Forced transverse vibrations of an elastically connected complex simply supported double-beam system

The present paper is devoted to analyzing undamped forced transverse vibrations of an elastically connected complex double-beam system. The problem is formulated and solved in the case of simply supported beams. The classical modal expansion method is applied to ascertain dynamic responses of beams due to arbitrarily distributed continuous loads. Several cases of particularly interesting excitation loadings are investigated. The action of stationary harmonic loads and moving forces is considered. In discussing vibrations caused by exciting harmonic forces, conditions of resonance and dynamic vibration absorption are determined. The beam-type dynamic absorber is a new concept of a continuous dynamic vibration absorber (CDVA), which can be applied to suppress excessive vibrations of corresponding beam systems. A numerical example is presented to illustrate the theoretical analysis.     

NON-LINEAR FORCED VIBRATIONS OF PLATES BY AN ASYMPTOTIC-NUMERICAL METHOD

Non-linear forced vibrations of thin elastic plates have been investigated by an asymptotic-numerical method (ANM). Various types of harmonic excitation forces such as distributed and concentrated are considered. Using the harmonic balance method and Hamilton's principle, the equation of motion is converted into an operational formulation. Based on the finite element method a starting point corresponding to a non-linear solution associated to a given frequency and amplitude of excitation is computed. Applying perturbation techniques in the vicinity of this solution, the non-linear governing equation obtained is transformed into a sequence of linear problems having the same stiffness matrix. Employing one matrix inversion, a large number of terms of the perturbation series of the displacement and frequency can be easily computed with a small computation time. Iterations of this method lead to a powerful path-following technique. Comprehensive numerical tests for forced vibrations of plates subjected to time-harmonic lateral excitations are reported.     

Overview of the New OMAH Technique for Scaling OMA Mode Shapes

Methods for scaling mode shapes determined by operational modal analysis (OMA) have been extensively investigated in the last years. A recent addition to the range of methods for scaling OMA mode shapes is the so-called OMAH technique, which is based on exciting the structure by harmonic forces applied by an actuator. By applying harmonic forces in at least one degree-of-freedom (DOF), and measuring the response in at least one response DOF, while using at least as many frequencies as the number of mode shapes to be scaled, the mode shape scaling (modal mass) of all modes of interest may be determined. In previous publications on the method the authors have proven that the technique is easy and robust to apply to both small scale and large scale structures. Also, it has been shown that the technique is capable of scaling highly coupled modes by using an extended multiple reference formulation. The present paper summarizes the theory of the OMAH method and gives recommendations of how to implement the method for best results. It is pointed out, as has been shown in previous papers, that the accuracy of the mode scaling is increased by using more than one response DOF, and by selecting DOFs with high mode shape coefficients. To determine the harmonic force and responses, it is recommended to use the three-parameter sine fit method. It is shown that by using this method, the measurement time can be kept short by using high sampling frequency and bandpass filtering whereas spectrum based methods require long measurement times. This means that even for structures with low natural frequencies, the extra measurement time for scaling the mode shapes can be kept relatively short.     

Transient analysis of nonlinear Euler–Bernoulli micro-beam with thermoelastic damping,via nonlinear normal modes

In this paper an Euler–Bernoulli model has been used for vibration analysis of micro-beams with large transverse deflection. Thermoelastic damping is considered to be the dominant damping mechanism and introduced as imaginary stiffness into the equation of motion by evaluating temperature profile as a function of lateral displacement. The obtained equation of motion is analyzed in the case of pure single mode motion by two methods; nonlinear normal mode theory and the Galerkin procedure. In contrast with the Galerkin procedure, nonlinear normal mode analysis introduces a nonconventional nonlinear damping term in modal oscillator which results in strong damping in case of large amplitude vibrations. Evaluated modal oscillators are solved using harmonic balance method and tackling damping terms introduced as an imaginary stiffness is discussed. It has been shown also that nonlinear modal analysis of micro-beam with thermoelastic damping predicts parameters such as inverse quality factor, and frequency shift, to have an extrema point at certain amplitude during transient response due to the mentioned nonlinear damping term; and the effect of system?s characteristics on this critical amplitude has also been discussed.     

Defect identification in rods subject to forced vibrations using the spatial wavelet transform

Use of the wavelet transform (WT) to study the forced vibrations in a rod in order to detect the presence of a defect is proposed. The axial vibrations in an inhomogeneous rod, produced by the application of different forces, are simulated by the Network Simulation Method and the obtained response is analyzed with the WT. An analysis of the detection and location of defects for several applied forces and defects generated by locally changing density or stiffness of the rod is presented. A noise test was carried out to check the robustness of the technique in real situations.     

The effect of higher harmonic forces on fatigue life of marine risers

Higher harmonic responses measured in flexible riser model tests conducted within the Norwegian Deepwater Programme (NDP) are studied to evaluate the effect of higher harmonics of force on the fatigue life of risers. A large third-harmonic contribution and sometimes a fifth-harmonic are observed for the majority of test cases. Since existing methods of fatigue life estimation account for the effect of hydrodynamic forces only at the frequency of vortex shedding, a new methodology is introduced which takes into account the influence of the higher harmonic components: First, a new response reconstruction technique is used to reconstruct the entire span-wise riser motion in both the cross-flow and in-line directions based on measurements at specific points along the length. Then, a force database obtained from forced in-line and cross-flow motions of rigid cylinders is used to estimate the higher harmonic components of the force along the entire riser length. A significant decrease in fatigue life is obtained when the higher harmonic components of the fluid forces are considered.     

On the forced response of waveguides using the wave and finite element method

The forced response of waveguides subjected to time harmonic loading is treated. The approach starts with the wave and finite element (WFE) method where a segment of the waveguide is modeled using traditional finite element methods. The mass and stiffness matrices of the segment are used to formulate an eigenvalue problem whose solution yields the wave properties of the waveguide. The WFE formulation is used to obtain the response of the waveguide to a convected harmonic pressure (CHP). Since the Fourier transform of the response to a general excitation is a linear combination of the responses to CHPs, the response to a general excitation can be obtained via an inverse Fourier transform process. This is evaluated analytically using contour integration and the residue theorem. Hence, the approach presented herein enables the response of a waveguide to general loading to be found by: (a) modeling a segment of the waveguide using finite element methods and post-processing it to obtain the wave characteristics, (b) using Fourier transform and contour integration to obtain the wave amplitudes and (c) using the wave amplitudes to find the response at any point in the waveguide. Numerical examples are presented.     

随机参数Duffing系统中的随机混沌及其延迟反馈控制

研究了谐和激励下含有界随机参数Duffing系统(简称随机Duffing系统)中的随机混沌及其延迟反馈控制问题.借助Gegenbauer多项式逼近理论，将随机Duffing系统转化为与其等效的确定性非线性系统.这样，随机Duffing系统在谐和激励下的混沌响应及其控制问题就可借等效的确定性非线性系统来研究.分析阐明了随机混沌的主要特点，并采用Wolf算法计算等效确定性非线性系统的最大Lyapunov指数，以判别随机Duffing系统的动力学行为.数值计算表明，恰当选取不同的反馈强度和延迟时间，可分别达到抑制或诱发系统混沌的目的，说明延迟反馈技术对随机混沌控制也是十分有效的. 关键词： 随机Duffing系统 延迟反馈控制 随机混沌 Gegenbauer多项式     

Improvement of the semi-analytical method, based on Hamilton's principle and spectral analysis, for determination of the geometrically non-linear response of thin straight structures. Part III: steady state periodic forced response of rectangular plates

In Parts I and II of this series of papers, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance, has been developed for beams and fully clamped rectangular plates.¹ Simple explicit formulae have been derived, which allowed, via the so-called first formulation, direct calculation of the basic function contributions to the first three non-linear mode shapes of clamped-clamped and clamped-simply supported beams, and the two first non-linear mode shapes of FCRP. Also, in Part I of this series of papers, this approach has been successively extended, in order to determine the amplitude-dependent deflection shapes associated with the non-linear steady state periodic forced response² of clamped-clamped beams, excited by a concentrated or a distributed harmonic force in the neighbourhood of the first resonance.This new approach has been applied in the present work to obtain the NLSSPFR formulation for FCRP, SSRP, and CCCSSRP, leading in each case to a non-linear system of coupled differential equations, which may be considered as a multi-dimensional form of the well-known Duffing equation. The single-mode assumption, and the harmonic balance method, have been used for both harmonic concentrated and distributed excitation forces, leading to one-dimensional non-linear frequency response functions of the plates considered. Comparisons have been made between the curves based on these functions, and the results available in the literature, showing a reasonable agreement, for finite but relatively small vibration amplitudes. A more accurate estimation of the FCRP non-linear frequency response functions has been obtained by the extension of the improved version of the semi-analytical model developed in Part I for the NLSSPFR of beams, to the case of FCRP, leading to explicit analytical expressions for the “multi-dimensional non-linear frequency response function”, depending on the forcing level, and the amplitude of the response induced in the range considered for the excitation frequency.     

Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements

This paper employs the numerical assembly method (NAM) to determine the “exact” frequency–response amplitudes of a multiple-span beam carrying a number of various concentrated elements and subjected to a harmonic force, and the exact natural frequencies and mode shapes of the beam for the case of zero harmonic force. First, the coefficient matrices for the intermediate concentrated elements, pinned support, applied force, left-end support and right-end support of a beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact dynamic response amplitude of the forced vibrating system corresponding to each specified exciting frequency of the harmonic force is determined by solving the simultaneous equations associated with the last overall coefficient matrix. The graph of dynamic response amplitudes versus various exciting frequencies gives the frequency–response curve for any point of a multiple-span beam carrying a number of various concentrated elements. For the case of zero harmonic force, the above-mentioned simultaneous equations reduce to an eigenvalue problem so that natural frequencies and mode shapes of the beam can also be obtained.     

Green׳s function method for piezoelectric energy harvesting beams

The development of validated mathematical models for piezoelectric harvesters is important as it provides predictive capabilities of their performance and insight to their coupled electromechanical behavior. Advanced solutions to these models allows for more realistic parameters to be considered. In this paper, we present a Fourier Transform–Green?s Function (FTGF) solution approach to the distributed parameter coupled electromechanical equations for a piezoelectric beam excited by an arbitrary external transverse force. This method, as opposed to modal analysis, allows for frequency-dependent material properties and damping coefficients to be considered. The special case of a harmonic base excitation is considered and closed-form expressions for the frequency response functions of the voltage generated by piezoelectric layer, relative tip displacement and local bending strain are obtained. Finally, the FTGF solution to these frequency response functions is compared with the modal analysis solution along with experimental data for validation.     

Modified independent modal space control of m.d.o.f. systems

The performance of the independent modal space control (IMSC) algorithm for structural vibration control is examined in this paper. Both the theoretical analysis and numerical simulation show that, for a multi-degree-of-freedom system, the modal control forces may increase the contributions of the vibration of higher modes (uncontrolled modes) to the system response if the IMSC algorithm is used to design a structural control system. Therefore, the responses of the controlled structure may be underestimated if the effects of control forces on the higher modes are not considered in the response analysis. A new control algorithm—modified independent modal space control (MIMSC) algorithm is proposed in this paper for eliminating the effect of modal control force on the uncontrolled modes. Numerical example shows that the structural responses can be effectively reduced when control system design is carried out based on the proposed algorithm. By comparing the simulated results obtained by the IMSC and MIMSC algorithms, it is found that, in order to achieve the same control objective, the proposed algorithm is more effective than IMSC since the modal control forces do not have any effect on the uncontrolled modes. In order to verify the effectiveness of the proposed algorithm, a practical example—active control design of UCLA Math-Science Building is presented and discussed.     

Dynamic analysis of ring-stiffened circular cylindrical shells

A numerical method is developed for the dynamic analysis of ring-stiffened circular cylindrical thin elastic shells. Only circular symmetric vibrations of the shell segments and radial and torsional vibrations of the rings are considered. The geometric and material properties of the shell segments and the rings may vary from segment to segment. Free vibrations or forced vibrations due to harmonic pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings. Forced vibrations due to transient pressure loading are treated with the aid of dynamic stiffness influence coefficients for shell segments and rings defined in the Laplace transform domain. The time domain response is then obtained by a numerical inversion of the transformed solution. The effect of external viscous or internal viscoelastic damping is also investigated by the proposed method. In all the cases, the dynamic problem is reduced to a static-like form and the “exact” solution of the problem is numerically obtained.     

Vibration analysis of a continuous system subject to generic forms of actuation forces and sensing devices

This work provides a general formulation to solve vibration problems for continuous systems with damping effects, including modal, transient, harmonic and spectrum response analyses. In modal analysis, the system eigenvalues and corresponding eigenfunctions can be determined. The orthogonal relations of eigenfunctions are shown. For transient, harmonic and spectrum analyses, the generic force/actuator functions and response/sensing operators are introduced, respectively, and used to derive the system response. The time domain response is obtained for transient analysis, the frequency response function is derived for harmonic analysis and statistical quantities of response variables due to random excitation are determined in spectrum analysis. The solution for each type of analysis can be formulated and expressed in a concise format in terms of generic force/actuator and response/sensor mode shape functions. In particular, one-dimensional beam and two-dimensional plate vibration analyses are illustrated by following the developed generic formulation. This work provides the complete analytical solutions of four types of vibration analyses for continuous systems and can be applied to other engineering structures as well.     

Non-linear vibrations of a harmonically excited autoparametric system

Harmonic forced vibration of a spring-mass-damper system with a parametrically excited pendulum hinged to the mass is investigated. Two types of restoring forces on the pendulum are considered. The method of harmonic balance is used to evaluate the system response. The results are also verified by numerical integration. Non-periodic system responses are possible if the excitation parameter is large. The performance of the pendulum as an absorber is also studied.     

光机靶室钢混结构基于动刚度等效的简化建模

钢混结构分离式有限元建模方法由于考虑了钢筋的实际分布,其计算精度较高,但建模过程复杂繁琐。开展了基于动刚度等效的钢混结构简化建模研究,该方法将钢筋和混凝土考虑为一种等效材料。研究中首先以典型钢混组合结构为例,以等效模型与分离式模型的模态特性等效为目标,建立了分部件的动刚度等效模型,然后将各部件等效模型组合成整体结构等效模型,数值仿真结果显示在相同载荷作用下,整体参考模型与整体等效模型的响应分布非常接近,而等效模型的规模远小于参考模型,说明了该等效建模方式的有效性。进一步将该等效建模方法应用于神光Ⅲ主机靶室钢混结构的动力学分析中,并计算了该巨型光机结构等效模型在实测基础加速度激励下的随机振动响应,靶球上各方向的响应实测结果与计算结果较为吻合。所采用的动刚度等效建模方法可应用在巨型光机结构动力学稳定性的进一步研究中。     

Dynamic displacement field reconstruction through a limited set of measurements: Application to plates

The paper presents a procedure for the identification of the full-field dynamic response of a structure from a limited set of experimental measurements. An iterative technique based on modal decomposition maps the displacement field of the vibrating structure by using experimental data in conjunction with the numerical model of the considered structure. Algebraic relationships between experimental measurements and equivalent modal loads allow the identification of the full-field dynamic response from few experimental data. This procedure is detailed for a plate structure subjected to a harmonic concentrated load.     

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.     

An analysis of the distortion effects of Coulomb damping on the vector plots of lightly damped systems

An analysis of the vector plot responses of lightly damped single degree-of-freedom systems with Coulomb damping has been made. The vector plots, as derived by using both an exact and an approximate method (the method of harmonic balance) are compared and it is shown that the distortion of the normally circular vector locus is due to the Coulomb damping. Although the vector plots of such systems are distorted it is also shown that the frequency gradient criterion is still applicable for location of a natural frequency even when the frictional force levels approach the excitation force levels. To permit estimation of the modal damping of these systems a criterion by means of which the limits of the useful frequency range can be specified is suggested. The criterion, which is based upon the quadrature input power necessary to excite the mode of vibration, is found to be equivalent to that obtained from the half power point theory when applied to linear systems.     

正交加筋板声透射的板梁理论模型研究

为了研究正交加筋板的声透射问题,基于经典薄板和梁振动理论,建立了正交加筋板声透射的板梁理论模型。首先通过分析加强筋的受迫弯曲和扭转运动,求得了平板和加强筋线接触之间的反力和反力矩,然后将其引入到平板振动控制方程中,得到了正交加筋板声振方程,最后采用空间谐波展开法求解该方程得到了传声损失的表达式;在此基础上,首先研究了无限大平板和单向加筋的隔声性能,通过与解析解及两种简化模型的计算结果作对比,验证了所建理论模型的有效性;并进一步研究了加筋形式对正交加筋板隔声性能的影响。结果表明:选择合适的加筋形式可以有效避开结构的隔声波谷。