Mathematical modelling of mechanical systems examples

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Mathematical modelling of mechanical systems examples

Mathematical Modelling of Engineering Problems MMEP is a top-rated international quarterly reporting the latest mathematical models and computer methods for scientific and engineering problems. Considering the significance of mathematical modelling in engineering design, we are committed to circulating the new developments in engineering science, and solve engineering problems with the most advanced mathematical and computer tools.

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mathematical modelling of mechanical systems examples

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Testing a dual-source heat pump. Evaluation of condensation heat transfer in air-cooled condenser by dominant flow criteria. Camaraza-Medina, Y. Operations of electric vehicle traction system.

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Transfer functions of nonloading cascaded elements Again, consider the two simple RC Tags: electrical gears mathematical mechanical modeling systems torque. Latest Highest Rated. The mathematical law governing mechanical systems is Newtons second law, while the basic laws governing the electrical circuits are Kirchhoffs laws.

Mathematical modeling and representation of a physical system

Spring, mass, damper and inverted pendulum are widely used devices to describe a large class of mechanical systems. Example Automobile suspension system. As the car moves along the road, the vertical displacements at the tires act as the motion excitation to the automobile suspension system.

The motion of this system consists of 4 a translational motion of the center of mass and a rotational motion about the center of mass, and can be simplified as a system with springs, mass and dampers shown below. Equivalent spring constants. Systems consisting of two springs in parallel and series, respectively.

Systems consisting of two dampers connected in parallel and series, respectively. Hence, 9 Example. Consider the spring-mass-dashpot system mounted on a massless cart. Let u, the displacement of the cart, be the input, and y, the displacement of the mass, be the output. Obtain the mathematical model of the SMD system.

Forces acting on the mass 10 By Newtons law Therefore, the system transfer function can be obtained as 11 Example. Mechanical system is shown below. Solution By Newtons second law, we have 12 Simplifying, we obtain 13 Taking the Laplace transforms of these two equations, and assuming zero initial conditions, Solving Equation 2 for X2 s and substituting it into Equation 1 and simplifying, we get 14 Example.While the previous page System Elements introduced the fundamental elements of translating mechanical systems, as well as their mathematical models, no actual systems were discussed.

This page discusses how the system elements can be included in larger systems, and how a system model can be developed. The actual solution of such models requires manipulation of the model into a useful form and is discussed elsewhere. Newton's second law states that an object accelerates in the direction of an applied force, and that this acceleration is inversely proportional to the force, or.

It is an inertial force that arises when you try to accelerate a mass. To visualize this consider pushing against a mass in the absence of friction with your hand in the positive direction. The inertial force is always in a direction opposite to the defined positive direction see the first example below. This formulation the sum of forces at a point is equal to zero also makes analogy to electrical systems easier the sum of all currents into a node is zero ; you needn't concern yourself with this now.

Equations of motion for translating mechanical system depend on the application of D'Alembert's law. Using this principle we say that the sum of force on an object is equal to zero. This inertial force is in the opposite direction from the defined positive direction. We can apply D'Alembert's law to develop equations of motion for translating mechanical systems through the use of free body diagrams. To do this we draw a free body diagram for each unknown position in a system. Consider a simple system with a mass that is separated from a wall by a spring and a dashpot.

The mass could represent a car, with the spring and dashpot representing the car's bumper. An external force is also shown. Only horizontal motion and forces are considered.

There is only one position in this system defined by the variable "x" that is positive to the right. This equation is in our standard form that has system outputs the unknown variables on the left hand side and system inputs the known variables on the right hand side.

We will often use "dot" notation, using one dot above a variable to denote differentiation:. In addition, we will also make it implicit that certain variables are functions of time and omit the " t " in equations. If we do so, the equations above become:. Recall that the definition of the positive direction is arbitrary. We could just as easilty have defined the positive direction to the left.

mathematical modelling of mechanical systems examples

Either set of equations in terms of x or in terms of y exactly, and identically, describes the behavior system. In the first case a positive force to the right will cause the system to start moving in the positive x direction to the right. In the second case a positive force to the right will cause the system to start moving in the negative y direction to the right. The definition of positive directions is arbitrary with respect to the behavior of the system.

mathematical modelling of mechanical systems examples

The example above was relatively simple, but this method applies to more complicated systems as well. The drawing above shows a system with two unknown positions, x 1 and x 2.A physical system is a system in which physical objects are connected to perform an objective. We cannot represent any physical system in its real form. Therefore, we have to make assumptions for analysis and synthesis of systems.

An idealized physical system is called a physical model. A physical system can be modeled in different ways depending upon the problem and required accuracy with which we have to deal.

We can model an electronic amplifier as an interconnection of linear lumped elements, and in case the stress is on distortion analysis then same can be pictured as nonlinear elements. When we have obtained the physical model of a physical system the next step is to obtain the mathematical model which is called the mathematical representation of the physical model. The process of drawing the block diagram for mechanical and electrical systems to find the performance and the transfer functions is called the mathematical modeling of the control system.

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Translational or Linear system : The motion that takes place along a straight line is called a translational motion. There are three different types of forces that we have to study.

Mathematical Modelling of Engineering Problems

Consider a body of mass 'M' and acceleration 'a' then according to newton's second law of motion:. A spring has potential energy. The restoring force of a spring is proportional to the displacement.

Rotational System :When the motion of a body takes place about a fixed axis, this type of motion is known as rotational motion. There are three types of torques that resist the rotational motion. Inertia Torque : The property of an element that stores the kinetic energy of rotational motion is called inertia J.

mathematical modelling of mechanical systems examples

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For example We can model an electronic amplifier as an interconnection of linear lumped elements, and in case the stress is on distortion analysis then same can be pictured as nonlinear elements. Before proceeding let us know, what is the meaning of modeling of the system? There are two types of physical system: Mechanical system. Electrical system.Mathematical models are an essential part for simulation and design of control systems.

The purpose of the mathematical model is to be a simplified representation of reality, to mimic the relevant features of the system being analyzed. Through mathematical modeling phenomena from real world are translated into a conceptual world. There are two main categories of mathematical modeling: theoretical and experimental modeling.

In order to be able to model the system in such a way, several simplifications have to be applied. For example, when modeling the suspension of a vehicle, we assume that the stiffness of the spring is constant, even if in reality is not. This means that the user has all the details concerning how the system works.

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The drawback of the simplification process is that it could ignore some relevant physical phenomena which could be critical for the control system design. Experimental modelingalso called system identificationis based on measurements.

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System which are modeled entirely based on experimental data input-output measurements are called black-box models. This means that the user can observe the response output of the model for a certain stimulus input but has no information about the internal mechanism principles.

Black-box models can be constructed as artificial neural networkstrained based on the input-output measurements of the system. In order to have a representative black-box model, based on artificial neural networks, we need a lot of data-set for training, which needs to cover all the possible working scenarios of the system.

A gray-box model is for example a transfer function or a state-space model. By using experimental data we can estimate the parameter K and T. In the table below we can summarize the characteristics of each type of mathematical models.

Even if theoretical modelingif done properly, delivers more information about the system being analyzed, experimental modeling could be the right method for modeling due to the following reasons:.

In our Systems Modeling category we are going to focus on theoretical modeling. This method makes the link between mathematics and physics and gives the user a clear understanding how mathematics is used and applied in real engineering topics.

I liked the mathematical modelling and the tabular representation. Good work by the author. Keep posting more.Muhammad Arif, PhD m. Obtain the transfer function of linear translational and rotational mechanical systems. Convert mechanical system into series and parallel circuit analogs. Given two springs with spring constant k1 and k2, obtain the equivalent spring constant keq for the two springs connected in:.

When the viscosity or drag is not negligible in a system, we often model them with the damping force. All the materials exhibit the property of Translational damping to some extent. If damping in the system is not enough then extra elements e. Dashpot are added to increase damping.

Developing Mathematical Models of Translating Mechanical Systems

Translational Damper. Analogies Between Electrical and Mechanical Components Mechanical systems, like electrical networks, have three passive, linear components. Newtons Second Law Newton's law of motion states that the algebraic sum of external forces acting on a rigid body in a given direction is equal to the product of the mass of the body and its acceleration in the same direction.

The law can be expressed as. The mechanical system requires just one differential equation, called the equation of motion, to describe it. Assume a positive direction of motion, for example, to the right.

This assumed positive direction of motion is similar to assuming a current direction in an electrical loop. First Step, draw a free-body diagram, placing on the body all forces that act on the body either in the direction of motion or opposite to it. Second Step, use Newtons law to form a differential equation of motion by summing the forces and setting the sum equal to zero.

Third Step, assuming zero initial conditions, we take the Laplace transform of the differential equation, separate the variables, and arrive at the transfer function. Example-1 a : Find the transfer function of the mechanical translational system given in the Figure.

Free Body Diagram. First step is to draw the free-body diagram. Place on the mass all forces felt by the mass. We assume the mass is traveling toward the right.The control systems can be represented with a set of mathematical equations known as mathematical model. These models are useful for analysis and design of control systems.

Analysis of control system means finding the output when we know the input and mathematical model. Design of control system means finding the mathematical model when we know the input and the output. Differential equation model is a time domain mathematical model of control systems.

Follow these steps for differential equation model. Get the differential equation in terms of input and output by eliminating the intermediate variable s.

Consider the following electrical system as shown in the following figure. This circuit consists of resistor, inductor and capacitor. All these electrical elements are connected in series. Transfer function model is an s-domain mathematical model of control systems. The Transfer function of a Linear Time Invariant LTI system is defined as the ratio of Laplace transform of output and Laplace transform of input by assuming all the initial conditions are zero.

Here, we represented an LTI system with a block having transfer function inside it. The above equation is a transfer function of the second order electrical system. The transfer function model of this system is shown below.

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Here, we show a second order electrical system with a block having the transfer function inside it. Control Systems - Mathematical Models Advertisements. Previous Page. Next Page. Previous Page Print Page.


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