Mathematical modelling of nervous systems

An electrochemical theory of the action potential is described, which takes into account the electric field produced by the ions themselves in the Debye layer. It is shown that under suitable conditions oscillations can arise, involving relatively large fluctuations of the concentrations of sodium and calcium at the internal surface of the neural membrane. It is possible in this way to account quantitatively for the general character, amplitude and frequency of the action potential, without postulating large variations in the ionic conductances of the membrane. Formulas for the available energy and information associated with a neuron are derived by a variational method. Problems which arise in the modelling of extended nervous systems are discussed, and methods are applied to model the experimental behavior of a ganglion of the leech.     

Mathematical modelling of hydraulic transients in complex systems

It is a very well-known fact that calculations necessary to analyze transient conditions in hydraulic systems are very time-consuming and difficult to organize, especially in complex systems. Nevertheless, such calculations are necessary to achieve efficiency and economy in the design and operation, as well as safety in these systems, since those objectives need precise calculations of pressures and flowrates. Suitable mathematical modeling of the different elements in a hydraulic system is necessary to get useful results, which help fulfill those objectives. In this paper, the mathematical model described in [1] is generalized to complex pressurized hydraulic systems. To model the behavior of the fluid within the ducts, use is made of the so-called elastic model, which is numerically solved by the methods of characteristics. Nevertheless, the main objective of this paper hinges on the treatment of the boundary conditions that allow developing a general model virtually representing every combination of elements at a given location of the system. The final objective is to provide the users with a powerful tool to devise the potential risks to which an installation may be exposed and to develop suitable protection strategies.     

Mathematical modelling of hydraulic transients in simple systems

Efficiency and economy in the design and operation of a hydraulic system, as well as its safety, are objectives needing precise calculations of pressures and flowrates within the system. The calculations are typically very time-consuming and, depending on the characteristics of the system, very complicated and difficult to organize. A suitable mathematical modelling of the different ingredients in a hydraulic system is necessary to get useful results, which help fulfill those objectives. In this paper, the mathematical modelling used to develop a computer program to simulate hydraulic transients in a simple system is described. The program (DYAGATS), developed by the authors, is currently being used by organizations and consultancies to simulate and, consequently, analyze hydraulic transients in water systems. It makes use of the so-called elastic model, also known as waterhammer, to model the behavior of the fluid within the pipes. Also, lump models for the different elements that introduce, damp, modify, absorb, etc., perturbations in the systems are presented in a unified treatment. The main objective is to provide users with a powerful tool to devise the potential risks to which an installation may be exposed and to develop suitable protection strategies.     

Mathematical modelling of rock bolt systems II

The main goal of the paper is to describe a reinforcement consisting of fully grouted bolts, which is applied to stabilizing underground openings and tunnels. After a variational formulation is given, the existence and uniqueness is proved. Some asymptotic results that make it possible to replace the real system with a continuous one more suitable for discretization are presented. Some other types of reinforcements and properties are studied.     

Mathematical modelling of rock bolt systems I

The main goal of the paper is to give a variational formulation of the behaviour of bolt systems in rock mass. The problem arises in geomechanics where bolt systems are applied to reinforce underground openings by inserting steel bars or cables. After giving a variational formulation, we prove the existence and uniqueness and some other properties.     

Mathematical modeling of complex mechanical systems

Mathematical modeling of mechanical systems based on multibody system models is a well tested approach. Generating the equations of motion for complex multibody systems with a large number of degrees of freedom is difficult with paper and pencil. For this reason methods for automatic equation generation have been developed. Most methods result in numerical equations of motion without explicit information about the parameters. In this paper a method is described resulting in symbolic equations of motion. The method allows also the determination of the constraint forces which are important for design purposes. The inverse problem of dynamics is also easily solved.     

Systematic dynamic modelling of mechanical systems containing kinematic loops

In this tutorial paper a systematic procedure is presented to obtain the dynamic models of mechanical systems containing kinematic loops, with a main emphasis on efficiency and with particular regard to robotic systems. The procedure retains a minimal set of generalized coordinates for the corresponding open loop structure, obtained by removing some additional constraints closing loops in the original structure, while adopting an efficient Newton-Euler formulation of the equations of motion. Two methods for including the loop closure equations are then discussed: the Lagrange multipliers method and the method based on an explicit solution of the constraint equations. In the first case the dynamic model is given in the form of an implicit Differential Algebraic Equations (DAE) system, while in the second case an Ordinary Differential Equations (ODE) system could be obtained.     

Mathematical modelling of transient vibration and acoustic radiation from impacted systems

Vibration response and acoustic radiation resulting from impactive excitation of beam- and plate-like structures are modelled. Mechanical energy absorption of an object is obtained through Fourier synthesis for any force-time history. Both, structural damping and acoustic radiation loss are accounted for in the equations of motion. Acoustic radiation is modelled as energy dissipation proportional to vibration velocity of the object. General expressions are obtained for vibration amplitudes and radiated power and energy.     

Mathematical modelling of the stochastic mechanical properties of wood and its extensibility at small scales

This paper investigates the relation between the uncertain mechanical properties of wood and its extensibility at the ultrastructural scale. A statistical approximation to the output of a multi-scale constitutive model is adopted to predict the extensibility of wood in the presence of parametric uncertainty. By means of this procedure, a very large number of computationally intensive fully-coupled multi-scale simulations are avoided. Following this approach, four different micromechanical parameters are chosen to assess their influence on the extensibility of the material under tensile loading conditions. These are the degree of cellulose crystallinity, the ultimate strain and Young’s modulus of the hemicellulose–lignin matrix, and the thickness of the amorphous cellulose layer which covers the periodic crystalline portions of cellulose. We believe that a better understanding of the mechanisms of deformation and extensibility in wood and in natural materials can pave the way for the development of new strategies to design more advanced materials in engineering structures.     

Structural instability of nonlinear plates modelling suspension bridges: Mathematical answers to some long-standing questions

We model the roadway of a suspension bridge as a thin rectangular plate and we study in detail its oscillating modes. The plate is assumed to be hinged on its short edges and free on its long edges. Two different kinds of oscillating modes are found: longitudinal modes and torsional modes. Then we analyze a fourth order hyperbolic-like equation describing the dynamics of the bridge. In order to emphasize the structural behavior we consider an isolated equation with no forcing and damping. Due to the nonlinear behavior of the cables and hangers, a structural instability appears. With a finite dimensional approximation we prove that the system remains stable at low energies while numerical results show that for larger energies the system becomes unstable. We analyze the energy thresholds of instability and we show that the model allows to give answers to several questions left open by the Tacoma collapse in 1940.     

A unified approach to the modelling of holonomic and nonholonomic mechanical systems

An alternative technique, called projection method, for solving constrained system problems is presented. This approach can be used to derive equations of motion of both holonomic and nonholonomic systems, and the dynamic equations can be expressed in generalized velocities and/or quasi-velocities. Compared against the other methods of classical mechanics (Lagrange's, Gibbs-Appell, Kane's,...), the present method turns out to be extraordinarily short, elementary and general. As such, it deserves to be promoted as a generally accepted method in academic and engineering applications. Three examples are reported to illustrate advantages of the technique     

Mathematical modelling of the flotation deinking process

The overall flotation deinking process can be divided into four basic microprocesses:

• 1.(1) collision or capture of an (ink) particle by an air bubble
• 2.(2) adhesion of an (ink) particle to the air bubble by sliding
• 3.(3) development of a three-phase contact at the air bubble/water/particle interface, and
• 4.(4) bubble/particle stability or instability after an aggregate is formed each of these microprocesses have an associated probability that they will occur successfully in a flotation cell.

In this paper, the associated probabilities of each microprocess are employed in the development of a kinetic- or population balance-type model of the overall flotation process. The overall model contains two kinetic constants: the first, k₁ governs the overall probability of a free ink particle successfully intercepting and adhering to an air bubble; the second, k₂ is a measure of the probability that a bubble/particle aggregate pair will become unstable and split to yield a “new” free ink particle.The solution to the kinetic model is presented in terms of k₁ and k₂, which are themselves functions of system parameters such as bubble and particle physical properties (e.g., diameter, density), fluid properties (e.g., viscosity, surface tension), etc. From this solution, a definition of a theoretical flotation efficiency, as well as other system performance parameters are presented.     

Mathematical modelling of spark-ignition engines

The flow field, scavenging efficiency, power output, heat transfer losses, and unburned hydrocarbon emissions have been numerically studied by means of a two-equation model of turbulence in a four-stroke, homogeneous-charge, spark-ignition engine. The engine is equipped with an intake valve, an exhaust valve, and a constant rate heat source which simulates the spark plug. Combustion has been modelled by means of a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression. The numerical results indicate that the intake stroke is characterized by the formation of two eddies which persist in the compression stroke. Turbulence is generated at the shear layers of the air jet drawn into the cylinder, but its level decreases in the compression stroke. Due to the heat released by the spark plug and the chemical reaction, a spherical flame kernel is formed. This kernel evolves into a cylindrical flame when the flame front reaches the piston. Fuel remains unburnt at the corner between the cylinder head and the cylinder wall due to heat transfer losses. The numerical results also indicate that despite uncertainties about the turbulence and heat transfer models, an engine model such as the one studied here can be used to understand the flow field, heat transfer losses, scavenging efficiency, and power output in conventional spark-ignition engines. Such capabilities are very helpful in the development and optimization stages of engines. For example, here the engine model thermal and scavenging efficiencies are 15.69% and 94%, respectively. The peak pressure is 33 atm and occurs at 6° ATDC. The unburnt hydrocarbon emissions are 7.41% of the total fuel admitted into the cylinder.