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For ball screw feed system, a sudden start or stop has a great influence on the transmission stiffness, so the axial stiffness mutation of feed system will occur. The purpose of this paper is to study the influence of acceleration on the transmission stiffness and dynamic characteristics of the ball screw feed system.

Taking the ball screw feed system as a research object, on the basis of the Hertz contact theory and the mixed element method, axial stiffness model and dynamic model are established. And the system stability was analyzed by the time history diagram and Phase-plane portrait diagram. The feed system was analyzed theoretically and experimentally, the experimental results are in good agreement with the model results.

Lead screw lead angle, preload, load and start acceleration affected ball-screw pair, bearing and transmission stiffness. And the load, nut contact stiffness, bearing contact stiffness, preload have a large effect on the transmission stiffness. The results show that a certain acceleration value will make the axial stiffness abrupt change.

This research provides a useful theoretical support for ensuring a good dynamic for the ball screw feed system.

The dynamic characteristics prediction and frequency-modulation of pipeline was an important work for the design of aircraft hydraulic structure.

A complex pipeline was deemed as a combination of several segments of straight-pipe-element (SPE). The 3D vibration equations of each SPE were established in their local coordinate system based on Timoshenko-beam model and Euler-beam model, respectively. The dynamic-stiffness-matrixes were deduced from the dispersion relation of these equations. According to the complex pipeline layout in the global coordinate system, a multi dynamic stiffness matrixes assembling (MDSMA) algorithm was carried out to establish the characteristic equations of the whole complex pipeline. The MDSMA solutions were verified to be consistent with experimental results.

The MDSMA method based on Timoshenko-Beam model was more suitable for the short span aviation pipeline and the vibration at high frequency stage (>350 Hz). The layout affected the pipeline's in-plane stiffness and out-plane stiffness, for the Z-shaped pipe, each order natural mode took place on the ZP and NP alternately. Reasonable designs of bending position and bending radius were effective means for complex pipeline frequency-modulation.

A new dynamic modeling method of aircraft complex pipeline was proposed to obtain the influence of pipeline layout parameters on dynamic characteristics.

Double-beam/column systems have drawn much attention in many engineering fields. This work aims to present the free and forced vibrations of a novel and complex double-column system with concentrated masses, axial loads and discrete viscoelastic supports subjected to the excitation of ground acceleration are solved by the extended Laplace transform method (ELTM).

In this work, the authors proposed an extended Laplace transform method (ELTM), which is an exact and explicit analytical method. Firstly, the mathematical model simulating the vibrations of the double-column system is reformulated with Dirac's delta function. Secondly, the exact and explicit mode shape solutions are obtained, based on which the natural frequencies and dynamic responses are obtained. An illustrating example is presented to show the validity of the proposed method. A parametric study is carried out to investigate the influences of the non-dimensional column stiffness ratio and the support stiffness ratio on the peak dynamic displacement and velocity.

It is shown that the proposed method can give exact and explicit solutions of the mode shapes and natural frequencies. It is found that the asynchronous vibrations of the proposed double-column systems can be implemented to efficiently dissipate seismic energy, as shown in the time-histories of displacement and velocity.

This research systematically studied the free and forced vibrations of the complex double-column system. The proposed extended ELTM is a general method. Its application to studying the energy dissipation capability implicates that the double-column system can be utilized to reduce responses in structures under earthquake attacks.

The proposed extended ELTM is original and powerful. Its application to study the complex double-columns system with discrete supports, concentrated masses and axial loads is novel.

The purpose of this paper is to address the practically important problem of the load dependence of transverse vibrations for helical springs. At the beginning, the author develops the equations for transverse vibrations of the axially loaded helical springs. The method is based on the concept of an equivalent column. Second, the author reveals the effect of axial load on the fundamental frequency of transverse vibrations and derive the explicit formulas for this frequency. The fundamental natural frequency of the transverse vibrations of the spring depends on the variable length of the spring. The reduction of frequency with the load is demonstrated. Finally, when the frequency nullifies, the side buckling spring occurs.

Helical springs constitute an integral part of many mechanical systems. A coil spring is a special form of spatially curved column. The center of each cross-section is located on a helix. The helix is a curve that winds around with a constant slope of the surface of a cylinder. An exact stability analysis based on the theory of spatially curved bars is complicated and difficult for further applications. Hence, in most engineering applications a concept of an equivalent column is introduced. The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The transverse vibration is represented by a differential equation of fourth order in place and second order in time. The solution of the undamped model equation could be obtained by separation of variables. The fundamental natural frequency of the transverse vibrations depends on the current length of the spring. Natural frequency is the function of the deflection and slenderness ratio. Is the fundamental natural frequency of transverse oscillations nullifies, the lateral buckling of the spring with the natural form occurs. The mode shape corresponds to the buckling of the spring with moment-free, simply supported ends. The mode corresponds to the buckling of the spring with clamped ends. The author finds the critical spring compression.

Buckling refers to the loss of stability up to the sudden and violent failure of seed straight bars or beams under the action of pressure forces, whose line of action is the column axis. The known results for the buckling of axially overloaded coil springs were found using the static stability criterion. The author uses an alternative approach method for studying the stability of the spring. This method is based on dynamic equations. In this paper, the author derives the equations for transverse vibrations of the pressure-loaded coil springs. The fundamental natural frequency of the transverse vibrations of the column is proved to be the certain function of the axial force, as well as the variable length of the spring. Is the fundamental natural frequency of transverse oscillations turns to be to zero, is the lateral buckling of the spring occurs.

The spring is substituted for the simplification of the basic equations by an equivalent column. Such a column must account for compressibility of axis and shear effects. The more accurate model is based on the equations of motion of loaded helical Timoshenko beams. The dimensionless for beams of circular cross-section and the number of parameters governing the problem is reduced to four (helix angle, helix index, Poisson coefficient, and axial strain) is to be derived. Unfortunately, that for the spatial beam models only numerical results could be obtained.

The closed form analytical formulas for fundamental natural frequency of the transverse vibrations of the column as function of the axial force, as well as the variable length of the spring are derived. The practically important formulas for lateral buckling of the spring are obtained.

In this paper, the author derives the new equations for transverse vibrations of the pressure-loaded coil springs. The author demonstrates that the fundamental natural frequency of the transverse vibrations of the column is the function of the axial force. For study of the stability of the spring the author uses an alternative approach method. This method is based on dynamic equations. The new, original expressions for lateral buckling of the spring are also obtained.

Moderately thick circular cylindrical shells are widely used as supporting structures or storage cavities in structural engineering, rock engineering, and aerospace engineering. In practical engineering, shells often work with micro-cracks or defects. The existence of micro-crack damage may result in the disturbance of dynamic behaviours and even induce accidental dynamic disasters. The free vibration frequency and mode are important parameters for the dynamic performance and damage identification analysis. In particular, stiffness weakening of the local damage region leads to significant changes in the vibration mode, which makes it difficult for the mesh generated in the conventional finite element method to capture a high-precision solution of the local oscillation.

In response to the above problems, this study developed an adaptive finite element method and a crack damage characterisation method for moderately thick circular cylindrical shells. By introducing the inverse power iteration method, error estimation, and mesh subdivision refinement technique for the analysis of finite element eigenvalue problems, an adaptive computation scheme was constructed for the free vibration problem of moderately thick circular cylindrical shells with circumferential crack damage.

Based on typical numerical examples, the established adaptive finite element solution for the free vibration of moderately thick circular cylindrical shells demonstrated its suitability for solving the high-precision free vibration frequency and mode of cylindrical shell structures. The any order frequency and mode shape of cracked cylindrical shells under the conditions of different ring wave numbers, crack locations, crack depths, and multiple cracks were successfully solved. The influences of the location, depth, and number of cracks on the disturbance of dynamic behaviours were analysed.

This study can be used as a reference for the adaptive finite element solution of free vibration of moderately thick circular cylindrical shells with cracks and lays the foundation for further development of a high-performance computation method suitable for the dynamic disturbance and damage identification analysis of general cracked structures.

This study aims to obtain earthquake responses of linear-elastic multi-span arch-frames by using exact curved beam formulations. For this purpose, the dynamic stiffness method (DSM) which uses exact mode shapes is applied to a three-span arch-frame considering axial extensibility, shear deformation and rotational inertia for both columns and curved beams. Using exact free vibration properties obtained from the DSM approach, the arch-frame model is simplified into an equivalent single degree of freedom (SDOF) system to perform earthquake response analysis.

The dynamic stiffness formulations of curved beams for free vibrations are validated by using the experimental data in the literature. The free vibrations of the arch-frame model are investigated for various span lengths, opening angle and column dimensions to observe their effects on the dynamic behaviour. The calculated natural frequencies via the DSM are presented in comparison with the results of the finite element method (FEM). The mode shapes are presented. The earthquake responses are calculated from the modal equation by using Runge-Kutta algorithm.

The displacement, base shear, acceleration and internal force time-histories that are obtained from the proposed approach are compared to the results of the finite element approach where a very good agreement is observed. For various span length, opening angle and column dimension values, the displacement and base shear time-histories of the arch-frame are presented. The results show that the proposed approach can be used as an effective tool to calculate earthquake responses of frame structures having curved beam elements.

The earthquake response of arch-frames consisting of curved beams and straight columns using exact formulations is obtained for the first time according to the best of the author’s knowledge. The DSM, which uses exact mode shapes and provides accurate free vibration analysis results considering each structural members as one element, is applied. The complicated structural system is simplified into an equivalent SDOF system using exact mode shapes obtained from the DSM and earthquake responses are calculated by solving the modal equation. The proposed approach is an important alternative to classical FEM for earthquake response analysis of frame structures having curved members.

The purpose of this paper is to propose and analyze the free vibration response of the spatial curved beams with variable curvature, torsion and cross section, in which all the effects of rotary inertia, shear and axial deformations can be considered.

The governing equations for free vibration response of the spatial curved beams are derived in matrix formats, considering the variable curvature, torsion and cross section. Frobenius’ scheme and the dynamic stiffness method are applied to solve these equations. A computer program is coded in Mathematica according to the proposed method.

To assess the validity of the proposed solution, a convergence study is carried out on a cylindrical helical spring with a variable circular cross section, and a comparison is made with the finite element method (FEM) results in ABAQUS. Further, the present model is used for reciprocal spiral rods with variable circular cross section in different boundary conditions, and the comparison with FEM results shows that only a limited number of terms in the results provide a relatively accurate solution.

The numerical results show that only a limited number of terms are needed in series solutions and in the Taylor expansion series to ensure an accurate solution. In addition, with a simple modification, the present formulation is easy to extend to analyze a more complicated model by combining with finite element solutions or analyze the transient responses and stochastic responses of spatial curved beams by Laplace transformation or Fourier transformation.

Purpose — Analysis of nonprismatic members has received a great deal attention from designers and engineers due to their ability in satisfaction of architectural and aesthetic necessities. Using these structural members in complex structures such as aircrafts, turbine blades and space vehicles, exact static and dynamic analyses of these members become more significant. Based on structural/mechanical principles, the purpose of this paper is to present a new method to evaluate exact structural matrices for nonprismatic Euler‐Bernoulli beam elements. Design/methodology/approach — Through introducing the concept of basic displacement functions (BDFs), it is shown that exact shape functions are derived in terms of BDFs. BDFs and their derivatives have structural interpretations; therefore, they are obtained via application of flexibility method. Unlike the conventional methods, which are almost categorized as displacement‐based methods, the flexibility basis of the method ensures the true satisfaction of equilibrium equations at any interior point of the element. Findings — The exact shape functions and consequently structural matrices are derived for general nonprismatic beam elements. Numerical examples are carried out to determine static deflection and natural frequencies, and the results are highly competent with the other methods in literature. Research limitations/implications — The method can be extended to structural analysis of curved beams, plates and shells as well. Moreover, it is possible to derive exact dynamic shape functions via BDFs by solving the governing equation for transverse vibration of beams. Theoretically, the method faces limitation in analysis of nonprismatic beams that converge to a point where cross‐sectional area and moment of inertia are equal to zero. Practical implications — The development of this idea, i.e. BDFs seems to lead to promotive novel approaches for structural analysis and could be a breaking point for developing new elements for plates and shells as it was shown for beam elements. Originality/value — The paper's introduction of special functions, namely BDFs and their application, in both static and dynamic analyses of structures, could be a breaking point in analysis procedures.

In this paper, a superconvergent patch recovery method is proposed for superconvergent solutions of modes in the finite element post-processing stage of variable geometrical Timoshenko beams. The proposed superconvergent patch recovery method improves the solution speed and accuracy of the finite element analysis of a curved beam. The free vibration and natural frequency of the beam were considered for studying forced vibrations and structural resonance. Beam vibration mode analysis was performed for high-precision vibration mode solutions and frequency values. The proposed method can be used to compute beam vibration modes of beams with different shapes and boundary conditions as well as variable cross sections and curvatures. The purpose of this paper is to address these issues.

An adaptive method was proposed to analyse the in-plane and out-of-plane free vibrations of the variable geometrical Timoshenko beams. In the post-processing stage of the displacement-based finite element method, the superconvergent patch recovery method and high-order shape function interpolation technique were used to obtain the superconvergent solution of mode (displacement). The superconvergent solution of mode was used to estimate the error of the finite element solution of mode in the energy form under the current mesh. Furthermore, an adaptive mesh refinement was proposed by mesh subdivision to derive an optimised mesh and accurate finite element solution to meet the preset error tolerance.

The results computed using the proposed algorithm were in good agreement with those computed using other high-precision algorithms, thus validating the accuracy of the proposed algorithm for beam analysis. The numerical analysis of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams helped verify that the solutions of frequencies were consistent with the results obtained using other specially developed methods. The proposed method is well suited for the mesh refinement analysis of a curved beam structure for analysing the changes in high-order vibration mode. The parts where the vibration mode changed significantly were locally densified; a relatively fine mesh division was adopted that validated the reliability of the mesh optimisation processing of the proposed algorithm.

The proposed algorithm can obtain high-precision vibration solutions of variable geometrical Timoshenko beams based on more optimized and reasonable meshes than the conventional finite element method. Furthermore, it can be used for vibration problems of parabolic curved beams, beams with variable cross sections and curvatures, elliptically curved beams and circularly curved beams. The proposed algorithm can be extended for application in superconvergent computation and adaptive analysis of finite element solutions of general structures and solid deformation fields and used for adaptive analysis of more complex plates, shells and three-dimensional structures. Additionally, this method can analyse the vibration and stability of curved members with crack damage to obtain high-precision vibration modes and instability modes under damage defects.

This study aims to perform dynamic response analysis of damaged rigid-frame bridges under multiple moving loads using analytical based transfer matrix method (TMM). The effects of crack depth, moving load velocity and damping on the dynamic response of the model are discussed. The dynamic amplifications are investigated for various damage scenarios in addition to displacement time-histories.

Timoshenko beam theory (TBT) and Rayleigh-Love bar theory (RLBT) are used for bending and axial vibrations, respectively. The cracks are modeled using rotational and extensional springs. The structure is simplified into an equivalent single degree of freedom (SDOF) system using exact mode shapes to perform forced vibration analysis according to moving load convoy.

The results are compared to experimental data from literature for different damaged beam under moving load scenarios where a good agreement is observed. The proposed approach is also verified using the results from previous studies for free vibration analysis of cracked frames as well as dynamic response of cracked beams subjected to moving load. The importance of using TBT and RLBT instead of Euler–Bernoulli beam theory (EBT) and classical bar theory (CBT) is revealed. The results show that peak dynamic response at mid-span of the beam is more sensitive to crack length when compared to moving load velocity and damping properties.

The combination of TMM and modal superposition is presented for dynamic response analysis of damaged rigid-frame bridges subjected to moving convoy loading. The effectiveness of transfer matrix formulations for the free vibration analysis of this model shows that proposed approach may be extended to free and forced vibration analysis of more complicated structures such as rigid-frame bridges supported by piles and having multiple cracks.