Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: A semi-analytical approach

The non-linear free and forced vibrations of simply supported thin circular cylindrical shells are investigated using Lagrange's equations and an improved transverse displacement expansion. The purpose of this approach was to provide engineers and designers with an easy method for determining the shell non-linear mode shapes, with their corresponding amplitude dependent non-linear frequencies. The Donnell non-linear shell theory has been used and the flexural deformations at large vibration amplitudes have been taken into account. The transverse displacement expansion has been made using two terms including both the driven and the axisymmetric modes, and satisfying the simply supported boundary conditions. The non-linear dynamic variational problem obtained by applying Lagrange's equations was then transformed into a static case by adopting the harmonic balance method. Minimisation of the energy functional with respect to the basic function contribution coefficients has led to a simple non-linear multi-modal equation, the solution of which gives in the case of a single mode assumption an expression for the non-linear frequencies which is much simpler than that derived from the non-linear partial differential equation obtained previously by several authors. Quantitative results based on the present approach have been computed and compared with experimental data. The good agreement found was very satisfactory, in comparison with previous old and recent theoretical approaches, based on sophisticated numerical methods, such as the finite element method (FEM), the method of normal forms (MNF), and analytical methods, such as the perturbation method.     

Effect of the geometry on the non-linear vibration of circular cylindrical shells

The non-linear vibration of simply supported, circular cylindrical shells is analysed. Geometric non-linearities due to finite-amplitude shell motion are considered by using Donnell's non-linear shallow-shell theory; the effect of viscous structural damping is taken into account. A discretization method based on a series expansion of an unlimited number of linear modes, including axisymmetric and asymmetric modes, following the Galerkin procedure, is developed. Both driven and companion modes are included, allowing for travelling-wave response of the shell. Axisymmetric modes are included because they are essential in simulating the inward mean deflection of the oscillation with respect to the equilibrium position. The fundamental role of the axisymmetric modes is confirmed and the role of higher order asymmetric modes is clarified in order to obtain the correct character of the circular cylindrical shell non-linearity. The effect of the geometric shell characteristics, i.e., radius, length and thickness, on the non-linear behaviour is analysed: very short or thick shells display a hardening non-linearity; conversely, a softening type non-linearity is found in a wide range of shell geometries.     

Free axisymmetric vibration of FGM circular plates

Based on the three-dimensional theory, this work presents a direct displacement method to investigate the free axisymmetric vibration of transversely isotropic circular plates, whose material is functionally graded and properties obey the exponential law along the thickness direction of the plate. Under two boundary conditions, the solution satisfies all basic equations and the Corresponding boundary condition at every point. Thus, it is three-dimensional exact. Numerical examples are presented and compared with previous works. The present method can also be extended to the case of arbitrary distribution of the material properties along the thickness direction of the plate.     

Free axisymmetric vibration of transversely isotropic laminated circular plates

Based on the three dimensional elastic theory, the state equation of the axisymmetric free vibration of transversely isotropic circular plates is established. Taking the advantage of finite Hankel transform, two exact solutions are derived for two boundary conditions, i. e. the rigid-slipping boundary and elastic simply supported boundary. Finally, numerical results are presented and compared with those of FEM. The project is supported by the National Natural Science Foundation of China and the Zhejiang Provincial Natural Science Foundation.     

Non-local effects on the non-linear modes of vibration of carbon nanotubes under electrostatic actuation

Carbon nanotubes (CNTs) based NEMS with electrostatic sensing/actuation may be employed as sensors, in situations where it is fundamental to understand their dynamic behaviour. Due to displacements that are large in comparison with the thickness and to the non-linearity of the electrostatic force, these CNT based NEMS operate in the non-linear regime. The knowledge of the modes of vibration of a CNT provides a picture of what one may expect from its dynamic behaviour not only in free, but also in forced vibrations. In this paper, the non-linear modes of vibration of CNTs actuated by electrostatic forces are investigated. For that purpose, a p-version finite element type formulation is implemented, leading to ordinary differential equations of motion in the time domain. The formulation takes into account non-local effects, which influence the inertia and the stiffness of CNTs, as well as the electrostatic actuation. The ordinary differential equations of motion are transformed into algebraic equations of motion via the harmonic balance method (HBM) and then solved by an arc-length continuation method. Several harmonics are considered in the HBM. The importance of non-local effects, combined with the geometrical non-linearity and with the action of the electrostatic force, is analysed. It is found that different combinations of these effects can result in alterations of the natural frequencies, variations in the degrees of softening or hardening, changes in the frequency content of the free vibrations, and alterations in the mode shapes of vibration. It is furthermore found that the small scale, here represented by the non-local theory, has an effect on interactions between the first and higher order modes which are induced by the geometrical and material non-linearities of the system.     

Static and free vibration analysis of straight and circular beams on elastic foundation

Static and free vibration analyses of straight and circular beams on elastic foundation are investigated. The Timoshenko beam theory is adopted in the derivation of the governing equation. Ordinary differential equations in scalar form obtained in the Laplace domain are solved numerically using the complementary functions method. The static and free vibration analyses of beams on elastic foundation are analyzed through various examples.     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

Simultaneous measurements of disk vibration and pressure fluctuation in turbulent flow developing in a model hard disk drive

The complex flow features inside hard disk drive models are investigated in an axisymmetric and a semi-open shroud configurations. For the axisymmetric case, we have employed both experimental and computational approaches. The experiment focuses on both flow dynamics and the disk vibration, where measurements of the fluctuating pressure and velocity are undertaken at some representative points. The correlation between the disk vibration and the fluctuating pressure in the turbulent flow between disks is evident from the spectral analysis. The experimentally observed fluctuating pressure and velocity are partly due to the disk vibration and its contribution could be estimated by comparing the experiment with the results of a large eddy simulation. For the semi-open shroud case, although the characteristic peaks attributable to the large-scale vortical structure are still observed in the power spectra, the pressure fluctuation and the disk vibration are suppressed when the arm is inserted.     

硬盘抗冲击振动特性的研究进展

硬盘驱动器是计算机最重要的外存储设备.由于硬盘抗冲击振动问题的重要性和复杂性,许多研究者对此问题从各个方面进行了研究.本文从硬盘的基本结构和工作原理入手,介绍了磁头/磁盘系统的静动态特性,分析了影响磁头/磁盘系统性能的重要因素及其原理,并对国内外有关提高硬盘抗冲击振动性能的研究进行了较为系统的回顾和分析,最后阐述了硬盘冲击振动的控制研究,分析了硬盘目前所面临的关键问题及其可能的解决方案.     

Flow-induced vibration of a circular cylinder subjected to wake interference at low Reynolds number

Two- and three-dimensional numerical simulations of the flow around two circular cylinders in tandem arrangements are performed. The upstream cylinder is fixed and the downstream cylinder is free to oscillate in the transverse direction, in response to the fluid loads. The Reynolds number is kept constant at 150 for the two-dimensional simulations and at 300 for the three-dimensional simulations, and the reduced velocity is varied by changing the structural stiffness. The in-line centre-to-centre distance is varied from 1.5 to 8.0 diameters, and the results are compared to that of a single isolated flexible cylinder with the same structural characteristics, m^?=2.0 and ζ=0.007. The calculations show that significant changes occur in the dynamic behaviour of the cylinders, when comparing the flow around the tandem arrangements to that around an isolated cylinder: for the tandem arrangements, the lock-in boundaries are wider, the maximum displacement amplitudes are greater and the amplitudes of vibration for high reduced velocities, outside the lock-in, are very significant. The main responsible for these changes appears to be the oscillatory flow in the gap between the cylinders.     

Study on the stability of drum brake non-linear low frequency vibration model

A 5-DOF non-linear model is presented to simulate the vibration of a drum brake at low frequency in the course of applying the brake. Analysis and calculation are carried out to illuminate that even when the friction coefficient is constant, the vibration and instability can occur with the combination of some specific parameters. And the stable-unstable area on the parameter plane on the condition of the combination of some specific parameters is presented to illuminate the effect of the structure parameter on the system stability.     

Periodic boundary layer near an axisymmetric stagnation point on a circular cylinder

An analysis is presented to investigate the time-mean characteristics of the laminar boundary layer near an axisymmetric stagnation point when the velocity of the oncoming flow relative to the body oscillates. Different solutions are obtained for the small and high values of the reduced frequency parameter. The range of Reynolds number considered was from 0.01 to 100. Numerical solutions for the velocity functions are presented, and the wall values of the velocity gradients are tabulated.     

Numerical study on bifurcations in the wake of a circular disk

Using Large-eddy simulation (LES), the dynamics in the wake of a circular disk with an aspect ratio of d/w = 5 is numerically studied. The circular disk is normal to the main flow, and Reynolds number ranges from 115 to 300. The first bifurcation is confirmed for Re = 120, leading to the steady state mode with a reflectional symmetry and a double-thread wake extending to the downstream. The Hopf bifurcation is found for Re = 152, and the planar symmetry is lost, which is different from that observed in the sphere wake; it is called the “reflectional-symmetry-breaking (RSB)” mode and the hairpin vortices in this mode are always shedding in a fixed orientation. The third bifurcation is captured for Re = 166, which is named the “standing wave (SW)” mode; the planar symmetry lost in RSB mode is recovered and the hairpin vortices are shedding in the oppositely sided orientations, unlike the ones observed in the sphere wake. The fourth bifurcation, referred to as “zigzag (ZZ)” mode, is observed for Re = 265 and the planar symmetry is lost again; the hairpin vortices are shedding in an irregular orientation and propagating in a zigzagged way; and a few small-scale structures begin to appear. Three different vortex shedding regimes are found in RSB, SW and ZZ modes, respectively. Results show that the recirculation region plays a significant role in the mode transitions, and the stagnation point of recirculation zone is conjectured to be the initial region causing the wake instability.     

Modeling of hard disk drives for shock and vibration analysis – consideration of nonlinearities and discontinuities

This paper reviews recent advances (mostly after year 2000) in shock and vibration analysis of hard disk drives (HDD) considering the presence of nonlinearities and discontinuities. Components and dynamic phenomena in HDD where effects of mechanical nonlinearity and discontinuities are significant are discussed, e.g., head actuator suspension, dimple and slider, head–disk interface, fluid dynamic bearing, spinning disks, and load/unload dynamics. Ways to model these nonlinearities and discontinuities are reviewed in detail. Our research on modeling an entire HDD in operating mode subject to shock and vibration using a flexible multibody dynamics formulation is also presented. The numerical simulation of the shock response of a 1-in. form factor HDD is presented. A half-sinusoidal acceleration shock is applied at the base of the HDD. Response of the flying height for different shock amplitudes and duration times is simulated.     

A study on the effect of electronic engine speed regulator on agricultural tractor ride vibration behavior

In this study, the effect of electronic speed adjustment on tractor ride vibration levels is examined. With normal pedal operation the engine rotational speed drops with an increasing load. The electronic regulator provides a constant speed mode of operation independent of the load. Vibration levels were measured under different operating conditions and surfaces. As a first series of tests, the tractor was driven on a conglomerate bituminous track at speeds of 20, 25 and 28 km/h. Vibration was measured upon the surface of the operator seat simultaneously in the x, y and z directions. The reference axis system was that defined by the ISO 2631-1 [1]. The weighted r.m.s. acceleration was found to be between 8% and 8.6% higher for the case where operation with electronic speed adjustment had been selected. Secondly, cultivating was chosen as the field task and the vibration was measured while the tractor was traversing a rough farm track at speeds of 6, 7.5 and 9 km/h. In this case, the vibration levels with automatic speed adjustment were between 4.3% and 8.6% lower than when driving with normal foot pedal operation. From the above results, we may infer that electronic speed regulation should not be used in transportation on asphalt country roads. On the contrary, it seems that electronic regulation has an advantage when used in typical field tasks such as cultivating.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

Separation impact of a circular disk floating on the surface of an ideal incompressible fluid of infinite depth

The three-dimensional mixed problem of the separation impact of a circular disk floating on the surface of an ideal incompressible unlimited fluid is considered. The position and shape of the contact area between the body and the fluid (and the separation zone) are not known and depend on the relation between the translational and angular velocities acquired by the disk upon impact. Because of this, the problem in question is nonlinear and belongs to the class of free-boundary problems. The problem is solved using the method of Hammerstein-type nonlinear boundary integral equations. This approach allows the fluid flow after impact and the unknown zone of separation of fluid particles to be determined simultaneously. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 76–86, July–August, 2009.     

The effect of end conditions on the vortex-induced vibration of cylinders

In the present investigation we study the effect of end conditions on the vortex-induced vibration of an elastically mounted rigid cylinder. This work was triggered by some initial controlled vibration experiments which showed that spanwise end conditions can have a large effect on measured fluid forces on a cylinder, and this suggested that some of the disparity amongst previous free vibration studies may possibly be attributed to differences in end conditions. In the principal experiments here, we are concerned with a vertical cylinder piercing the clean free surface of a water channel, and attached to a carriage system mounted atop the channel. The upper end of the submerged cylinder is thus the free surface, while the lower end is manipulated to yield three different conditions, namely: an attached endplate; an unattached endplate fixed to the channel floor (with a variable gap between cylinder and plate); and a condition of no endplate at all. Interestingly, we find that the free vibration response for the attached and unattached endplate cases were nearly identical. One expectation was that the case without an endplate would lead to a flow around the end of the body, modifying the vortex dynamics, and thereby reducing the correlation of the induced fluid forces on the body. Surprisingly, over the entire response plot, the vibration amplitude is markedly higher in the absence of an endplate, with the exception of the peak amplitude, which remains nearly unchanged. Unexpectedly, the vibrations become much more steady at flow velocities in the vicinity of the peak response, if the endplate is removed. In a further set of experiments, we undertake controlled vibration, where we vary the gap between cylinder and endplate. We discover a large discontinuous jump in the magnitude of fluid excitation, when the gap exceeds 15% of a diameter. For larger gaps, the fluid excitation becomes independent of the gap size, effectively equivalent to having no plate at all. This study is consistent with some of the disparity between the character of vibration response plots in previous studies, if one takes into account the particular end conditions chosen in those studies.     

A new method for the non-linear deflection analysis of an infinite beam resting on a non-linear elastic foundation

The aim of this paper is to develop a new method of analyzing the non-linear deflection behavior of an infinite beam on a non-linear elastic foundation. Non-linear beam problems have traditionally been dealt with by semi-analytical approaches that involve small perturbations or by numerical methods, such as the non-linear finite element method. In this paper, in contrast, a transformed non-linear integral equation that governs non-linear beam deflection behavior is formulated to develop a new method for non-linear solutions. The proposed method requires an iteration to solve non-linear problems, but is fairly simple and straightforward to apply. It also converges quickly, whereas traditional non-linear solution procedures are generally quite complex in application. Mathematical analysis of the proposed method is performed. In addition, illustrative examples are presented to demonstrate the validity of the method developed in the present study.     

Effect of geometric imperfections on non-linear stability of circular cylindrical shells conveying fluid

Circular cylindrical shells conveying incompressible flow are addressed in this study; they lose stability by divergence when the flow velocity reaches a critical value. The divergence is strongly subcritical, becoming supercritical for larger amplitudes. Therefore the shell, if perturbed from the initial configuration, has severe deformations causing failure much before the critical velocity predicted by the linear threshold. Both Donnell's non-linear theory retaining in-plane displacements and the Sanders-Koiter non-linear theory are used for the shell. The fluid is modelled by potential flow theory but the effect of steady viscous forces is taken into account. Geometric imperfections are introduced and fully studied. Non-classical boundary conditions are used to simulate the conditions of experimental tests in a water tunnel. Comparison of numerical and experimental results is performed.