Characteristic Orthogonal Polynomials Research Articles

Investigations on the aerothermoelastic properties of composite laminated cylindrical shells with elastic boundaries in supersonic airflow based on the Rayleigh–Ritz method

This paper is devoted to investigate the aerothermoelastic flutter and thermal buckling characteristics of composite laminated cylindrical shells with elastic boundary conditions. In the structural modeling, Donnell's shell theory is employed, and the supersonic piston theory is applied to evaluate the aerodynamic pressure. The elastic boundary constraints for the cylindrical shell are simulated by a series of distributed artificial springs. The shape functions of the laminated cylindrical shell with elastic boundary conditions are composed of characteristic orthogonal polynomials which are derived by the Rayleigh–Ritz method. The equation of motion of the structural system is established using Hamilton's principle. The high accuracy of the present method is verified by comparing the natural frequencies and flutter bounds with published results. Based on the frequency domain method, the aerothermoelastic properties of the laminated cylindrical shell with elastic boundary conditions are analyzed. The influences of different types of spring stiffnesses on the flutter and thermal buckling bounds are studied in detail. In addition, the influences of several parameters including the length to radius ratio, ply angle and boundary condition on the aerothermoelastic behaviors of the laminated cylindrical shell with elastic boundary conditions are researched.

A unified Gram-Schmidt-Ritz formulation for vibration and active flutter control analysis of honeycomb sandwich plate with general elastic support

An admissible function constructed by orthogonal polynomials is presented to investigate the vibration and active flutter control of Honeycomb sandwich plate with general elastic support in a supersonic airflow. The control strategies based on the displacement feedback and LQR method are applied by means of pasting the piezoelectric material on the surface of the plate structure. Employing the first-order piston theory to describe the aerodynamic loads, the governing equation of plate system with the piezoelectric actuator is established based on the Hamilton principle. The mode shapes are obtained using the admissible functions, which are a set of characteristic orthogonal polynomials generated directly by employing the Gram-Schmidt process, and the general elastic constraint is modeled using the artificial spring technology. The frequency and mode shape under different boundary are calculated by Rayleigh-Ritz method. The validity of the proposed approach is confirmed by comparing the results with those obtained from FEM and literatures. The phenomenon of mode jumping is observed with the increase of the aerodynamic pressure. The study of active control shows that, the increasement of displacement feedback gain improves the critical dynamic pressure.

Vibration characteristics and thermoelastic analysis of the supersonic composite laminated panel with arbitrary elastic boundaries

The novelty of this study is to present a theoretical approach to investigate the dynamic behaviors of the laminated panels under arbitrary elastic boundary conditions. The motion equations of panels considering the first order piston theory are derived using Hamilton principle. A solution of computing the vibration characteristics of a panel with arbitrary elastic boundaries is proposed based on the Rayleigh–Ritz method, in which the admissible functions are constructed by a set of characteristic orthogonal polynomials employing the Gram–Schmidt process; the support boundary is modeled by introducing the technology of artificial springs. The effects of spring stiffness and different boundaries on the dynamic characteristics and thermal aeroelastic behaviors are presented in detail. Numerical results show that the small ply angle and large spring stiffness are helpful to improve the aerodynamic stability. Multiple new phenomena have been observed, e.g. the phenomena of mode jumping and the alterations of coupled mode orders. Moreover, the thermal loads and aerodynamic loads play an opposite effect on the stability of composite panels.

Free and forced vibration analysis of arbitrarily supported rectangular plate systems with attachments and openings

Rectangular plates with different kinds of attachments (continuous or lumped) and various opening shapes are main constitutive parts of almost all engineering structures, e.g. aircrafts, bridges, buildings, ships, offshore structures, etc. Therefore, an assessment of their free and forced responses is generally very important for safe and rational structural design. In this paper, a range of different vibration problems inherent to rectangular plate systems, that can be solved by the energy based assumed mode method is considered. The concept of assumed mode method is outlined together with application of the mode superposition method to forced response calculation for plates under concentrated harmonic forces or enforced boundary displacement. Furthermore, complete mathematical model for vibration analysis of plate structures carrying arbitrary number of spring-mass systems is developed, based on the receptance method application. The plate is modelled by the Mindlin thick plate theory and Timoshenko beam theory is applied for stiffeners. The eigenvalue problem represented with a multi-degree-of-freedom system equation is formulated by Lagrange's equation of motion, while characteristic orthogonal polynomials having the properties of Timoshenko beam functions and satisfying the specified edge constraints are used as approximation functions. The corresponding in-house software is developed and dynamic responses of rectangular plate systems having different sets of edge constraints are analysed. Comparisons of the results with existing solutions and general finite element (FE) software are included, and very good agreement is achieved.

An admissible function for vibration and flutter studies of FG cylindrical shells with arbitrary edge conditions using characteristic orthogonal polynomials

A general approach for the vibration and aeroelastic stability of the functionally graded cylindrical shell with arbitrary boundary conditions is firstly presented. The Sanders' shell theory, a steady-state heat transfer equation and the piston theory are employed to establish the motion equation, where the thermo-mechanical properties of material are set to be location- and temperature- dependent. The orthogonal polynomials series generated by employing the Gram–Schmidt process are taken as the admissible functions to express the general formulations of displacement. Moreover, the artificial spring technique is introduced to simulate the elastic constraints imposed on the cylinders' edges. The frequency equations are derived considering the strain energy of artificial springs during the Rayleigh–Ritz procedure, and the motion equation of cylindrical shells subjected to combined thermal and aerodynamic loads is established based on the Hamilton principle. A few comparisons for the frequency and critical flutter pressure are performed to validate the proposed approach. The influences of the volume fraction, thermal gradient, boundary conditions and spring stiffness on the flutter characteristics are highlighted. This paper overcomes the limitations of previous vibration and flutter studies which are confined to the structure under simply supported or clamped boundaries.

Vibration characteristics and flutter analysis of a composite laminated plate with a store

The effects of an external store on the flutter characteristics of a composite laminated plate in a supersonic flow are investigated. The Dirac function is used to formulate the interaction between the plate and the store. The first-order piston theory is used to describe the aerodynamic load. The governing equation of the composite laminated plate with an external store is established based on the Hamilton principle. The mode shapes are constructed by the admissible functions which are a set of characteristic orthogonal polynomials generated directly by the Gram-Schmidt process, and the boundary constraint is modeled as the artificial springs. The frequency and mode shapes of the plate under different boundaries are determined by the Rayleigh-Ritz method. The validity of the proposed approach is confirmed by comparing the results with those obtained from the finite element method (FEM). The effects of the mounting position, the center of gravity position and the mounting points spacing of the external store on the flutter boundary are discussed for both the simply supported and cantilever plates, respectively, which correspond to the two installation sites of the external store, i.e., the belly and wings of the aircraft.

ANALYTICAL BENDING SOLUTION OF ALL CLAMPED ISOTROPIC RECTANGULAR PLATE ON WINKLER’S FOUNDATION USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL

The analytical bending solution of all clamped rectangular plate on Winkler foundation using characteristic orthogonal polynomials (COPs) was studied. This was achieved by partially integrating the governing differential equation of rectangular plate on elastic foundation four times with respect to its independents x and y axis. The foundation was assumed to be homogeneous, elastic and isotropic. The governing differential equation was non-dimensionalised to make it consistent. The deflection polynomials functions were formulated. Thereafter, the Galerkin’s works method was applied to the governing differential equation of the plate on Winkler foundation to obtain the deflection coefficient, . Numerical example was presented at the end to compare the results obtained by this method and those from earlier studies. The percentage difference obtained for central deflection of all clamped rectangular plate loaded with UDL using the method and earlier research works for K = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are: 0.000042, 0.000052, 0.000043, -0.000011, -0.000068, 0.000001, 0.000001, 0.000001, -0.000001, 0.000000, 0.000001. 0.000033, 0.000035, 0.000033, -0.000018, -0.000072, -0.000003, -0.000003, -0.000003, 0.000002, -0.000002, -0.000001. The result showed that an easy to use and understandable model was developed for determination of deflections of all clamped rectangular plates on Winkler’s elastic foundation using principle of COPs. http://dx.doi.org/10.4314/njt.v36i3.9

Nonlocal elasticity in plates using novel trial functions

This study presents the Ritz formulation, which is based on boundary characteristic orthogonal polynomials (BCOPs), for the two-phase integro-differential form of the Eringen's nonlocal elasticity model. This approach is named the nonlocal Ritz method (NL-RM). This feature greatly reduces the computational cost compared to the nonlocal finite-element method (NL-FEM). Another advantage of this approach is that, unlike NL-FEM, the nonlocal mass and stiffness matrices are independent of the mesh distribution. Here, these formulations are applied to study the static-bending and free-dynamic analyses of the Kirchhoff plate model. In this paper, novel 2D BCOPs of the plate are derived as coordinate functions. These polynomials are generated using a modified Gram-Schmidt process and satisfy the given geometrical boundary conditions as well as the natural boundary conditions. The accuracy and convergence of the presented model, demonstrated through several numerical examples, are discussed. A concise argument on the advantages of NL-RM compared to NL-FEM is also provided.

DYNAMICS OF RECTANGULAR ORTHOTROPIC PLATES USING CHARACTERISTIC ORTHOGONAL POLYNOMIAL – GALERKIN’S METHODS

The assumed deflection shapes used in the approximate methods such as in the Galerkin’s method were normally formulated by inspection and sometimes by trial and error, until recently, when a systematic method of constructing such a function in the form of Characteristic Orthogonal Polynomial (COPs) was developed by Bhat in 1985. In the vibrational analyses of orthotropic rectangular plates with different boundary conditions, the study used the characteristic orthogonal polynomial theory to obtain satisfactory approximate shape functions for these plates. These functions were applied to Galerkin indirect varational method to obtain new set of fundamental natural frequencies for these plates. The results were reasonable when compared with those in the previous work. All round simply supported thin rectangular plate (SSSS), rectangular clamped plated (CCCC) and rectangular plate with one edge clamped and all others edges simply supported (CSSS) gave 5.172, 9.429 and 6.202 natural frequencies in rad /sec respectively at 0.05%, 0.0% and 22.93% difference with the previous[3] results5.170rad/sec, 9.429rad/sec and 8.048rad/sec for SSSS, CCCC and CSSS. For others like: rectangular plate with one edge simply supported and all other edges clamped (CCSC), rectangular plate simply supported at two opposite sides and clamped at the others (CSCS) and rectangular plate clamped at two adjacent sides and simply supported at the others (CCSS) with no available results, their natural frequencies obtained are 8.041rad/sec, 6.272rad/sec and 7.106rad/sec respectively. http://dx.doi.org/10.4314/njt.v36i1.8

Study on the Use of Boundary Characteristic Orthogonal Polynomials in Determining the Fundamental Frequency of Rectangular Plates with Bidirectional Linear Thickness Variation

Free vibration of rectangular plates with linearly varying thickness is considered. Plates are important structural components used in aircrafts, bridges etc. and hence for their safe design, thorough vibration analysis is important. Many research and practical applications use plates of variable thickness due to economical usage of the material and increased strength. Vibration response of these types of plates is different from plates of uniform thickness, which makes their analysis critical. In this study, Boundary Characteristic Orthogonal Polynomials (BCOPs) in one and two dimensions have been used to obtain deflection shape function. The first member of the series is generated using the boundary conditions, in this case all edges clamped, which satisfies both the geometric boundary conditions and the natural boundary conditions. Gram Schmidt Orthogonalization for polynomials is used to generate the higher members of the shape function which only satisfy the geometric boundary conditions. Thickness variation considered for the plate is linear in both x and y direction. Natural frequencies were obtained by using Rayleigh-Ritz method. Natural frequencies were calculated by varying taper parameters for both directions and compared with those obtained with the case of uniform thickness. Natural frequencies were also found comparable with those obtained from Finite Element Analysis by using ANSYS.

The sensibility on dynamic characteristics of pre-pressure thin-wall pipe under elastic boundary conditions

Consideration is given to dynamic behavior of cylindrical pressure pipe with elastic boundary conditions. Based on Sanders’ shell theory and Hamilton principle, the system equations are established for integrating the uniform distributed pressure into the elastic boundary condition. In the analytical formulation, the Rayleigh–Ritz method with a set of displacement shape functions is used to deduce mass, damping, and stiffness matrices of the pipe system. The displacements in three directions are represented by the characteristic orthogonal polynomial series and trigonometric functions which are satisfied with the elastic boundary conditions, which are represented as four sets of independent springs placed at the ends including three sets of linear springs and one set of rotational spring. The pressure pipe always suffers a uniform distributed pressure in radial direction. To verify the accuracy and reliability of the present method, several numerical examples with classical boundary condition, including free and simply supported supports are listed and comparisons are made with open literature. Then the influences of boundary restraint stiffness and the distributed pressure on natural frequency and the forced vibration response are studied: The natural frequencies increase significantly as the restraint stiffness or the distributed pressure increases. Compared to the rotational spring stiffness, the stiffnesses of axial, radial, and circumferential springs have more significant effect on natural frequency. And the lower modes are more sensitive on restraint stiffness than higher modes. But the variation of natural frequency with respect to the spring stiffness decreases monotonically with the increasing distributed pressure. The forced vibration response is also affected by the restraint stiffness.

Simplified dynamic analysis of stepped thickness rectangular plate structures by the assumed mode method

In this article, the assumed mode method is applied to simplified dynamic analysis of stepped thickness rectangular Mindlin plates and stiffened panels with arbitrary boundary conditions. The natural and frequency responses of stepped thickness plate structures subjected to harmonic point excitation force and enforced acceleration at boundaries, respectively, are considered. Potential and kinetic energies of the system are formulated and used to derive eigenvalue problem utilizing Lagrange’s equation of motion, and mode superposition method is further used for forced response assessment. Characteristic orthogonal polynomials having the property of Timoshenko beam functions are used for the assumed modes. Numerical examples analysing vibration of stepped thickness plate structures with different topologies and various sets of boundary conditions are provided. Numerical results are compared with the results from the relevant literature and finite element solutions obtained by a general finite element tool, and a very good agreement is achieved. Hence, it is expected that stepped rectangular plate structures satisfying the prescribed criteria regarding natural and frequency responses can be efficiently designed based on the proposed method.

Free vibration analysis of stiffened panels with lumped mass and stiffness attachments

Stiffened panels are basic constitutive members of ships and offshore structures, and in practice they often have different mass and stiffness attachments, which significantly influence their dynamic response. In this paper, a numerical procedure is presented for the free vibration analysis of stiffened panels with arbitrary sets of boundary conditions and carrying multiple lumped mass and stiffness attachments. It is based on the assumed mode method, where characteristic orthogonal polynomials having the properties of Timoshenko beam functions and satisfying the specified edge constraints are used as approximation functions. The Mindlin theory is applied for plate and the Timoshenko beam theory for stiffeners. The total potential and kinetic energies of the system are formulated in a convenient manner and further applied to derive an eigenvalue problem by means of Lagrange's equation of motion. Based on the developed numerical procedure, an in-house code is developed and is applied to a free vibration analysis of bare plates and stiffened panels carrying lumped masses and locally supported by pillars or springs. Comparisons of the results with those available in the literature and FEA solutions confirm the high accuracy and practical applicability of the presented procedure.

Studies on global analytical mode for a three-axis attitude stabilized spacecraft by using the Rayleigh–Ritz method

With extending the Rayleigh–Ritz procedure to study the hub-plate system, the characteristics of global analytical modes are addressed for a typical rigid–flexible coupling dynamic system, i.e., a three-axis attitude stabilized spacecraft installed with a pair of solar arrays. The displacement field of the solar arrays is expressed as a series of admissible functions which is a set of characteristic orthogonal polynomials generated directly by employing Gram–Schmidt process. The rigid body motion of spacecraft is represented by the product of constant and generalized coordinate. Then, through Rayleigh–Ritz procedure, the eigenvalue equation of the three-axis attitude stabilized spacecraft installed with a pair of solar arrays is derived. Solving this eigenvalue equation, the frequencies and analytical expressions of global modes for the flexible spacecraft are obtained. To validate the present analysis, comparisons between the results of the present method and ANSYS software are performed and very good agreement is achieved. The convergence studies demonstrate the high accuracy, excellent convergence and high efficiency of the present approach. Finally, the method is applied to study the characteristics of global modes of the flexible spacecraft.

Nonlinear Dynamic Behavior Analysis of Pressure Thin-Wall Pipe Segment with Supported Clearance at Both Ends

An analysis of nonlinear behaviors of pressure thin-wall pipe segment with supported clearance at both ends was presented in this paper. The model of pressure thin-wall pipe segment with supported clearance was established by assuming the restraint condition as the work of springs in the deformation directions. Based on Sanders shell theory, Galerkin method was utilized to discretize the energy equations, external excitation, and nonlinear restraint forces. And the nonlinear governing equations of motion were derived by using Lagrange equation. The displacements in three directions were represented by the characteristic orthogonal polynomial series and trigonometric functions. The effects of supporting stiffness and supported clearance on dynamic behavior of pipe wall were discussed. The results show that the existence of supported clearance may lead to the changing of stiffness of the pipe vibration system and the dynamic behaviors of the pipe system show nonlinearity and become more complex; for example, the amplitude-frequency curve of the foundation frequency showed hard nonlinear phenomenon. The chaos and bifurcation may emerge at some region of the values of stiffness and clearance, which means that the responses of the pressure thin-wall pipe segment would be more complex, including periodic motion, times periodic motion, and quasiperiodic or chaotic motions.

A variational Ritz formulation for vibration analysis of thick quadrilateral laminated plates

In the present study, a variational Ritz approach for vibration analysis of thick arbitrarily quadrilateral laminated plates, based on the trigonometric shear deformation theory (TSDT) is developed. In this theory, shear stresses are vanished at the top and bottom surfaces of the laminate and shear correction factors are no longer required. A general straight-sided quadrilateral domain is mapped into a square domain in the computational space using a four-node master plate, employing a geometric transformation. The displacement field components are approximated by sets of beam characteristic orthogonal polynomials generated using the Gram–Schmidt procedure. The use of Ritz method allows a high spectral accuracy and faster convergence rates than local methods such as finite element. The algorithm developed is quite general, free of shear locking and can be used to obtain natural frequencies and modal shapes of laminated plates having various material parameters, geometrical planforms, length-to-thickness ratios and any combinations of free, simply supported and clamped edge support conditions.Through several numerical examples, the capability, efficiency and accuracy of the formulation are demonstrated. Convergence studies and comparison with other existing solutions in the literature suggest that the present algorithm is robust and computationally efficient.

A new simple shear deformation theory for free vibration analysis of isotropic and FG plates under different boundary conditions

Purpose– A new simple parametric shear deformation theory applicable to isotropic and functionally graded plates is developed. This new theory has five degrees of freedom, provides parabolic transverse shear strains across the thickness direction and hence, it does not need shear correction factor. Moreover, zero-traction boundary conditions on the top and bottom surfaces of the plate are satisfied rigorously. The paper aims to discuss these issues.Design/methodology/approach– Material properties are temperature-dependent and vary continuously through the thickness according to a power law distribution. The plate is assumed to be initially stressed by a temperature rise through the thickness. The energy functional of the system is obtained using Hamilton’s principle. Free vibration frequencies are then calculated using a set of characteristic orthogonal polynomials and by applying Ritz method for different boundary conditions.Findings– In the light of good performance of the present theory for all boundary conditions considered, it can be considered as an excellent alternative to some two-dimensional (2D) theories for approximating the tedious and time consuming three-dimensional plate problems.Originality/value– To the best of the authors’ knowledge and according to literature survey, almost all published higher order shear deformation theories have been limited to simply supported boundary conditions and without taking into account the thermal stresses effects. The existing 2D shear deformation theories of Reddy, Karama and Touratier can be easily recovered. Furthermore, this feature can be highly appreciated in an iterative design process where a large number of derived plate models can be tested by selecting only two parameters in a simple polynomial function which is computationally efficient. Finally, new results are presented to show the effect of material variation, and temperature rise on natural frequencies of the FG plate for different boundary conditions.

Vibration and buckling analyses of nanobeams embedded in an elastic medium

Boundary characteristic orthogonal polynomials are used as shape functions in the Rayleigh–Ritz method to investigate vibration and buckling of nanobeams embedded in an elastic medium. The present formulation is based on the nonlocal Euler–Bernoulli beam theory. The eigen value equation is developed for the buckling and vibration analyses. The orthogonal property of these polynomials makes the computation easier with less computational effort. It is observed that the frequency and critical buckling load parameters are dependent on the temperature, elastic medium, small scale coefficient, and length-to-diameter ratio. These observations are useful in the mechanical design of devices that use carbon nanotubes.

Vibration of beams using novel boundary characteristic orthogonal polynomials satisfying all boundary conditions

Boundary characteristic orthogonal polynomials proposed by the author in 1985 have been used in the Rayleigh Ritz method extensively in order to obtain natural frequencies of vibrating plates with different boundary conditions. The method used products of the characteristic orthogonal polynomials along the two directions of the plate. The first member of the boundary characteristic orthogonal polynomials set satisfied all the boundary conditions of the vibrating beam, including the natural conditions. However, the higher members of the set satisfied only the geometry boundary conditions. In this study, a modified Gram–Schmidt orthogonalization method is presented where all the members of the orthogonal set of polynomials satisfy all the boundary conditions including the natural boundary conditions. Furthermore, the exact solution of the beam differential equation is expressed in the form of a generalized Fourier series in terms of the set of new boundary characteristic orthogonal polynomials which forms an eigenvalue problem that can provide the natural frequencies and the corresponding normal modes of the beam more accurately.

Inspection of a Rectangular Plate Dynamics Under a Moving Mass With Varying Velocity Utilizing BCOPs

This research is dedicated to the inspection of a thin rectangular plate dynamic behavior traversed by an accelerated moving mass. BCOPs (boundary characteristic orthogonal polynomials) are utilized to treat the constitutive equation of plate vibration for different boundary conditions. Comprehensive parametric surveys are carried out to shed light on the effects of the plate fixities and aspect ratios as well as the moving mass weight, velocity and acceleration on the plate DAF (dynamic amplification factor). The convenience of adopting the presented solution dealing with various plate fixity cases makes it a superior approach comparing with eigenfunction expansion method.