Non-ideal Boundary Conditions Research Articles

Using the Method of Base Components for a Heuristic Solution to the Diffraction Problem on a Half-Plane with Nonideal Boundary Conditions

Abstract—Using the method of base components, a heuristic solution to the diffraction problem on a semitransparent half-plane has been determined. The heuristic formulas have been verified via numerical solution. We have a semitransparency function for describing the transformation of the heuristic formulas upon variation in the transparency parameter of a scatterer.

Nonlinear vibrations of a beam with non-ideal boundary conditions and stochastic excitations - experiments, modeling and simulations

This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to stochastic broadband excitations. Broadband excitations are performed experimentally in order to show the hardening effect and the enlargement of the response peak in the vicinity of the primary resonance, and to detect the presence of secondary peaks resulting from the harmonics generated by the primary resonance.Using the decomposition of a random signal into a multi-harmonic periodic equivalent signal, the shooting method and the Harmonic Balance Method (HBM) are used to simulate the response of the nonlinear clamped-clamped beam subjected to a stochastic excitation. The latter is modeled as a multi-harmonic periodic equivalent signal in order to be used by both the shooting method and the HBM. A periodogram strategy is used to ensure a good estimate of the resulting Power Spectral Density (PSD). The two nonlinear methods are compared in terms of computation time and precision. A specific attention is paid on correct use of these methods.Finally, comparison between experiments and numerical simulations are performed for different levels of stochastic excitations. Good correlations are observed thus validating the global nonlinear proposed strategy.

Free vibration analysis of Euler-Bernoulli beams with non-ideal clamped boundary conditions by using Padé approximation

Using the non-ideal boundary condition model, which is a linear combination of ideal simply supported and ideal clamped boundary conditions, the equation governing the free vibration of Euler-Bernoulli beams produces nonlinear rational functions that relate natural frequencies with the weighting factors of the non-ideal boundary condition model. The natural frequencies in practice are numerically computed by using a standard root-finding method with suitable initial guesses. In the present study, the nonlinear rational functions are approximated using Pade approximants to get analytical formulations of natural frequencies as functions of the weighting factors. Numerical examples are provided for cantilever and beams clamped at both ends with non-ideal boundary conditions. The formulas in most cases are accurate enough to get the natural frequencies up to two-digit accuracy. Those approximations can be easily utilized as starting values in the root finding method to avoid ambiguities in selecting initial guesses.

Analytical and experimental study on deformation of thin-walled panel with non-ideal boundary conditions

Based on the Kirchhoff-Love's assumptions, shell equations with non-ideal boundary conditions are established to describe the deformation of thin-walled aircraft panel resulting from positioning variation and clamping force in the assembly process. The shell equations are simplified by the theory of functions of a complex variable, and the analytical solutions are presented by the Fourier series approach and Galerkin method. Meanwhile, the potential functions are introduced to calculate the particular solution, which successfully avoids employing specialized displacement functions with different types of loads. A coordinate transformation model is also developed to transform the actual displacement and rotation boundary constraint in a Cartesian coordinate system into the non-ideal boundary in the curvilinear coordinate system. To verify the general applicability of the proposed method, analytical calculation, the finite element (FE) simulation and experiment in real application have been performed taking into account simply supported, ideal boundary and non-ideal boundary conditions, respectively. The results have shown the good performance of the accuracy of presented solution.

Dynamic Response of a Damped Euler–Bernoulli Beam Having Elastically Restrained Boundary Supports

While doing vibration analysis in terms of the dynamic response of a beam, the end conditions, in most of the cases, are assumed to be fixed, simply supported, free or sliding imposing an ideal condition of end support displacement. But, in real engineering structures, the end supports are non-ideal (for example welded, riveted etc.) and allow certain degree of translational or rotational motion of the support ends. To simulate the effect of real engineering boundary states, a combination of linear and torsional springs has been considered at the two boundaries of the beam structure. The theoretical analysis of free undamped beam structure with elastically restrained end conditions has been undertaken using approximation of the modal displacement as a Fourier cosine series to generate natural frequencies and corresponding mode shape equations. The effect of variation in stiffness values of the boundary springs on the vibration characteristics (natural frequency, mode shapes and dynamic response) has been presented for vibration conditions for the free undamped beam structure. Thereafter, forced undamped and damped (taking combined effects of both viscous and structural damping) beam structures have been theoretically analysed to generate response equation. Finally, numerical assessment has been undertaken to present the effect of damping ratio, forcing frequency and force magnitude on the response of a nearly clamped beam and has been compared with the response of an undamped beam under similar forcing and boundary condition. The results indicate considerable effects of real boundary condition, on the vibration characteristics and dynamic response of beam structures thus cannot be ignored in design of real engineering structures. Further, the presence of damping does not change the vibration characteristics beam structures, but is useful in achieving steady-state condition of response for a harmonic force input. The technique presented provides a simplified and convenient tool for obtaining vibration response for real engineering boundary supports. Further, the beam model can be extended for analysis of plate structures considering non-ideal boundary conditions.

Source and boundary condition effects on unconfined and confined vertically distributed turbulent plumes

Plumes generated by vertically distributed sources of buoyancy have been observed to have substantially lower entrainment coefficients than their equivalent-geometry constant buoyancy flux plumes. Two differences between distributed and localized sources of buoyancy are the presence of a wall shear stress at the source and that non-ideal source conditions are distributed over the whole height of the enclosure for a vertically distributed source. Herein the impact of non-ideal source and boundary conditions on vertically distributed plumes is analysed. It is shown that, at small heights, the plume volume flow rate is significantly influenced by the wall-source volume flux. At larger heights the wall-source buoyancy is greater than the mean plume buoyancy, creating a non-self-similar horizontal buoyancy distribution within the plume. Recent experiments into the behaviour of a vertically distributed source of buoyancy in a confined region have also shown that the plume partially detrains in the stratified region of the enclosure. This detrainment has not been observed for constant buoyancy flux plumes in a confined region. Although models have been proposed to quantify the detrainment process, it is still unclear why vertically distributed buoyancy sources detrain while constant buoyancy flux plumes do not in the same physical geometry. The impact of source and boundary effects on previously published experiments on vertically distributed plumes are reviewed and the possible implications for plume entrainment and detrainment are discussed.

Non-linear vibrations of a beam with non-ideal boundary conditions and uncertainties – Modeling, numerical simulations and experiments

Abstract This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to Harmonic excitations and epistemic uncertainties. These uncertainties are propagated in order to calculate the dynamic response of the nonlinear structure via a coupling between the Harmonic Balance Method (HBM) and a non-intrusive Polynomial Chaos Expansion (PCE). The system studied is a clamped-clamped steel beam. First of all, increasing and decreasing swept sine experiments are performed in order to show the hardening effect in the vicinity of the primary resonance, and to extract the experimental multi-Harmonic frequency response of the structure. Secondly, the Harmonic Balance Method (HBM) is used alongside a continuation process to simulate the deterministic response of the nonlinear clamped-clamped beam. Good correlations were observed with the experiments, on the condition of updating the model for each excitation level. Finally, the effects of the epistemic uncertainties on the variability of the nonlinear response are investigated using a non-intrusive Polynomial Chaos Expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology based on a phase criterion was developed in order to allow the PCE analysis to be performed despite the presence of bifurcations in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations. Then, the stochastic numerical results are shown to envelope the experimental responses for each excitation level without the need for model updating, validating the nonlinear stochastic methodology as a whole.

Geometrically Nonlinear Transient Response of Laminated Plates with Flexible Supports

Laminated plates are loading-bearing components that are generally connected to flexible pads and exhibit complicated mechanical responses. To investigate the geometrically nonlinear transient responses of a laminated plate with flexible pad supports, a varied constraint reaction model and a systematic numerical procedure are presented in this paper. The flexible pad supports of the plate were treated as viscoelastic boundary conditions, wherein the strip-type pad per unit length was modeled as a cantilever beam. The nonlinear Kelvin–Voigt model was developed to simulate the nonlinear viscoelastic behaviors of the flexible pads. The dynamically varied constraint reactions generated by the viscoelastic supports, which depend upon the displacement and velocity of the nodes along the plate edge, were determined by the deflection and slope equations of the beam theory used, and they were applied on the plate edges by using the nonlinear load functions. Thus, the dynamical responses of the laminated plate with viscoelastic supports were obtained. Numerical results show that the present method can effectively treat the geometrically nonlinear transient response of the laminated plate with viscoelastic supports, and it is essential to consider the effects of non-ideal boundary conditions in the nonlinear transient analysis.

Flexural Vibration Analysis of Nonuniform Double‐Beam System with General Boundary and Coupling Conditions

In this paper, an analytical modeling approach for the flexural vibration analysis of the nonuniform double‐beam system is proposed via an improved Fourier series method, in which both types of translational and rotational springs are introduced to account for the mechanical coupling on the interface as well as boundary restraints. Energy formulation is employed for the dynamic description of the coupling system. With the aim to treat the varying thickness across the beam in a unified pattern, the relevant variables are all expanded into Fourier series. Supplementary terms with the smoothed characteristics are introduced to the standard Fourier series for the construction of displacement admissible function for each beam. In conjunction with the Rayleigh–Ritz procedure, the transverse modal characteristics of nonuniform double‐beam system can be obtained by solving a standard eigenvalue problem. Instead of solving the certain value of nonideal boundary conditions, the continuous spring stiffnesses of the boundary conditions are considered, and the rotational restrains are introduced in the coupling beam interface. Numerical results are then presented to demonstrate the reliability of the current model and study the influence of various parameters, such as taper ratio, boundary, and coupling strength on the free vibration characteristics, with the emphasis put on the rotational restraining coefficients on the beam interface. This work can provide an efficient modeling framework for the vibration characteristics study of the complex double‐beam system, especially with arbitrary varying thickness and coupling stiffness.

Dynamic measurement of the thermal conductivity of phase change materials in the liquid phase near the melting point

This paper deals with the determination of the thermal conductivity of liquid-phase phase change materials (PCMs) near the melting point by studying the evolution process of the solid-liquid interface. A theoretical model was proposed which enables the thermal conductivity to be determined from the displacement characteristics of interface during a melting process. Experiments were performed using a well-designed testing system for two kinds of inorganic PCMs, and the experimental results agreed with the literature values within a range of ±6%. The sources of measurement error were studied from the temporal variation characteristics of the experimental results. It is found that the non-ideal boundary conditions of the testing system and the onset of the natural convection were the major causes of the measurement error. To improve the accuracy of the experimental results, numerical techniques were utilized to simulate the melting process of the experiments. The calibrated values of the thermal conductivity were iteratively obtained from the comparisons of interface movement curves between the experiments and simulations, and the discrepancies between the calibrated results and the reference values were limited to ±3%.

Using one-dimensional waveguide resonators to measure phase velocities in bubbly liquids.

Resonator techniques can be successfully used to extract effective medium properties from dispersive materials. However, in some cases the dispersion can cause modes to repeat. If repeated modes are not taken into account, the useful range of the resonator technique is limited. A resonance tube containing tethered balloons is used to create a dispersive effective medium. Resonator measurements show that modes do repeat. Direct measurement of the mode shapes allows exploitation of all longitudinal radially symmetric modes and expands the frequency range of the technique. A theoretical model is also used to predict when modes repeat. For the presented data set this method increases the measurement range from below 160 Hz to 3000 Hz excluding the stop band where resonances are damped. A means to account for non-ideal resonator boundary conditions often found in highly dispersive systems is discussed.

Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions

In this paper, free transverse vibration of two parallel beams connected through Winkler type elastic layer is investigated. Euler-Bernoulli beam hypothesis has been applied and it is assumed that boundary conditions of upper and lower beams are similar while arbitrary without any limitation even for non-ideal boundary conditions. Material properties and cross-section geometry of beams could be different from each other. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli-Fourier method. Explicit expressions are derived for the natural frequencies. In order to verify accuracy of results, the problem once again solved using modified Adomian decomposition method. Comparison between results indicates excellent accuracy of proposed formulation for any arbitrary boundary conditions. Derived explicit formulation is simplest method to determine natural frequencies of double-beam systems with high level of accuracy in comparison with other methods in literature.

Vibrations of fluid conveying microbeams under non-ideal boundary conditions

This study aimed to present the effects of non-ideal boundary conditions (BCs) on fundamental parametric resonance behavior of fluid conveying clamped microbeams. Non-ideal BCs are modelled by using the weighting factor (k). Equations of motion are obtained by using the Hamilton’s Principle. A perturbation technique, method of multiple scales, is applied to solve the non-linear equations of motions. In this study, frequency-response curves of fundamental parametric resonance are plotted and the effects of non-ideal BCs are shown. Besides, instability areas of microbeams under ideal and non-ideal BCs are investigated by considering different system parameters. Numerical results show that instability areas significantly changed by the effect of non-ideal BCs.

Free vibrations of fluid conveying microbeams under non-ideal boundary conditions

In this study, vibration analysis of fluid conveying microbeams under non-ideal boundary conditions (BCs) is performed. The objective of the present paper is to describe the effects of non-ideal BCs on linear vibrations of fluid conveying microbeams. Non-ideal BCs are modeled as a linear combination of ideal clamped and ideal simply supported boundary conditions by using the weighting factor (k). Non-ideal clamped and non-ideal simply supported beams are both considered to show the effects of BCs. Equations of motion of the beam under the effect of moving fluid are obtained by using Hamilton principle. Method of multiple scales which is one of the perturbation techniques is applied to the governing linear equation of motion. Approximate solutions of the linear equation are obtained and the effects of system parameters and non-ideal BCs on natural frequencies are presented. Results indicate that, natural frequencies of fluid conveying microbeam changed significantly by varying the weighting factor k. This change is more remarkable for clamped microbeams rather than simply supported ones.

Experiments and numerical simulations of nonlinear vibrations of a beam with non-ideal boundary conditions and uncertainties

This paper presents experiments and numerical simulations of a nonlinear clamped-clamped beam subjected to external excitations and uncertainties. A coupling of the Harmonic Balance Method (HBM) with a non-intrusive Polynomial Chaos Expansion (PCE) is proposed in order to determine the dynamic response of the nonlinear structure. The system studied is a steel beam bonded on both sides to a heavy steel block. First of all, increasing and decreasing swept sine experiments are performed in order to show the presence of jumping points in the vicinity of the primary resonance. From the swept-sine experiments, we use a signal processing tool to extract the experimental multi-harmonic frequency response of the structure. Secondly, a numerical method, called the Harmonic Balance Method (HBM), is used to simulate the deterministic vibrational stationary response of the nonlinear clamped-clamped beam. The input of the HBM is either a mono-harmonic signal to update the nonlinearities of the clamped-clamped beam or a multi-harmonic experimental signal to show the effects of the non-ideal input signal on the response. Finally, the quantification of uncertainty effects on the variability of the nonlinear multi-harmonic response is investigated by using a non-intrusive polynomial chaos expansion (PCE) alongside the Harmonic Balance Method (HBM). A new methodology was developed to split the simulated solutions into various intervals (i.e. various branches of continuous nonlinear solutions) in order to allow the PCE analysis to be performed despite the presence of returning points in the nonlinear response. The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo simulations.

Thermal effects on nonlocal vibrational characteristics of nanobeams with non-ideal boundary conditions

In this manuscript, the small scale and thermal effects on vibration behavior of preloaded nanobeams with non-ideal boundary conditions are investigated. The boundary conditions are assumed to allow small deflections and moments and the concept of non-ideal boundary conditions is applied to the nonlocal beam problem. Governing equations are derived through Hamilton\' s principle and then are solved applying Lindstedt-Poincare technique to derive fundamental natural frequencies. The good agreement between the results of this research and those available in literature validated the presented approach. The influence of various parameters including nonlocal parameter, thermal effect, perturbation parameter, aspect ratio and pre-stress load on free vibration behavior of the nanobeams are discussed in details.

A two-dimensional vibration analysis of piezoelectrically actuated microbeam with nonideal boundary conditions

In this paper, the influences of nonideal boundary conditions (due to flexibility) on the primary resonant behavior of a piezoelectrically actuated microbeam have been studied, for the first time. The structure has been assumed to treat as an Euler–Bernoulli beam, considering the effects of geometric nonlinearity. In this work, the general nonideal supports have been modeled as a the combination of horizontal, vertical and rotational springs, simultaneously. Allocating particular values to the stiffness of these springs provides the mathematical models for the majority of boundary conditions. This consideration leads to use a two-dimensional analysis of the multiple scales method instead of previous works' method (one-dimensional analysis). If one neglects the nonideal effects, then this paper would be an effort to solve the two-dimensional equations of motion without a need of a combination of these equations using the shortening or stretching effect. Letting the nonideal effects equal to zero and comparing their results with the results of previous approaches have been demonstrated the accuracy of the two-dimensional solutions. The results have been identified the unique effects of constraining and stiffening of boundaries in horizontal, vertical and rotational directions. This means that it is inaccurate to suppose the nonideality of supports only in one or two of these directions like as previous works. The findings are of vital importance as a better prediction of the frequency response for the nonideal supports. Furthermore, the main findings of this effort can help to choose appropriate boundary conditions for desired systems.

Excitation and dynamics of liquid tin micrometer droplet generation

The dynamics of capillary breakup-based droplet generation are studied for an excitation system based on a tunable piezoelectrically actuated oscillating piston, which generates acoustic pressure waves at the dispenser nozzle. First, the non-ideal pressure boundary conditions of droplet breakup are measured using a fast response pressure probe. A structural analysis shows that the axial modes of the excitation system are the main reasons for the resonance peaks in the pressure response. Second, a correlation between the nozzle inlet pressure and the droplet timing jitter is established with the help of experiments and a droplet formation model. With decreasing wave number, the growth rate of the main excitation decreases, while noise contributions with wave numbers with higher growth rates lead to a non-deterministic structure of the droplet train. A highly coherent and monodisperse droplet stream is obtained when the excitation system is tuned to generate high acoustic pressures at the desired operation frequency and when the noise level on the jet is limited. The jet velocity, hence droplet spacing for a set frequency is then adjusted by varying the reservoir pressure, according to the trade-off between lowest wave number and acceptable timing jitter.

Effect of the equivalent stiffness of flexible supports on the MEMS cantilever-based sensors

Small deflections and moments exist on the boundaries of the flexible supports of MEMS cantilever-based sensors. The qualitative dynamical behavior is significantly affected by the non-ideal boundary conditions. Based on the Boussinesq’s and Cerruti’s displacement equations and the principle of energy equivalence, rigorous theoretical solutions of the normal, tangential and rotational equivalent stiffness are presented in this paper. The proposed model is validated by the fact that the theoretical results remained a good situation consistency with the reported experimental data. The variations of the equivalent stiffness with the geometric dimensions of the cross section of the flexible supports are investigated. It is observed that when the “Hard material” is used as the supports’ materials, e.g. Silicon carbide, the equivalent stiffness is large. Yet the equivalent stiffness is small for the “Soft material”, e.g. Silicon and Poly-silicon. In addition, by employing the method of multiple time scales, the non-dimensional differential partial equation of the motion as well as coupled boundary conditions are analytically solved. The results indicate that the resonance frequencies decrease with the flexible supports, however, increase due to the nonlinearity mechanical spring.

Evolution of the ultrasonic resonance modes in a three-layer structure with change of material and interface adhesion properties

The quantitative non-destructive evaluation (NDE) of interface adhesion has long been a challenge for the safe use of bonding structures. It is difficult to predict the adhesion resistance force between adhesive and adhered material without performing destructive testing. Ultrasonic approach seems to be the only potential way for its NDE based on the reason of mechanical nature of the problem. Different ultrasonic techniques, such as bulk wave echography, reflection resonance, and Lamb guided waves, have been used to evaluate the interface adhesion strength. But no direct relation between the interfacial bonding strength and the ultrasonic measurement has been established. The most used compression wave echography and resonance at normal incidence are less sensitive to the interface condition, except for a disbond. It is essential that the interface should be excited with a shear stress component to increase the measurement sensibility. But it is not easy to excite the interface by using shear waves in experiment, while the use of guided waves will encounter the problems of high attenuation and mode selection as all modes are not sensitive to a certain interface in a bonded structure. A previous study has shown that the V (z) inversion technique can be used to perform a multimode measurement on a layered structure, where both compression and shear stress resonance occur. This method has the advantage in using a simple experimental setup working at the normal incidence with a focus transducer of large angular aperture. The inversed angular-frequency reflectance function R(; f) gives the resonance modes which are equivalent to the Lamb type guided modes, while it is a local determination of the wave mode, thus the difficulty in guided wave measurement above mentioned can be avoided. The first part of the paper contains the development of the theoretical model for wave propagation in a multilayered structure where three-layer sandwich bonded structures can be considered as a particular case. A weak interfacial adhesion is described by two interface compression and shear stiffness parameters, namely km and kt. By integrating the transfer matrix formalism under the non-ideal boundary conditions, the plane wave angular (incident angle) and frequency reflection coefficient function R(; f) for a liquid immersed asymmetric metal-adhesive-metal three-layer and its dispersion curves of guided mode waves with or without charge are calculated. It is confirmed that the evolutions of the reflection zeros (mode resonances) correspond to the dispersion curves of the guided waves of the same structure without charge. Furthermore, the resonance modes observed in R(; f) can be considered as a combination of the respective Lamb modes of the top and bottom single metal layers coupled through the modes conditioned by the middle adhesive layer and the its interface conditions. The second part of the paper shows the behaviors of the resonance modes by changing the parameters related to the bonding strength. The acoustical impedance, the mass density and the thickness of the adhesive layer, which are related to the cohesive property, and the shear interfacial stiffness coefficient kt which conditions the adhesive property, are changed respectively to observe the resonance mode evolutions. The mode evolutions due to each parameter are analyzed and differentiated. It can be concluded that the change in the adhesion strength of the bonding structure does not affect significantly the modes belonging to those inherent to the two adhered aluminum layers, while the coupling modes will be shifted in frequency and exchange with or replace the said inherent modes. It is expected that the obtained results in this study will be of significance for quantitatively characterizing the interfacial properties of an adhesively bonded layered structure by using the V (z) inversion technique.