Characteristic Orthogonal Polynomials Research Articles

Aerothermoelastic Analysis of Conical Shell in Supersonic Flow

The aerothermoelastic behavior of a conical shell in supersonic flow is studied in the paper. According to Love’s first approximation shell theory, the kinetic energy and strain energy of the conical shell are expressed and the aerodynamic model is established by using the linear piston theory with a curvature correction term. By taking the characteristic orthogonal polynomial series as the admissible functions, the mode function of conical shell under different boundary conditions can be obtained using the Rayleigh–Ritz method. Then, the dynamic model of the conical shell is derived by using the Lagrange equation. Based on the model, variations in the natural frequencies with respect to temperature and free-stream static pressure are analyzed. Additionally, the effects of the length-to-radius ratio, the thickness-to-radius ratio, and semi-vertex angle, as well as the thermal and aerodynamic loads on the aerothermoelastic stability of the structure are investigated in detail.

Vibration isolation performance of a rectangular panel with high-static-low-dynamic stiffness supports

In this paper, the nonlinear vibration of a corner-supported rectangular thin panel with high static low dynamic stiffness (HSLDS) isolator supports is investigated. The dynamic model of the panel clamped at corners is established and the frequency equations are derived by employing the Rayleigh-Ritz procedure. The characteristic orthogonal polynomial series is introduced as the assumed modes and the equations of motions are obtained. The dynamic response of the system is analytically calculated for the first time by applying the complexification-averaging method. And the approximation solutions are verified with the numerical and finite element method. The isolation performance of the system is evaluated by the displacement transmissibility and the effects of the parameters are investigated for optimal design. The comparison with the isolation performance of linear elastic-supported panel illustrates that introducing the HSLD isolators has a superior effect on reducing the peak transmissibility in the low-frequency band and provides a wide isolation range for the system.

Deflection of an Isotropic All-Round Simply Supported Rectangular Plates Incorporating Shear Effect Using Characteristic Orthogonal Polynomial Function

The breadth and dexterity with which plates are used in the vast majority of engineering structures necessitates an ever-increasing and deeper study focus on plate strength and stiffness at the ultimate and serviceability limit states of response. From Kirchhoff’s hypothesis, thin plates when subjected to transverse loading, bend and experience transverse deflections that are typically minor in comparison to the plate thickness. However, for thicker plates there is an observed limitation in the application of Kirchhoff’s hypothesis, as this theory ignore the effect of transverse shear on the deformation of plates. This study therefore analytically examines the effect of induced shear on the deflection indices of plates with varying aspect ratios, using the characteristic orthogonal polynomial function. Result obtained shows a close agreement between present study and Kirchhoff’s hypothesis for membrane and thin plates. However, a significant difference was observed for moderately thick plates and thick plates, which clearly shows the effect of transverse shear as the plate thickness increases, which further validates the limitations of Kirchhoff’s hypothesis for moderately thick as well as thick plates. For an aspect ratio of 1.0 – 2.0 at 0.1 interval, results obtained indicated a percentage difference in deflection between the Present study and Kirchhoff’s hypothesis to range between -0.040 – 3.508%, 0.527 – 3.552%, 4.266 – 5.858%, and 13.980 – 17.011% for membrane, stiffened, moderately thick, and thick plates respectively. The validation of the Kirchhoff’s hypothesis for membrane as well as stiffened plates by the present study, indicates the suitability of the application of the characteristic orthogonal polynomial function in the evaluation of the deflection of plates regardless of thickness.

Natural Characteristics Analysis for the Spacecraft Equipped with Constructed Cantilever Solar Panels

The power series polynomial constraining method is proposed in this paper. The dynamical model of the cantilever plate can be established by applying the constraint, which is different from the traditional polynomial. Firstly, the characteristic orthogonal polynomial was used to describe the displacement field of the rectangular plate of which all edges are free. Then the four-sided free plate was equivalent to cantilever plate by power series multiplier constraint method. The characteristic equation of the constructed cantilever plate was obtained by the Rayleigh–Ritz method. Natural frequencies and modal shapes of the plate were obtained by solving the characteristic equation. Next, the proposed method was adopted to establish dynamical model of a pair of solar panels clamped on the central platform symmetrically. The convergence of the proposed method was verified by comparing the calculated results of the cantilever plate with that of the finite element software ANSYS 15.0. The optimum order the power series polynomial was obtained by comparing different results. The analysis of the dynamical characteristics of the cantilever plate and the spacecraft demonstrates the validation of the proposed method. This method can provide a new idea for the plate with local edge constrained.

Analysis of vibration characteristics of stators of electrical machines under actual boundary

This paper presents a general method for the studies of the vibration characteristics of an electrical machine stator with encased construction under practical boundary conditions. In this paper, artificial springs are introduced at both ends of the stator, and arbitrary boundary conditions can be simulated by adjusting the spring stiffness, including the support conditions in practical applications of the stator. In order to cooperate with the solution of the vibration characteristics of the stator with arbitrary edges, the Gram-Schmidt method is employed to construct characteristic orthogonal polynomial series to simulate the mode shape functions under arbitrary boundary conditions. The analysis is based on the Novozhilov shell theory, which is more applicable to the construction and actual operating environment of stators of electric vehicle drive motors, and takes into account moment of inertia neglected in the past. Further, the effects of teeth, windings, frame and cooling ribs are incorporated in the analysis, and the teeth with complicated shapes are considered as longitudinal ribs of unlimited cross-section, with bending and torsional coupling motion and warping deformation considered. The Rayleigh-Ritz method is used to build a unified dynamic analysis model of the stator of an electric machine, which can provide information on the radial, circumferential and axial vibrations of the stator with higher time efficiency, and a series of comparison studies and experiments are performed to demonstrate the general validity and accuracy of the model. On this basis, the possibility of construction optimization of the stator is discussed to improve the vibration and noise performance while reducing the cost.

Vibration Analysis of Rotating Combined Thin-Walled Shells With Multiple Conical Segments

Abstract In this paper, vibration analysis of rotating combined thin-walled shells with multiple conical segments has been carried out. Considering the centrifugal force, Coriolis force, and initial hoop tension due to rotation, the elastic strain energy and kinetic energy of a single rotating conical shell are expressed based on Love’s first approximation theory. The artificial springs are introduced to simulate the connections of adjacent conical shells and the boundaries of the rotating combined thin-walled shells. Taking characteristic orthogonal polynomial series as the admissible functions, the Rayleigh–Ritz method is employed to derive the frequency equations of the combined shell and corresponding vibration characteristics are then obtained. Given that the cylindrical shell and annular plates can be regarded as conical shells with semi-vertex angles of 0 deg and 90 deg, respectively, the solution given is also available for the vibration analysis of rotating combined thin-walled shells comprised of any segments of cylindrical, conical shell, and annular plate. As examples of rotating combined thin-walled shells with two and five segments, vibrations of rotating conical–conical joined shell and cylindrical–conical–cylindrical–conical–cylindrical joined shell are investigated in the paper. Traveling wave frequencies and corresponding mode shapes are shown, and the effects of rotating speed, circumferential wave number, spring stiffness, and semi-vertex angles on the vibration behavior are given in detail.

A 2D-sampling optimization method for buckling layup design of doubly-curved laminated composite shallow shells

A two-dimensional sampling optimization (2DSO) method is presented for buckling layup design of doubly-curved laminated composite shallow shells. The lamination parameters (LP) are utilized to represent distances of stacking sequences in composite laminates. Firstly, random continuous variables are generated that are distributed uniformly in LP design domain. Following that, stacking sequences are determined based on uniformly distributed LP. In terms of these stacking sequences, objective values are calculated, and a few best points are identified. Afterwards, the local sampling optimization is carried out iteratively in the vicinity of the best points until convergence is achieved. Lastly, a sequential permutation search (SPS) or a layerwise optimization approach (LOA) is served as the local optimization solver to target the final optimum. In addition, new Ritz solutions are derived for buckling of moderately thick doubly-curved laminated composite shallow shells using characteristic orthogonal polynomials based on a higher-order shear deformation theory (HSDT). Buckling loads of doubly-curved laminated composite shallow shells with different geometries, under multiple boundary conditions, subjected to different loadings, are optimized. Performances of 2DSO are compared to LOA, SPS, and genetic algorithm (GA), results illustrating that 2DSO outperforms LOA and SPS, and has similar robustness to GA while efficiency is multiplied several times.

General Solution of Thick Plates on Winkler-Pasternak Foundation Using Orthogonal Polynomial Displacement Functions

A solution for thick plates on Winkler-Pasternak foundation using characteristic orthogonal polynomials (COPs) displacement functions was presented in this study. A shear deformation function was developed from Soldatos’ (1992) trigonometric function and the Ritz energy approach was used to determine the total potential energy of the thick plate on elastic foundation. This method presents a simple and efficient method where only definite integration is required to obtain solutions. Results for non-dimensional in-plane and out-of-plane displacements for simply supported (SSSS) thick plates were presented and was observed to converge with that of other third-order shear deformation theories (TSDT) available in literature. From the study, it was observed that the in-plane and out-of-plane displacements reduced generally as the foundation modulus of the soil increased and decreases as the span-depth ratio increased for each aspect ratio. Both the Winkler and Pasternak foundation parameters influenced the results obtained as the foundation modulus is dependent on these parameters.

Dynamic modeling and orbit maneuvering response analysis for a three-axis attitude stabilized large scale flexible spacecraft installed with hinged solar arrays

An analytical dynamic model for a three-axis attitude stabilized large scale flexible spacecraft (LSFS) installed with hinged solar arrays is established. The natural property of the LSFS is investigated by simplifying it as a rigid central body hinged with two sets of multi-panel structures in this paper. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraints at hinges. For the model of LSFS, the Rayleigh-Ritz method is employed to obtain analytical global modes which are explicitly functions of space coordinates and effective for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. Moreover, the method is applied to investigate the parametric influence on natural characteristics for the LSFS. The paper develops a novel method to obtain rigid-flexible coupling global modes and construct the low-dimensional dynamical model for LSFS and study the rigid-flexible coupling dynamic response of in-orbit LSFS with complicated external excitation.

Investigation on flutter instability of magnetic-electric-thermo-elastic functionally graded plates in the supersonic airflow with any yawed angle

This paper proposes a general approach to deal with flutter instability of the magnetic-electric-thermo-elastic functionally graded plates (METE-FGPPs) subject to supersonic airflow with any yawed angle. The unified method is established on the basis of the first order shear deformation theory and the supersonic piston theory. Various boundary conditions, two porosity distributions and three temperature distributions are taken into consideration. The material properties and temperature gradient distribution changing continuously along the thickness are evaluated according to Voigt model and the formulations of motion equations of METE-FGPPs can be derived by adopting Rayleigh-Ritz method. In order to satisfy various boundary conditions easily, the physical spring technology is used. The displacement functions can be expressed by using the characteristic orthogonal polynomials to improve the convergence of the numerical calculations. Firstly, the convergence of the present method is confirmed by verifying the convergence of the physical spring and truncated numbers. Then, the versatility, stability and accuracy are verified by some numerical examples and experiments. Finally, the influences of the key parameters, including power law indexes, boundary conditions, magnetic potential, electric voltage, temperature and yawed flow angles etc, on the flutter instability of the METE-FGPPs are discussed in detail.

Study on analytical global modes for a multi-panel structure connected with flexible hinges

A dynamic modeling method for a multi-panel structure connected with flexible hinges is proposed in this paper. For the solar array composed of several flexible rectangular panels, the global explicit modal functions of spatial coordinates are obtained by employing the Rayleigh-Ritz method, which are in favor of deriving the lower-dimension discrete dynamic equations and the design of controllers for the system in the further research. The Gram-Schmidt characteristic orthogonal polynomials are adopted to construct the flexible deformation expressions of the solar panel. The Lagrange multipliers are introduced to describe the constraint at flexible hinges. The eigen equation of the multi-panel structure is derived to obtain the natural frequencies and corresponding global modes of the system through the Rayleigh-Ritz procedure. Considering the natural frequency calculated from the finite element method as a reference value, the correctness of the modeling process is validated by comparing results with those obtained from the finite element method. Moreover, a high accuracy of the proposed method are confirmed by a very good agreement between results from ANSYS and the theoretical method proposed here. The proposed method derives analytical global modes of the whole system which can be conveniently used to design the controller and work out the nonlinear dynamic model and with a few degree-of-freedoms directly. Finally, the interesting mode interchanged phenomenon is observed with the variation of the stiffness of the rotation spring at flexible hinges.

Free vibration analysis of rotating thin-walled cylindrical shells with hard coating based on Rayleigh-Ritz method

This paper focuses on the free vibration analyses of the rotating hard-coating cylindrical shells with various boundary conditions and the effects of hard coating on the vibration behaviors of the rotating shells. To accurately predict more natural characteristics, the characteristic orthogonal polynomials are taken as the admissible displacement functions. Considering the influences of Coriolis force, centrifugal force and initial hoop tension caused by rotation, the equations of motion of the shells are established by the use of the Rayleigh-Ritz method. Based on the state vector method, an efficient method is developed to solve the equations. By comparing with the results of both the finite element analysis and published literatures, the high accuracy and good convergence of the proposed model are verified. In addition, the effects of the boundary conditions, parameters of hard coating, rotating speed and number of circumferential waves on the vibration behaviors of the hard-coating shells are evaluated. This study may provide a reference for the application of hard-coating damping treatment to the vibration suppression of rotating thin-walled structures.

Solution of Rectangular Plate on Winkler’s Elastic Foundation Using Characteristic Orthogonal Polynomials

Solution of Rectangular Plate on Winkler’s Elastic Foundation Using Characteristic Orthogonal Polynomials

Applying the methods of BCOP and matrix exponential in the vibration analysis of multi-span composite plates under opposing moving inertia loads

The dynamic response of multi-span composite plates excited by moving inertia loads is scrutinized within this study. The masses are traveling on a two-lane trajectory in opposing direction with constant velocity. Two major types of opposing moving masses are comprehensively assessed here, including a case that two masses start to move in opposing direction simultaneously, and the other position when the second mass starts to move in the opposite direction just the first one left the plate. All case studies are presented for two cases of boundary conditions SFSF and SSSS. The complete solution of governing differential equations for the structure is achieved by a novel combination of two methods as Boundary Characteristic Orthogonal Polynomials (BCOP) and the robust numerical procedure of matrix exponential. Utilizing the features of orthogonal polynomials through the BCOP method, the natural frequencies would be extracted by proceeding a free vibration analysis. Having established the state-space formulation for the problem, the temporal response of equations would be derived using an iterative numerical procedure based on the matrix exponential functions. A comprehensive parametric study is conducted through numerical examples to show that the important parameters such as mass ratio, aspect ratio, velocity parameter as well as the parameter of the distance between lanes how could affect the dynamic behavior of FRP plates, under two-lane opposing moving masses. By introducing a conversion factor, the effect of inertia terms on the dynamic behavior of composite plats is evaluated.

Investigation on global analytic modes for a three-axis attitude stabilized spacecraft with jointed panels

The natural property of a three-axis attitude stabilized spacecraft is investigated by simplifying it as a rigid central body jointed with 4 solar panels in this paper. For the spacecraft composed of a rigid central body and flexible panels, the Rayleigh-Ritz method is employed to obtain global modes which are explicitly functions of space coordinates and are useful for the modeling of lower-order discrete dynamic equations and design of active vibration controllers for the system. The Gram-Schmidt process is used to construct a set of characteristic orthogonal polynomials as the displacement field of the solar panel. Lagrange multipliers are introduced to describe the constraint at joints. The characteristic equation of the whole system is derived through the Rayleigh-Ritz procedure. Then, natural frequencies and corresponding global modes of the spacecraft with jointed panels are obtained. Taking the natural frequency obtained from the finite element method as a reference value, the process of getting natural frequencies is validated by comparing results obtained with those from the finite element method. Moreover, an excellent convergence and high accuracy of the present method is demonstrated by a very good agreement between results from ANSYS and theoretical method proposed here. Finally, parametric studies on characteristics of the spacecraft are conducted. The interesting mode shift phenomenon is observed when parameters of the spacecraft are changed.

The thermal vibration characteristics of the functionally graded porous stepped cylindrical shell by using characteristic orthogonal polynomials

This paper presents a general approach for dealing with the thermal vibration characteristics of functionally graded porous stepped cylindrical shell (FGP-SCS) by employing characteristic orthogonal polynomials. The thermal vibration characteristics including free vibration, steady-state and transient response are investigated in detail. The governing equations are derived by Rayleigh-Ritz method and the displacement admissible functions are denoted by using characteristic orthogonal polynomials. The FGP-SCS is divided into N segments with different thickness and the coupling of adjacent segment is realized by connecting springs. The boundary conditions of FGP-SCS can be determined easily by selecting the proper stiffness values of boundary springs and the boundary conditions including classical and elastic boundary conditions. Three temperature distributions including uniform, linear and nonlinear distributions and two porosity distributions including even and uneven porosity distributions are taken into account. Meantime, the influences of some key parameters including power law index, porosity volume fraction, temperature change, geometric structure on the thermal vibration behaviors of FGP-SCS are investigated in detail.

On the Effect of Functionally Graded Materials on Resonances of Rotating Beams

Radially rotating beams attached to a rigid stem occur in several important engineering applications. Some examples include helicopter blades, turbine blades and certain aerospace applications. In most studies the beams have been treated as homogeneous. Here, with a goal of system improvement, non-homogeneous beams made of functionally graded materials are explored. The effects on the natural frequencies of the system are investigated. Euler-Bernoulli theory, including an axial stiffening effect and variations of both Young's modulus and density, is employed. An assumed mode approach is utilized, with the modes taken to be beam characteristic orthogonal polynomials. Results are obtained via Rayleigh-Ritz method and are compared for both the homogeneous and non-homogeneous cases. It was found, for example, that allowing Young's modulus and density to vary by approximately 2.15 and 1.15 times, respectively, leads to an increase of 23% in the lowest bending rotating natural frequency of the beam.

Simplification through regression analysis on the dynamic response of plates with arbitrary boundary conditions excited by moving inertia load

Dynamic response of a thin rectangular plate traversed by a moving inertia load with arbitrary boundary condition is investigated through this paper. The inertia effect of mass is considered and relevant formulation is established based on the full-term of acceleration, employing the method of Boundary Characteristic Orthogonal Polynomials, BCOP. To acquire the complete solution of partial differential equations governing on the plate, the Galerkin method is used to separate the temporal function from the spatial one. The problem is formulated in the state space and applying the numerical method of Matrix Exponential the complete solution would be achieved. In the numerical studies, a comprehensive parametric study is performed for both cases of loading when inertia effect is included or neglected. Several mass and aspect ratios for the plate with major types of boundary conditions CCCC, SSSS, CFCF and SFSF are accounted for presenting the results. Dynamic amplification factor against velocity parameter is scrutinized within many graphs alongside with a time history analysis of dynamic deflection for the plate's mid-span. Investigating on the dynamic response concludes to the critical boundary condition upon moving mass. By introducing a conversion factor, the margin of inertia and the critical velocity where happened would be achieved, then through a regression analysis a curve fitting model of polynomials is proposed. Corresponding coefficients testify the goodness of fit for such regression which are reported within tables. Referring to this simplified model of conversion factor pertaining to the specific boundary condition, it would be possible to handle the problem in moving load case without undertaking the complexities arisen from inertia contribution into the formulation. Having derived the factor from simplified model which has been calculated for a specific mass and velocity ratio, then multiplying into the moving load response, the complete solution for moving mass would be achieved.

Boundary characteristic orthogonal polynomials method in the vibration analysis of multi-span plates acting upon a moving mass

The Boundary Characteristic Orthogonal Polynomials (BCOP) method is used in this study in order to analyze multi-span plates traversed by a moving inertia load traveling on an arbitrary path with constant velocity. The plate is assumed to be free from any support at the longitudinal edges and the spans are made by simply supported constraints at width, i.e. SFSF. The plate's mode shapes are generated by the BCOP method while the boundary condition is satisfied over all computational modes. A free vibration analysis is done in order to find natural frequency. The governing differential equations of motion are derived by Hamilton's principle and the solution in the time domain is found by using the Matrix Exponential method after modeling the problem in state space. All of the convective inertia terms are included in the acceleration derivatives and the responses are presented both for the load moving on the plate's surface ignoring/including the mass inertia effect. A comprehensive parametric study on the plate's mid-spans is carried out for the single, two- and three-span plates, investigating Dynamic Amplification Factor (DAF) versus non-dimensional velocity (V). The effect of mass and aspect ratio along with the location of reference point of calculation on the dynamic behavior of a multi-span plate is investigated and many graphs are generated as spectra. One can easily find the critical velocity as well as the peak deflection for each case study by introducing a corrective factor. The solution under moving mass excitation is obtained by the factor if the same response for moving load is known.

Characteristic Orthogonal Polynomial based Galerkin Method for Solving Fractional Order Delay Differential Equations

Characteristic Orthogonal Polynomial based Galerkin Method for Solving Fractional Order Delay Differential Equations