Admissible Functions Research Articles

Free Vibration Analysis of Arbitrary‐Shaped Plates Based on the Improved Rayleigh–Ritz Method

In this investigation, an improved Rayleigh–Ritz method is put forward to analyze the free vibration characteristics of arbitrary‐shaped plates for the traditional Rayleigh–Ritz method which is difficult to solve. By expanding the domain of admissible functions out of the structural domain to form a rectangular domain, the admissible functions of arbitrary‐shaped plates can be described conveniently by selecting the appropriate admissible functions. Adopting the spring model to simulate the general boundary conditions, the problems of vibration of the arbitrary plate domain can be solved perfectly. Then, a numerical method is introduced to figure out the structure strain energy, kinetic energy, and elastic potential energy of the boundary. Finally, comparing the result with the simulation results and reference examples, the accuracy and convergence of this method are testified. Therefore, an effective new method is proposed for the guidance of the related research and practical engineering problems.

Inhomogeneous bonding state modeling for vibration analysis of explosive clad pipe

Early detection of damage bonding state such as insufficient bonding strength and interface partial contact defect for the explosive clad pipe is crucial in order to avoid sudden failure and even catastrophic accidents. A generalized and efficient model of the explosive clad pipe can reveal the relationship between bonding state and vibration characteristics, and provide foundations and priory knowledge for bonding state detection by signal processing technique. In this paper, the slender explosive clad pipe is regarded as two parallel elastic beams continuously joined by an elastic layer, and the elastic layer is capable to describe the non-uniform bonding state. By taking the characteristic beam modal functions as the admissible functions, the Rayleigh-Ritz method is employed to derive the dynamic model which enables one to consider inhomogeneous system and any boundary conditions. Then, the proposed model is validated by both numerical results and experiment. Parametric studies are carried out to investigate the effects of bonding strength and the length of partial contact defect on the natural frequency and forced response of the explosive clad pipe. A potential method for identifying the bonding quality of the explosive clad pipe is also discussed in this paper.

Dynamic interaction between elastic plate and transversely isotropic poroelastic medium

In this paper, dynamic response of an elastic circular plate, under axisymmetric time-harmonic vertical loading, resting on a transversely isotropic poroelastic half-space is investigated. The plate-half-space contact surface is assumed to be smooth and fully permeable. The discretization techniques are employed to solve the unknown normal traction at the contact surface based on the solution of flexibility equations. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results corresponding to the deflections of a circular plate, with different degrees of flexibility, resting on a transversely isotropic poroelastic half-space are presented.

A Semianalytical Three‐Dimensional Elasticity Solution for Vibrations of Orthotropic Plates with Arbitrary Boundary Conditions

A semianalytical three‐dimensional (3D) elasticity solution for the vibration of the orthotropic plate is presented under arbitrary boundary conditions. Three‐dimensional (3D) elasticity theory provides the theoretical support for the energy function of orthotropic plates. The orthotropic plates which have the arbitrary boundary condition are realized by the way of arranging three sets of linear springs at the edges. With the aim of eliminating the nonsmooth phenomenon at the edges, the admissible displacement function of an orthotropic plate is expressed with a modified Fourier series solution. Under this framework, a change that occurs on the boundary conditions only needs to modify the boundary parameters of the orthotropic plate, without the need for new derivation, thus greatly saving the modeling time. The convergence and accuracy of the proposed method are better than those of the published literature. Lastly, the new vibration results and parametric research of thick orthotropic plates as well as the geometric parameter are also presented.

Free Vibration Analysis of Functionally Graded Spherical Torus Structure with Uniform Variable Thickness along Axial Direction

Based on the Ritz method, this paper focused on the free vibration of functionally graded (FG) spherical torus with uniform variable thickness along axial direction under different boundary conditions. The first‐order shear deformation theory (FSDT) is employed to formulate the analytical model. The method involves partitioning of the spherical torus structure into proper shell segments in order to satisfy the computing requirement of high‐order vibration responses according to the domain decomposition method. The two adjacent segments are connected by using the penalty method, where penalty parameters are defined by the artificial springs; the continuity condition and different boundary conditions can be obtained by assigning the appropriate values of springs. The displacement functions’ components are double mixed series, in which Fourier series and unified Jacobi polynomials, respectively, represent displacement function along circumferential direction and axial direction. Then the Ritz method is used to obtain final solutions. The numerical results obtained by the proposed method show great agreement with previously published literatures and those from the finite element program ABAQUS. The effects of boundary conditions and geometric parameters on the vibration responses of the structure are also presented. The most novelty of this paper is to generalize the selection of admissible displacement functions by using Jacobi polynomial.

Applications of third order differential subordination and superordination involving generalized struve function

In the present paper, by making use of the linear operator associated with generalized Struve functions suitable classes of admissible functions are investigated and the dual properties of the third-order differential subordinations are presented. As a consequence, various sandwich-type results are established for a class of univalent analytic functions involving generalized Struve functions. Relevant connections of the new results presented here with those that were considered in earlier works are pointed out.

Kato's inequalities for admissible functions to quasilinear elliptic operators <i>A </i>

Kato's inequalities for admissible functions to quasilinear elliptic operators <i>A </i>

Analytical solution for free vibration of multi-span continuous anisotropic plates by the perturbation method

Accurately determining the natural frequencies and mode shapes of a structural floor is an essential step to assess the floor\'s human-induced vibration serviceability. In the theoretical analysis, the prestressed concrete floor can be idealized as a multi-span continuous anisotropic plate. This paper presents a new analytical approach to determine the natural frequencies and mode shapes of a multi-span continuous orthotropic plate. The suggested approach is based on the combined modal and perturbation method, which differs from other approaches as it decomposes the admissible functions defining the mode shapes by considering the intermodal coupling. The implementation of this technique is simple, requiring no tedious mathematical calculations. The perturbation solution is validated with the numerical results.

Free Vibration of Rectangular Plates with Porosity Distributions under Complex Boundary Constraints

Free vibration of rectangular plates with three kinds of porosity distributions and different boundary constraints has been performed by means of a semianalytical method. The distribution of porous varies along the thickness of the plate, in which the mechanical properties are defined by open‐cell metal foam. Regardless of boundary conditions, displacement admissible functions are represented by combination of standard cosine Fourier series and auxiliary sine series. The kinetic energy and potential energy of plates are also expressed on the basis of first‐order shear deformation theory (FSDT) and displacement admissible functions. Finally, the coefficients in the Fourier series which determine natural frequencies and modal shape are derived by means of the Rayleigh–Ritz method. Convergence and dependability of the current method are verified by comparing with the results of FEM and related literatures. In addition, some new results considering geometry parameters under classical and elastic boundary constraints are listed. The effects of geometry parameters, material parameters, and boundary constraints have been discussed in detail.

Free vibrations of functionally graded porous rectangular plate with uniform elastic boundary conditions

In this paper, the free vibrations of functionally graded porous (FGP) rectangular plate with uniform elastic boundary conditions is investigated by means of an improved Fourier series method (IFSM). It is assumed that the distributions of porosity are uniform or non-uniformly along a certain direction and three types of the porosity distribution are considered, among which material property of two non-uniform porous distributions was expressed as the simple cosine. The size of the pore in a rectangular plate is determined by the porosity coefficients. Using the first-order shear deformation theory(FSDT), the energy expression of FGP rectangular plate is created. In order to obtain the admissible function of displacement for functionally graded porous rectangular plate, the IFSM is employed. Then, the Rayleigh-Ritz method is used to solve coefficients in the Fourier series which determine natural frequencies and modal shapes. Convergence and comparative research are performed to prove the convergence, reliability and accuracy of the current method. On this foundation, some new results covering the influence of the geometrical parameters subject to classical and elastic boundary condition are presented, and the parametric studies are also investigated in detail, which can provide a reference for future research by other researchers.

Free vibration analysis of FGM plates based on Rayleigh-Ritz method

Rayleigh-Ritz method was introduced to analyze the vibration characteristics of functionally graded material (FGM) rectangular plate with complex boundary conditions. The Improved Fourier series was chosen as the admissible function for its great property to be used universally in various boundary conditions. The virtual spring model was adopted to simulate the complex boundary conditions. The strain and kinetic energy of the structure and elastic potential energy of the border were easily calculated by using relevant formula. Finally, comparing the result with the references, the accuracy and convergence can be proved.

Control Digital System Function Set Defining While Debugging Tests Development

Digital control systems are considered, the functioning of which can be represented as a sequence of functions from a finite alphabet. For such systems projects debugging by simulation it is necessary to generate some set of tests for the applying on the simulated system to verify that it is functioning correctly. This paper is devoted to the development of test sets for function successions correctness. It is shown that on admissible function successions partly defined semigroup exists. There exists also word set on limited alphabet of functions, and they could be defined by some right liner grammar. Admissible successions are formally described by the graph of functions. Such a graph defines admissible functions for all digital system states. Digital system function set development is proposed in a way that admissible function successions could be defined as a graph. If the admissibility of two functions fulfillment one after another depends on previously fulfilled functions and the digital system internal state, then some functions should be divided into several subfunctions. The method of such a process is described. Developed graph of functions together with input interaction set for each digital system function define specification for digital system behavior. Proposed method is illustrated on the drawing machine control digital system functions development. The method of test set development on graph function is proposed.

Modelling a spade rudder as a hollow two-way tapered Kirchhoff’s plate : free dry and wet vibration study with numerical verification

A semi-analytical approach to free dry and wet vibration of a trapezoidal, 2-way tapered, pivoted hollow spade rudder is presented. The rudder is modeled as a hollow Kirchhoff’s plate, with a NACA0018 profile chord section. The rudder pivot is modeled as a combination of a translational spring and a rotational spring. The span-wise and chord-wise non-uniform beam vibration is first analyzed by the Rayleigh-Ritz method, to establish the non-uniform beam modeshapes, which act as admissible functions to the Galerkin’s method for plate vibration. Eigenvalue analysis generates the plate natural frequencies and the plate modeshapes. Alternately, uniform beam modeshapes themselves are used as admissible functions into the Galerkin’s method. Frequencies are analyzed for various pivot positions, taper ratios, and NACA sections. For the wet vibration, constant strength source distribution technique is used to generate the added mass of a 2D aerofoil. Also, 3D panel method is used to generate the modal added masses, and hence the wet natural frequencies. The added mass coefficient is generated for various aerofoil fineness ratios, pivot fixities, taper ratios, aspect ratios.

A note on the Blaschke-Petkantschin formula, Riesz distributions, and Drury’s identity

Abstract The Blaschke-Petkantschin formula is a variant of the polar decomposition of the k-fold Lebesgue measure on ℝ n in terms of the corresponding measures on k-dimensional linear subspaces of ℝ n . We suggest a new elementary proof of this famous formula and discuss its connection with Riesz distributions associated with fractional powers of the Cayley-Laplace operator on matrix spaces. Another application of our proof is the celebrated Drury identity that plays a key role in the study of mapping properties of the Radon-John k-plane transforms. Our proof gives precise meaning to the constants in Drury’s identity and to the class of admissible functions.

Thermodynamic calculation of spin scaling functions

Critical phenomena theory centers on the scaled thermodynamic potential per spin ϕ(β,h)=|t|pY(h|t|−q), with inverse temperature β=1/T, h=−βH, ordering field H, reduced temperature t=t(β), critical exponents p and q, and function Y(z) of z=h|t|−q. I discuss calculating Y(z) with the information geometry of thermodynamics. Scaled solutions are found to obtain with three admissible functions t(β): 1) t=e−Jβ, 2) t=β−1, and 3) t=βC−β, where J and βC are constants. For p=q, information geometry yields Y(z)=1+z2, consistent with the one-dimensional (1D) ferromagnetic Ising model.

Geometrically nonlinear vibration of laminated composite cylindrical thin shells with non-continuous elastic boundary conditions

The geometrically nonlinear forced vibration response of non-continuous elastic-supported laminated composite thin cylindrical shells is investigated in this paper. Two kinds of non-continuous elastic supports are simulated by using artificial springs, which are point and arc constraints, respectively. By using a set of Chebyshev polynomials as the admissible displacement function, the nonlinear differential equation of motion of the shell subjected to periodic radial point loading is obtained through the Lagrange equations, in which the geometric nonlinearity is considered by using Donnell’s nonlinear shell theory. Then, these equations are solved by using the numerical method to obtain nonlinear amplitude–frequency response curves. The numerical results illustrate the effects of spring stiffness and constraint range on the nonlinear forced vibration of points-supported and arcs-supported laminated composite cylindrical shells. The results reveal that the geometric nonlinearity of the shell can be changed by adjusting the values of support stiffness and distribution areas of support, and the values of circumferential and radial stiffness have a more significant influence on amplitude–frequency response than the axial and torsional stiffness.

Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line

For nonuniform exponentially bounded evolution families on the half-line we introduce a class of Banach function spaces on which we define nonuniform evolution semigroups. We completely characterize nonuniform exponential stability in terms of invertibility of the corresponding generators. We emphasize that in particular our results apply to all linear differential equations with bounded operator and finite Lyapunov exponent.

A modeling method for vibration analysis of cracked beam with arbitrary boundary condition

This paper establishes a cracked Timoshenko beams model to investigate the vibration behavior based on the ultraspherical polynomials. Timoshenko beam theory is applied to model the free vibration analysis of the cracked beam and the numerical results are obtained by using ultraspherical orthogonal polynomials. The boundary conditions of both ends of the cracked beam are modeled as the elastic spring and the beam is divided into two parts by the crack section, and continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of fracture mechanics theory. Ignoring the influence of boundary conditions, displacements admissible functions of cracked Timoshenko beam can be set up as ultraspherical orthogonal polynomials.The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method (FEM). In addition, the effects of flexibility coefficient on the natural frequencies are also investigated by using the proposed method. Numerical examples are given for free vibration analysis of cracked beams with various boundary conditions, which may be provided as reference data for future study.

A solution set-based entropy principle for constitutive modeling in mechanics

Entropy principles based on thermodynamic consistency requirements are widely used for constitutive modeling in continuum mechanics, providing physical constraints on a priori unknown constitutive functions. The well-known M\"uller-Liu procedure is based on Liu's lemma for linear systems. While the M\"uller-Liu algorithm works well for basic models with simple constitutive dependencies, it cannot take into account nonlinear relationships that exist between higher derivatives of the fields in the cases of more complex constitutive dependencies. The current contribution presents a general solution set-based procedure, which, for a model system of differential equations, respects the geometry of the solution manifold, and yields a set of constraint equations on the unknown constitutive functions, which are necessary and sufficient conditions for the entropy production to stay nonnegative for any solution. Similarly to the M\"uller-Liu procedure, the solution set approach is algorithmic, its output being a set of constraint equations and a residual entropy inequality. The solution set method is applicable to virtually any physical model, allows for arbitrary initially postulated forms of the constitutive dependencies, and does not use artificial constructs like Lagrange multipliers. A Maple implementation makes the solution set method computationally straightforward and useful for the constitutive modeling of complex systems. Several computational examples are considered, in particular, models of gas, anisotropic fluid, and granular flow dynamics. The resulting constitutive function forms are analyzed, and comparisons are provided. It is shown how the solution set entropy principle can yield classification problems, leading to several complementary sets of admissible constitutive functions; such problems have not previously appeared in the constitutive modeling literature.

A modified series solution for free vibration analyses of moderately thick functionally graded porous (FGP) deep curved and straight beams

As a novel class of weight-efficient engineering materials, the functionally graded porous (FGP) beam structures have great potential value. However, the current research on it is relatively small. Based on this research status, the aim of this paper is establishing a unified analytical model to study the vibration behavior of moderately thick functionally graded porous deep curved and straight beam with general boundary conditions. The first-order beam theory which considering the influence of shear deformation, inertia rotary and deepness term are adopted in the formulation. The theoretical solution model is obtained by means of modified series solution which core soul is using the modified Fourier series including a standard cosine Fourier series with two auxiliary terms to expand the admissible function. This fact gives the opportunity to derive the exact solution for FGP beam with general boundary conditions by utilizing a reasonable spring stiffness value at both ends. A series of numerical examples show that the current model has superior convergence characteristics, computational accuracy and stability. On this basis, a series of innovative results are also highlighted in the text, which may be providing basic data for other algorithm research in the future.