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The dynamic behavior of processes in nature can be roughly represented by the following general model of linear dynamic behavior: The Laplace transform is a very useful mathematical tool for the analysis of linear dynamic systems. In fact, the Laplace transform allows solving linear differential equations by transforming them into algebraic equations, thereby facilitating their study. Once the behavior of dynamic systems has been studied, one can proceed to design and analyze control systems in a simple way. The control system design process In order to design an automatic control system, it is required to know the process to be controlled, that is, to know the differential equation that describes its behavior, using physical, chemical and / or electrical laws. This differential equation is called a process model. Once the model is in place, the controller can be designed.
CARLOS ANDRES SANCHEZ VALDES

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The transfer function in a rotating mechanical system

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The transfer function in a rotating mechanical system

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The use of the Laplace Transform offers the possibility of representing dynamic processes in a simple way by means of a mathematical model. The advantage is that it replaces the differential equations in the time-variable by algebraic equations in the complex variable s which does not depend on time.
-Osmar Caballero.

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m (d^2 y)/(dt^2 )+b dy/dt+ky(t)=r(t)

"m" is the mass, "b" is the coefficient of friction, "k" is the spring constant, "y(t)" is the displacement and "r(t)" is the applied force

M(s^2 Y(s)-sy(0)-y’(0))+b(sY(s)-sy(0)-y’(0))+KY(s)=R(s)

considering---> Y(0)= 0, y(0)=0

Ms^2 Y(s)+ bs Y(s)+KY(s)=R(s)

the transfer function is:

Y(s)/R(s) = 1/Ms^2+bs+K

uses and characteristics:
·         The transfer function is given as the quotient of two polynomials in the Laplace complex variable s, one, N (s) (numerator) and the other D (s) (denominator).

·         The degree of the denominator of the transfer function is the order of the system.

·         The polynomial of the denominator, D (s), is called the characteristic equation of the system.

·         Different systems can share the same transfer function, so it does not provide information about its internal structure.
·         Knowing the transfer function of a system, it is possible to study its output for different types of inputs.
·         The polynomial of the denominator of the transfer function, D (s), is called a characteristic function, since it determines, through the values ​​of its coefficients, the physical characteristics of the elements that make up the system.

Isaias sierra

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Investigación del uso de la función de transferencia

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MATHEMATICAL MODELING OF THREE SERIES REACTORS

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MATHEMATICAL MODELING OF THREE SERIES REACTORS

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It is made up of a spring, a mass, and a shock absorber.
Where X (t) is the force, which has been the input, that pushes the mass down, and the forces opposing this movement are the shock absorber and the spring .
Therefore, we will obtain that when applying the force the mass is displaced, with which Y (t) will be the displacement

Applying the Laplace transform, we will obtain the transfer function for this basic translational mechanical system.

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Un sistema de control es un conjunto de dispositivos que se encarga de gestionar el comportamiento de otro sistema.

Por ejemplo, un sistema de control en una vivienda podría controlar la temperatura y humedad de la misma. Un sistema de control en un vehículo podría controlar el frenado ante unas determinadas circunstancias. El sistema de control de una fábrica de alimentos podría controlar la concentración exacta de un determinado aditivo, etc...
Todos estos procesos requieren o requirieron de la herramienta Laplace para fabricarlos.
Porque al desarrollar dicho sistema es necesario conocer todas las leyes que lo rigen así como el comportamiento de lo que vayamos a realizar ya sea mecánico, químico, eléctrico, etc.
En base a lo planteado anteriormente surgen varias ecuaciones diferenciales que son las que se denominan modelo del proceso y para trabajar con estos modelos es necesario el uso de la transformada de Laplace.

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Joel Gonzalez
An optical system's optical transfer function (OTF) determines how different spatial frequencies are handled by the system, such as a camera, microscope, human eye, or projector. Optical engineers use this term to explain how optics project light from an object or scene onto a photographic film, detector array, retina, screen, or the next item in the optical transmission chain. The modulation transfer function (MTF) is a variation that ignores phase effects yet is often similar to the OTF.
An ideal lens system
The optical transfer function is similar to the modulation transfer function since a flawless lens system will offer a high contrast projection without altering the periodic pattern. Typically, the contrast will decrease progressively until it reaches zero at a point indicated by the optics' resolution. The optical transfer function represented in the right hand figure is for a flawless, non-aberrated, f/4 optical imaging system employed at a visible wavelength of 500 nm.

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Funcion de transferencia de un tanque pulmón 

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Funcion de transferencia de un tanque pulmón 

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Consider that the input of the system is the applied force F and the displacement of the mass x - the output of the system. The differential equation relating these two variables is:

F=M  ((d^2)x/dt^2) + f (dx/dt) + kx

Taking the Laplace transform for both sides of the above equation and assuming the initial conditions equal to zero we have:

F(s) = (Ms^2 + f s + k) * (s)

the transfer function of the system is given by:

G(s) = X(s) / F(s) = 1 / Ms^2 + f s + k

The response of the system in the time domain is obtained by taking the inverse transform of X(s),

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"FUNCIÓN DE TRANSFERENCIA DEL SISTEMA DE AMORTIGUACIÓN DE UN AUTOMÓVIL"

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"FUNCIÓN DE TRANSFERENCIA DEL SISTEMA DE AMORTIGUACIÓN DE UN AUTOMÓVIL"

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Transfer function in a Mass-Spring and Damping system

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Transfer function in a Mass-Spring and Damping system

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Briefly, a transfer function is a linear mathematical function that uses the Laplace transform and allows it to represent the dynamic and stationary behavior of any system.

In this system, there is a displacement of the mass x(t) with reference x0.

Considering the diagram, applying Newton's law, we obtain:
M(d^2x/dt^2) + c(dx/dt) + kx = F

Then, applying the Laplace transform:
Ms^2X(s) + csX(s) + kX(s) = F(s)

The transfer function of the system would be as follows:
G(s) = X(s)/F(s) = (1/M)/(s^2 + (c/M)s + k/M)

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Función de transferencia del nivel en un tanque

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Función de transferencia del nivel en un tanque

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Flujo que entra - Flujo que sale= Acumulamiento

Donde
q_i (t)- q_o (t)=A dh(t)/dt

R=h(t)/(q_o (t) )

q_i (t)-1/R h(t)=A dh(t)/dt

Usando Laplace

Q_i (s)-1/R H(s)=AsH(s)

Q_i (s)=H(s)(As+1/R)

Función de trasnferencia  (Q_i (s))/H(s) =1/((As+1/R) )=R/((ARs+1) )

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The transfer function of a thermoemter can be derived as follows.  If the temperature of the water in the container is θ1  and the indicated temperature is  θ0  f the ratio of heat flow in the thermometer through its walls is:

q= θi  - θo / R

where R is the thermal resistance of the turner-netro wall.  The indicated temperature rises to a ratio of:

where R is the thermal resistance of the turner-netro wall.  The indicated temperature rises to a ratio of:

(aθo/dt) = (1/c) q

where c is the capacity of the thermometer.  Therefore, the transfer function that relates the temperature of the water in the container and the indicated temperature is:

G(s)=  (θo/θi)  (s)= 1 / Rcs+1