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The behaviour of mechanical structures in low frequencies is strongly affected by the existence of the boundary conditions. It is not usually possible to provide ideal boundary conditions, i.e. simply supported or clamped, for structures. Therefore the real structures are mostly constrained by elastic supports. Constructing an accurate mathematical...
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... supports, or boundary conditions, play an important role in a structure’s dynamic behaviour and must be considered carefully when constructing mathematical or numerical models. In reality, the supports of structures are not rigid enough, and they show flexibility to some degree. The flexibility of the supports can be modelled as elastic boundary conditions. In order to have an accurate model of a structure, the knowledge of the support parameters is essential. The support parameters can be identified by using experimental results. The sensitivity method is one of the most widely used approaches in determining boundary condition parameters. 1 In this method the difference between model predictions and test observations is defined as an objective function. An iterative process is then adopted, and the objective function is mini- mized by using the sensitivity approach. It should be noted that the sensitivity of higher natural frequencies to support parameters is low, which results in convergence problems in the optimization procedure. 2 In the characteristic equation method the boundary support parameters are identified by solving the nonlinear characteristic equations. In this method, which was adopted by Ahmadian et al., the number of characteristic equations formed is equal to the number of measured natural frequencies. The boundary condition parameters are then identified by simultaneously solving the characteristic equations. 3 Waters et al. and Wang and Yang adopted the static flexibility measurements and identified the boundary conditions of a tapered beam. 4, 5 They modelled the beam as a uniform rigid beam that was constrained by collocated equivalent trans- lational and rotational springs. The boundary conditions are identified by quasi-static stiffness measurements obtained from impact tests. This paper deals with the support parameter identification of a rectangular plate constrained in its edges by an elastic boundary condition. The boundary condition contains structural damping. The solution method proposed by Li et al. is adopted to analyse the free vibration of the beam. 6 The analysis leads to obtaining the natural frequencies and damping ratios of the plate. An identification approach is proposed based on the solution presented by Li et al. and by using the measured modal properties (i.e. natural frequencies and damping ratios). 6 The proposed method is verified by using simulated and experimental results. The next section considers the free vibration analysis of an elastically supported plate. Figure 1 shows an elastically supported rectangular plate, which is constrained by lateral and torsional springs. It is considered that the elastic boundary condition contains structural damping. The governing differential equation for the free vibration of the rectangular plate is expressed in Eq. ...
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Citations
... For the study of boundary stiffness identification, the boundary conditions are usually identified based on the dynamic behaviors of flexible structures. Ahmadian [8,9] proposed an approach to determine the boundary parameters of elastically supported plates by solving the reduced order characteristic equation based on natural frequency. Similar work was also performed by Mao [10], who verified the effectiveness of the method through the meshing models with different boundary conditions. ...
Boundary supporting stiffness is an important structural parameter for status monitoring and defect detection. In this paper, a novel Fourier series-Neural network hybrid approach is proposed to identify the boundary-supporting stiffness of a flexible beam. The transverse displacements of the flexible beam under arbitrary boundary conditions are described using the admissible function constructed by the Fourier series with supplementary terms. The modal characteristics corresponding to the different boundary-supporting stiffness are solved by the energy principle and Rayleigh-Ritz procedures. The effect of the boundary supporting stiffness on the natural frequencies is analyzed. On this basis, the PSO-BP (particle swarm optimization backpropagation) neural network is established to describe the relationship between boundary supporting stiffness and the natural frequencies. The results show that the neural network built by the proposed method has a good effect on stiffness identification and performs better than the common network. By further comparing the results output from the proposed model with those obtained by the finite element method and the experiment, the effectiveness and feasibility of the proposed model in practical application are demonstrated.
... In the 2000's, Li and Yu [13] proposed a simple formula for natural frequencies of plates with uniformly restrained edges, and Li and others [14] presented a series solution for rectangular plates with general elastic boundary supports. Eftekhari and Jafari [15] used a variational approach for vibration of variable thickness plates with elastic edges, and Ahmadian and Esfandiar [16] attempted to identify the elastic boundary condition. ...
... Using [C] which is obtained by substituting Eq. (16) into the four sets of node coordinates, W is transformed as ...
This paper aims to present comprehensive lists of accurate natural frequencies of isotropic thin free rectangular plates constrained only by translational springs distributed uniformly on the edges. Analytical and numerical approaches are employed to study the free vibration of the plates. The first approach is an extension of Ritz method, and second approach is the finite element method coded by the author. Numerical examples cover rectangular plates from totally free plate to totally simply supported on all edges. Convergence and comparison studies are made to establish accuracy of these solutions. Nine numerical examples are presented in terms of different of elastic constraints, and the lowest six frequency parameters are provided in the examples with different aspect ratios.
... Ahmadian et al. 4 proposed a boundary condition identification method according to solving reduced-order characteristic equations, and verified the method using a plate with elastic supports. Utilizing the optimization toolbox in MATLAB, Ahmadian et al. 5 proposed another boundary condition identification method based on the residuals between measured and theoretical modal parameters (i.e., natural frequencies and damping ratios), and verified the method by boundary identifications of a simulative square plate supported by springs and an experimental steel plate supported by rubber seal. Two identification methods of boundary conditions were investigated by Pabst and Hagedorn 6 , i.e., a direct identification method based on the characteristic equation using natural frequencies, and an iterative method based on sensitivity analysis of natural frequencies and modal shapes, and then the two methods were validated by a cantilever beam and a rectangular plate with torsional stiffness, respectively. ...
... The results in Figs. [5][6][7] show that the identifiable dimensionless parameters listed in Table 1 can be identified by using the first four natural frequencies. Moreover, the noise level will affect the identification results, i.e., the higher the noise level is, the greater the identified error of the boundary parameters is. ...
... reason for choosing a smaller identifiable variable is: (A) natural frequencies can be used to identify the boundary condition when its sensitivity is large enough; (B) the method proposed in this paper is applicable when the initial model is reasonably close to the final identified model. (5) The proposed method can be used to identify specific boundary conditions. For different boundary conditions, all the boundary parameters can be identified for BC I; only some boundary parameters can be identified for BC II and BC III. ...
The actual boundary conditions of cantilever-like structures might be non-ideally clamped in engineering practice, and they can also vary with time due to damage or aging. Precise modelling of boundary conditions, in which both the boundary stiffness and the boundary mass should be modelled correctly, might be one of the most significant aspects in dynamic analysis and testing for such structures. However, only the boundary stiffness was considered in the most existing methods. In this paper, a boundary condition modelling and identification method for cantilever-like structures is proposed to precisely model both the boundary stiffness and the boundary mass using sensitivity analysis of natural frequencies. The boundary conditions of a cantileverlike structure can be parameterized by constant mass, constant rotational inertia, constant translational stiffness, and constant rotational stiffness. The relationship between natural frequencies and boundary parameters is deduced according to the vibration equation for the lateral vibration of a non-uniform beam. Then, an iterative identification formulation is established using the sensitivity analysis of natural frequencies with respect to the boundary parameters. The regularization technique is also used to solve the potential ill-posed problem in the identification procedure. Numerical simulations and experiments are performed to validate the feasibility and accuracy of the proposed method. Results show that the proposed method can be utilized to precisely model the boundary parameters of a cantilever-like structure.
... By minimizing the differences between experimental and predicted modal properties (i.e. natural frequencies and damping ratios), Ahmadian et al. [14] proposed another boundary condition identification method for plate with elastic support using optimization toolbox in MATLAB, and verified the method by boundary identification of simulative square plate supported by springs and experimental steel plate supported by rubber seal. Similarly, Mao et al. [15] proposed a boundary condition identification method based on an iterative model updating strategy by minimizing the differences between experimental and predicted modal properties (i.e. ...
This paper investigates a boundary condition identification method for tapered beam with the specific flexible boundaries using static flexibility measurements. The specific flexible boundaries are modeled by two translational springs with a particular interval which are connected at one end of the tapered beam, and the purpose of this paper is just to identify the stiffnesses of the two translational springs. According to the static equilibrium equation, it is proved that the static flexibility of the beam is a function of the flexural rigidity of the beam at its constrained end and the stiffnesses of the two translational springs. Then, using three different static flexibility measurements, a set of linear equations are established to identify the stiffnesses of the two translational springs. Finally, the feasibility and effectiveness of the proposed method are demonstrated using both simulative and experimental examples.
The purpose of this paper is to investigate the effect of boundary flexibility on the performance of piezoelectric vibration energy harvester (PVEH) beam systems, which has not been studied comprehensively in the literature despite its importance. The coupled electromechanical equations of motion of a piezoelectric cantilever beam with a tip mass are established, with the base boundary constrained by translational and rotational springs. An exact closed-form solution of the frequency response function (FRF) of the PVEH is obtained by the distributed transfer function method (DTFM). The DTFM is a systematic powerful tool for the dynamic analysis of distributed parameter continua with non-classical boundary conditions, intermediate constraints, coupled fields, and non-proportional damping without adding much complexity to the solution formulation. Moreover, the DTFM computes the derivatives of the response, that is, the strains, which are required in the electromechanical coupling formulation, simultaneously without any differentiation. Numerical results showing the effects of boundary flexibility on energy harvesting efficiency are presented. A first-order rational function relating the boundary stiffness parameters and the harvesting efficiency is determined by nonlinear curve fitting of the calculated data. Physical insights and applicability of this analytical function for end-of-line quality check of the boundary of PVEH are discussed.
This paper deals with the vibration modeling of a glass panel, with emphasis on smartphone applications, and attempts to identify the elastic boundary condition to support the glass panel. This study is motivated to develop a more-efficient vibration function of the glass panel in smartphones. First, based on the classical plate theory, the vibration behavior is analyzed with consideration of the elastic translational and rotational springs distributed along the panel edge. The accuracy of applying the energy method (Ritz method) is verified by comparing the analytical results with the experimental results by a modal analysis technique, and it is shown that the plate theory used for crystalline materials (i.e. metals and plastics) is applicable also to the glass (non-crystalline) plates. Second, the elastic constants of such distributed springs are identified by a combination of the genetic algorithm and the present Ritz method, and the design approach is summarized for developing more effective use of the vibrating function on glass panels in the smartphones and other tablet terminals.
This article proposes a novel damage detection method based on the sensitivity analysis and chaotic moth-flame-invasive weed optimization (CMF-IWO), which is utilized to simultaneously identify the damage of structural elements and bearings. First, the sensitivity coefficients of eigenvalues to the damage factors of structural elements and bearings are deduced, the regularization technology is used to solve the problem of equation undetermined, meanwhile, the modal strain energy-based index is utilized to detect the damage locations, and the regularization objective function is constructed to quantify the damage severity. Then, for the subsequent procedure of damage detection, CMF-IWO is proposed based on moth-flame optimization and invasive weed optimization as well as chaos theory, reverse learning, and evolutional strategy. The optimization effectiveness of the hybrid algorithm is verified by five benchmark functions and a damage identification numerical example of a simply supported beam; the results demonstrate it is of great global search ability and higher convergence efficiency. After that, a numerical example of an 8-span continuous beam and an experimental reinforced concrete plate are both adopted to evaluate the proposed damage identification method. The results of the numerical example indicate that the proposed method can locate and quantify the damage of structural elements and bearings with high accuracy. Furthermore, the outcomes of the experimental example show that despite the existence of some errors and uncertain factors, the method still obtains an acceptable result. Generally speaking, the proposed method is proved that it is of good feasibility.
Structural strain responses are extensively applied in the field of structural health monitoring (SHM); however, previous studies have focused on the damage identification methods based on strain responses. This paper focuses on measurement technology and proposes a new strain measurement method. Four strain gauges are set in a cross direction on the surface of the structure. Then, the Wheatstone full-bridge connection is used to measure the combined strain of the four strain gauges. The peak picking method is applied to estimate the strain modal shape and frequency of the structure under ambient excitation. The strain mode difference is used as a damage identification index. Finally, clamped plate experiments, including single-damage, multi-damage and damage distance experiments, are successfully implemented. Compared with traditional half-bridge strain measurements, the results show that the cross-direction strain measurement method is more sensitive to minor damage and has higher identification accuracy.
