Bubnov variation equations, which depend on the structure of displacement functions. It is found that the exact analytical solutions of the problem correspond to oscillation modes without shear strains. It is shown that it is possible to select such solutions from them which correspond to trigonometric functions with a zero harmonic in one direction. These solutions describe only flexural oscillation modes of the plate, and the results obtained are equivalent to those given by the classical Kirchhoff model known in the theory of rods, plates, and shells.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 461–488, July–August, 2005.     

Predicting nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring

This paper is focused on nonlinear dynamic response of internal cantilever beam system on a steadily rotating ring via a nonlinear dynamic model. The analytical approximate solutions to the oscillation motion are obtained by combining Newton linearization with Galerkin's method. Numerical solutions could be obtained by using the shooting method on the exact governing equation. Compared with numerical solutions, the approximate analytical solutions here show excellent accuracy and rapid convergence. Two different kinds of oscillating internal cantilever beam system on a steadily rotating ring are investigated by using the analytical approximate solutions. These include symmetric vibration through three equilibrium points, and asymmetric vibration through the only trivial equilibrium point. The effects of geometric and physical parameters on dynamic response are useful and can be easily applied to design practical engineering structures. In particular, the ring angular velocity plays a significant role on the period and periodic solution of the beam oscillation. In conclusion, the analytical approximate solutions presented here are sufficiently precise for a wide range of oscillation amplitudes.     

Phenomenological model of interaction of a plate with a flow

In the present paper, a finite-dimensional phenomenological model of unsteady interaction of a rigid plate with a flow is proposed. It is assumed that the plate performs translational motion across the flow. The internal dynamics of the flow is modeled by the attached second order dynamical system. It is shown that the model allows satisfactory agreement with experimental data. With the developed model an inverse problem of dynamics is examined for the situation where the plate performing uniform translational motion at some moment begins uniform deceleration and finally stops. It is shown that for sufficiently large values of the plate acceleration for some time range the flow does not resist the motion of the plate but “accelerates” it. It is shown also that the equations of motion in the context of the proposed model can be reduced to the integro-differential form, and comparison with the known model of S. M. Belotserkovsky is performed. The structural resemblance of the motion equations for a body in flow in both models is noted. The domain of applicability of the quasi-stationary model is examined. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 43–62, 2005.