Structural Vibration Problems Research Articles

Symplectic analytical solutions for free vibration of elastically line-hinged orthotropic rectangular plates with rotationally restrained edges

The elastically line-hinged orthotropic rectangular plates with rotationally restrained edges are commonly used in engineering applications such as deployable structures. However, it is intractable to obtain the analytical solutions for the free vibration problems of such structures owing to the challenges in processing the high-order partial differential equations. Here, we make a first attempt to deal with such issues within the Hamilton system-based symplectic space. The problem of a plate is transformed into the symplectic space from the Euclidean space, and the subplates that may be analyzed analytically by the symplectic superposition method are then obtained with the division of the entire plate. The complex boundary and connection forms are achieved by enforcing mechanical quantities with undetermined expansion coefficients on the edges of the subplates. By integrating the solutions of the subplates, the final solution of an elastically line-hinged orthotropic rectangular plate with rotationally restrained edges is accessible. The proposed solution scheme is performed with rational derivations, with no requirement for pre-defined solution forms. Comprehensive results with validations under various boundary and hinge connection cases are presented. Moreover, detailed parametric investigations are conducted, which could facilitate the engineering design of deployable structures.

Analysis of internal pressure and vibration of a flexible expansion ratio elongation nozzle shape

The nozzle made of flexible materials can expand and elongate its profile at different flight altitudes, greatly improving the engine’s endurance and ballistic accuracy. However, this also introduces the problem of structural vibration in the nozzle, which adversely affects engine performance. This paper introduces a high-precision hyperelastic material constitutive model for the flexible elongation nozzle and uses numerical simulation to conduct a detailed analysis of the internal pressure and structural vibration under aerodynamic pressure loads. As the flight altitude increases, the pressure magnitude and distribution on the inner wall of the same elongated flexible expansion section do not change. The pressure distribution on the inner wall of the flexible elongation expansion section exhibits certain nonlinear characteristics, and its magnitude decreases gradually along the axial position. The vibration of the elongated expansion nozzle profile becomes more intense with the increase in flight altitude. At high characteristic flight altitudes, the elongation length of the longer expansion section approaches an undamped vibration state.

Static and Dynamic Analysis of Coupled Vibration of Suspension Bridge Structure Under Earthquake Action

This article investigates the coupled vibration problem of suspension bridge structures under earthquake action, and conducts static and dynamic analysis using computer algorithms. In terms of static analysis, complex structures are divided into many small elements, and the stress and deformation of each element are calculated through finite element analysis, thereby obtaining the static characteristics of the entire structure. In terms of dynamic analysis, the seismic response analysis method based on time history considers seismic action as a time-varying external excitation, and calculates and simulates the dynamic response of the suspension bridge structure. In order to achieve efficient and accurate calculations, this article also adopts optimization algorithms and numerical calculation methods. Through a comprehensive study of statics and dynamics analysis, the key characteristics and regularities of coupled vibration of suspension bridge structures under earthquake action have been obtained. These research results not only provide important theoretical guidance and technical support for related engineering practices, but also provide useful references and inspirations for dynamic analysis and optimization design of similar structures. Meanwhile, using computer algorithms and simulation software for static and dynamic analysis can significantly reduce analysis costs and time, improve analysis accuracy and efficiency, and have broad application prospects.

A refined mode superposition method for dynamical responses of an underwater cylindrical shell with substructures

To overcome the difficulties of time-consuming and inefficient in the response calculations of the submerged cylindrical shell with internal substructures, this paper firstly presents a refined mode superposition method. In view of the necessity and difficulties of directly and quickly obtaining the frequencies and modes of structures submerged in fluid (named the wet modes), the wet modes are expanded by the modes in vacuum (dry modes) and solved by the energy method. The added kinetic energy of the fluid is calculated via boundary integration, and the Lagrange equations of the second kind is applied to the fluid-structure coupling equations. Then the wet modes are solved by eigenvalue calculation and the modal mass and stiffness of each order wet mode are obtained. Finally, they are used for establishing a mode superposition approach for response calculations. The accuracy of the present method is verified by ANSYS software. In this method, all the required data are obtained from the structural analysis and the traditional complicated fluid force modeling is no longer required. Thus it has the advantages of high computational efficiency and applicability. Compared to the traditional semi-analytic model, this modeling methodology has broad application potential for vibration problems of complex underwater structures since the structural dry modes can be solved efficiently by commercial software. It also has practical value as a theoretical reference for developing mode-superposition-based calculations for fluid-structure problems using commercial software.

SPECTRAL DYNAMIC STIFFNESS METHOD FOR THE FLUTTER PROBLEM OF COMBINED PLATES

At present time, the spectral dynamic stiffness method is being actively developed as an alternative to the finite element method for vibration and stability problems of composite structures from beams, rods, plates and shells. This approach, based on exact solutions of governing differential equations, makes it possible to more effectively study the problem in the medium and high frequency ranges, and gives analytical expressions for natural modes. It is proposed to use the advantages of this method to study the problems of dynamic stability and flutter of an orthotropic composite plate in a supersonic gas flow. Using the linear approximation of piston theory, solution of the problem is investigated according to the Galerkin method on the basis of the eigenforms of a composite plate in vacuum. According to this approach the boundary value problem is reduced to a homogeneous infinite linear algebraic system of equations with coefficients are depending from physical-mechanical and geometrical parameters of the problem. The frequency parameter is included in the system linearly, that allows us to reduce eigenproblem for infinite system to the problem of determining the eigenvalues and vectors of a matrix. The convergence of the Galerkin method depending on the number of basis functions is studied numerically. It is shown that the first 16 eigenforms provide the good convergence of the method. Examples of numerical implementation are given, obtained solution allow us to study the dependence of the critical velocity from the properties of the material and geometry of the combined plate.

AT-PINN: Advanced time-marching physics-informed neural network for structural vibration analysis

Solving partial differential equations through deep learning has recently received wide attention, with physics-informed neural networks (PINNs) being successfully used and showing great potential. This study focuses on the development of an efficient PINN approach for structural vibration analysis in “long-duration” simulation that is still a technical but unresolved issue of PINN. The accuracies of the standard PINN (STD-PINN) and conventional time-marching PINN (CT-PINN) methods in solving vibration equations, especially free-vibration equations, are shown to decrease to varying degrees with the simulation time. To resolve this problem, an advanced time-marching PINN (AT-PINN) approach is proposed. This method is used to solve structural vibration problems over successive time segments by adopting four key techniques: normalization of the spatiotemporal domain in each time segment, a reactivating optimization algorithm, transfer learning and the sine activation function. To illustrate the advantages of the AT-PINN approach, numerical simulations for the forced and free vibration analysis of strings, beams and plates are performed. In addition, the vibration analysis of plates under multi-physics loads is also studied. The results show that the AT-PINN approach can provide accurate solutions with lower computational cost even in long-duration simulation. The techniques adopted are verified to effectively avoid the offset of the spatiotemporal domain, reduce the accumulative error and enhance the training efficiency. The present one overcomes the drawback of the existing PINN methods and is expected to become an effective method for solving time-dependent partial differential equations in long-duration simulation.

Free Vibration and Sound Insulation of FG-CNTRC Sandwich Plates with Different Cores

The core layer is very important in studying the vibration and noise problems of sandwich structures. The present paper studied the free vibration and sound insulation characteristics of FG-CNTRC sandwich plates with FG-CNT face sheets and different cores including the soft core (epoxy), hard core (aluminum) and porous metal foam core with different porosity distributions. By Hamilton’s principle the dynamic equations of FG-CNTRC sandwich plates are formulated based on the third-order shear deformation theory (TSDT), and the sound insulation expression is given under the excitation of the sound pressure. The theoretical solution of natural frequency and sound insulation are compared with the finite element simulation results and validated. The natural frequency of the square sandwich plate are compared with the rectangular case. The free vibration and sound insulation performance of FG-CNTRC sandwich plates are analyzed and discussed, and parametric studies are performed to evaluate the effects of CNT volume fraction, CNT distribution patterns, CNT orientation angles, porosity coefficient and incident angles on the vibro-acoustic characteristics of the sandwich plate. Some new results are presented to be used as references for the further research.

Non-stationary non-Gaussian random vibration analysis of Duffing systems based on explicit time-domain method

Non-stationary non-Gaussian random vibration problems of structures are challenging and drawing increasing attention. In the present study, firstly, an explicit time-domain method (ETDM) is proposed to determine the higher-order response statistics of linear systems subjected to non-stationary non-Gaussian random excitations, in which the first four orders of cumulants of dynamic responses are directly formulated through the cumulant operation rule based on the explicit expressions of responses. Secondly, an equivalent linearization – explicit time-domain method (EL-ETDM) is further developed to solve the non-stationary non-Gaussian random vibration problems of Duffing systems, in which the equivalent linear system is derived discarding the assumption of Gaussian response, and the corresponding higher-order cumulant analyses of the linearized system are accomplished by the efficient ETDM. The present approach can account for non-Gaussian random excitations with arbitrary forms, and two specific applications to the Poisson white noise and the square form of Gaussian random process are investigated. Four numerical examples are presented to demonstrate the effectiveness of the proposed methods.

Energy spectral element method for longitudinal acoustic propagation in ducts

Sound field behavior in an acoustic enclosure is an important part of the design of transportation vehicle passenger cabin, concert halls, conference rooms, and etc. Different analysis methods are available and have strengths and weaknesses. Low frequency band envelopes and negligible absorption in the walls can be modeled by Modal Analysis or Finite Element Method. However, as the frequency band increases, both methods become computationally intensive and Statistical Energy Analysis or the Sabine model can be an efficient approach. However, these methods do not take into account any spatial variation within the enclosure. The Energy Flow Analysis (EFA) solution for acoustic enclosures can be done analytically. In this paper, the Energy Spectral Element Method (ESEM) is formulated and applied to predict the spatial distribution of energy flux and density of acoustic ducts at high frequencies. ESEM is a matrix methodology based on EFA to solve acoustic and structural vibration problems. In this work, numerical models involving simple and coupled one-dimensional acoustic ducts are generated by ESEM, and the results are compared with energy densities calculated from the pressure fields predicted by the Spectral Element Method (SEM).

Finite Element Modeling and Vibration Control of Plates with Active Constrained Layer Damping Treatment

An enhanced lightness and thinness is the inevitable trend of modern industrial production, which will also lead to prominent low-frequency vibration problems in the associated structure. To solve the vibration problem of thin plate structures in various engineering fields, the active constrained layer damping (ACLD) thin plate structure is taken as the research object to study vibration control. Based on the FEM method, energy method, and Hamilton principle, the dynamic model of an ACLD thin plate structure is derived, in which the Golla-Hughes-McTavish (GHM) model is used to characterize the damping characteristics of the viscoelastic layer, and the equivalent Rayleigh damping is used to characterize the damping characteristics of the base layer. The order of the model is reduced based on the high-precision physical condensation method and balance reduction method, and the model has good controllability and observability. An LQR controller is designed to actively control the ACLD sheet, and the controller parameters and piezoelectric sheet parameters are optimized. The results show that the finite element model established in this paper is accurate under different boundary conditions, and the model can still accurately and reliably describe the dynamic characteristics of the original system in the time and frequency domain after using the joint reduction method. Under different excitation and boundary conditions, LQR control can effectively suppress structural vibration. Considering the performance and cost balance, the most suitable control parameter for the system is: Q-matrix coefficient is between 1 × 104 and 1 × 105, the R-matrix coefficient is between 1 and 10, and the thickness of the piezoelectric plate is 0.5 mm.

Concurrent optimization method of shape and topology for natural vibration problem of shell structures

In this paper, a concurrent optimization method of shape and topology for natural vibration design of shell structures is presented. The fundamental vibration eigenvalue is maximized under volume constraints for shape and topology optimization. The free-form optimization method for shells and the solid isotropic material with penalization (SIMP) method are respectively employed and combined effectively for shape and topology design, in which shape and fictitious homogenized-density variations are used as design variables and simultaneously determined. In other words, the optimal topology is determined in the variable design surface optimized by shape optimization. The optimal design problem is formulated as a distributed-parameter optimization problem, and the sensitivity functions with respect to shape and density variations are theoretically derived. Both the optimal shape and density variations are determined with the unified H1 gradient method, where the sensitivity functions are respectively applied as the Robin condition to the design surface in order to determine the optimal shape and topology. We also show an improved method for the spurious (or localized) mode problem of shell structures, which often occurs in the lower density region in topology optimization for vibration design. With the proposed method, shell structure for light weight and improved low-order vibration characteristics can be obtained without any design parameterization, free of numerical instabilities such as checkerboard pattern and zigzag-shape problems.

An investigation on the structural vibrations of multi-rotor passenger drones

In order to solve the structural damage problem of the first generation of large multi-rotor manned drones, the present work has designed to study the structural vibration problems of multi-rotor drones. On a small multi-rotor drone, the laser vibration meter verified the reliability of acceleration sensor measurement of vibration and found that circular shape carbon fiber arms have strong damping abilities, with the strongest vibration in the Z-axis direction. To improve the design of the second generation of large multi-rotor manned drones, elliptical shape carbon fiber arms were employed instead of circular arms. Experiments showed that the main vibrations of the large multi-rotor manned drone’s arm are low-frequency vibrations below 200Hz, producing mainly torsional and bending modes, and the elliptical carbon fiber arms significantly reduce vibrations in the Z-axis direction. This study provides experimental data support for multi-rotor manned drones and further presents an improvement strategy for suppressing the vibrations of the multi-rotor manned drones.

A high order Newton method to solve vibration problem of composite structures considering fractional derivative Zener model

This paper presents a numerical method to solve the non-linear vibration problem of composite structures made of many unidirectional fibers/matrix layers. To take into account damping properties of structures, fractional derivative Zener model for complex moduli is used in each layer. The frequency dependence of these models leads to a non-linear vibration problem. To solve it, a High Order Newton (HON) solver based on homotopy and perturbation techniques is proposed. Thus, the initial non-linear problem turns into a set of linear systems, easier to solve. Numerical results obtained by the HON solver are compared with experimental and numerical results of literature.

Structured model order reduction for vibro-acoustic problems using interpolation and balancing methods

Vibration and dissipation in vibro-acoustic systems can be assessed using frequency response analysis. Evaluating a frequency sweep on a full-order model can be very costly, so model order reduction methods are employed to compute cheap-to-evaluate surrogates. This work compares structure-preserving model reduction methods based on rational interpolation and balanced truncation with a specific focus on their applicability to vibro-acoustic systems. Such models typically exhibit a second-order structure and their material properties as well as their excitation may be depending on the driving frequency. We demonstrate the effectiveness of all considered methods in terms of their accuracy and computational cost by applying them to numerical models of vibro-acoustic systems depicting structural vibration, sound transmission, acoustic scattering, and poroelastic problems. The results of the experiments are extensively discussed to derive guidelines for the choice of model reduction methods based on the problem setting.

Surrogate modeling in structural vibration problems with dynamic mode decomposition

Solving large-scale vibration problems presents a difficult challenge for complicated geometries that is only addressable through the use of large-scale compute resources. These problems not only require extensive compute time and cost, but are also expensive in engineering hours for modeling and analysis. While optimization techniques can help limit engineering time in the loop, the computational requirements of these models have made applying traditional optimization techniques to this class of problem untenable. Dynamic Mode Decomposition (DMD) is an approach that can help bridge this gap by building high-fidelity surrogate models allowing for the inline development of a surrogate model to dramatically reduce the computational time and complexity of a problem. This approach enables fast frequency sweeping while capturing the dominant dynamics of the model. DMD has historically been applied to other high-data volume models such as computational fluid dynamics, medical imaging and controls, as well as less structured problems including sociology and market behavior. This paper will cover the fundamentals of Dynamic Mode Decomposition, the extension of the theory to apply the technique to vibration problems, the demonstration of a potential workflow, and show how this technique performs on simple test cases illustrating performance speed ups for classic problems from existing literature.

Aeroelastic Experimental Investigation of Hyperbolic Paraboloid Membrane Structures in Normal and Typhoon Winds

The lightweight and flexible membrane structure of roofs are susceptible to wind loads with the risk of damage and failure. Compared with uniform and low-level turbulence flow cases (i.e., normal winds) that have been well investigated, the wind-induced vibration problem of membrane structures in high-level turbulence flows such as typhoons has been paid little attention. To address the gap, this paper aimed at investigating the aerodynamic behavior of hyperbolic paraboloid membrane structures in normal and typhoon winds by a series of wind tunnel tests. Some distinct wind characteristics of upcoming normal and typhoon flows in terms of vertical profiles of wind velocity, turbulence intensity, and power spectrum density of fluctuating winds were well simulated in an automatically controlled wind tunnel. The aeroelastic behavior of a scaled model was analyzed and discussed in terms of displacement time-history responses, probability distribution characteristics, and dynamic characteristics including the natural frequency, mode shape, and damping ratio. Results show that the increasing suction in a typhoon leads to significant growth in maximum deformations and more risks to suffer from aeroelastic instability. Non-Gaussian characteristics appear more remarkable with skewness and kurtosis increasing almost two-fold in typhoons. Structural modal parameters are influenced by both turbulence intensity and wind velocity. This study provides basic insights into the deficiency of dynamic response of membrane structures in typhoons, and promotes the applications of membrane structures in green buildings.

Numerical and experimental correlation of a catamaran’s vibration modes using supervised machine learning

Modal parameters are crucial in investigating structural vibration problems and virtual sensing of vibration responses. This work proposes machine learning (ML) models such as Decision Tree, Random Forest, Adaboost Trees and RUSboost Trees to correlate 13 finite element (FE) modal shapes with the experimental ones. A catamaran’s FE model was assembled and used as a case study. Firstly, a full FE model with 93591 DOFs was reduced to 174 DOFs using SEREP. Secondly, as the experimental mode shapes have a low spatial resolution (only six measured DOFs), another goal of this work is to expand these 6 DOFs to the 174 DOFs of the reduced FE model, that had not been measured. The ML models were used to correlate and expand the experimental mode shapes, thus, solving their lack of spatial resolution. As a result, the ML models identified the most relevant parameters for the scaling process of each of the 13 numerical modes using a supervised database. Adaboost Trees and Random Forest were used to correlate and expand five mode shapes each and RUSboost Trees were used to correlate and expand three mode shapes. In conclusion, Adaboost Trees and Random Forest models showed better performance during the correlation process.

A Generalized Method for Dispersion Analysis of Guided Waves in Multilayered Anisotropic Magneto-Electro-Elastic Structures

Based on the symplectic structure of the Hamiltonian matrix, the precise integration method (PIM), and the Wittrick–Williams (W-W) algorithm, a generalized method for computing the dispersion curves of guided waves in multilayered anisotropic magneto-electro-elastic (MEE) structures for different types of mechanical, electrical, and magnetical boundaries is developed. A strictly theoretical analysis shows that the W-W algorithm cannot be applied directly to the MEE structure. This is because a block of the Hamiltonian matrix is not positive definite for MEE structures so that the eigenvalue count of the sublayer is not zero when the divided sublayer is sufficiently thin. To overcome this difficulty, based on the symplectic structure of the Hamiltonian matrix, a symplectic transformation is introduced to ensure that the W-W algorithm can be applied conveniently to solve wave propagation problems in multilayered anisotropic MEE structures. The application of the PIM based on the mixed energy matrix to solve the wave equation can ensure the stability and efficiency of the method, and all eigenfrequencies are found without the possibility of any being missed using the W-W algorithm. This research provides the necessary insight to apply the W-W algorithm in wave propagation and vibration problems of MEE structures.

Equivalent linearization method improved by higher order statistics in modal space for geometrically nonlinear vibrations

In this paper, an equivalent linearization method considering higher order statistics based on nonlinear reduced order modeling techniques is proposed for the geometrically nonlinear random vibration problems of complex structures. Nonlinear reduced order models of the structures are constructed by leveraging the nonlinear analysis capabilities of commercial finite element codes, and an improvement to the Stiffness Evaluation Procedure method for determining the stiffness coefficients is achieved through the equivalence relation, leading to a notable reduction in computational cost with no further requirements for commercial finite element codes. The nonlinear terms are twice regulated equivalent, and then a regulated form of the stiffness coefficients is derived to introduce higher order statistics into the equations of motion. Then a linearized system obtained by the criterion of force error minimization is used to predict the random response of the original nonlinear system. The nonlinear problems are solved by the linearized system in the modal space, increasing computational efficiency significantly. Higher order statistical information of the response is introduced to improve the accuracy. Typical examples are used to verify the effectiveness of the proposed method, while its applicability is further demonstrated via the analysis of turbulence-excited composite laminates.

Influences of elastic foundations on the nonlinear free vibration of composite shells containing carbon nanotubes within shear deformation theory

In this work, the solution of nonlinear free vibration problem of composite shells structures containing carbon nanotubes (CNTs) resting on elastic soils within shear deformation theory (ST) is presented. After modeling the mechanical properties of nanocomposite shell structures containing CNTs and elastic soils, the basic relations, and governing equations of double curved shell structures within the ST are established considering the geometric nonlinearity. The frequencies of nonlinear and linear free vibrations and their ratios for inhomogeneous nanocomposite structures on the soils within the ST are obtained using perturbation method for the first time. After checking the methodology of the research, the effects of soils, nonlinearity, shear strains and patterns of CNT on the frequency-amplitude dependence of nanocomposite shell structures for various geometric parameters are carried out.