ANALYTICAL AND NUMERICAL STUDY ON SPATIAL FREE VIBRATION OF NON-SYMMETRIC THIN-WALLED CURVED BEAMS

For spatial free vibration of non-symmetric thin-walled circular curved beams, an accurate displacement field is introduced by defining all displacement parameters at the centroidal axis and three total potential energy functionals are consistently derived by degenerating the potential energy for the elastic continuum to that for thin-walled curved beams. The closed-form solutions are newly obtained for in-plane and out-of-plane free vibration analysis of monosymmetric curved beams respectively. Also, two thin-walled curved beam elements are developed using the third and fifth order Hermitian polynomials. In order to illustrate the accuracy and the practical usefulness of the present method, analytical and numerical solutions by this study are presented and compared with previously published results or solutions by ABAQUS' the shell element. Particularly, effects of the thickness curvature as well as the inextensional condition are investigated on free vibration of curved beams with monosymmetric and non-symmetric cross-sections.     

MODELLING OF TRANSVERSE VIBRATION OF SHORT BEAMS FOR CRACK DETECTION AND MEASUREMENT OF CRACK EXTENSION

The method of detection of location of crack in beams based on frequency measurements is extended here to short beams, taking into account the effects of shear deformation and rotational inertia through the Timoshenko beam theory and representing the crack by a rotational spring. Methods for solving both forward (determination of frequencies of beams knowing the crack parameters) and inverse (determination of crack location knowing the natural frequencies) problems are included. A method to estimate crack extension from a change in the first natural frequency is presented. Both numerical and experimental studies are given to demonstrate the accuracy of the methods. The accuracy of the results is quite encouraging.     

三合板梁弯曲问题

一.引言 三合板的弯曲问题在航空工业上有广泛的应用。近十年来出现了不少关于这方面的理论,但是这些理论都是以線性问题观点来处理的,在实际结果与理论出入太大时,一般都认为是材料的品质和结构的缺点的问题。本文的理论分析了非線性的因素,无论是線性及由線性扩展到非線性范围;而且根据本文的理论结果得到了很好的实验证明。     

VIBRATION AND BUCKLING OF INITIALLY STRESSED CURVED BEAMS

Equations of motion for curved beams in a general state of non-uniform initial stresses are derived using the principle of virtual work. The equations are adjusted to a generic expression by using appropriate transformations. The free vibration behaviours of the curved beams subjected to a combination of uniform initial tensile of compressive stresses and uniform initial bending stress are examined. The Galerkin method is employed in obtaining accurate values of free frequencies and initial buckling stresses. The curved beam is assumed to be vibrating in its plane. Natural frequencies and initial buckling stresses for hinged supported curved beams are presented for validation. Effects of arc segment angles, elastic foundation, and initial stresses on the natural frequencies are investigated. Effects of arc segment angles, elastic foundation, and initial bending stresses on the initial buckling stresses are explored in this paper.     

PARAMETRIC INSTABILITY OF MULTI-LAYERED SANDWICH BEAMS

The present work deals with the parametric instability of multi-layered symmetric sandwich beams with alternate elastic and viscoelastic layers, subjected to periodic axial load. The equations of motion and boundary conditions are derived by applying Hamilton's principle, and the general Galerkin method is utilized to convert the equations of motion into a set of coupled Hill's equations with complex coefficients in the time domain. Numerical results are obtained for beams having, three, five and seven layers. The effects of the shear and core-thickness parameters as well as the core loss factor upon the regions of parametric instability are considered. Zones of instability are obtained for the cases in which beams with various number of layers have the same weight or size or flexural rigidity. Finally, for the criteria of constant size and flexural rigidity as well as constant weight and flexural rigidity, the effects of various parameters on the stability of the system are studied.     

三合板的一般弹性理论

一.引言 本文所指之三合板是一种轻质板材,在航空工业上近年来有广泛的应用。板由三层材料构成:外层为二硬薄钣(金属薄钣),夹心为较轻软之材料(如泡状醋酸赛璐珞、软木、或蜂巢状金属材料)。前者称为表钣,后者称为夹心,表钣与夹心胶合成一综合板。三合板之形变与应力之计算在十多年来曾引起力学家与工     

VIBRATION AND BUCKLING OF COMPOSITE THIN-WALLED BEAMS WITH SHEAR DEFORMABILITY

In this paper, a theoretical model is developed for the dynamic analysis of composite thin-walled beams with open or closed cross-sections. The present model incorporates, in a full form, the shear flexibility (bending and warping shear) as well as a state of initial stresses. This allows to study the free vibration and buckling problems in a unified fashion. An analytical solution of the developed equations is obtained for the case of simply supported thin-walled beams. Numerical examples are given to demonstrate the importance of the shear flexibility on the vibration and buckling behavior of the considered structures.     

THE SWING VIBRATION, TRANSVERSE OSCILLATION OF CRACKED ROTOR AND THE INTERMITTENCE CHAOS

This paper studies the non-linear dynamic response of a cracked rotor by taking the swing vibration of disc into consideration. The results show that if a small crack appears, the frequency of transverse oscillation is synchronous with rotating speed ratio (Ω), and the frequency of swing vibration is N Ω (N=1,2,…). As the crack increases, the response becomes chaotic in some range of Ω. The deeper the crack is, the wider the chaotic range of Ω is. Routes to chaos include intermittence to chaos and quasi-period to chaos. When the crack is fairly deep, some new resonance regions develop. In these regions, the response becomes infinity rapidly. The appearance of intermittence chaos is induced by the frequent frustration of stable oscillation, which is resulted from the continuous increase of swing amplitude. Unbalance parameter U is effective in suppressing chaos. Crack angle β cannot affect the essence of response, but can influence the amplitude of synchronous response.     

OPTIMIZATION OF ANISOTROPIC SANDWICH BEAMS FOR HIGHER SOUND TRANSMISSION LOSS

This paper presents a study on the optimization of sound transmission loss across anisotropic sandwich beams. It has been found in earlier studies that there is a significant increase in the sound transmission loss for sandwich beams with anisotropic materials compared to those with isotropic ones. The optimization studies presented in this work further validate this concept. The material and geometric properties of the structure are treated as the design variables with the objective to maximize the sound transmission loss across the beam. Appropriate constraints are imposed to maintain material and structural integrity.     

FREE VIBRATION ANALYSIS OF NON-UNIFORM BEAMS WITH AN ARBITRARY NUMBER OF CRACKS AND CONCENTRATED MASSES

An exact approach for free vibration analysis of a non-uniform beam with an arbitrary number of cracks and concentrated masses is proposed. A model of massless rotational spring is adopted to describe the local flexibility induced by cracks in the beam. Using the fundamental solutions and recurrence formulas developed in this paper, the mode shape function of vibration of a non-uniform beam with an arbitrary number of cracks and concentrated masses can be easily determined. The main advantage of the proposed method is that the eigenvalue equation of a non-uniform beam with any kind of two end supports, any finite number of cracks and concentrated masses can be conveniently determined from a second order determinant. As a consequence, the decrease in the determinant order as compared with previously developed procedures leads to significant savings in the computational effort and cost associated with dynamic analysis of non-uniform beams with cracks. Numerical examples are given to illustrate the proposed method and to study the effect of cracks on the natural frequencies and mode shapes of cracked beams.     

平顶高斯光束与超高斯光束传输特性的比较研究

对平顶高斯光束与超高斯光束的传输特性作了研究和比较,并给出了典型的数值计算例．所得的主要结论是有相同M²因子和束宽的平顶高斯光束与超高斯光束在通过光学系统时,具有非常相似的包括光强分布和光阑衍射效应在内的传输变换特性． 关键词：     

FUNDAMENTAL FREQUENCY OF CRACKED BEAMS IN BENDING VIBRATIONS: AN ANALYTICAL APPROACH

This paper presents an analytical approach to the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations. The flexibility influence function method used to solve the problem leads to an eigenvalue problem formulated in integral form. The influence of the crack was represented by an elastic rotational spring connecting the two segments of the beam at the cracked section. In solving the problem, closed-form expressions for the approximated values of the fundamental frequency of cracked Euler-Bernoulli beams in bending vibrations are reached. The results obtained agree with those numerically obtained by the finite element method.     

MODELLING AND VIBRATION CONTROL OF BEAMS WITH PARTIALLY DEBONDED ACTIVE CONSTRAINED LAYER DAMPING PATCH

A detailed model for the beams with partially debonded active constraining damping (ACLD) treatment is presented. In this model, the transverse displacement of the constraining layer is considered to be non-identical to that of the host structure. In the perfect bonding region, the viscoelastic core is modelled to carry both peel and shear stresses, while in the debonding area, it is assumed that no peel and shear stresses be transferred between the host beam and the constraining layer. The adhesive layer between the piezoelectric sensor and the host beam is also considered in this model. In active control, the positive position feedback control is employed to control the first mode of the beam. Based on this model, the incompatibility of the transverse displacements of the active constraining layer and the host beam is investigated. The passive and active damping behaviors of the ACLD patch with different thicknesses, locations and lengths are examined. Moreover, the effects of debonding of the damping layer on both passive and active control are examined via a simulation example. The results show that the incompatibility of the transverse displacements is remarkable in the regions near the ends of the ACLD patch especially for the high order vibration modes. It is found that a thinner damping layer may lead to larger shear strain and consequently results in a larger passive and active damping. In addition to the thickness of the damping layer, its length and location are also key factors to the hybrid control. The numerical results unveil that edge debonding can lead to a reduction of both passive and active damping, and the hybrid damping may be more sensitive to the debonding of the damping layer than the passive damping.     

NON-DESTRUCTIVE EVALUATION OF HORIZONTAL CRACK DETECTION IN BEAMS USING TRANSVERSE IMPACT

Numerical and experimental studies for crack detection in beam employing transverse impact are presented. In the numerical study, a beam model of wave propagation is developed to calculate the time history of beam response before, over and after the crack region. It is expected that the resulting wave in the beam will be scattered by the crack and will carry information on the location and geometry of the crack. Experiments using a scanning laser vibrometer on specimens containing simulated crack are then conducted to verify the numerical results. Comparison study between the numerical results and experimental observations are conducted; good correlation between theory and experiment is observed. The beam model of wave propagation and adaptive multilayer perceptron networks (MLP) are then used for inverse identification of crack parameters (i.e., crack location, depth and length) in the beams. Time-domain displacement responses calculated using the present beam model containing predetermined crack parameters are used as training data for the MLP. Once the MLP is trained, the MLP networks are then employed for inverse determination of an unknown crack in a beam using experimental displacement responses measured with a scanning laser vibrometer. Examples show that the procedure performs well for the determination of a wide range of values for the location, depth and length of the crack.     

VIBRATION AND STABILITY OF ROTATING POLAR ORTHOTROPIC ANNULAR DISKS SUBJECTED TO A STATIONARY CONCENTRATED TRANSVERSE LOAD

In the present paper, natural frequencies and stability of a spinning polar orthotropic disk subjected to a stationary concentrated transverse load are investigated. The analysis of the free vibration of a spinning disk is performed first to find natural frequencies and corresponding vibration modes. The resulting eigenfunctions obtained from the free vibration are used as deflection functions of the forced vibration of a disk where the load is modelled as a mass-spring-dashpot system fixed in space. By using the Galerkin approximation method, eigenvalues of the whole system are determined. Results show that disks with higher values of modulus ratios or the Poisson ratios have higher natural frequencies, and the stability of the whole system can be improved by raising the value of the modulus ratio or lowering that of the Poisson ratio.     

EFFECT OF PHASE ON HUMAN RESPONSES TO VERTICAL WHOLE-BODY VIBRATION AND SHOCK—ANALYTICAL INVESTIGATION

The effect of the “phase” on human responses to vertical whole-body vibration and shock has been investigated analytically using alternative methods of predicting subjective responses (using r.m.s., VDV and various frequency weightings). Two types of phase have been investigated: the effect of the relative phase between two frequency components in the input stimulus, and the phase response of the human body. Continuous vibrations and shocks, based on half-sine and one-and-a-half-sine accelerations, each of which had two frequency components, were used as input stimuli. For the continuous vibrations, an effect of relative phase was found for the vibration dose value (VDV) when the ratio between two frequency components was three: about 12% variation in the VDV of the unweighted acceleration was possible by changing the relative phase. The effect of the phase response of the body represented by frequency weightings was most significant when the frequencies of two sinusoidal components were about 3 and 9 Hz. With shocks, the effect of relative phase was observed for all stimuli used. The variation in the r.m.s. acceleration and in the VDV caused by variations in the relative phase varied between 3 and 100%, depending on the nature of stimulus and the frequency weighting. The phase of the frequency weightings had a different effect on the r.m.s. and the VDV.     

ANALYSIS OF THE FLEXURAL VIBRATION OF A THIN-PLATE BOX USING A COMBINATION OF FINITE ELEMENT ANALYSIS AND ANALYTICAL IMPEDANCES

Many practical built-up thin-plate structures, e.g., a modern car body, are essentially assemblies of numerous thin plates joined at their edges. The plates are so thin that they invariably support the weight of the structure and machinery using their substantial in-plane stiffness. Consequently, vibrational power injected into the structure from sources mounted at these stiff points is controlled by high impedance long-wavelength in-plane waves in the plates. As the long in-plane waves propagate around the structure, they impinge upon the numerous structural joints at which short-wavelength flexural waves are generated in adjoining plates. These flexural waves have much lower impedance than the in-plane waves. Hence, the vibration of thin-plate structures excited at their stiff points develops into a mixture of long in-plane waves and short flexural waves. In a previous paper by the same authors, a numerically efficient finite element analysis which accommodated only the long in-plane waves was used to predict the forced response of a six-sided thin-plate box at the stiff points. This paper takes that finite element analysis and, drawing on theory developed in two additional papers by the same authors, couples analytical impedances to it in order to represent the short flexural waves generated at the structural joints. The parameters needed to define these analytical impedances are identified. The vibration of the impedances are used to calculate estimates of the mean-square flexural vibration of the box sides which compare modestly with laboratory measurements. The method should have merit in predicting the vibration of built-up thin-plate structures in the so-called “mid-frequency” region where the modal density of the long waves is too low to allow confident application of statistical energy analysis, yet the modal density of the short flexural waves is too high to allow efficient finite element analysis.     

双束流激发的纵向尾场与纵向不稳定性的研究

 用解析的方法研究了同向和反向运动束流激发的纵向尾场,并且给出了线性近似下束团沿环不均匀分布情形下的双束流纵向多束团不稳定性的计算公式,得出了双束流不稳定性与高频腔(或其它类腔结构)在环上位置及模的对称性有关的结论。     

FINITE ELEMENT FORMULATION AND ACTIVE VIBRATION CONTROL STUDY ON BEAMS USING SMART CONSTRAINED LAYER DAMPING (SCLD) TREATMENT

This work deals with the active vibration control of beams with smart constrained layer damping (SCLD) treatment. SCLD design consists of viscoelastic shear layer sandwiched between two layers of piezoelectric sensors and actuator. This composite SCLD when bonded to a vibrating structure acts as a smart treatment. The sensor piezoelectric layer measures the vibration response of the structure and a feedback controller is provided which regulates the axial deformation of the piezoelectric actuator (constraining layer), thereby providing adjustable and significant damping in the structure. The damping offered by SCLD treatment has two components, active action and passive action. The active action is transmitted from the piezoelectric actuator to the host structure through the viscoelastic layer. The passive action is through the shear deformation in the viscoelastic layer. The active action apart from providing direct active control also adjusts the passive action by regulating the shear deformation in the structure. The passive damping component of this design eliminates spillover, reduces power consumption, improves robustness and reliability of the system, and reduces vibration response at high-frequency ranges where active damping is difficult to implement. A beam finite element model has been developed based on Timoshenko's beam theory with partially covered SCLD. The Golla-Hughes-McTavish (GHM) method has been used to model the viscoelastic layer. The dissipation co-ordinates, defined using GHM approach, describe the frequency-dependent viscoelastic material properties. Models of PCLD and purely active systems could be obtained as a special case of SCLD. Using linear quadratic regulator (LQR) optimal control, the effects of the SCLD on vibration suppression performance and control effort requirements are investigated. The effects of the viscoelastic layer thickness and material properties on the vibration control performance are investigated.     

AN ANALYTICAL AND NUMERICAL STUDY OF A MODIFIED VAN DER POL OSCILLATOR

A three-dimensional system of differential equations that models an electronic oscillator is considered. The equations allow a variety of periodic orbits that originate from a degenerate Hopf bifurcation, which is analytically studied. Numerical results are presented that show the existence of saddle-node cusps of periodic orbits, as well as period-doubling bifurcations, that result in the coexistence of multiple “canard” orbits if one of the parameters is small. The presence of chaotic attractors is also detected.