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The paper shows an example of performed optimization of sizes in terms of welding costs in a characteristic loaded welded joint. Hence, in the first stage, the variables and constant parameters are defined, and mathematical shape of the optimization function is determined. The following stage of the procedure defines and places the most important constraint functions that limit the design of structures, that the technologist and the designer should take into account. Subsequently, a mathematical optimization model of the problem is derived, that is efficiently solved by a proposed method of geometric programming. Further, a mathematically based thorough optimization algorithm is developed of the proposed method, with a main set of equations defining the problem that are valid under certain conditions. Thus, the primary task of optimization is reduced to the dual task through a corresponding function, which is easier to solve than the primary task of the optimized objective function. The main reason for this is a derived set of linear equations. Apparently, a correlation is used between the optimal primary vector that minimizes the objective function and the dual vector that maximizes the dual function. The method is illustrated on a computational practical example with a different number of constraint functions. It is shown that for the case of a lower level of complexity, a solution is reached through an appropriate maximization of the dual function by mathematical analysis and differential calculus.

The vehicle presents a dynamic system performing vibrations under the influence of certain stimulus. Information driver receives from the environment and the vehicle is: macro and micro texture of the road, vibrations and noise of the vehicle's subsystem and so on. The road is usually evaluated according to its macro and micro texture. Microroughness of the roads is of great importance for the study of vehicle's oscillatory movements, reliable computation of vehicle systems, safety etc. Therefore, a special attention must be given to them during the analysis of dynamics, strength and safety of the vehicle. Microroughness that characterize the microprofile of the road in longitudinal and transverse direction have random character and can be established by experimental study of the road with utilization of devices. Accordingly, a variety of devices have been developed for measurement of macro- and micro profiles of the road as well as numerous mathematical and experimental procedures, which with certain approximation satisfy the given requirements, and with the increasing use of computer technology.

In this paper, for the case of gas oscillator two characteristic thermodynamic change of state, and for them defined functional dependence of the force from moving the piston in a nonlinear form were considered. In this paper small oscillations are observed and the linearization is performed by neglecting nonlinear term as the small size at isothermal change and with development obtained function in Taylor series for adiabatic change. Based on the analysis of analog oscillator of mechanical system, or the equivalent scheme of problems, characteristics oscillation parameters and the motion law of the observed thermodynamic system are defined. With this procedure greatly simplifying for problem solution was achieved with respect to classical procedure. At the end of the paper, an analysis of the obtained results were done and appropriate electrical analogy was setting. This is used for the presentation of the problem through appropriate electric circuits and analogy mathematical relations, which involve voltage and electric currents as electric sizes. Also, it was shown that the electrical analogy can be used for all mechanical and thermodynamic processes which describes by homogeneous second-order differential equations which are the most common in technical practice.

Starting from the expression for the thrust in the field of full utilization cross section scrapings and tool life, an expression is derived for the maximum required thrust of universal machine. Then, using a working diagram the analysis of the main features of the simultaneous utilization of machines was performed and determined the optimal area of its utilization for given optimal diameter of treatment. Based on this, the well-known machine using its structural details and the corresponding function workability, derived relations to determine the optimal coefficient slenderness of scrapings suitable for practical use. In this case we think that is known critical material for work piece. Considering the critical and authoritative material of the work piece, based on the expression for the cutting speed was determined by the characteristic constant workability as the basis for establishing optimum tool material which is adequate for optimum regime. Both obtained relation can be considered as general model that can be applied directly to solving setting problems. Also, given the possibilities of practical application of the presented relations, especially in the case of other typical kinds of treatment. Finally, the model is verified on a one calculation example from practice for a Specific machine tool, where certain important characteristics of the optimal treatment are defined.

In a semi-ideal gas, entropy changes cannot be determined through the medium specific heat capacity in a manner as determined by the change of internal energy and enthalpy, i.e. the amount of heat exchanged. Taking this into account, the authors conducted two models through which it is possible to determine the change in the specific entropy of a semi-ideal gas for arbitrary temperature interval using the spread sheet method, using the mean values of the appropriate functions. The idea is to replace integration, which occurs here in evitably, with mean values of the previous functions. The models are derived based on the functional dependence of the actual specific heat capacity on the temperature. The theorem used is that of the mean value of a function as well as the mathematical properties of the definite integral. The mean value of a fractional function is determined via its integrand while the logarithmic functions were performed by applying a suitable transformation of the differential calculus. The relations derived relation, using the computer program, have enabled the design of appropriate thermodynamic tables through which it is possible to determine the change in entropy of arbitrary state changes in an efficient and rational manner, without the use of calculus or finished forms. In this way, the change in the entropy of a semi-ideal gas is determined for an arbitrary temperature interval using the method which is analogous to that applied in determining the change of internal energy and enthalpy or the amount of heat exchanged, which was the goal of the work. Verification of the proposed method for both the above functions was performed for a a few characteristic semi-ideal gases where change c(T) is significant, for the three adopted temperature intervals, for the characteristic change of state. This was compared to the results of the classical integral and the proposed method through the prepared tables. In certain or special cases, it is possible to apply the presented method also in determining the change in entropy of the real gas. Apart from that, the paper shows that the change in entropy for the observed characteristic case can be represented or graphically determined using the planimetric method of diagrams with suitably selected coordinates.

In this paper for complex processes in the field of production mechanical engineering, that describes with the product of step and exponential functions, unknown parameters are calculated. The starting complex function is reduced by suitable method to linear form and then using Taylor's formula developed in series surrounding center point of the experiment. Mathematical model obtained without factors interaction was compared with a similar regression functions. After some mathematical transformations this model allows to determination unknown parameters in the starting function. To simplify the experimental determination of the coefficients of regression and to simplify the analysis, linear model of first order with interactions in coded form was used. Transformation to natural coordinates is performed by using adequate equations. Verification of the suggested method was performed using characteristic experiment in the field of metal deformation. For well-known law of resistance of metals at hot plastic deformation, as a function of influencing factors, all unknown parameters were determined in two ways: by comparing corresponding linear functions and using available derived models. The regression function of prior thermo mechanical process technology was determined experimentally using orthogonal multiple factor first order plan. Also, required statistical analysis and testing of this model were performed. With graphical interpretation of obtained results in logarithmic coordinates, well fittings models and experimental results were shown. Finally, some possibilities for practical application of the proposed methods and models in the field of production engineering, especially in the case of complex processes, were suggested. This method, compared to other methods used for solving setting problem, is faster, more efficient and more reliable.

Starting from the definition of the average specific heat capacity for chosen temperature range, the analytic dependence between the real and the mean specific heat capacities is obtained using differential and integral calculation. The obtained relation in differential form for the defined temperature range allows for the problem to be solved directly, without any special restrictions on its use. Using the obtained relation, a general model in the form of a polynomial of arbitrary degree in the function of temperature was derived, which has more suitable and faster practical application and is more general in character than the existing model. New graphical method for solving the problem is obtained based on differential geometry and using the derived equation. This may also have practical significance since many problems in thermodynamics are solved analytically and graphically. This result was used in order to obtain the amount of specific heat exchanged using an analytical model or a planimetric method. In addition, this graphical solution was used for the construction of the diagram showing the dependence between the specific heat exchanged and temperature. This diagram also gives a simple graphical procedure for the calculation of the real and the average specific heat capacity for arbitrary temperature or temperature interval. The confirmation for all graphic constructions is obtained using the differential properties between thermodynamic units. In order for the graphical solutions presented to be applicable in practice, suitable ratio coefficients have been determined for all cases. Verification of the model presented, as well as the possibilities of its application, were given using several characteristic examples of semi-ideal and real gas. Apart from linear and non-linear functions in the form of polynomials, the exponential function of the dependence between specific heat capacities and temperature was also analysed in this process.

Starting from the experimental concentration-time ( cA,t diagram this work gives the construction of the rate of reaction-time (rA,t diagram using the pure graphic method. The diagram was constructed based on the constructed tangents in arbitrary points of the starting diagram by drawing lines parallel to them in the predetermined pole. The evidence of the construction was derived using differential geometry, i.e. the main theorem of differential calculus. Differential properties between the observed values were used in the method. Starting from the analytic relations rA = rA(t) and cA = cA(t), which can be very complex (polynomes of the n-th order), and, eliminating time t in order to give a full description of the process, we obtain the analytical relation rA = rA(cA), which is then graphically represented. Hoewever, this elimination of time can also be done graphically, in a relatively simple way. After that, through the use of the integral calculus, it was shown that concentration increase in a time interval is proportional to the (rA,t) diagram surface area. Using a similar procedure, further in the paper, it was shown that the time increase is proportional to the (1/rA, cA) diagram surface area. In order for the method to be applicable in practice, we have derived relations for appropriate coefficients of proportionality. Verification of the method is illustrated by the two characteristic examples from chemical kinetics at different monotonies of the starting experimental functions.

The subject of the paper is reliability design in polymerization processes that occur in reactors of a chemical industry. The designed model is used to determine the characteristics and indicators of reliability, which enabled the determination of basic factors that result in a poor development of a process. This would reduce the anticipated losses through the ability to control them, as well as enabling the improvement of the quality of production, which is the major goal of the paper. The reliability analysis and calculation uses the deductive method based on designing of a scheme for fault tree analysis of a system based on inductive conclusions. It involves the use standard logical symbols and rules of Boolean algebra and mathematical logic. The paper eventually gives the results of the work in the form of quantitative and qualitative reliability analysis of the observed process, which served to obtain complete information on the probability of top event in the process, as well as objective decision making and alternative solutions.

Starting from the fact that the real mechanism in a chemical equation takes places through a certain number of radicals which participate in simultaneous reactions and initiate chain reactions according to a particular pattern, the aim of this study is to determine their number in the first couple of steps of the reaction. Based on this, the numbers of radicals were determined in the general case, in the form of linear difference equations, which, by certain mathematical transformations, were reduced to one equation that satisfies a particular numeric series, entirely defined if its first members are known. The equation obtained was solved by a common method developed in the theory of numeric series, in which its solutions represent the number of radicals in an arbitrary step of the reaction observed, in the analytical form. In the final part of the study, the method was tested and verified using two characteristic examples from general chemistry. The study also gives a suggestion of a more efficient procedure by reducing the difference equation to a lower order.

The problem of calculating stoichiometric coefficients in a chemical equation can be solved by standard methods and the method of multidimensional vector space, but good knowledge of vector algebra is required. In this paper, the authors proposed a matrix method and other treatment of the problem was given as the authors' own interpretation. A matrix was formed in the form of base using all the elements which take place in a chemical reaction, after which the matrixes of all the chemical compounds were determined based on numerical indexes and element symbols. This approach enables the setting of a principal matrix equation based on a mathematical approach. The solutions of this matrix equation are the desired stoichiometric coefficients that form a balanced equation. A new approach to tabular solving is presented. This method, compared to existing standard methods, is faster, simpler, and more effective, especially for complex chemical equations. The method was tasted on examples from inorganic chemistry and metallurgy.

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Copyright of Chemical Industry / Hemijska Industrija is the property of Association of Chemical Engineers and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)