An Analytical Model Predicting Fixed Index Assembly Machine Performance

1972 ◽  
Vol 94 (2) ◽  
pp. 419-424
Author(s):  
W. B. Heginbotham ◽  
G. E. Rippon
Keyword(s):  
Mathematical Model ◽  
Analytical Model ◽  
Digital Computer ◽  
Matrix Operator ◽  
Operator Technique ◽  
Machine Performance ◽  
Simplified Method ◽  
The Matrix ◽  
Operating Methods

A simplified method of analyzing the performance of fixed index assembly machines using the matrix operator technique is described. This method can be used widely for exploring the effects of operating methods for assembly machines, and an example is quoted where each workhead has a different reliability. By using a digital computer, an exact mathematical model can be built which will enable studies to be made about machine performance when all the head parameters and method of operation are different for each individual head.

scholarly journals MATHEMATICAL MODEL OF THE PLAE RESONANCE VIBRATION SYSTEM BASED ON THE MATRIX-OPERATOR METHOD

2019 ◽  
Vol 2 (46) ◽  
pp. 25-32
Author(s):  
D. Vasilchuk ◽  
◽  
D. Semenets ◽  
V. Romanusha ◽  
B. Kobylianskyi ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Operator Method ◽  
Resonance Vibration ◽  
Matrix Operator ◽  
Vibration System ◽  
The Matrix

A simplified method of temperature control maximizing productivity of the batch reactor; The effect of inertia of the cooling system

1982 ◽  
Vol 47 (2) ◽  
pp. 454-464 ◽  
Author(s):  
František Jiráček ◽  
Josef Horák
Keyword(s):  
Mathematical Model ◽  
Temperature Control ◽  
Cooling System ◽  
Batch Reactor ◽  
Exothermic Reaction ◽  
Cooling Capacity ◽  
Variable Activity ◽  
Simplified Method ◽  
Opening And Closing ◽  
Time Of Operation

The effect has been studied of the inertia of the cooling system on the reliability of control of the temperature of the reaction mixture. The study has been made using a mathematical model of the batch reactor with an exothermic reaction. The temperature has been controlled by a two-level controller opening and closing the flow of the coolant. The aim of the control has been to maintain a constant value of the degree of utilization of the cooling capacity of the reactor. The instantaneous value of the degree of utilization has been assessed from the ratio of times for which the cooling system is idle to the time of operation. The reliability of control has been studied for variable activity of the catalyst.

A NUMERICAL PROOF THAT CERTAIN CELLS FOLLOW THE TRAILS OF THE DIFFUSIONS OF SOME CHEMICALS IN THE EXTRACELLULAR MATRIX

2021 ◽  
pp. 2150027
Author(s):  
İREM ÇAY ◽  
SERDAL PAMUK
Keyword(s):  
Mathematical Model ◽  
Extracellular Matrix ◽  
Tumor Angiogenesis ◽  
Numerical Solutions ◽  
Extra Cellular Matrix ◽  
The Matrix ◽  
Good Agreement ◽  
Cellular Matrix

In this work, we obtain the numerical solutions of a 2D mathematical model of tumor angiogenesis originally presented in [Pamuk S, ÇAY İ, Sazci A, A 2D mathematical model for tumor angiogenesis: The roles of certain cells in the extra cellular matrix, Math Biosci 306:32–48, 2018] to numerically prove that the certain cells, the endothelials (EC), pericytes (PC) and macrophages (MC) follow the trails of the diffusions of some chemicals in the extracellular matrix (ECM) which is, in fact, inhomogeneous. This leads to branching, the sprouting of a new neovessel from an existing vessel. Therefore, anastomosis occurs between these sprouts. In our figures we do see these branching and anastomosis, which show the fact that the cells diffuse according to the structure of the ECM. As a result, one sees that our results are in good agreement with the biological facts about the movements of certain cells in the Matrix.

scholarly journals Mathematical model of methane driven by hydraulic fracturing in gassy coal seams

2020 ◽  
Vol 38 (3-4) ◽  
pp. 127-147
Author(s):  
Weiyong Lu ◽  
Bingxiang Huang
Keyword(s):  
Mathematical Model ◽  
Hydraulic Fracturing ◽  
Pore Pressure ◽  
Hydraulic Fracture ◽  
Water Saturation ◽  
Reduction Rate ◽  
Space Time ◽  
Matrix Block ◽  
The Matrix ◽  
Pore Pressure Gradient

During hydraulic fracturing in gassy coal, methane is driven by hydraulic fracturing. However, its mathematical model has not been established yet. Based on the theory of ‘dual-porosity and dual-permeability’ fluid seepage, a mathematical model is established, with the cleat structure, main hydraulic fracture and methane driven by hydraulic fracturing considered simultaneously. With the help of the COMSOL Multiphysics software, the numerical solution of the mathematical model is obtained. In addition, the space–time rules of water and methane saturation, pore pressure and its gradient are obtained. It is concluded that (1) along the direction of the methane driven by hydraulic fracturing, the pore pressure at the cleat demonstrates a trend of first decreasing and later increasing. The pore pressure gradient exhibits certain regional characteristics along the direction of the methane driven by hydraulic fracturing. (2) Along the direction of the methane driven by hydraulic fracturing, the water saturation exhibits a decreasing trend; however, near the cleat or hydraulic fracture, the water saturation first increases and later decreases. The water saturation in the central region of the coal matrix block is smaller than that of its surrounding region, while the saturation of water in the entire matrix block is greater than that in the cleat or hydraulic fracture surrounding the matrix block. The water saturation at the same space point increases gradually with the time progression. The space–time distribution rules of methane saturation are contrary to those of the water saturation. (3) The free methane driven by hydraulic fracturing includes the original free methane and the free methane desorbed from the adsorption methane. The reduction rate of the adsorption methane is larger than that of free methane.

A Study of Transient Response of Flexible Rotors in High Speed Turbomachinery

1992 ◽  
Author(s):  
V. Pavelic ◽  
R. S. Amano
Keyword(s):  
Mathematical Model ◽  
Computer Program ◽  
Analytical Model ◽  
High Speed ◽  
Angular Speed ◽  
Journal Bearings ◽  
Hydrodynamic Characteristics ◽  
Flexible Assembly ◽  
Lagrange’S Equation ◽  
Equations Of Motions

In many applications the design operating range of the turbomachinery may be well above the rotor first critical speed which leads to the problem of insuring that the turbomachinery performs with a stable, low-level amplitude of vibration. Under certain conditions of high speed and loading the rotor system can start orbiting in its bearing at a rate which is less than the rotor angular speed, and this phenomena is commonly known as whirling or whipping action. This whipping action may produce additional undesirable dynamic loads on the overall flexible assembly and eventually destroy the rotor. Some of this action is also transient in nature. Whirling is a self-exited vibration caused mainly by the fluid bearings and by the internal friction damping of the rotor. To understand this occurrence, a general dynamic mathematical model was derived considering also the complete viscous characteristic of hydrodynamic journal bearings. The general equations of motions of the system are obtained from Lagrange’s equation of motion. The system kinetic, potential, and dissipation functions are determined based on the generalized coordinates of the system. The journal displacements are related to the overall dynamics of the rotor using deformable bearings. The loads acting at the journals of the shaft are integrated from the fluid film pressure distribution in the journal bearings using mobility method. A unique mathematical model is formulated and solved. This model includes the elastic and inertial properties of the flexible rotor, the elastic, damping and inertial properties of supports and the hydrodynamic characteristics of the journal bearings. The equations of motions result in a system of nonlinear second order differential equations which are solved by using finite difference method. The solution of the equations of motions is used to plot maps of motion of journal centers. A computer program was implemented to aid in the solution of the system of equations and to verify analytical model. The computer program used test data available in literature and the results were compared to be very good. The analytical model and results obtained in this study can be of great help to designers of high speed turbomachinery.

scholarly journals Simulation of the Vibratory Behavior of Slender Shafts Subject to Transverse Loads Moving in the Axial Direction

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
F. Bucchi ◽  
P. Forte
Keyword(s):  
Mathematical Model ◽  
Manufacturing Industry ◽  
Axial Direction ◽  
Transverse Load ◽  
Design Phase ◽  
Vibration Level ◽  
Machine Performance ◽  
Transverse Loads ◽  
Spatial Interval ◽  
Influential Parameters

In various machines of the manufacturing industry, and in particular in paper converting machinery, there are shafts operating under conditions similar to that of a slender beam subjected to a transverse load moving in the axial direction. This condition can lead to vibrations and consequent deterioration of the machine performance and of the product quality. The problem has been theoretically studied in the literature since the 1990s. While shaft mass and stiffness are universally considered among the most influential parameters on its vibratory behavior, less obvious and not investigated in the literature is the influence of the spatial interval between two successive loads, an aspect that should be considered in the shaft design phase. In fact, if that is less than the length of the shaft, i.e., if there is more than one transverse load on the shaft at a given time, the vibration level may decrease with respect to the single-load configuration. This work describes the development of a mathematical model of a slender shaft hinged at its ends, representing the rotor of a paper roll perforating unit, with the SW Mathematica. The effect of a load moving axially at a given speed followed by similar loads after given spatial intervals was simulated investigating the influence of speed and load interval on shaft vibrations and resonance. The results showed how reducing the load interval can lead to a reduction of the shaft vibration which is a useful indication on possible design corrective actions.

scholarly journals Symmetric Determinants and the Cayley and Capelli Operator

1948 ◽  
Vol 8 (2) ◽  
pp. 76-86 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Invariant Theory ◽  
Differential Operators ◽  
Matrix Operator ◽  
Linear Combinations ◽  
Classical Invariant Theory ◽  
Independent Variables ◽  
The Matrix ◽  
Classical Invariant ◽  
Initial Discovery ◽  
Special Case

The result obtained by Lars Gårding, who uses the Cayley operator upon a symmetric matrix, is of considerable interest. The operator Ω = |∂/∂xij|, which is obtained on replacing the n2 elements of a determinant |xij by their corresponding differential operators and forming the corresponding n-rowed determinant, is fundamental in the classical invariant theory. After the initial discovery in 1845 by Cayley further progress was made forty years later by Capelli who considered the minors and linear combinations (polarized forms) of minors of the same order belonging to the whole determinant Ω: but in all this investigation the n2 elements xij were regarded as independent variables. The apparently special case, undertaken by Gårding when xij = xji and the matrix [xij] is symmetric, is essentially a new departure: and it is significant to have learnt from Professor A. C. Aitken in March this year 1946, that he too was finding the symmetrical matrix operator [∂/∂xij] of importance and has already written on the matter.

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