Method Of Orthogonal Polynomials Research Articles

The Vibration Characteristics Analysis of Fiber-Reinforced Thin-Walled Conical–Cylindrical Composite Shells with Artificial Spring Technique

In this paper, the vibration characteristics (free vibration and forced vibration) of fiber-reinforced thin-walled conical–cylindrical composite shells (FTCCS) are analyzed by combining theory and experiment. Based on the classical shell theory and Kirchhoff–Love assumption, the overall structure of the FTCCS is theoretically modeled. The artificial spring technology is used to simulate the arbitrary boundary conditions at the joint, which is divided into a main spring and a secondary spring. At the same time, the displacement functions of the two sub-structure shells of the FTCCS are established by the orthogonal polynomial method, and the energy function of the FTCCS is constructed according to the continuity condition of the bolt joint. Then, the vibration characteristics of the FTCCS are solved using the Rayleigh–Ritz method and the modal superposition method. Finally, the TC300/epoxy resin-based fiber-reinforced thin-walled conical–cylindrical composite shells are selected as the research object, and the rationality of the mathematical model is verified by pulse, frequency sweeping and resonance excitation tests. The experimental results show that the errors between the natural frequency and the resonance displacement calculated by the theory and the experimental results are 1.75–4.76% and 3.79–9.28%, which confirms the rationality of the proposed model. By changing the length of the shell, the lamination angle and the semi-cone angle of the conical shell, the vibration characteristics parameters under different conditions are calculated to evaluate the influence of three physical parameters on the vibration characteristics of the FTCCS.

Rayleigh waves in viscoelastic piezoelectric half-space with cladding structures: An analytic Legendre-Laguerre polynomial method

Piezoelectric materials are widely used in surface acoustic wave devices. Many piezoelectric materials themselves have viscoelastic properties, and their surface wave characteristics, especially the attenuation characteristics still needs to be explored. This article proposes a Legendre-Laguerre orthogonal polynomial method to solve the Rayleigh wave problems in a viscoelastic piezoelectric half space with a covering layer. The proposed method compensates for the inherent shortcomings of traditional Laguerre polynomials in solving layered half spaces: the normal stress and electric displacement are discontinuous. The correctness of the method was verified through literature comparison and finite element simulation. At the same time, by utilizing the orthogonality of the Legendre-Laguerre polynomial, the integral analytical formula encountered in the solution process is derived, which improves the computational efficiency by more than ten times. Through the analysis and discussion of the dispersion and attenuation curves, it is found that the piezoelectric effect can suppress the attenuation of Rayleigh waves; the piezoelectric and viscous properties of the covering layer mainly affect attenuation at high frequencies, while those of the half space layer mainly affect attenuation at low frequencies of high-order modes.

Modal identification from turbulence response based on improved frequency domain decomposition

Turbulence excitation is an unavoidable form of excitation in flutter flight tests, and it is also a necessary and effective excitation method in high-speed flights, dives, other high-risk test, and high-frequency modal information mining. However, because the turbulence excitation signals are unmeasurable in the time domain, although the turbulence response contains rich and valuable flutter test information, the randomness and low quality of the data often cause difficulties in modal analysis. Therefore, an improved frequency domain decomposition algorithm for turbulence response processing is proposed in this paper. First, due to the irrelevancy of the atmospheric excitation, the power spectral density function matrix of the multi-channel turbulence response is subjected to singular value decomposition. There is a mathematical relationship between the maximum singular value curve and the system frequency response function. Second, the modal assurance criteria are used to calculate the maximum singular value of a single-degree-of-freedom system. Finally, an orthogonal polynomial method is applied to fit the maximum singular value curve, and the system identification is performed directly in the frequency domain. The simulated data and a certain type of aircraft flutter flight test data are used to verify the proposed method, and the results confirm the effectiveness and engineering applicability of the method developed in this work.

Factoring determinants and applications to number theory

Products of shifted characteristic polynomials, and ratios of such products, averaged over the classical compact groups are of great interest to number theorists as they model similar averages of [Formula: see text]-functions in families with the same symmetry type as the compact group. We use Toeplitz and Toeplitz plus Hankel operators and the identities of Borodin–Okounkov–Case–Geronimo, and Basor–Ehrhardt to prove that, in certain cases, these unitary averages factor as polynomials into averages over the symplectic group and the orthogonal group. Building on these identities we present new proofs of the exact formulas for these averages where the “swap” terms that are characteristic of the number theoretic averages occur from the Fredholm expansions of the determinants of the appropriate Hankel operator. This is the fourth different proof of the formula for the averages of ratios of products of shifted characteristic polynomials; the other proofs are based on supersymmetry; symmetric function theory, and orthogonal polynomial methods from random matrix theory.

103 Effects of partial or total replacement of dried distiller’s grain with soybean meal on growth performance, carcass characteristics, sera urea nitrogen, and net energy utilization in finishing beef steers

Abstract The objective of this study was to determine if partial or complete replacement of dried distiller’s grains plus solubles (DDGS) with solvent-extracted soybean meal (SBM) influences growth performance, efficiency of dietary net energy utilization, sera urea-N (SUN) concentrations, or carcass traits in finishing beef steers. Continental × British steers [initial shrunk body weight (BW) = 382 ± 37 kg] were used in a 139-d experiment. Treatments included: DDGS fed at 20% DM (13.4% CP; DDGS), SBM replacing 50% of DDGS (14.4% CP; SBM50), and SBM replacing 100% of DDGS (15.4% CP; SBM100). Steers (n = 189) were weighed and allotted to 1 of 24 pens (n = 8 steers per pen; 8 pens per treatment) at the Ruminant Nutrition Center near Brookings, SD. Whole blood was collected and harvested as sera on d 77, 105, and 139 to determine SUN concentration. Growth and carcass data were analyzed as a randomized complete block design using the GLIMMIX procedure of SAS 9.4 with treatment and block (initial BW grouping) as fixed effects; SUN data were analyzed using repeated measures, pen was the experimental unit. Inclusion level was evaluated for linear and quadratic components by the method of orthogonal polynomials. Final BW was calculated by dividing hot carcass weight (HCW) by 0.625. No differences were noted (P ≥ 0.18) for final BW, average daily gain, dry matter intake, or feed efficiency. Dietary treatment tended to affect gain to feed ratio (G:F; quadratic; P = 0.10) and SBM increased the apparent efficiency of energy capture. Treatment had no effect (P ≥ 0.22) on HCW, rib fat, marbling score, USDA yield grade, or final BW adjusted to 28% empty body fat. Complete replacement of DDGS with SBM increased rib eye area by 1% (linear; P = 0.02), but decreased dressing percentage (linear, P = 0.03). Distribution of USDA quality or yield grades were unaffected by treatment (P ≥ 0.36). No treatment × day interaction was observed for SUN (P = 0.20). However, day (P < 0.01) and treatment (P < 0.01) effects were observed. Initial samples (d 77) differed between DDGS and SBM treatments, with the latter having greater SUN values (P < 0.01). On d 105, the SBM100 treatment had greater SUN than DDGS, and SBM50 was intermediate (P < 0.01). On d 139, steers from SBM100 had elevated increased SUN compared with SBM50 and DDGS treatments (P < 0.01). Net energy values calculated from animal performance for DDGS agreed closely with tabular estimates. In this experiment, replacement of DDGS with SBM increased the observed to expected ratio of dietary NE, increased muscularity and tended to decrease DP with no other observed effects on growth performance or carcass traits.

Vibrations characteristics analysis of metal composite 17–4PH material extrusion additively manufactured parts under amplitude dependence property

The ultra-fuse 17–4 PH stainless-steel filament has emerged as a promising new material in the 3D printing field during this past decade. Since the introduction recently of the 17–4 PH stainless-steel filament in the material extrusion process, extensive research has been carried out on the mechanical performance (tensile strength, plasticity, and surface roughness) of the metal composite 17–4 PH stainless-steel material extrusion additively manufactured parts, however few of them mainly focused on the dynamic performance of the 17–4 PH stainless-steel material extrusion additively manufactured parts thus leaving a slight void, therefore to fill this gap a dynamic linear model of a metal composite filament ultra-fuse 17–4 PH stainless-steel material extrusion additively manufactured parts has been established in this study considering the amplitude dependence. Material extrusion additively manufactured plates have been built in x and z-direction, and based on the Jones-Nelson model an investigation of the natural frequency and response and damping ratio of ultra-fuse metal composite 17–4 PH metal composite material extrusion specimens has been completed. Equations of the proposed dynamic model have been solved by the combination of the orthogonal polynomial method and energy method. The Newton-Raphson iteration has been employed to predict the natural frequencies and responses of the ultra-fuse 17–4 PH stainless-steel material extrusion manufactured parts. The results indicate that the predicted damping characteristics of the ultra-fuse metal composite 17–4 PH metal composite material extrusion parts have a good consistency with the experimental ones, meanwhile, when amplitude vibration increases from [0.25 g…2.00 g], the response of the ultra-fuse 17–4 PH metal composite material extrusion parts increase as well both for x and z-direction samples.

The orthogonal polynomials method using Gegenbauer polynomials to solve mixed integral equations with a Carleman kernel

<abstract> <p>The orthogonal polynomials approach with Gegenbauer polynomials is an effective tool for analyzing mixed integral equations (MIEs) due to their orthogonality qualities. This article reviewed recent breakthroughs in the use of Gegenbauer polynomials to solve mixed integral problems. Previous authors studied the problem with a continuous kernel that combined both Volterra (V) and Fredholm (F) components; however, in this paper, we focused on a singular Carleman kernel. The kernel of FI was measured with respect to position in the space <inline-formula><tex-math id="M1">\begin{document}$ {L}_{2}[-\mathrm{1, 1}], $\end{document}</tex-math></inline-formula> while the kernel of Ⅵ was considered as a function of time in the space <inline-formula><tex-math id="M2">\begin{document}$ C[0, T], T < 1 $\end{document}</tex-math></inline-formula>. The existence of a unique solution was discussed in <inline-formula><tex-math id="M3">\begin{document}$ {L}_{2}\left[-\mathrm{1, 1}\right]\times C\left[0, T\right] $\end{document}</tex-math></inline-formula> space. The solution and its error stability were both investigated and commented on. Finally, numerical examples were reviewed, and their estimated errors were assessed using Maple (2022) software.</p> </abstract>

FREE VIBRATION MODELING IN A FUNCTIONALLY GRADED HOLLOW CYLINDER USING THE LEGENDRE POLYNOMIAL APPROACH

Introduction: The building industry is under increasing pressure to maximize performance while reducing the costs and the environmental impact. To solve this problem, a new type of materials, i.e., functionally graded materials (FGMs), are proposed. These materials have the advantage of being able to withstand harsh environments without losing their properties. Purpose of the study: The paper aims to further extend the understanding of the propagation modes and characteristics of guided waves in FGM cylinders with infinite lengths. In the course of the study, we analyzed a cylindrical shell composed of three annular layers, each separated by a gradient layer across the wall thickness. A modeling tool based on the Legendre orthogonal polynomial method is proposed in the paper. Methods: The method applied results in an eigenvalue/eigenvector problem. The boundary conditions are integrated into the constitutive equations of guided wave propagation. The phase velocity and normalized frequency dispersion curves are calculated. Besides, the displacement distributions and stress field profiles for a functionally graded cylinder with various graded indices in both modes (axisymmetric and symmetric) are calculated and discussed. The results show a constant fluctuation of effective FGM material. Results: It was found that the phase velocity curves of the same mode decrease as the exponents of the power law increase. In addition, the boundary conditions have a greater impact on the normal stresses. The accuracy and effectiveness of the improved orthogonal polynomial method are demonstrated through a comparison of the exact solution obtained by an analytical-numerical method and our numerical results.

Моделювання вимушених гармонічних коливань прямокутного фундамента на водонасиченій шаруватій основі

A numerical study of the dynamic contact problem is carried out by the method of orthogonal polynomials. А symbolic solution was found for the double transformants of the vertical displacements of the layered two-phase poroelastic liquid saturated base under the action of arbitrary vertical contact pressures. Numerical analysis was performed for the Rayleigh function and the components of displacements in the kernels of integrable functions. An example of calculating double integrals necessary to determine the contact pressures of two-phase soil under the base of the foundation is given.

Hidden ring crack in a rotating hollow cylinder under torsion

We consider the impact of a ring crack within a rotating hollow cylinder of fixed height under axisymmetric (torsion) loading. The form of the displacement is obtained from the equation of motion using the Fourier sin transform. The displacement jump over the crack is obtained from the boundary condition on the tangential stress, formulated as a singular integral equation which is solved by the method of orthogonal polynomials. The stress intensity factors on the opposing crack surfaces are calculated. The dependence of the crack extension on the problem geometry is investigated, including the impact of the crack’s location, cylinder’s height, torsion loading and rotation frequency. Possible extensions of the model to cover fatigue cracking are considered. Whether the presented model can be used to develop a practical test for detecting cracks is investigated.

Rayleigh Waves Propagating in the Functionally Graded One-Dimensional Hexagonal Quasicrystal Half-Space

For the manufacturing and optimization of quasicrystal structures, Rayleigh waves propagating in the functionally graded one-dimensional hexagonal quasicrystal half-space are investigated. The analytical Laguerre orthogonal polynomial method is employed to solve dynamic equations of wave motion, which greatly improves the computational efficiency. Dispersion curves and displacement distributions are illustrated. The influences of the phonon–phason coupling effect, inhomogeneity, and quasiperiodic direction on wave characteristics are analyzed. Some new results are obtained: (1) Compared with the classical Laguerre polynomial method, the improvement in computational efficiency of the analytical Laguerre polynomial method is more than 99%. (2) The energy penetration depth of phason modes is greater than that of phonon modes. The results lay a theoretical foundation for designing and optimizing SAW devices.

Formulation and antibacterial effectiveness of fermented soybean husk extracts from industrial waste of tempe as eco-friendly hand sanitizer gel

This study aims to determine the formulation and antibacterial effectiveness of fermented soybean seed husk extract from tempe industrial waste as an environmentally friendly hand sanitizer gel. This type of research is an experimental study with variations in fermented soybean husk extract concentration, namely 0%, 5%, 15%, 20%, 50%, and 100%. Observational data were tested by analysis of variance ANOVA at a 5% level using Co-Stat software. Suppose there is a significant difference, a further test of Honest Significant Difference (BNJ) (organoleptic test), a further test of Orthogonal Polynomial Method (MOP) (spreadability, pH, viscosity), descriptive analysis test (flavonoid test, homogeneity test, antibacterial activity test is carried out) and test the total number of germs). The results showed that the fermented extract of soybean seed husk was positive for flavonoids, characterized by a bright yellow color change when added with 10% NaOH. The concentration of fermented soybean husk extract significantly affected the hand sanitizer gel's organoleptic properties, dispersion, total pH, and viscosity. Based on the organoleptic test, the panelists preferred the clear gel color in the F1 treatment (0%) and the thick, slightly liquid gel form with a slightly scented soybean husk in the F3 treatment (10%). The concentration treatment of soybean husk fermented extract (50%) was the best treatment based on good homogeneity, total pH 4.35, viscosity 2405 cPs, following SNI No.06-2588, and spreadability of 4.47 cm which was close to hand gel. For commercial sanitizers, the diameter of the inhibition zone against Escherichia coli bacteria ATCC 2922 and Staphylococcus aureus ATCC 2923 is 14 mm and 12 mm.

The effect of kurut wood (Dysoxylum parasiticum) and jackfruit wood (Artocarpus heterophyllus) on the natural fermentation of palm juice and palm sugar quality

This study aims to determine the effect of the concentration of kurut wood and jackfruit wood on the yield, pH, and percentage of reducing sugars of palm sugar. This experiment used a completely randomized design (CRD) which consists of 6 treatments, i.e., 0 g/L, 3 g/L, 4 g/L, 5 g/L, 6 g/L, and 7 g/L. The data were analyzed by Analysis of Variance (ANOVA) using Co-Stat software and posthoc analysis for chemical score using Orthogonal Polynomial Method (MOP). Statistical analysis of chemical value shows that the increase in the concentration of kurut wood and jackfruit wood (0 g/L – 7 g/L) significantly affects the decrease of the total microbe and percentage of reducing sugars, and the increase of the amount of yield, palm juice, and sugar pH. The 7 g/L treatment was the best, with different analysis results for kurut wood and jackfruit wood. In kurut wood, the total microbes counted 5.46 CFU/ml; yield 495.33g; pH 6.72; and reducing sugar 0.96. In jackfruit wood, total microbes counted 5.6 CFU/ml; yield 386.33g; pH 6.68; and reducing sugar 0.85.

On a Solution of a Third Kind Mixed Integro-Differential Equation with Singular Kernel Using Orthogonal Polynomial Method

This paper deals with the solution of a third kind mixed integro-differential equation (MIDE) in displacement type in space L 2 − 1 , 1 × C 0 , T , T < 1 . The singular kernel is modified to take a logarithmic form, while the kernels of time are continuous and positive functions. Using the separation of variables technique, we have a system of Fredholm integral equations (FIEs) that can be transformed into an algebraic system after using orthogonal polynomials. In all the previous researchers’ works, the time periods were divided, and the mixed equation transformed into an algebraic system of FIEs. While when using the separation method, we are able to obtain FIE with time coefficients, and these functions are described as an integral operator in time. Thus, we can study the behavior of the solution with the time dimension in a broader and deeper than the previous one. Some examples are given and discussed to show the performance and efficiency of the proposed methods.

ПЛОСКА ЗАДАЧА ПРО КОЛИВАННЯ БАЛКОВОЇ ПЛИТИ НА ПРЯМОКУТНІЙ ОСНОВІ

he plane contact problem of a beam’s oscillations on a rectangular elastic base is considered. The vertical edges of the rectangular base are in conditions of the nonfriction contact, the lower edge is fixed, and a beam (beam plate) with free ends is attached to the upper edge. The normal loading is applied to the beam and harmonically changes in time. To solve the boundary valued problem for the elastic base the integral transform method is applied. The apparatus of the Green’s function is used to construct the solution for the boundary valued problem for a beam. The displacements of the rectangular base and the deflection of the beam were found. The interface condition between the base and the plate is used to derive the integral singular equation relatively the dynamical contact stress. The orthogonal polynomial method was used to solve the integral equation. The investigation of the oscillations’ frequency influence on the deflection of beam and elastic rectangular base’s displacements and stress was conducted.

A fundamental approach: E-polarized electromagnetic wave diffraction by two dimensional arbitrary-shaped objects with impedance boundary condition

Abstract In the present study, a new methodology in computational electromagnetics is developed for two-dimensional arbitrarily-shaped objects with impedance boundary conditions. The proposed approach investigates the E-polarized electromagnetic diffraction by a two-dimensional object with the Leontovich boundary condition. The scattered electric and magnetic fields are expressed as the convolution integral of the corresponding Green’s function and the current induced on the obstacle surface. After obtaining integral equations by applying the boundary condition, the integral equations are solved as in the case of the method of auxiliary sources (MAS) which is a well-known method in computational electrodynamics. The results are compared with first, different methods such as the method of moments (MoM), orthogonal polynomials (OP), and second, different boundary conditions such as Dirichlet, Neumann, and fractional boundary conditions. Some results are also obtained for the different shape scatterers at some values of the surface impedance.

Characteristics of complete circumferential guided wave in a piezoelectric semiconductor cylindrical shell

In this paper, the characteristics of circumferential Lamb-like wave in a cylindrical shell of N-type piezoelectric semiconductor are investigated by employing the analytic Legendre orthogonal polynomial method (ALOPM). Based on the linear phenomenological theory, the governing differential equations are derived for different electrical boundary conditions, and transformed into a matrix eigenvalues problem via using the ALOPM. The dispersion solutions for a piezoelectric semiconductor cylindrical shell with big radius-thickness ratio obtained from the ALOPM are verified by comparing with the results from the global matrix method (GMM). Three-dimensional spectrum is plotted for a better observation. The spectrum of piezoelectric semiconductor is symmetrically distributed about the frequency axis, which is different from the piezoelectric dielectric material. The effects of different electrical boundary conditions, steady-state carrier density and radius-thickness ratio on phase velocity dispersion curves are illustrated. The amplitude distributions of displacement, electric potential and perturbation of the carrier density are studied. The effects of steady-state carrier density on the stress distribution are also illustrated. Non-propagating wave modes with low attenuation and high phase velocity value are found. The corresponding results presented in this work can be used to optimize the design of piezoelectric semiconductor acoustic wave devices.

Size-dependent and piezoelectric effects on SH wave propagation in functionally graded plates

This paper is concerned with the propagation of shear horizontal waves in a functionally graded piezoelectric small-scaled plate. In the context of the modified couple stress theory, the Legendre orthogonal polynomial method is utilized to derive the solutions based on the explicit functionally graded model. Four different electric boundary conditions are considered. The obtained results are compared with those by the global matrix method and the advantages of the polynomial method are indicated. Furthermore, the couple stress effect and piezoelectric effect on the SH waves are investigated. Numerical results indicate that the couple stress and piezoelectric coupling effects on SH propagation are mutually inhibited.

Hertzian and adhesive plane models of contact of two inhomogeneous elastic bodies

Previous study of contact of power-law graded materials concerned the contact of a rigid body (punch) with an elastic inhomogeneous foundation whose inhomogeneity is characterised by the Young modulus varying with depth as a power function. This paper models Hertzian and adhesive contact of two elastic inhomogeneous power-law graded bodies with different exponents. The problem is governed by an integral equation with two different power kernels. A nonstandard method of Gegenbauer orthogonal polynomials for its solution is proposed. It leads to an infinite system of linear algebraic equations of a special structure. The integral representations of the system coefficients are evaluated, and the properties of the system are studied. It is shown that if the exponents coincide, the infinite system admits a simple exact solution that corresponds to the case when the Young moduli are different but the exponents are the same. Formulas for the length of the contact zone, the pressure distribution and the surface normal displacements of the contacting bodies are obtained in the form convenient for computations. Effects of the mismatch in the Young moduli exponents are studied. A comparative analysis of the Hertzian and adhesive contact models clarifies the effects of the surface energy density on the contact pressure, the contact zone size and the profile of the contacting bodies outside the contact area.

An Efficient Orthogonal Polynomial Method for Auxetic Structure Analysis with Epistemic Uncertainties

Traditional approaches used for analyzing the mechanical properties of auxetic structures are commonly based on deterministic techniques, where the effects of uncertainties are neglected. However, uncertainty is widely presented in auxetic structures, which may affect their mechanical properties greatly. The evidence theory has a strong ability to deal with uncertainties; thus, it is introduced for the modelling of epistemic uncertainties in auxetic structures. For the response analysis of a typical double-V negative Poisson’s ratio (NPR) structure with epistemic uncertainty, a new sequence-sampling-based arbitrary orthogonal polynomial (SS-AOP) expansion is proposed by introducing arbitrary orthogonal polynomial theory and the sequential sampling strategy. In SS-AOP, a sampling technique is developed to calculate the coefficient of AOP expansion. In particular, the candidate points for sampling are generated using the Gauss points associated with the optimal Gauss weight function for each evidence variable, and the sequential-sampling technique is introduced to select the sampling points from candidate points. By using the SS-AOP, the number of sampling points needed for establishing AOP expansion can be effectively reduced; thus, the efficiency of the AOP expansion method can be improved without sacrificing accuracy. The proposed SS-AOP is thoroughly investigated through comparison to the Gaussian quadrature-based AOP method, the Latin-hypercube-sampling-based AOP (LHS-AOP) method and the optimal Latin-hypercube-sampling-based AOP (OLHS-AOP) method.