Modal Superposition Technique Research Articles

N-sided polygonal cell-based smoothed finite element method (nCS-FEM) based on Wachspress shape function for modal analysis

In this article, an n-sided polygonal cell-based smoothed finite element (nCS-FEM) based on the Wachspress shape function is formulated for free and forced vibration of solid structures. Cell-based smoothing domains are constructed in n-sided polygonal element mesh. Using the gradient smoothing technique, the computation of the strain-displacement matrix requires only the value of the Wachspress shape function, not the derivatives of the shape function or the mapping relationship. The Lanczos algorithm is used to solve eigenvalue problems that produces vibration modes of a given structure, and the vibration modes and modal superposition techniques are used to obtain transient dynamic responses of structures subjected to arbitrary dynamics forces. The nCS-FEM is applied to solve a number of solid structures for modal analysis and compared with the Polygonal FEM, nES-FEM and ABAQUS, it is found that nCS-FEM has high precision, high efficiency and super convergence.

Nonlinear vibration characteristics of composite pyramidal truss sandwich cylindrical shell panels with amplitude dependence

Recently studies on the bending, compression, buckling, and other static issues of composite pyramidal truss sandwich cylindrical shell (CPTSCS) panels have been reported. However, rare literature has explored the nonlinear vibration characteristics of the CPTSCS panels. In this paper, their amplitude-dependent vibration properties are studied both experimentally and theoretically. First, a fabrication flowchart is proposed to obtain such panel specimens, and a pulse excitation method via a hammer and sweeping frequency and fixed excitation techniques provided by a base vibration shaker are employed to measure the nonlinear variation of vibration parameters. As a result, the amplitude-dependent vibration phenomenon is confirmed to exist in both the pyramidal truss core and composite skins. After the nonlinear assumptions of Young's moduli, shear moduli, and loss factors of the pyramidal truss core and composite skins are defined based on Jones-Nelson nonlinear material theory, complex modulus method, Gibson equivalent principle, etc., a novel theoretical model is developed to predict the nonlinear natural frequencies, damping ratios, and resonant responses under different base excitation energies, in which the first order shear deformation theory, Rayleigh Ritz variation method, modal strain energy approach, and modal superposition technique are adopted to solve key vibration parameters. The pre-defined amplitude-dependent fitting coefficients that are vital to the completeness of the current model are further identified using the measured frequency and damping data. Also, detailed tests are done to make sure that both the proposed model and the predictions are correct. Finally, the effects of critical geometric and material parameters of the CPTSCS panels on the amplitude-dependent vibration characteristics are evaluated based on the calculated dynamic results, with some practical suggestions being refined for improving the vibration suppression capability of such advanced composite structures.

Free and forced vibrations of composite cylindrical–cylindrical​ shells with partial bolt loosening connections: Theoretical and experimental investigation

In the present study, the free and forced vibrations of composite cylindrical–cylindrical shells with partial bolt loosening connections are investigated both experimentally and theoretically. Initially, a theoretical model is proposed based on the assumption of primary and secondary connection springs with reduced stiffnesses related to different loosening positions of connection bolts. The equation of motion is derived for predicting the fundamental frequencies, mode shapes, and vibration displacement responses on the basis of the Rayleigh–Ritz method, the Jacobian polynomial approach, the modal superposition technique, etc. To ensure the calculation accuracy of the present model, a convergence study is conducted to determine the truncation number. Also, to give a solid validation of the model, some detailed tests, such as the pulse, sweeping-frequency, and resonant excitations, are conducted on a combined shell specimen, which is obtained by assembling two different composite cylindrical shells via twelve M8 bolts, with a torque wrench being adopted to control the torques in the connection bolts to achieve the desired partial bolt loosening effect. By comparing the calculated and tested results, it can be found that the present model is reliable to predict the structural dynamic parameters because the calculation errors of natural frequencies and resonant displacements are less than 7.4 and 11.9 %, respectively. Finally, the impact of critical bolt loosening parameters on vibration characteristics of the combined shells is evaluated, with some important conclusions being highlighted to illustrate the reduction mechanism of dynamic properties as the partial bolt loosening degree and number increase or the loosening pattern becomes more complex.

Response of laminated glass elements subject to dynamic loadings using a monolithic model and a stress effective Young’s modulus

Due to the brittle nature of the glass, to study the response of glass elements subjected to dynamics loadings is of great interest, mainly when human injury resulting from glass breakage may occur. Laminated glass consists of two or more layers of monolithic glass and one or more viscoelastic interlayers. In this paper, a technique to predict the dynamic response (deflections and stresses) of laminated glass elements using a linear elastic monolithic model is proposed and validated. In a first step, the modal parameters and the dynamic response of a monolithic model are calculated using the modal superposition technique. Then, the modal parameters and the modal coordinates of the laminated glass element are estimated using the equations derived in this paper. Finally, the stresses in time and frequency domain are estimated using an effective stress Young’s modulus.

Large amplitude forced vibrations of Timoshenko beams using coupled displacement field method

Forced vibration behaviour of a Timoshenko beam, with a concentrated mass at the centre, subjected to a concentrated harmonic force is investigated using the coupled displacement field method. The expressions for the transverse displacement were deduced from the assumed admissible function having a single term undetermined coefficient for a total rotation of the beam using the coupled displacement field method. The equation of motion for the forced vibration condition is derived using the principle of conservation of total energy. The Newmark method is employed to solve the equation of motion to obtain the dynamic response of the Timoshenko beam. The dynamic forced vibration response for the two boundary conditions, simply supported and clamped-clamped, is computed by applying the mode superposition technique for the obtained coupled displacement field equations. The authors also carried out the finite element analysis using ANSYS 21 software to validate the results obtained by the coupled displacement field method, and the results are also validated with the existing literature wherever possible. The obtained results are very closely matching with the FEM and other published results.

The Impact of the Cracks on the Harmonic Response of Stiffened Steel Plates

Stiffened plates are employed in various engineering structures including aerospace, civil, naval, etc. These structures may be subjected to harmonic loads. Therefore, it is essential to understand the harmonic response of a cracked stiffened structure. This study presents the effects of the crack location on the free vibration and harmonic response of the stiffened plates. For this purpose, horizontal and vertical cracks are modelled throughout the fixed from two-side structure. The analyses are performed via the Finite Element Method and the Mode Superposition technique. The natural frequencies and the displacement responses of the cracked and healthy structures are obtained and interpreted. It is concluded that a horizontal crack can decrease the third natural frequency by 60% as it is located in the center of the stiffened plate that is fixed from its non-stiffened edges. Thus, the displacement response of the stiffened structures is increased by 18% in the presence of the side horizontal or vertical cracks, which are located at the free ends of the stiffened structures no matter which boundary condition is considered.

Active vibration suppression of wind turbine blades integrated with piezoelectric sensors

Abstract As the wind turbine size gets larger, the optimal design of blades, which is a major source of energy for the wind turbines and also the cause of loads, is becoming more important than anything else. Therefore, reducing the load on the blade should be the top priority in designing a blade. In this article, we studied the vibration control of the stiffened wind blades subjected to a wind load with piezoelectric sensors and actuators to mitigate fluctuations in loading and adding damping to the blade. The model is a laminated composite blade with a shear web and the PZT piezomaterial layers embedded on the top and bottom surfaces act as a sensor and actuator, respectively. A uniformly distributed external wind load is assumed over the entire plate surface for simplicity. The first-order shear deformation (FSDT) theory is adopted, and Hamilton’s principle is used to derive the finite element equation of motion. The modal superposition technique and the Newmark- β \beta method are used in numerical analysis to calculate the dynamic response. Using the constant gain negative velocity feedback control algorithm, vibration characteristics and transient responses are compared. Furthermore, vibration control at various locations of the shear webs subjected to an external load is discussed in detail. Through various calculation results performed in this study, this article proposes a method of designing a blade that can reduce the load by actively responding to the external load acting on the wind turbine blade.

Assessment of wave induced higher order resonant vibrations of ships at forward speed

An efficient nonlinear time domain method computed higher order springing induced vertical bending vibrations of ships in waves. A weakly nonlinear time domain approach obtained the hydrodynamic response of the elastic hull girder, and a linear finite element model based on Timoshenko’s beam theory calculated its structural response. Coupling of the fully nonlinear stationary forward speed problem with the weakly nonlinear elastic body seakeeping problem constituted the major progress. We demonstrated that accounting for the nonlinear stationary forward speed problem significantly affected the prediction of springing-induced vibrations. Rigid body motions were computed via a nonlinear equations of motion. Elastic vibrations were computed using the modal superposition technique based on linear elastic motion equations. Radiation forces of the moving and vibrating hull structure (rigid body and elastic) were computed via convolution integrals, and Froude–Krylov and hydrostatic forces were combined and integrated over the instantaneous wetted surface. A waterline integral accounted for nonlinear effects of radiation and diffraction forces due to the changing wetted surface. A two-way coupling algorithm ensured accurate convergence. Comparisons of the numerically calculated midship vertical bending moment of a large container ship with experimental results showed good agreement. Investigations of the subject container ship advancing at forward speeds of 15 and 22kn in regular head waves focused on second, third, and fourth order springing.

A fast mode superposition algorithm and its application to the analysis of bridges under moving loads

Modal superposition (MS) techniques are widely used in linear dynamic analyses but the study of large structures subject to actions such as long vehicle convoys is not currently feasible. This work presents an algorithm called MS5 embedded in the Python library MDyn that combines an efficient indexing strategy to deactivate specific structural nodes and movements with a novel modal truncation based on a new dynamic participation factor and the vectorisation of the MS algorithm. The MS5 is compared with the conventional MS method, the analytical solution and the commercial finite element software ABAQUS in the dynamic analysis of a short-span beam bridge and of a large cable-stayed bridge under different load scenarios. The results obtained with MS5 are almost identical to other methods, but it is on average 9 times faster than the standard MS method. The proposed algorithm is applied to the study of critical traffic actions in both structures, observing important dynamic amplification effects for certain convoy arrangements that are able to trigger resonant responses in the deck. MS5 is also applicable to other line-like structures such as towers, masts, etc.

Simulation of vertical dynamic vehicle–track interaction using a three-dimensional slab track model

When improving track design, a better understanding of the track’s damage modes is needed, and the railway industry is then dependent on the availability of accurate simulations of the dynamic vehicle–track interaction. In the present study, the vertical dynamic interaction between a travelling railway vehicle and a slab track is simulated in the time domain by using an extended state-space vector approach. A three-dimensional slab track model is launched where the rails are modelled using Rayleigh–Timoshenko beam elements and the concrete panel and roadbed are represented by using either shell or solid finite elements. From the parameterised track model, which is developed in Abaqus using Python scripts, the system matrices are exported to Matlab where the simulation of the dynamic vehicle–track interaction is performed. A complex-valued modal superposition technique is employed, which reduces the computational cost of the simulation. In a post-processing step, calculated wheel–rail contact forces from the dynamic analysis are used as input to the Abaqus track model where various track responses are evaluated. In particular, the time history of principal stresses is determined at critical locations in the concrete panel. Also the influence of the speed of the vehicle on the wheel–rail contact forces, and the influence of a transverse culvert below the track (modelled as a local increase of the foundation bedding modulus) on the track stiffness variation at the rail level, are investigated. A mesh convergence study for a range of track responses has been conducted including investigations of when to use linear or quadratic elements and shell or solid elements. Finally, the presented three-dimensional models have been compared to an alternative two-dimensional model to determine in what situations a two-dimensional model is sufficient.

A Novel Approach for Evaluating Nonstationary Response of Dynamic Systems to Stochastic Excitation

The transient state of a dynamic system, such as offshore structures, to random excitation is always nonstationary. Many studies have contributed to evaluating response covariances at the transient state of a linear multi-degree-of-freedom (MDOF) system to random excitations, but a closed-form solution was not available unless the excitation was assumed to be a physically unrealizable white noise process. This study derives explicit, closed-form solutions for the response covariances at the transient state by using a pole-residue (PR) approach operated in the Laplace domain when the excitations are assumed to be stationary random processes described by physically realizable spectral density functions. By using the PR method, we can analytically solve the triple integral in evaluating the nonstationary response covariance. As this approach uses the poles and residues of system transfer functions, rather than the conventional mode superposition technique, the method is applicable to MDOF systems with non-classical damping models. Particular application of the proposed method is demonstrated for multi-story shear buildings to stochastic ground acceleration characterized by the Kanai-Tajimi spectral density function model, and a numerical example is provided to illustrate the detailed steps. No numerical integrations are required for computing the response covariances as the exact closed-form solution has been derived. The correctness of the proposed method is numerically verified by Monte Carlo simulations.

A novel capacitive mass sensor using an open-loop controlled microcantilever

This article presents a method for quantitative estimation of the mass of an entity attached to the surface of an electrostatically actuated clamped-free microbeam implemented as a mass sensor. For this investigation, the microbeam is modeled as a Euler–Bernoulli beam taking into account the effects of electrostatic nonlinearity, viscous energy dissipation together with the effect of the fringing field capacitance. A modal superposition technique is used to simulate the dynamic response of the microcantilever. The dynamic pull-in voltage of microcantilever is evaluated using a developed modal. The present dynamic pull-in voltage results are compared with the existing results and on the basis of comparison, the accuracy of the developed model is ascertained. For mass quantification, firstly, an input shaping technique is employed to stabilize the microbeam at the specified position and then voltage perturbation is induced to make the system sensitive to inertial change. A parasitic mass expressed as a fraction of the total mass of the beam is then added resulting in an increase in the amplitude of vibration and hence an alteration in the capacitance. An empirical relation between the added mass and the change in capacitance is proposed.

Dynamic Stress and Displacement Fields on Concrete Gravity Dams due to Harmonic Ground Motion

The partial differential equations (PDE) of equilibrium governing the natural vibrations of concrete gravity dams were derived in this work such that the fluid structure interactions were accounted for. The displacement formulation is a system of two coupled PDE in two unknown displacement components. For seismic ground motion assumed to be horizontal harmonic motion whose amplitude and period are known, the system of two coupled PDEs were solved subject to the boundary conditions using the method of undetermined parameters. In applying the method of undetermined parameters to the PDE, displacement shape functions constructed to satisfy the displacement boundary conditions were used in assuming the trial dynamic displacement fields in terms of two unknown parameters that were determined by substitution into the governing equations. Conditions for the trial dynamic displacement fields to be solutions to the governing PDE were sought by solving the resulting system of equations. The problem reduced to an algebraic eigenvalue eigenvector problem which was solved for nontrivial cases to obtain the characteristic frequency equation and the eigenvalues. Modal superposition technique was employed to obtain the general solution for the displacement fields. The use of the displacement boundary conditions at the upstream face and Fourier series theory yielded the dynamic displacement field components, and the dynamic stress fields. The maximum value of the hydrodynamic pressure distribution on the upstream face is found to occur at the bottom of the dam and is found mathematically to be a convergent series of infinite terms. The maximum hydrodynamic force was calculated by integration of the hydrodynamic pressure distribution over the upstream face of the dam, and found to be a convergent series. Values of the maximum hydrodynamic force computed in this work agree with solutions from the technical literature.

Numerical analysis of violent hydroelastic problems based on a mixed MPS-mode superposition method

The traditional potential-flow-based hydroelasticity computation is unable to directly simulate the violent free surface deformation and slamming, and has a lack of mutual interactions between rigid-body and flexible motion parts. In this paper, the modified MPS (Moving Particle Semi-implicit) and modal superposition methods are used for the time-domain hydroelasticity computation of cylindrical shell and non-uniform beam structures involving violent free surface deformation and slamming. The improved modal superposition technique is revisited and extended to derive the time-domain governing equations for flexible shell and non-uniform beam motions, which couples rigid-body and flexible modes and considers the mutual interaction between them. The modified MPS method developed by the authors previously is first validated by the rigid cylinder water problem, and then the flexible cylindrical shell water entry and ship hull structure slamming are simulated. Relatively good agreement has been obtained between current results and the available numerical and experimental data, which shows the feasibility of the proposed model for the hydroelasticity computation involving violent free surface deformation.

Multi-objective optimisation of transition zones between slab track and ballasted track using a genetic algorithm

The vertical dynamic vehicle–track interaction in a transition between ballasted track and slab track is simulated in the time domain using an extended state-space vector approach. A complex-valued modal superposition technique is applied for the linear, time-invariant and non-periodic finite element model of the railway track. By considering a multi-objective optimisation problem solved by a genetic algorithm, the maximum dynamic loads on the track structure are minimised with respect to the selected design variables. To reduce the risk of long-term degradation of track geometry due to ballast/subgrade settlement, the transition zone is designed to minimise the influence of the track stiffness gradient between the two different track forms. The methodology is demonstrated by minimising the maximum wheel–rail contact force and the maximum pressure between sleeper/panel and foundation, while the selected design variables are distributions of rail pad stiffness and sleeper spacing adjacent to the transition. From the solution of the optimisation problem, non-dominated fronts are obtained illustrating potential for a significant reduction of the dynamic loads. It is shown that the optimised design leads to a more uniform distribution of load on the foundation reducing the risk of differential track settlement. The influences of the length of the transition zone and direction of travel on the maximum dynamic loads are investigated. Prescribed irregularities in longitudinal level may be accounted for but have been neglected in the optimisation as the optimised design would be more influenced by the given irregularity than by the stiffness gradient.

2-D Equivalent Source Method for a Finite Periodically Loaded Multiconductor Transmission Line Analysis With an Application to EBG Power Distribution Networks

A novel theoretical model based on an equivalent source method and a modal superposition technique is proposed to derive a dyadic Green's function for two-dimensional (2-D) finite periodically loaded multiconductor transmission lines (MTLs) to efficiently analyze electromagnetic band gap (EBG) structures embedded in power distribution networks (PDNs) for noise suppression. Many of such structures can be easily modeled using an MTL model with periodically loaded series and shunt components. By treating each periodic load as an equivalent local current source at the center of each unit cell, based on equivalent source theory, a finite uniform 2-D MTL excited by a group of equally spaced equivalent current sources is obtained. The amplitude of these local exciting sources may be analytically evaluated for the modes in the structure. Upon identifying these local equivalent current sources, a simple analytical expression for the 2-D dyadic Green's function is derived based on the modal superposition technique. For validation, a mushroom-type EBG structure embedded in a typical PDN is examined using the proposed approach and comparison with full-wave simulations and measurements are presented.

Reduced order finite element formulations for vibration reduction using piezoelectric shunt damping

The present work proposes an original reduced order model for prediction of passive reduction of structural vibration by means of shunted piezoelectric patches. The problem consists of an elastic structure with surface-mounted piezoelectric patches. The piezoelectric elements are connected to a resonant shunt circuits in order to damp specific resonant frequencies of the structure. An efficient electromechanical finite element formulation for the dynamic analysis of the problem is first presented. The classical modal superposition techniques using the system eigenvectors with all patches short-circuited or open-circuited are then recalled. An advanced reduced order models using two new modal projection bases able to solve the problem at lower cost are developed: (i) the combined basis formed by both the short-circuited and open-circuited modes, and (ii) the coupled basis formed by the electromechanical modes that take into account the effect of the inductances of the electrical shunt circuits. Various numerical and experimental results are presented in order to validate and illustrate the efficiency of the proposed new finite element reduced order formulations.

Fatigue assessment of a high-speed railway composite steel-concrete bridge by the hot-spot stress method

PurposeThe purpose of this paper is to investigate the fatigue performance of a welded detail from a composite steel-concrete railway twin girder bridge caused by a passenger train circulating at varying speeds, by identifying the dynamic amplification scenarios induced by resonance. For this purpose, the hotspot stress method is used, instead of the traditional nominal stress methods.Design/methodology/approachThis paper assesses the fatigue behavior of a welded connection considering critical stress concentration locations (hotspot). Finite element analysis (FEA) is applied, utilizing both a global and a local submodel, made compatible by displacements field interpolation. The dynamic response is obtained through the modal superposition method. Stress cycles are extracted with the rainflow counting method and the fatigue damage is calculated with Palmgren-Miner’s rule. The feasibility of five submodels with different mesh densities, i.e. 1, 2, 4, 8 and 20 mm is verified.FindingsAn increase in the fatigue damage due to the resonance effect was found for the train traveling at a speed of 225 km/h. A good agreement between the computed fatigue damage for the submodels is achieved. However, a non-monotonic hotspot stress/fatigue damage vs mesh density convergence was observed with a peak observed for the 4 mm model, which endorses the mesh sensitivity that could occur when using the surface stress extrapolation detailed rules specified in the standards for the hotspot stress method.Originality/valueAdvanced dynamic analyses are proposed to obtain local stresses in order to apply a local method for the fatigue assessment of a bridge’s structure subjected to high-speed railway traffic on the basis of the mode superposition technique resulting in much less computing times.

Development of an efficient approach for fatigue crack initiation and propagation analysis of bridge critical details using the modal superposition technique

The fatigue damage assessment of large bridges is highly conditioned by the required computational high demands. Generally, in order to overcome the multi-scale problem, global and local models are needed to properly account for both global structural behaviour and the local nature of the fatigue damage. The analysis of such structural problems using direct time-integration algorithms is impracticable in most of the cases, which leads to the necessity of developing alternative methodologies in order to increase the computational efficiency and the accuracy of fatigue cracking assessments. In this respect, effective computational algorithms based on the modal superposition technique have been proposed and implemented in previous works. Overall, such workflow considers the interaction between the global and local models combined with the application of the modal stress intensity factor concept. Aiming at performing an efficient and accurate assessment of the fatigue damage, firstly, combining the Fracture Mechanics principles and crack propagation laws, the crack propagation phase in a complex bridge detail is analysed. In this regard, the present paper aims at proposing relevant improvements to the above-mentioned methodology, namely: i) the refinement of the implemented submodelling techniques in order to increase the accuracy of stress and strain fields computation and allow to account for smaller initial crack lengths; ii) the analysis and limitation of the considered number of vibration modes to the relevant ones for the local dynamic response; and iii) the implementation of a parallel computing approach for the calculation of the modal stress intensity factors related to the vibration modes defined in ii). The fatigue assessment procedures were applied to an assumed cracked welded detail of a recent railway composite bowstring bridge located in Portugal. Also, since the assumption of a pre-existing crack may lead to very conservative predictions, the modal superposition technique is further extended to evaluate the fatigue crack initiation phase, demonstrating the safety of the analysed case study in the absence of existing defects.

Simulation of vertical dynamic vehicle–track interaction using a two-dimensional slab track model

ABSTRACTThe vertical dynamic interaction between a railway vehicle and a slab track is simulated in the time domain using an extended state-space vector approach in combination with a complex-valued modal superposition technique for the linear, time-invariant and two-dimensional track model. Wheel–rail contact forces, bending moments in the concrete panel and load distributions on the supporting foundation are evaluated. Two generic slab track models including one or two layers of concrete slabs are presented. The upper layer containing the discrete slab panels is described by decoupled beams of finite length, while the lower layer is a continuous beam. Both the rail and concrete layers are modelled using Rayleigh–Timoshenko beam theory. Rail receptances for the two slab track models are compared with the receptance of a traditional ballasted track. The described procedure is demonstrated by two application examples involving: (i) the periodic response due to the rail seat passing frequency as influenced by the vehicle speed and a foundation stiffness gradient and (ii) the transient response due to a local rail irregularity (dipped welded joint).