Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section

A finite element method for studying non-linear free torsional vibrations of thin-walled beams with bisymmetric open cross-section is presented. The non-linearity of the problem arises from axial loads generated at moderately large amplitude torsional vibrations due to immovability of end supports. The derivation of the fundamental differential equation of the problem is based on the classical assumption of a thin-walled beam with a non-deformable cross-section. The non-linear eigenvalue problem is solved iteratively by series of linear eigenvalue problems until the required accuracy is obtained. Non-linear frequencies, fundamental mode shapes and axial loads computed for various amplitude of torsional vibrations of thin-walled I beams are included.     

Experimental investigations of the response of a cracked rotor to periodic axial excitation

In this paper, the coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Steady state unbalance response of a cracked rotor with a single centrally situated crack subjected to periodic axial impulses is investigated experimentally. The cracked rotor is excited axially using an electrodynamic exciter at a frequency equal to its bending natural frequency in both non-rotating and rotating conditions. The resulting time domain and frequency domain signals of the cracked rotor are studied. Spectral response of the cracked rotor with and without axial excitation is found to be distinctively different. When excited axially, it shows prominent presence of rotor bending natural frequency. However for an uncracked rotor, the response is similar with or without axial excitation. It is thus proposed that the response of the rotor to axial impulse excitation could be used for more reliable diagnosis of rotor cracks.     

Optimal design of thin-walled I beams for a given natural frequency of torsional vibrations

A method of extremum weight design of thin-walled I beams for a given natural frequency of torsional vibrations is presented. The effects of warping stresses and constant axial loads are taken into account. The optimality condition for only one (except for the web height) dimension of the cross-section, variable along the axis of the beam, is derived by using Pontryagin's maximum principle. The solution of the problem formulated, with account also taken of the additional geometrical conditions, is obtained in an iterative way. Some numerical examples of optimal design of an I beam with variable flange width, for a specified fundamental frequency, are given.     

An approximate method for determining the static deflection and natural frequency of a cracked beam

This paper provides an approximate method to determine the stiffness and the fundamental frequency of a cracked beam. The cracked beam is first represented as an un-cracked beam with equivalent reduced sections around the cracks. The effect of the cracks is explained, visualised and quantified using the equivalence concept developed for stepped beams with periodically variable cross-sections. Then an alternative expression of the improved Rayleigh method is provided to calculate the natural frequencies of a beam with a variable stiffness distribution along its length. As the method is insensitive to the assumed mode shapes, it avoids the difficulty in choosing appropriate mode shapes and yields accurate results. This is shown using several examples to compare the results determined using the proposed method and the Finite Element method (FEM). The method greatly simplifies the calculation of cracked beams with complicated configurations, such as a beam with several cracks, a cracked beam with concentrated masses, a beam with cracks close to each other, and a beam with periodically distributed cracks.     

Free vibration analysis of beams and frames with multiple cracks for damage detection

The problem of calculating the natural frequencies of beams with multiple cracks and frames with cracked beams is studied. The natural frequencies are obtained using a new method in which a rotational spring model is used to represent the cracks. For beams, dynamic stiffness matrices of order 4 are obtained in a recursive manner, according to the number of cracks, by applying partial Gaussian elimination. The Wittrick–Williams algorithm is used to compute the natural frequencies in the resulting transcendental eigenvalue problem. Published numerical examples for cracked beams are used for validation. The global dynamic stiffness matrix of a frame with multiply cracked members is then assembled. A published two bay frame example is used to evaluate the new method. The effect of changing the location of a crack in a two bay two storey frame is studied numerically, giving insight into the inverse problem of damage detection.     

Flexural vibration of non-uniform beams having double-edge breathing cracks

Flexural vibration of non-uniform Rayleigh beams having single-edge and double-edge cracks is presented in this paper. Asymmetric double-edge cracks are formed as thin transverse slots with different depths at the same location of opposite surfaces. The cracks are modelled as breathing since the bending of the beam makes the cracks open and close in accordance with the direction of external moments. The presented crack model is used for single-edge cracks and double-edge cracks having different depth combinations. The energy method is used in the vibration analysis of the cracked beams. The consumed energy caused by the cracks opening and closing is obtained along the beam's length together with the contribution of tensile and compressive stress fields that come into existence during the bending. The total energy is evaluated for the Rayleigh-Ritz approximation method in analysing the vibration of the beam. Examples are presented on simply supported beams having uniform width and cantilever beams which are tapered. Good agreements are obtained when the results from the present method are compared with the results of Chondros et al. and the results of the commercial finite element program, Ansys^©. The effects of breathing in addition to crack depth's asymmetry and crack positions on the natural frequency ratios are presented in graphics.     

Mathematical model of the dynamics of micropolar elastic thin beams. Free and forced vibrations

A method of hypotheses has been developed to construct a mathematical model of micropolar elastic thin beams. The method is based on the asymptotic properties of the solution ofan initial boundary value problem in a thin rectangle within the micropolar theory of elasticity with independent displacement and rotation fields. An applied model of the dynamics of micropolar elastic thin beams was constructed in which transverse shear strains and related strains are taken into account. The constructed dynamics model was used to solve problems of free and forced vibrations of a micropolar beam. Free vibration frequencies and modes, forced vibration amplitudes, and resonance conditions were determined. The obtained numerical calculation results show the specific features of free vibrations of thin beams. Micropolar thin beams have a free vibration frequency which is almost independent of the thin beam size, but depends only on the physical and inertial properties of the micropolar material. It is shown for the micropolar material that the free vibration frequency values of beams can be readily adjusted and hence a large vibration frequency separation can be achieved, which is important for studying resonance.     

Optimal design of beams under flexural vibration

The minimum weight design of a cantilever beam in flexural vibration is considered. The aim is the maximization of a given natural bending frequency (usually the first) for a given beam weight or equivalently the minimization of beam weight for a specified value of a natural frequency. The beams considered are of rectangular section and are subject, in a range of cases presented, to a variety of constraints on lower and upper bounds on the cross-section dimensions or to the specification of a point mass at the end of the beam. Simple bending theory is regarded as applicable to the problem. A variational statement of the problem is made and the necessary conditions for a minimum are obtained as a system of non-linear equations which are solved numerically. Results are given in the form of tables and of figures showing computed optimum profiles. Some experiments on a sample set of beams of equal mass are described briefly. The optimum profile beam was found to have the greatest fundamental frequency, in support of the theoretical predictions.     

IDENTIFICATION OF CRACK LOCATION IN VIBRATING BEAMS FROM CHANGES IN NODE POSITIONS

It is known that the effect of a single crack in an axially vibrating thin rod is to cause the nodes of the mode shapes move toward the crack. This paper is an analytical/experimental investigation of the analogous problem for a thin beam in bending vibration. The monotonicity property linking changes in node position and crack location does not hold in the bending case. The analysis of the direct problem, however, shows that the direction by which nodal points move may be useful for predicting damage location. Analytical results agree well with experimental tests performed on cracked steel beams.     

A method for determining the location and depth of cracks in double-cracked beams

The influence of two transverse open cracks on the antiresonances of a double cracked cantilever beam is investigated both analytically and experimentally. It is shown that there is a shift in the antiresonances of the cracked beam depending on the location and size of the cracks. These antiresonance changes, complementary with natural frequency changes, can be used as additional information carrier for crack identification in double cracked beams. Experimental results from tests on plexiglas beams damaged at different locations and different magnitudes are found to be in good agreement with theoretical predictions. Based on the results of the present work, an efficient prediction scheme for crack localization and characterization in double cracked beams is proposed.     

Thermoelastic damping in torsion microresonators with coupling effect between torsion and bending

Predicting thermoelastic damping (TED) is crucial in the design of high Q MEMS resonators. In the past, there have been few works on analytical modeling of thermoelastic damping in torsion microresonators. This could be related to the assumption of pure torsional mode for the supporting beams in the torsion devices. The pure torsional modes of rectangular supporting beams involve no local volume change, and therefore, they do not suffer any thermoelastic loss. However, the coupled motion of torsion and bending usually exists in the torsion microresonator when it is not excited by pure torque. The bending component of the coupled motion causes flexural vibrations of supporting beams which may result in significant thermoelastic damping for the microresonator. This paper presents an analytical model for thermoelastic damping in torsion microresonators with the coupling effect between torsion and bending. The theory derives a dynamic model for torsion microresonators considering the coupling effect, and approximates the thermoelastic damping by assuming the energy loss to occur only in supporting beams of flexural vibrations. The thermoelastic damping obtained by the present model is compared to the measured internal friction of single paddle oscillators. It is found that thermoelastic damping contributes significantly to internal friction for the case of the higher modes at room temperature. The present model is validated by comparing its results with the finite-element method (FEM) solutions. The effects of structural dimensions and other parameters on thermoelastic damping are investigated for the representative case of torsion microresonators.     

A SEMI-ANALYTICAL APPROACH TO THE NON-LINEAR DYNAMIC RESPONSE PROBLEM OF BEAMS AT LARGE VIBRATION AMPLITUDES, PART II: MULTIMODE APPROACH TO THE STEADY STATE FORCED PERIODIC RESPONSE

The semi-analytical approach to the non-linear dynamic response of beams based on multimode analysis has been presented in Part I of this series of papers (Azrar et al., 1999 Journal of Sound and Vibration224, 183-207 [1]). The mathematical formulation of the problem and single mode analysis have been studied. The objective of this paper is to take advantage of applying this semi-analytical approach to the large amplitude forced vibrations of beams. Various types of excitation forces such as harmonic distributed and concentrated loads are considered. The governing equation of motion is obtained and can be considered as a multi-dimensional form of the Duffing equation. Using the harmonic balance method, the equation of motion is converted into non-linear algebraic form. Techniques of solution based on iterative-incremental procedures are presented. The non-linear frequency and the non-linear modes are determined at large amplitudes of vibration. The basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated. The percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution. Also, the percentage contributions of the higher modes to the bending moment near to the clamps are given, in order to determine accurately the error introduced in the non-linear bending stress estimated by different approximations. Solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency. The non-linear deflection shapes in various regions of the solution, the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes. The numerical results obtained here for the non-linear forced response are compared with those from the linear theory, with available non-linear results, based on various approaches, and with the single mode analysis.     

ANALYSIS OF THE RESPONSE OF A CRACKED JEFFCOTT ROTOR TO AXIAL EXCITATION

The coupling of lateral and longitudinal vibrations due to the presence of transverse surface crack in a rotor is explored. Steady state unbalance response of a Jeffcott rotor with a single centrally situated crack subjected to periodic axial impulses is studied. Partial opening of crack is considered and the stress intensity factor at the crack tip is used to decide the extent of crack opening. A crack in a rotor is known to introduce coupling between lateral and longitudinal vibrations. Therefore, lateral vibration response of a cracked rotor to axial impulses is studied in detail. Spectral analysis of response to periodic multiple axial impulses shows the presence of rotor bending natural frequency as well as side bands around impulse excitation frequency and its harmonics due to modulations caused by rotor running frequency. It is concluded that the above approach can prove to be a useful tool in detecting cracks in rotors.     

Variation in the stability of a rotating blade disk with a local crack defect

The effect of a near root local blade crack on the stability of a grouped blade disk is investigated in this paper. A bladed disk comprised of periodically shrouded blades is used to simulate the coupled periodic structure. The blade crack is modeled using the local flexibility with coupling terms. The mode localization phenomenon introduced by the blade crack on the longitudinal and bending vibrations in the rotating blades has also been considered. Using the Galerkin's method, the imperturbation equations of a bladed disk in which one of the blades is cracked, subject to fluctuations in the rotation speed, can be derived. Employing the multiple scales method, the boundaries of the instability zones in the mistuned turbo blade system are approximated. Numerical results indicate that an additional unstable zone is introduced near the localization frequency and the regions of unstable zones are varied with the crack size and fluctuations in disk speed.     

CRACK IDENTIFICATION IN VIBRATING BEAMS USING THE ENERGY METHOD

An energy-based numerical model is developed to investigate the influence of cracks on structural dynamic characteristics during the vibration of a beam with open crack(s). Upon the determination of strain energy in the cracked beam, the equivalent bending stiffness over the beam length is computed. The cracked beam is then taken as a continuous system with varying moment of intertia, and equations of transverse vibration are obtained for a rectangular beam containing one or two cracks. Galerkin's method is applied to solve for the frequencies and vibration modes. To identify the crack, the frequency contours with respect to crack depth and location are defined and plotted. The intersection of contours from different modes could be used to identify the crack location and depth.     

Local vibrations of one-dimensional model of dislocation

Changes in the frequency spectrum of the vibrations of a one-dimensional model of a crystal, which is perturbed by dislocations, are determined by the method described in paper [10]. The influence of an external homogeneous stress on the frequency of the local vibrations is qualitatively evaluated.The author thanks assist. prof. O. Litzman for valuable discussions and help in solving the problem.     

Nonlinear free vibrations of beams in space due to internal resonance

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.     

Mode shapes analysis of a cracked beam and its application for crack detection

In this paper, mode shapes of a cracked beam with a rectangular cross section beam are analysed using finite element method. The 3D beam element is applied for this finite element analysis. The influence of the coupling mechanism between horizontal bending and vertical bending vibrations due to the crack on the mode shapes is investigated. Due to the coupling mechanism the mode shapes of a beam change from plane curves to space curves. Thus, the existence of the crack can be detected based on the mode shapes: when the mode shapes are space curves there is a crack in the beam. Also, when there is a crack, the mode shapes have distortions or sharp changes at the crack position. Thus, the position of the crack can be determined as a position at which the mode shapes exhibit such distortions or sharp changes. While in previous studies using 2D beam element, distortions in the mode shapes caused by a small crack could not be detected, these distortions in the case using the 3D beam element can be amplified and inspected clearly by using the projections of the mode shapes on appropriate planes. The quantitative analysis is also implemented to relate the size and position of the crack with the observed coupled modes. These results can be applied for crack detection of a beam. In this paper, the stiffness matrix of a cracked element obtained from fracture mechanics is presented and numerical simulations of three case studies are provided.     

幂函数形式连续变截面梁振动的弯曲固有频率分析*

本文使用微分方程解析法求解变截面梁固有频率。首先,建立变截面梁模型,其中截面面积和惯性矩均按幂次函数变化。得到变截面梁自由振动时挠度的解析表达式,并获得不同边界条件下梁弯曲振动的固有频率方程。其中惯性矩所对应幂指数与截面面积的幂指数的差值为4时,可得自振频率方程的精确形式;而幂指数差值不等于4时,给出近似解法。其次,对4种具体的变截面梁求解不同边界下的自振频率,并与瑞利-里兹法所得的自振频率解比较。验证精确解法结果的正确性,并发现近似解法结果的相对偏差在5%以内。该解析方法较瑞利-里兹法具有能快速求解的特点,且易于分析截面参数对梁固有频率的影响。由算例可得,边界和其他参数不变时,梁的同阶次无量纲自振频率随着幂次指数的增加而增加。几何参数中仅截面形状参数改变时,随着形状参数的增加,梁的同阶次无量纲自振频率随之减小,但固定-自由梁的第一阶自振频率除外。     

Dynamic behaviour of structures in large frequency range by continuous element methods

The continuous element method is presented in the context of the harmonic response of beam assemblies. A general formulation is described from the displacement solution of the elementary problem. A direct computation of elementary dynamic stiffness matrices is presented. In the present formulation, distributed loadings are taken into account. In the case of more complex geometries for which many coupling phenomena occur, an explicit formulation is no more conceivable. In this case, a numerical approach is presented. This approach allows an algorithmic computation of exact dynamic stiffness matrices. This method, called “Numerical Continuous Element”, allows one to consider the coupled vibrations of curved beams and those of helical beams. The validation of this numerical method is achieved by comparisons with the harmonic response of various beams obtained by a finite element approach. Finally, a comparison between eigenfrequencies obtained experimentally and numerically for a straight beam and a helical beam has been made to evaluate the performances of the method.