Fundamental Solutions of a Normal Force in Semi-Infinite Coating Materials Part II: Solution of Force Acting in the Interior of Substrate and at Interface

2011 ◽  
Vol 243-249 ◽  
pp. 5935-5940
Author(s):  
Zhen Yang ◽  
Shi Qun Guo

In this work, the spatial theoretical solutions of concentrated normal forces acting in the substrate and at the interface of semi-infinite coating materials have been deduced based on the image point method. The explicit solution is given in the series of the displacement functions corresponding to each image point. It is found that all displacement functions can be deduced by the order of the image point. Numerical analysis has been carried out to verify the theoretical deductions. It is found that the accurate enough theoretical solution can be obtained by only taking the displacement functions corresponding to the first several image points into account. The fundamental solutions for a concentrated force acting at interface of two bonded semi-infinite bodies is also deduced, which is necessary in the analysis of coating materials by using these theoretical solution as the fundamental solution.

2011 ◽  
Vol 243-249 ◽  
pp. 5917-5922
Author(s):  
Zhen Yang ◽  
Shi Qun Guo

The problem of a normal force acting in the thin layer of semi-infinite coating materials has been studied by adopting the image point technology in this paper. The explicit fundamental solution is given in the series of the displacement functions corresponding to each image point. The displacement functions corresponding to the higher order image points can be determined from those to the lower ones by the recurrence formula presented in this paper. Numerical analysis by means of the finite element method has been carried out to verify the theoretical results. It is found that the accurate enough theoretical solution can be obtained by only taking the displacement functions corresponding to the first several image points into account.


2021 ◽  
Vol 21 (3) ◽  
pp. 239-246
Author(s):  
A. V. Galaburdin

Introduction. A method for solving the problem on the action of a normal force moving on an infinite plate according to an arbitrary law is considered. This method and the results obtained can be used to study the effect of a moving load on various structures.Materials and Methods. An original method for solving problems of the action of a normal force moving arbitrarily along a freeform open curve on an infinite plate resting on an elastic base, is developed. For this purpose, a fundamental solution to the differential equation of the dynamics of a plate resting on an elastic base is used. It is assumed that the movement of force begins at a sufficiently distant moment in time. Therefore, there are no initial conditions in this formulation of the problem. When determining the fundamental solution, the Fourier transform is performed in time. When the Fourier transform is inverted, the image is expanded in terms of the transformation parameter into a series in Hermite polynomials.Results. The solution to the problem on an infinite plate resting on an elastic base, along which a concentrated force moves at a variable speed, is presented. A smooth open curve, consisting of straight lines and arcs of circles, was considered as a trajectory. The behavior of the components of the displacement vector and the stress tensor at the location of the moving force is studied, as well as the process of wave energy propagation, for which the change in the Umov-Poynting energy flux density vector is considered. The effect of the speed and acceleration of the force movement on the displacements, stresses and propagation of elastic waves is investigated. The influence of the force trajectory shape on the stress-strain state of the plate and on the nature of the propagation of elastic waves is studied. The results indicate that the method is quite stable within a wide range of changes in the speed of force movement.Discussion and Conclusions. The calculations have shown that the most significant factor affecting the stress-strain states of the plate and the propagation of elastic wave energy near the concentrated force is the speed of its movement. These results will be useful under studying dynamic processes generated by a moving load.


2013 ◽  
Vol 690-693 ◽  
pp. 1919-1922
Author(s):  
Xiao Zeng Wang

A circular ring is often used in the engineering. There are many applications in the tunnels, subways, and down hole. The purpose of this paper is to obtain the theoretical and numerical solution of the deflection of the circular ring under action of a concentrated force. Adopting the method of numerical analysis and Castigliano’s theory, the max deflection and the 4th equivalent stress of ring are developed. The results of the research indicate that the deflection of ring is the linear function of the concentrated force which is in accord with the elasticity body assumptions, that is Hook’s law. The theoretical solution of the deflection of ring is almost the same as the numerical one of the ring. So it is feasible that the numerical model of the ring is used to analyze the deflection of the eccentric ring.


2021 ◽  
Author(s):  
Xiaocui Wang ◽  
◽  
Runlan Wang ◽  
Bo Huang ◽  
Jiliang Mo ◽  
...  

In this work, a comparative study is performed to investigate the influence of time-varying normal forces on the friction properties and friction-induced stick-slip vibration by experimental and theoretical methods. In the experiments, constant and harmonic-varying normal forces are applied, respectively. The measured vibration signals under two loading forms are compared in both time and frequency domains. In addition, mathematical tools such as phase space reconstruction and Fourier spectra are used to reveal the science behind the complicated dynamic behaviour. It can be found that the friction system shows steady stick-slip vibration, and the main frequency does not vary with the magnitude of the constant normal force, but the size of limit cycle increases with the magnitude of the constant normal force. In contrast, the friction system harmonic normal force shows complicated behaviour, for example, higher-frequency larger-amplitude vibration occurs as the frequency of the normal force increases. The interesting findings offer a new way for controlling friction-induced stick-slip vibration in engineering applications.


Author(s):  
T.G. Ergashev ◽  
A. Hasanov

In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella’s hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green’s function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella’s hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem. В данной работе мы исследуем задачу Холмгрена для многомерного эллиптического уравнения с несколькими сингулярными коэффициентами. Мы используем фундаментальное решение уравнения, содержащее гипергеометрическую функцию Лауричеллы от многих переменных. Затем методом «abc» доказывается единственность решения проблемы Холмгрена. Применяя метод функции Грина, мы можем найти решение задачи в явном виде. Более того, формулы разложения и суммирования, формулы дифференцирования и некоторые смежные соотношения для гипергеометрических функций Лауричеллы от многих переменных были использованы для нахождения явного решения поставленной задачи.


2020 ◽  
Author(s):  
Banuvathy Rajakumar ◽  
Varadhan SKM

AbstractBackgroundThe human hand plays a crucial role in accomplishing activities of daily living. The contribution of each finger in the human hand is remarkably unique in establishing object stabilization. According to the mechanical advantage hypothesis, the little finger tends to exert a greater normal force than the ring finger during a supination moment production task to stabilize the object. Similarly, during pronation, the index finger produces more normal force when compared with the middle finger. Hence, the central nervous system employs the peripheral fingers for torque generation to establish the equilibrium as they have a mechanical advantage of longer moment arms for normal force. In our study, we tested whether the mechanical advantage hypothesis is supported in a task in which the contribution of thumb was artificially reduced. We also computed the safety margin of the individual fingers and thumb.MethodologyFifteen participants used five-finger prismatic precision grip to hold a custom-built handle with a vertical railing on the thumb side. A slider platform was placed on the railing such that the thumb sensor could move either up or down. There were two experimental conditions. In the “Fixed” condition, the slider was mechanically fixed, and hence the thumb sensor could not move. In the “Free” condition, the slider platform on which the thumb sensor was placed could freely move. In both conditions, the instruction was to grasp and hold the handle (and the platform) in static equilibrium. We recorded tangential and normal forces of all the fingers.ResultsThe distribution of fingertip forces and moments changed depending on whether the thumb platform was movable (or not). In the free condition, the drop in the tangential force of thumb was counteracted by an increase in the normal force of the ring and little finger. Critically, the normal forces of the ring and little finger were statistically equivalent. The safety margin of the index and middle finger did not show a significant drop in the free condition when compared to fixed condition.ConclusionWe conclude that our results does not support the mechanical advantage hypothesis at least for the specific mechanical task considered in our study. In the free condition, the normal force of little finger was comparable to the normal force of the ring finger. Also, the safety margin of the thumb and ring finger increased to prevent slipping of the thumb platform and to maintain the handle in static equilibrium during the free condition. However, the rise in the safety margin of the ring finger was not compensated by a drop in the safety margin of the index and middle finger.


2015 ◽  
Author(s):  
Melissa E. Tribou ◽  
Geoffrey Swain

Ship hull grooming is proposed as an environmentally friendly method of controlling fouling on ship hulls. It is defined as the frequent and gentle cleaning of a coating when the ship is idle to prevent the Establishment of fouling. Prior research by Tribou and Swain has evaluated the effectiveness of different methods and the frequency of grooming on different types of ship hull coatings. It was found that vertical rotating cup style Brushes provided the best method to maintain the coatings in a smooth and fouling free condition. This study investigated brush design and operational parameters in relationship to normal forces imparted by the brushes to the surface. A brush stiffness factor was developed and the independent variables for brush design non-dimensionalized for the normal force. A load cell was used to measure the forces imparted by different brushes and the models were validated using these non-dimensional terms. The knowledge gained by these studies will be used to optimize brush design for the implementation of grooming.


2014 ◽  
Vol 752 ◽  
pp. 649-669 ◽  
Author(s):  
L. Divaret ◽  
O. Cadot ◽  
P. Moussou ◽  
O. Doaré

AbstractThis work aims to improve understanding of the damping induced by an axial flow on a rigid cylinder undergoing small lateral oscillations within the framework of the quasistatic assumption. The study focuses on the normal force exerted on the cylinder for a Reynolds number of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=24\, 000$ (based on the cylinder diameter and axial flow velocity). Both dynamic and static approaches are investigated. With the static approach, fluid forces, pressure distributions and velocity fields are measured for different yaw angles and cylinder lengths in a wind tunnel. It is found that for yaw angles smaller than $5{^\circ }$, the normal force varies linearly with the angle and is fully dominated by its lift component. The lift originates from the high pressure coefficient at the front of the cylinder, which is found to depend linearly on the angle, and from a base pressure coefficient that remains close to zero independent of the yaw angle. At the base, a flow deficit and two counter-rotating vortices are observed. A numerical simulation using a $k\mbox{--}\omega $ shear stress transport turbulence model confirms the static experimental results. A dynamic experiment conducted in a water tunnel brings out damping-rate values during free oscillations of the cylinder. As expected from the linear dependence of the normal force on the yaw angle observed with the static approach, the damping rate increases linearly with the axial flow velocity. Satisfactory agreement is found between the two approaches.


Author(s):  
Dirk Drees ◽  
Satish Achanta

Friction at different force, length, and time scales is of great interest in tribology. The mechanical, chemical, and physical (atomic) interactions, each operating at their own time length and force scale, makes friction complex. This work is an attempt to improve the understanding of friction at normal forces ranging from nN up to N. This investigation was carried out under reciprocating ball-on-flat sliding conditions on engineering surfaces like diamond-like carbon (DLC) and dual phase steel. The test equipments used for this investigation are an atomic force microscope, a microtribometer, and a macrofretting tester. It was observed that for a hard/hard tribocouple like DLC/Si3N4, the variation in the coefficient of friction is negligible whereas the variation is large when the tribocouple is hard / soft like in dual phase steel / Si3N4. By changing the surface roughness of the material, the dependence of friction on normal force could be altered or manipulated.


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