Modelling and analysis of the nonlinear string-mass structure of the vibration absorber

ABSTRACT

In this paper a nonlinear string-mass structure of the vibration absorber is analyzed. This structure is convenient to be installed in vibration damping systems of high buildings for their protection in the case of earthquake. The considered string-mass structure contains a translator movable mass connected with two strings. Due to nonlinear geometric properties of the system the motion of the mass is described with a strong nonlinear second order differential equation. In the paper the approximate procedure for solving of the nonlinear equation of motion is developed. Based on the solution the influence of the string preloading force, slider mass and friction force on the vibration property of the string-mass system is investigated. It is concluded that variation of the preloading string force may be applied as a control parameter for vibration absorption and as the regulator of vibration decay time.     

The vibration of a rotating heavy non-uniform thread and its stability

The linear vibration of a heavy non-uniform thread is investigated for different boundary conditions at the ends and taking an arbitrary additional tension into account. The thread is assumed to be ideal and inextensible and the motion takes place in a plane which may rotate about the vertical axis at constant angular velocity. A general scheme for solving the initial- and boundary value problem is proposed. Attention is focused mainly on the effective computation of the natural frequencies and mode of vibration. Given specific parametric types of mass distribution for the thread, sufficiently complete families of solutions describing the principal modes of vibration are constructed. Based on these families, stability and instability domains are constructed effectively, in terms of the system parameters, for a plane vibration of a rotating heavy thread subject to concentrated tension. New mechanical effects, of possible interest in practice, are observed and discussed.     

The Displacements Produced in an Elastic Half-space by a Suddenly Applied Surface Force

The problem considered here is that of the sudden applicationof a constant force to the surface of a homogeneous, isotropicelastic half-space. The case of a normally applied force isconsidered under axially-symmetric conditions. This force isassumed to be either concentrated at a point or distributedover a circular disc. The method of solution is based on theuse of integral transforms and differs in some respects frommethods previously used for this type of problem. The displacementsat all points of the solid are calculated in terms of finiteintegrals. These integrals are evaluated numerically for selectedinterior points of the half-space for point loading, and forpoints on the axis of symmetry in the case of distributed loading.An expansion of the integrals which is valid for large valuesof the time is also obtained.     

Forced vibration analysis of a Timoshenko beam featuring bending-torsion on Pasternak foundation

Free and forced vibration analysis of a Timoshenko beam on viscoelastic Pasternak foundation featuring coupling between flapwise bending and torsional vibrations is studied in this article. The system motion is described through a coupled set of three partial differential equations. The differential transform method, DTM, as an efficient mathematical technique is adopted to obtain the natural frequencies and the mode shapes. The system force response is assessed for a moving concentrated load with a constant velocity. Two different methods are studied and applied in obtaining forced vibration response of the system: (1) the same time functions, STF, by setting out the orthogonality conditions derived in this article and (2) the different time functions, DTF. The difference between the responses of the system is assessed by applying STF and DTF for a constant moving load. The effects of some parameters on the system response are probed. A numerical example is solved to validate the results obtained here with the available ones and a close agreement is found. It is observed that the time functions in DTF and STF are almost identical for transverse displacement and bending angle and are significant for torsion angle, recommending the application of DTF when the bending-torsion coupling is of concern.     

Steady soliton-like solutions of Euler's equations with intrinsic force fields

Steady solutions of a system of Euler's equations when there are intrinsic force interactions of electrical and gravitational type are analysed. Particular attention is devoted to the class of soliton “particle-like” solutions with concentrated charge and mass. Periodic steady solutions for the case when the intrinsic gravitational field has non-zero density at infinity is also considered.     

Determination of the Steady-State Response of Viscoelastically Point-Supported Rectangular Orthotropic Plates

In the present study, the steady-state response to a sinusoidally varying concentrated force acting at the center of a rectangular point-supported orthotropic elastic plate is analyzed. The equation of the plate is represented by finite-difference expressions. By using the ordinary finite-difference technique, the problem is reduced to the solution of a system of algebraic equations. The influence of mechanical properties of the plate material on the vibration modes and the steady-state response of the plate is investigated numerically. Also, the effect of location of the point supports is studied. The problems considered are solved within the framework of the Kirchhoff-Love hypothesis.     

具有复杂边界条件的杆的振动分析

本文研究一端带有集中质量并支以弹簧另一端作支承运动的杆的纵向振动.由于这个问题的边界条件比较复杂,且要考虑阻尼,因此本文只求稳态周期解.首先分析线性系统;然后考虑材料非线性,用摄动法求具有非线性边界条件的非线性方程的近似解析解.     

A Mathematical Model for Trapping Skinning in Polymers

When saturated polymer films are desorbed, a thin skin of glassy polymer can form at the exposed surface, inhibiting desorption. In addition, trapping skinning, in which an increase in the force driving the desorption decreases the accumulated flux, can also occur. These behaviors cannot be described by the simple Fickian diffusion equation. The mathematical model presented for the system is a moving boundary-value problem with a set of coupled partial differential equations that cannot be solved by similarity variables. Therefore, integral equation techniques are used to obtain asymptotic estimates for the solution. It is shown that although increasing the driving force will increase the instantaneous flux, the time of accumulation will decrease, thus reducing the overall flux. In addition, the model is shown to exhibit sharp fronts moving with constant speed, another distinctive feature of non-Fickian polymer-penetrant systems.     

An energy finite element method for high frequency vibration analysis of beams with axial force

In order to predict the high frequency vibration response of beams with axial force, an energy finite element method (EFEM) is developed. An Euler–Bernoulli beam with constant axial force is considered. The energy density and energy intensity of the beam with axial force are introduced, and the relationship between the wavenumber and group velocity is derived, then the energy density governing equation is established. In addition, the impedance of beam with axial force is derived to calculate the input power, and the energy density is evaluated by solving the governing equation using finite element method. Finally, the feasibility and correctness of the present EFEM are verified by comparing the results obtained by the EFEM with the corresponding exact solutions. The effects of axial force on the energy density response and the limitations of EFEM are also discussed.     

On the deformation of an elastic half-space with a thin slit for mixed conditions on its boundary

The problem of the state of stress and strain in an elastic half-space with a cutout in the shape of a circular slot is solved by Kelvin's method /1/. The conditions on the slot are satisfied in a suitable manner by selecting the scalar and vector mass force potentials as generalized functions concentrated at the slot. The problem reduces to a system of Fredholm integral equations of the second kind in a semi-infinite interval. The solution for an elastic space with a slot is obtained in final form in the limiting case, which enables an estimate to be made of the magnitude of the settling of the earth's surface as a result of oil or gas deposit development.     

On the non-ideal vibrating structures

In this work, analysis of the response to vibrations of a small building equipped with an electromechanical vibration absorber is investigated. The case of harmonic excitation with constant or time dependent frequencies is considered. The interaction between the structure and the energy source is analyzed via the Sommerfeld effect inside the resonance region. The resonance capture and the vibration reduction are displayed by the time history displacement of the last story. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic behavior on the Milne problem with a force term

This paper studies the existence, uniqueness and asymptotic behavior of the solution for a half-space linearized stationary Boltzmann equation with an external force term, in the case of a specified incoming distribution at the boundary and a given mass flux. Without the external force, the solution of the stationary Boltzmann equation has been proved to tend toward a constant state, which is independent of the space variable. Due to the presence of the external force, we show that the solution tends to some function which depends on the space variable.     

正交各向异性半平面内作用集中载荷的弹性解及边界元法常单元基本公式

本文采用镜相法,推导出了正交各向异性半平面作用集中载荷的理论解,给出了常单元系数矩阵表达式,为采用边界元法求解半平面问题提供了必要的公式.特解表达形式简洁,对边界元间接法常单元和高次单元各积分均可求出其原函数,可避免计算程序中的定积分数值计算过程.     

Dynamic analysis and suppressing chaotic impacts of a two-degree-of-freedom oscillator with a clearance

A two-degree-of-freedom impact oscillator is considered. The maximum displacement of one of the masses is limited to a threshold value by the symmetrical rigid stops. Impacts between the mass and the stops are described by an instantaneous coefficient of restitution. Dynamics of the system is studied with special attention to periodic-impact motions and bifurcations. Period-one double-impact symmetrical motion and transcendental impact Poincaré map of the system is derived analytically. Stability and local bifurcations of the period-one double-impact symmetrical motions are analyzed by using the impact Poincaré map. The Lyapunov exponents in the vibratory system with impacts are calculated by using the transcendental impact map. The influence of the clearance and excitation frequency on symmetrical double-impact periodic motion and bifurcations is analyzed. A series of other periodic-impact motions are found and the corresponding bifurcations are analyzed. For smaller values of clearance, period-one double-impact symmetrical motion usually undergoes pitchfork bifurcation with decrease in the forcing frequency. For large values of the clearance, period-one double-impact symmetrical motion undergoes Neimark–Sacker bifurcation with decrease in the forcing frequency. The chattering-impact vibration and the sticking phenomena are found to occur in the region of low forcing frequency, which enlarges the adverse effects such as high noise levels, wear and tear and so on. These imply that the dynamic behavior of this system is very rich and complex, varying from different types of periodic motions to chaos, even chattering-impacting vibration and sticking. Chaotic-impact motions are suppressed to minimize the adverse effects by using external driving force, delay feedback and feedback-based method of period pulse.     

Elimination of tall building vibration resulting from ground motions

In this paper, the vibration problems of tall buildings are considered. The focus is on vibration caused by earthquakes, semi–seismic phenomena and ground vibrations of other origins. The construction consists of the main system and a vibration eliminator (passive tuned mass damper – pendulum type) which is attuned to the first eigenfrequency of the main structure. The analysis focuses on elimination of structure vibration caused by horizontal components of ground motions, while the functioning of the eliminator is simultaneously influenced by the vertical component (parametric effect – the possibility of improper functioning of the device). The vertical periodic movement of the support point can cause changes of the vibration eliminator's stiffness. In such a case parametric excitation occurs in the system, which signifies that parametric resonance may appear. The numerical analysis of the problem was performed with the Newmark method in conjunction with FEM. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Natural frequencies and mode shapes of flexible mechanisms by a transfer matrix method

Transfer matrices are used to calculate the natural frequencies and corresponding mode shapes of planar elastic mechanisms. Mechanism motion is discretized by dividing the motion into a finite number of intervals over which the mechanism is considered to be an instantaneous structure with constant geometry. A transfer matrix relating the continuous, non-zero components of the state vector across a pin joint is developed. An elastic four-bar mechanism is analyzed using a lumped mass model.     

在集中力和集中力偶作用下不同弹性材料圆形界面的裂纹问题

运用推广的Schwarz延拓原理结合对复应力函数的奇性主部分析,求解一类有集中荷载的平面弹性问题,十分有效。文[1]用此方法研究了同种材料的弹性问题。本文把它推广于在集中力和集中力偶作用下不同弹性材料的圆形界面上有多条裂纹的情形,求出了几种典型情况复应力函数的封闭解,算出了应力强度因子,并由此导出一系列特殊解答,其中两个在文[1]、[6]中找到一致结果。     

Potential flow past a porous body with a core of different permeability

It is well known that a uniform flow past a non-permeable rigid body does not exert a total force upon the surface of the body, however this is not the case when the body is permeable. Power et. al. (1984, 1986) first solved the problem of uniform potential flow past a two-dimensional permeable circular cylinder, with constant permeability, and found that the exterior flow exerts a drag force upon the surface of the cylinder independent of its size and secondly the problem when the uniform potential flow past a porous sphere, with constant permeability, in this case the exterior flow exerts a drag force on the sphere which is linearly dependent on the radius of the sphere. Here we will present the solution of two problems, a uniform potential flow past a porous circular cylinder and past a porous sphere, for each case the porous body is composed of two materials with different permeabilities. In both cases the total force exerted by the exterior flow upon the body is dependent on the thickness of the porous materials, and in the limit when the two permeabilities are equal, the previous results, circular cylinder and sphere, with constant permeability, are recovered. Atlhough, the mathematics involved in the solution of the present problem is simple, due to the nice boundary geometry of the bodies, the final expression for the total force found in each case is quite interesting on the way it depends on the permeability relation, in particular, in the limiting cases of a porous body with solid or hollow core.     

A novel cutting force modelling method for cylindrical end mill

This paper proposes a new and simplified method for the calibration of cutting force coefficients and cutter runout parameters for cylindrical end milling using the instantaneous cutting forces measured instead of average ones. The calibration procedure is derived for a mechanistic cutting force model in which the cutting force coefficients are expressed as the power functions of instantaneous uncut chip thickness (IUCT). The derivations are firstly performed by establishing mathematical relationships between instantaneous cutting forces and IUCT. Then, nonlinear algorithms are proposed to solve the established nonlinear contradiction equations. The typical features of this new calibration method lie in twofold. On the one hand, all derivations are directly based on the tangential, radial and axial cutting force components transformed from the forces which are measured in the workpiece Cartesian coordinate system. This transformation makes the calibration procedure very simple and efficient. On the other hand, only a single cutting test is needed to be performed for calibrating the cutting force coefficients that are valid over a wide range of cutting conditions. The effectiveness of the proposed method in developing cutting force model is demonstrated experimentally with a series of verification cutting tests.