An analytical solution to heat conduction with a moving heat source

This paper deals with an analytical solution to heat conduction in the medium subjected to a moving heat source. It evaluates the temperature distribution around a rectangular shape source moving at a constant speed along the axis of a bar. The transient temperature field from a moving heat source was analyzed using a Fourier series procedure. The most interesting result of the theory, is the derivation of a single formula capable of predicting the cooling time and cooling rate with a fairly good accuracy for ranges of temperature. Because of the passage of the heat source, the rise of temperature produced at a given near the source, tends to rapidly become constant. Several sample problems are discussed and illustrated, and comparisons with numerical approaches where these can also be used made. The results show that these solutions are in good agreement with the numerical results.     

An existence condition for the inverse dynamics solution of a slewing Euler-Bernoulli beam

Causal inverse dynamics problem of a slewing flexible beam has been identified as being ill-posed because it violates the stability of solution. Such problem has been studied intensively for the past three decades due to its application to the tip tracking control of flexible-link manipulators. A well known remedy has been to modify the location of the boundary point, or the output function in tracking control. This paper re-examines the problem by analyzing a closed form solution. Such a solution is made possible by truncating the assumed modes beam deflection model after the first mode. An existence condition for the solution is established, which links to the deflection mode shape, payload inertia, and the location of boundary point. This condition indicates that for certain system parameters, causal solutions may exist. Existing remedies as suggested by many published results can be explained by the condition. Extension to a multi-mode model is discussed and examined through numerical simulations.     

Combined analytical and numerical solution for an elastically supported Timoshenko beam to a moving load

A combined analytical and numerical method is presented to get a response for an elastically supported Timoshenko beam to a moving load. The analytical steady-state solution as a particular solution is established and summed with the numerically calculated homogeneous solution. A steady-state solution is sought analytically through the direct application of the Fourier transform to the moving step load. It is shown to be a compact formula composed of exponential and sinusoidal functions depending on the load velocity. The homogeneous solution is established numerically to cancel out the discontinuities and the inconsistent boundary and initial conditions of the steady-state solution. The discontinuities produced by the steady-state solution are removed using the physical characteristics related to the bar wave. Some response curves are shown to compare the beam motions at different load velocities.     

An approximate solution to a problem of pseudo-steady flow of strain-hardening material

An approximate technique is presented for estimating an external force in steady or pseudo-steady flow of strain-hardening material, for which the deformation can be simplified to a rigid-body motion mechanism. The rigid blocks are separated from each other by layers of the material undergoing intensive shear. The strain-hardening effect is incorporated into the analysis by assuming the flow shear stress as an exponential function of equivalent strain. This technique, similar in its nature to the upper bound approach of limit analysis, cannot be rigorously regarded as such since the distribution of the shear stress and the boundaries in pseudo-steady state problems cannot be specified a priori. It is shown that the rate of work-dissipation within a shear layer depends on the total increase of the tangential velocity and is independent of its thickness, as in the case of perfectly plastic solids. The results of computations for a pseudo-steady pyramidal indentation of a strain-hardening solid are presented and compared to the experimentally measured forces reported by other authors. Computations for a pseudo-steady flattening of a tetrahedral asperity are also briefly reported.     

An analytical solution of a finite bearing with an inclined journal

A solution of Reynolds' equation for a finite bearing with an inclined journal is attempted. A solution (pressure) is expanded as a series of circular trigonometric functions of β′, the coordinate corresponding to circumferential distances, with coefficients as functions of z, the coordinate corresponding to axial distances. Substitution of such a series for pressure in the equation gives an infinite set of coupled ordinary differential equations with varying coefficients. Forms of solutions of this set of equations are assumed in terms of arbitrary constants and with restrictions of continuity at zero values of the inclinations. The arbitrary constants are evaluated using differential equations and boundary conditions. The integrations of pressure for forces and torques are converted into single integrations which are readily evaluated by digital computer.     

An analytical solution for nonlinear dynamics of a viscoelastic beam-heavy mass system

The aim of this study is to develop an approximate analytic solution for nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam with an attached heavy intra-span mass. A geometric nonlinearity due to midplane stretching is considered and Newton’s second law of motion along with Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model, are employed to derive the nonlinear equations of motion. The method of multiple timescales is applied directly to the governing equations of motion, and nonlinear natural frequencies and vibration responses of the system are obtained analytically. Regarding the resonance case, the limit-cycle of the response is formulated analytically. A parametric study is conducted in order to highlight the influences of the system parameters. The main objective is to examine how the vibration response of a plain (i.e. without additional adornment) beam is modified by the presence of a heavy mass, attached somewhere along the beam length.     

An improved closed-form solution to interfacial stresses in plated beams using a two-stage approach

Shear stress and normal stress in the thickness direction at interfaces (referred to as interfacial shear and transverse normal stresses, respectively, hereafter) have played a significant role in understanding the premature debonding failure of beams strengthened by bonding steel/composite plates at their tension surfaces. Due to the occurrence of dissimilar materials and the abrupt change of cross-section, the stress distribution at plate ends becomes singular and is hence considerably complicated. Extensive experimental and analytical analyses have been undertaken to investigate this problem. Large discrepancies have been found from various studies, particularly from experimental results due to the well-acknowledged difficulty in measuring interfacial stresses. Numerical analyses, e.g. 2-D or 3-D finite-element analysis (FEA), may predict accurate results, but they demand laborious work on meshing and sensitivity analysis. Analytical solutions, in particular those in a closed form, are more desirable by engineering practitioners, as they can be readily incorporated into design equations. This paper reports an improved closed-form solution to interfacial stresses in plated beams using a two-stage approach. In this solution, beams and bonded plates can be further divided into a number of sub-layers to facilitate the inclusion of steel bars or multiple laminae. Thermal effects may also be considered by using equivalent mechanical loads, i.e. equivalent axial loads and end moments. Numerical examples are presented to show interfacial stresses in concrete or cast iron beams bonded with steel or FRP plates under mechanical and/or thermal loads. The effect of including steel reinforcements with various ratios in the RC beam on interfacial stresses is also investigated. Compared with previously published analytical results, this one improves the accuracy of predicting the transverse normal stresses in both adhesive-beam and plate-adhesive interfaces and the solution is in a closed form.     

Dynamic response analysis of rotating composite-VEM thin-walled beams incorporating viscoelastic materials in the time domain

This paper addresses the analytical modeling and dynamic response of the advanced composite rotating blade modeled as thin-walled beams and incorporating viscoelastic material. The blade model incorporates non-classical features such as anisotropy, transverse shear, rotary inertia and includes the centrifugal and coriolis force fields. The dual technology including structural tailoring and passive damping technology is implemented in order to enhance the vibrational characteristics of the blade. Whereas structural tailoring methodology uses the directionality properties of advanced composite materials, the passive material technology exploits the damping capabilities of viscoelastic material (VEM) embedded into the host structure. The VEM layer damping treatment is modeled by using the Golla-Hughes-McTavish (GHM) method, which is employed to account for the frequency-dependent characteristics of the VEM. The case of VEM spread over the entire span of the structure is considered. The displayed numerical results provide a comprehensive picture of the synergistic implications of both techniques, namely, the tailoring and damping technology on the dynamic response of a rotating thin-walled beam exposed to external time-dependent excitations.     

An analytical solution of the ECAE process of bi-layer strip using a Bezier type streamline

Co-extrusion of strips of different strength and ductility is a process in the production of bimetal rods of desirable physical and mechanical properties. Equal channel angular extrusion (ECAE) is an innovative process for applying severe plastic deformation to materials. The present study is concerned with the ECAE of different bi-layer strips. Developing a kinematically admissible velocity field using a Bezier-type streamline is the aim of present research. In this analysis, using the mentioned velocity field and the upper bound theorem, the extrusion force as well as the strain distribution in the deformation zone of the ECAE process are predicted. The solution takes into account the die profile, friction conditions, strength, arrangement and thickness ratio of the layer materials. Experimental results of ECAE of aluminum/copper and aluminum/mild steel bi-layer strips showed a good agreement with the predicted results.

     

An approximate solution for the contact area and elastic compliance of a smooth punch of arbitrary shape

An approximate solution is developed for the contact area and the load-penetiation relation for frictionless indentation of the elastic half-space by a punch of arbitrary profile. The method makes use of a previous result to the effect that the contact area in this problem is that which maximizes the total indenting force. An estimate of this force is obtained by applying the reciprocal theorem to the solution for indentation by a flat punch of the same plan-form, for which an approximate solution has recently been developed by Fabrikant. The method is illustrated using an example, the results of which are compared with a direct numerical solution using Hartnett's algorithm.     

Minimizing the total weighted completion time on a single machine under linear deterioration

This paper investigates a single-machine problem in which processing times of jobs are start time dependent and the aim is to minimize the total weighted completion time. Recent research has shown the complexity of this problem to be NP-hard; however, no optimal or heuristic algorithms have been proposed. In this paper, we explore the exact solution and propose several heuristic algorithms derived based on the impacts of model parameters. The effects of normal processing times and deterioration rates are also studied.     

An analytical solution for inhomogeneous strain within cylinders of silicon and effect on quantum band structure under the double-punch test

An analytical solution for inhomogeneous strain distributions within a finite cylinder of silicon under the double-punch test is obtained. The stress function method is employed and a new expression for the stress function is proposed so that all of the governing equations and the boundary conditions are satisfied exactly. The solution for isotropic cylinders under the double-punch test is recovered as a special case. Numerical results show that the strain singularities are usually developed near two end surfaces, but the strain distributions are relatively uniform in the central part of the cylinder. The largest tensile strain is always induced along the axis of loading. In addition, based on the envelope-function method of energy-band theory and quantum mechanics, the effect of external loads on the valence-band structure of silicon is analyzed. The spin–orbit interaction is considered. It is found that external load under the double-punch test can alter considerably the quantum behavior of energy-band structure of silicon, which manifests in the change of the constant-energy surfaces of the heavy-hole band, the light-hole band, the split-off band and the corresponding conductivity masses. The present study provides an alternative method to investigate the electro-optic properties of strained silicon.     

General temperature rise solution for a moving plane heat source problem in surface grinding

In the present study, the general solutions for a transient state as well as for the temperature rise formed everywhere in the workpiece due to a rectangular-shaped moving plane heat source arising at the grinding zone are derived. The present analysis starts from a point heat source solution by applying the method of separation of variables to a three-dimensional heat conduction problem. Because the workpiece moving velocity is quite small, the convective term related to the workpiece velocity is first excluded from the heat conduction equation. This workpiece velocity effect will be included in the model by slightly modifying the coordinate variable in the sliding direction shown in the solution of the point heat source. Therefore, the general three-dimensional solution of the stationary temperature rise can be expressed in an integral form as a function of the product value of the unknown initial condition and the particular solution of temperature rise. The unknown initial temperature rise in the solution can be replaced by the point heat source due to frictional that multiplying the product of the Dirac delta functions defined for three directions. Using the definition of the Dirac delta function, the temperature rise solution for a point heat source can thus be obtained. This solution is further extended to obtain the moving and uniform heat sources arising in a rectangular grinding zone. A comparison among the experimental result and the theoretical results predicted by the present model and Jaeger’s model [Jaeger JC (1942) Proc Roy Soc, NSW 76:203–224.] show that the present model is quite accurate and is generally superior to Jaeger’s model; it can be applied to predict the three-dimensional temperature rise distributions in the workpiece.     

Detection in near-field domain of biomolecules adsorbed on a single metallic nanoparticle

In this study, neurodegeneration phenomena were investigated, by performing third harmonic generation imaging measurements on the nematode Caenorhabditis elegans, in vivo. The in vivo, precise identification of the contour of the degenerating neurons in the posterior part of the nematode and the monitoring, in real time, of the progression of degeneration in the worm, through third harmonic generation imaging measurements, were achieved. Femtosecond laser pulses (1028 nm) were utilized for excitation. Thus, the THG image contrast modality comprises a powerful diagnostic tool, providing valuable information and offering new insights into morphological changes and complex developmental processes in live biological specimens.     

Uniaxial strength asymmetry in cellular materials: An analytical model

The ability to predict failure of cellular materials depends on the knowledge of microstructural mechanisms that contribute to macroscopic behavior. In this paper, we develop microstructural models to examine the mechanisms responsible for differences in tensile and compressive strength observed in cellular materials. We limit our analyses to those materials that fail by the same mechanism (yielding or microcracking) in tension and compression. Using both a honeycomb and an open-cell foam model, we demonstrate that density-dependent, compression-strong strength asymmetry arises when two conditions are met: the cell wall material has a higher yield strength in compression than in tension, and the cell walls are loaded simultaneously by axial forces and bending moments. Our models predict that strength asymmetry (defined as the ratio of compressive to tensile yield strength of the cellular material) increases with relative density (as observed in real materials such as rigid polyurethane foams and trabecular bone), and that strength asymmetry is more pronounced in anisotropic materials (where oblique struts are more closely aligned with the direction of loading).     

Potential solution for sloshing in a horizontally moving rectangular tank and design of tank velocity profile

Journal of Mechanical Science and Technology - Sloshing makes it challenging to transport liquid-containing systems fast. Accurate prediction and effective suppression of sloshing are essential in...     

Error analysis to minimize cross-axis couplings in 6-DOF motion systems with a single moving part

In multi-axis motion control, cross-axis couplings cause error force and position disturbances in an axis when a desired motion is generated along another axis. Different from the parasitic errors that result from the imperfections of the mechanical bearings and reference surfaces, cross-axis perturbations are caused by errors that occur both statically (geometrical errors) and dynamically (in the transient responses) and are more prevalent in air-bearing and magnetic-levitation (maglev) stages. The parasitic errors are heavily dependent on the sizes of the stage's mechanical components, while the cross-axis perturbations depend significantly on the mover's speed and acceleration. For stages using permanent magnets (PMs) and Lorentz coils, the causes of off-axis forces include 1) errors in the coil turns' straightness, perpendicularity, and parallelism of the motor axes, and 2) errors in the local magnetizations and PMs' fringing effects. The purpose of this paper is to analyze the topologies of 6-degree-of-freedom (6-DOF) single-moving-part stages to minimize cross-axis couplings. The outcome is a stage configuration with reduced couplings and cross-axis perturbations. This is supported by experimental results performed on a newly developed 6-DOF maglev laser-interferometer stage. Its achieved root-mean-square (rms) positioning noise and minimum step size in XY are 3 nm and 10 nm, respectively. Its achieved resolution in out-of-plane rotations is 0.1 μrad. In addition to the analysis supported by these results, this paper introduces a new measure to represent cross-axis perturbations and to compare the effects of couplings in multi-axis positioning. This measure is entitled the cross-coupling quantity (CCQ) and calculated from the displacement of the stage in the axis of interest, the peak time of the response, and the peak-to-peak (p-p) error in the perturbed axis.     

Heat checking in the contact zone of a bearing seal (a two-dimensional model of a single moving asperity)

When two bodies are in sliding contact under heavy loads, local high temperature may occur as a result of excessive frictional heating near the contacting surfaces. Because of a combination of thermal heating and the mechanical load, the material may crack in the neighborhood of the contact zone. This phenomenon is called heat checking. It commonly occurs in mechanical seals and brakes. In recent years there has been increased emphasis on finding a solution. In this paper a simple model is proposed to determine those parameters which can be optimized to control the heat checking of the materials.Because of the size difference between the contact area and the seal, the mathematical model is represented by a half-space subjected to a fast-moving load which is distributed over a small area. The load comes from a combination of an arbitrarily distributed heat source and mechanical loads of pressure and friction. The general solutions are expressed in the form of integral equations with determined Green's functions. Numerical results for fracture criteria are then obtained using a computer program.For the current problem, a nominal pressure of 365 MPa (53000 Ibf in^?2) and a corresponding friction of 183 MPa (26500 Ibf in^?2) are used. The induced heat source is 1.5 × 10⁷ in Ibf in^?2 s^?1 (1.70 × 10⁶Jm^?2s^?1). Such a load results in fracture where the crack is first initiated immediately beneath the surface at the trailing edge of the moving load.     

Performance modeling of a manufacturing cell with a single material transporter: An approximation

We consider a flexible manufacturing system with a number of workstations, a single material transporter, and a common storage space of finite capacity. The material handling delay times are explicitly considered in the model and assumed to follow a two-stage Coxian distribution. The material processing times on a workstation also have a two-stage Coxian distribution. The routing of parts within the system follows a Markov chain. An approximate performance model is developed and the results are compared with the exact or simulation results. We also investigate how this performance model compares to a simulation with deterministic routing and processing times. Finally, we study the effect, on the performance measures, of ignoring the material transporter or of modeling the transporter as a central server with aggregation of routing information.