Comparisons of experimental and numerical frequencies for classical beams carrying a mass in-span

A frequency analysis of an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented. The dimensionless frequency equation for ten combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. The resulting transcendental frequency equations are numerically solved. A parametric study on the effects of the mass and its location for each respective case is presented. To verify the validity of the transcendental equations, the results for the fixed-fixed cases are compared with those obtained experimentally. On the other hand, approximate results are given, using the Rayleigh’s method with two static deflection shape functions. The effects of the position and magnitude of the mass, as well as comparisons of the different results obtained analytically, are investigated and discussed. The comparisons clearly show that the eigenfrequencies of the beam–mass system can be accurately predicted by solving the transcendental equation, whereas the closed-form Rayleigh’s expression is suggested for a quick estimation of fundamental frequency.     

Composite beam–columns with interlayer slip—Approximate analysis

An approximate second order analysis procedure for composite beam–columns with interlayer slip subjected to transverse loading and axial compressive loads is developed. The magnification factors to be applied to the first order solutions in order to estimate the deflections and internal forces obtained by the second order analysis approach are presented. The method of applying magnification factors to internal axial forces is discussed. The approximate second order analysis procedure is developed for the four Euler cases with various transverse load conditions. The procedure is applied to and the accuracy is illustrated for simply supported partially beam–columns of steel and concrete, and timber and concrete with different bending stiffness and interlayer slip properties. The deflections and internal forces obtained by the approximate method compared extremely well, except for slip forces in case of very flexible shear connectors, with those obtained by the more rigorous second order analysis approach for different composite action (partial interaction) parameters (shear connector stiffness values). The study also shows that the magnification factor associated with the deflections can be utilized to estimate also the internal actions, except shear forces in case of very flexible shear connectors, in the second order case with minimal error for simply supported beam–columns. Thus, for members with shear connector stiffness of structural significance the proposed approximate method can be used in general for simply supported beam–columns. For other boundary and loading conditions, the approximate method needs to be re-evaluated. The approach of using one magnification factor greatly simplifies the analysis task for those components.     

Analytical aspects in boundary integral equation formulation for the generalized linear micropolar thermoelasticity

A formulation of the boundary integral equation method for generalized linear micropolar thermoelasticity is given. Fundamental solutions, in Laplace transform domain, of the corresponding differential equations are obtained. The initial, mixed boundary value problem is considered as an example illustrating the BIE formulation. The results are applicable to the generalized thermoelasticity theories: Lord–Shulman with one relaxation time, Green–Lindsay with two relaxation times, Green–Naghdi, theory of type II without energy dissipation, Green–Naghdi, theory of type III and Chandrasekharaiah and Tzou theory with dual-phase lag, as well as to the dynamic coupled theory.     

Vibratory characteristics of flexural non-uniform Euler–Bernoulli beams carrying an arbitrary number of spring–mass systems

A new exact method for the analysis of free flexural vibrations of non-uniform multi-step Euler–Bernoulli beams carrying an arbitrary number of single-degree-of-freedom and two-degree-of-freedom spring–mass systems is presented in this paper. The closed-form solutions for free vibrations of non-uniform Euler–Bernoulli beams are derived for five important cases. Then, using the massless equivalent springs to replace the spring–mass systems and the fundamental solutions developed in this paper, the frequency equation for free flexural vibrations of a multi-step non-uniform beam with any kind of support configurations and carrying an arbitrary number of spring–mass systems can be conveniently established from a second-order determinant. The proposed method is computationally efficient due to the significant decrease in the determinant order as compared with previously developed procedures.     

Iterative Step-Like Reconstruction of Stratified Dielectric Media from Multifrequency Reflected-Field Data

We describe in this paper a frequency domain microwave method to reconstruct the permittivity profile of layered dielectric half-space. The inversion algorithm is based on the iterative Newton–Kantorovitch procedure applied to the Riccati nonlinear differential equation. To obtain a correction term to the initial profile, the integral equation relating to a small change of the profile function and corresponding change of the reflection coefficient is solved twice, with different systems of expanding functions for the unknown profile. The first set of functions is utilized to determine positions of the reflecting interfaces between layers. The second set represented by the Heaviside step functions provides the profile correction term completing the iteration. The proposed approach is valid for the essentially band-limited input data, that is, when both low- and high-frequency information is not available. This alleviates practical implementation especially from the viewpoint of antenna design, dielectric rods fed by TEM horns are used as transmitting/receiving antennas. Some numerical and experimental examples are presented for the input data in the frequency range of 1–4GHz, which demonstrate validity and robustness of the method for the complicated highly contrasted dielectric profiles.     

Free vibration of elastic helicoidal shells

An elastic helicoidal structure modelled as a plate twisted around its axis is studied in this paper. Accurate strain–displacement relationships for the shell are derived by the Green strain tensor in general shell theory and first-order shear deformation theory. An energy equilibrium equation of free vibration is introduced by the principle of virtual work. Applying the Rayleigh–Ritz method, an analytical eigenvalue equation is formulated and solved via an efficient computational approach for vibration characteristics of the helicoidal structure. A set of normalized orthogonal polynomials generated by the Gram–Schmidt procedure is presented to approximate the admissible functions. The first polynomial is taken as a kinematically compliant geometric equation of boundary conditions of the shell. The convergence and the accuracy of the present method, and the effects of geometric parameters and boundary conditions on vibration of the helicoidal structure are investigated.     

Analytical solutions for buckling of multi-step non-uniform columns with arbitrary distribution of flexural stiffness or axial distributed loading

In this paper, the function for describing the distribution of flexural stiffness K(x) of a non-uniform column is arbitrary, and the distribution of axial distributed loading N(x) acting on the column is expressed as a function of K(x) and vice versa. The governing equation for buckling of a one-step non-uniform column is reduced to a differential equation of the second-order without the first-order derivative by means of variable transformation. Then, this kind of differential equation is reduced to Bessel equations and other solvable equations for 14 cases. The analytical buckling solutions of one-step non-uniform columns are thus found. Then the obtained analytical solutions are used to derive the eigenvalue equation for buckling of a multi-step non-uniform column for several boundary supports by using the transfer matrix method. A numerical example shows that the proposed procedure is an efficient method for buckling analysis of multi-step non-uniform columns.     

Development of the meshless Hermite–Cloud method for structural mechanics applications

In this work, a novel true meshless numerical technique is proposed. It is termed the Hermite–Cloud method and is based on the classical reproducing kernel particle method except that a fixed reproducing kernel approximation is used instead. Another distinction is that the point collocation technique is used for the discretization of the governing partial differential equations. In this method, the Hermite theorem is employed for the construction of the interpolation functions. Through the constructed Hermite-type interpolation functions, we are able to generate the expressions of approximate solutions of both the unknown functions and the first-order derivatives, in a direct manner. A set of auxiliary conditions have also been developed so as to construct a complete set of PDEs with mixed Dirichlet and Neumann boundary conditions. Through several structural analysis examples, it is shown that the numerical results at the scattered discrete points generated by the Hermite–Cloud method are distinctly improved, for both the approximate solutions as well as the first-order derivatives.     

Free vibration of Timoshenko beam with finite mass rigid tip load and flexural–torsional coupling

In this paper, the free vibration of a cantilever Timoshenko beam with a rigid tip mass is analyzed. The mass center of the attached mass need not be coincident with its attachment point to the beam. As a result, the beam can be exposed to both torsional and planar elastic bending deformations. The analysis begins with deriving the governing equations of motion of the system and the corresponding boundary conditions using Hamilton's principle. Next, the derived formulation is transformed into an equivalent dimensionless form. Then, the separation of variables method is utilized to provide the frequency equation of the system. This equation is solved numerically, and the dependency of natural frequencies on various parameters of the tip mass is discussed. Explicit expressions for mode shapes and orthogonality condition are also obtained. Finally, the results obtained by the application of the Timoshenko beam model are compared with those of three other beam models, i.e. Euler–Bernoulli, shear and Rayleigh beam models. In this way, the effects of shear deformation and rotary inertia in the response of the beam are evaluated.     

空间滚柱凸轮曲面建模及虚拟样机计算处理方法

若无行之有效的计算和处理方法，直接应用大型CAD及CAE软件进行空间滚柱凸轮曲面的三维建模，及凸轮机构虚拟样机的运动学、动力学分析是很难的。论文应用广义矢量微分函数来描述展开后的滚轮凸轮机构，解决的空间滚轮凸轮机构凸轮曲线的正、反面设计问题，并在实际应用中取得了显著的效果。     

Pulse loading shape effects on pressure–impulse diagram of an elastic–plastic, single-degree-of-freedom structural model

Pulse loading shape effects on the dynamical response of an elastic–plastic, single-degree-of-freedom (SDOF) structural model are studied in the present paper. Two dimensionless parameters are introduced to classify the studied problem into elastic, elastic–plastic and rigid–plastic structural responses. When structural damage is controlled by maximum structural deflection, a characteristic curve in loading parameter space can be used to define an isodamage curve, i.e., pressure–impulse diagram, which is generally loading-shape-dependent. Dimensionless loading parameters, termed as effective loading parameters, are introduced in the present paper to give unique loading-shape-independent pressure–impulse diagram for each response category.     

Non-equilibrium theory of hydrodynamic lubrication

A new variational method is developed for the solution of the Navier Stokes equations which describe the unsteady flow between two infinite plates approaching or receding from each other symmetrically. The partial differential equations of the system are reduced to ordinary non-linear differential equations by using similarity transformations containing the Reynolds number as a parameter. The application of the governing principle of dissipative processes reduces the non-linear differential equation to an algebraic equation which can be solved for the required value of R and thus the flow characteristics can be studied. The velocity functions are obtained for various values of R ranging from 0.01 to 1.0.     

Thermal and non-Newtonian analysis on mixed liquid–solid lubrication

A mixed liquid–solid lubrication theory is proposed which concerns the effect of solid particle, liquid lubricant and rubbing surface topography. Especially, it focuses on the circumstances when particle diameter, surface composite roughness and oil film thickness are in the same order of magnitude. A mathematical model containing Reynolds equation, particle load carrying equation, asperity contact equation and heat balance equation is constructed to simulate the mixed liquid–solid lubrication. Moreover, the introduction of non-Newtonian constitutive equation and the rheological parameters related to heat and pressure makes the model closer to practical application. Some typical examples have been analyzed to explore the characteristics of mixed liquid–solid lubrication. In these examples, the effects of the mixed liquid–solid lubricant, the particle diameter and mass concentration, the surface composite roughness, and the material properties are discussed. The simulating results are accordant with early experimental researches, which indicated that the mathematical model is in agreement with the practical mixed liquid–solid lubrication. The input parameters in the examples can be adjusted to adapt to versatile applications.     

Time–frequency ratio-based blind separation methods for attenuated and time-delayed sources

We propose two types of time–frequency (TF) blind source separation (BSS) methods suited to attenuated and delayed (AD) mixtures. These approaches, inspired from a method that we previously developed for linear instantaneous (LI) mixtures, almost only require each source to occur alone in a tiny TF zone, i.e. they set very limited constraints on the source sparsity and overlap, unlike various previously reported TF-BSS methods. Our approaches consist in identifying the columns of the (filtered permuted) mixing matrix in TF zones where these methods detect that a single source occurs, using TIme–Frequency Ratios Of Mixtures (hence their name TIFROM). We thus identify columns of scale coefficients and time shifts. The detection stage for time shifts uses regression lines associated to the above-mentioned TF ratios of mixtures. The detection stage for scale coefficients uses the variance of these TF ratios of mixtures, either in Constant-Time or in Constant-Frequency analysis zones. This yields two alternative BSS methods, which are resp. called AD-TIFROM-CT and AD-TIFROM-CF. These methods are especially suited to non-stationary sources. We derive their performance from many tests performed with AD mixtures of speech signals. This demonstrates that they yield major SNR improvements, i.e. about 45 dB with optimum parameters for time shifts ranging from 0 to 20 samples and above 18 dB for 200-sample time shifts.     

Mechanical and microstructural analysis on the superplastic deformation behavior of Ti–6Al–4V Alloy

The present investigation has been made to study the superplastic deformation behavior of Ti–6Al–4V alloy based on the theory of inelastic deformation, and to analyze the boundary sliding characteristics using transmission electron microscopy. Flow characteristics for the microstructures of 2.5–16 μm grain sizes were analyzed by the load relaxation tests at various temperatures ranging from 600 to 927°C. The results showed that at relatively low temperatures such as 600°C the grain matrix deformation was dominant and found to be consistent with the state equation based on the dislocation dynamics. On the contrary, above the temperature of 800°C, the grain boundary sliding became dominant resulting in the change of curvature in the stress–strain rate curves, which was more pronounced in the finer microstructures. However, the deformation mode changes from grain boundary sliding to grain matrix deformation with the increase in grain size as evidenced by transmission electron microscopy.     

Improved flexural–torsional stability analysis of thin-walled composite beam and exact stiffness matrix

A simple but efficient method to evaluate the exact element stiffness matrix is newly presented in order to perform the spatially coupled stability analysis of thin-walled composite beams with symmetric and arbitrary laminations subjected to a compressive force. For this, the general bifurcation-type buckling theory of thin-walled composite beam is developed based on the energy functional, which is consistently obtained corresponding to semitangential rotations and semitangential moments. A numerical procedure is proposed by deriving a generalized eigenvalue problem associated with 14 displacement parameters, which produces both complex eigenvalues and multiple zero eigenvalues. Then the exact displacement functions are constructed by combining eigenvectors and polynomial solutions corresponding to non-zero and zero eigenvalues, respectively. Consequently exact element stiffness matrices are evaluated by applying member force–displacement relationships to these displacement functions. As a special case, the analytical solutions for buckling loads of unidirectional and cross-ply laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, the numerical, analytical, and the finite element solutions using the Hermitian beam elements are presented and compared with those from ABAQUS's shell elements. The effects of fiber orientation and the Wagner effect on the coupled buckling loads are also investigated intensively.     

The envelope order spectrum based on generalized demodulation time–frequency analysis and its application to gear fault diagnosis

The generalized demodulation time–frequency analysis is a novel signal processing method, which is particularly suitable for the processing of multi-component amplitude-modulated and frequency-modulated (AM–FM) signals as it can decompose a multi-component signal into a set of single-component signals whose instantaneous frequencies own physical meaning. While fault occurs in gear, the vibration signals measured from gearbox would exactly display AM–FM characteristics. Therefore, targeting the modulation feature of gear vibration signal in run-ups and run-downs, a fault diagnosis method in which generalized demodulation time–frequency analysis and envelope order spectrum technique are combined is put forward and applied to the transient analysis of gear vibration signal. Firstly the multi-component vibration signal of gear is decomposed into some mono-component signals using the generalized demodulation time–frequency analysis approach; secondly the envelope analysis is performed to each single-component signal; thirdly each envelope signal is re-sampled in angle domain; finally the spectrum analysis is applied to each re-sampled signal and the corresponding envelope order spectrum can be obtained. Furthermore, the gear working condition can be identified according to the envelope order spectrum. The analysis results from the simulation and experimental signals show that the proposed algorithm was effective in gear fault diagnosis.     

Optimum design of structures subjected to follower forces

In this paper, shape optimization is used to optimize the critical load of an Euler–Bernoulli cantilever beam with constant volume subjected to a tangential compressive tip load and/or a tangential compressive load arbitrarily distributed along the beam. This is achieved by varying appropriately the beam cross-section, thus its stiffness and mass properties, along its length, so that the critical load reaches its maximum or a prescribed value. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds, which, together with large slenderness ratios, ensure the validity of the Euler–Bernoulli theory and the serviceability of the beam. The evaluation of the objective function requires the solution of the dynamic stability problem of a cantilever beam with variable cross-section. This problem is solved using the analog equation method (AEM) of Katsikadelis for the fourth-order hyperbolic differential equation with variable coefficients, together with a simple and direct iterative method for the evaluation of the critical load based on the eigenvalue sensitivity. Besides its accuracy, this method overcomes the shortcoming of a possible FEM solution, which would require resizing of the elements and re-computation of their stiffness properties during the optimization process. Example problems of various types of follower forces are presented, which illustrate the method and demonstrate its applicability and efficiency.     

Thermoelastohydrodynamic analysis of the static performance of tilting-pad journal bearings with the Newton–Raphson method

Thermoelastohydrodynamic lubrication (TEHD) analysis is presented to investigate the static performance of tilting-pad journal bearings. A completely numerical solution is obtained. The Newton–Raphson method is employed to predict the bearing characteristics of the hydrodynamic pressure, the eccentricity and the pad attitude angles simultaneously. For the temperature calculation, three-dimensional (3D) energy equations for the fluid under each pad and 3D heat transfer equations for the pads are solved using a sequential sweeping method. The elastic deformation and thermal expansion of each pad are calculated with the 20-node isoparametric finite element method. It is found that the Newton–Raphson method is a smart and efficient method. The results show that the elastic deformation due to the hydrodynamic pressure and the influence of the temperature elevation play an important role in the calculated bearing system.     

Plane strain hot rolling of slab and strip

A numerical solution of the steady-state plane strain hot rolling problem with maximum friction is found in the range from the thick slab to the thin strip rolling geometries. The problem is kinematically determinate, and it is solved on the physical and hodograph planes using the finite-difference approximation of hyperbolic differential equations of the plane strain rigid–plastic flow theory. A sequence of the boundary value problems defined by the slipline field mode is treated as a non-linear vector equation which is solved by Broyden’s secant method. Six versions of the FORTRAN PC programs are written for a wide range of the slab/strip rolling geometries. The solution for the thin strip rolling with large roll radius and small strip thickness reduction is approached to Prandtl’s thin plastic layer with homogeneous velocity field.