Mathematical Modelling of Mechanical and Electrical Systems Prepared By: Div-ME02 NAME Enrolment No. Antala Viral J 130010119004 Bhanderi Sagar S 130010119009 Bhingaradiya Summit 140013119002

Mathematical Modeling Mathematical modeling seeks to gain an understanding of science through the use of mathematical models on HP computers.

Mathematical Models Think how systems behave with time when subject to some disturbances. In order to understand the behavior of systems, mathematical models are required. Mathematical models are equations which describe the relationship between the input and output of a system. The basis for any mathematical model is provided by the fundamental physical laws that govern the behavior of the system.

Mathematical Modeling Is often used in place of experiments when experiments are too large, too expensive, too dangerous, or too time consuming. Can be useful in “what if” studies; e.g. to investigate the use of pathogens (viruses, bacteria) to control an insect population. Is a modern tool for scientific investigation.

Building Blocks Systems can be made up from a range of building blocks. Each building block is considered to have a single property or function. Example: an electric circuit system which is made up from blocks which represent the behaviour of resistance, capacitance, and inductor, respectively. By combining these building blocks a variety of electrical circuit systems can be built up and the overall input-output relationship can be obtained. A system built in this way is called a lumped parameter system.

Mechanical Systems Translational Mass Spring Damper/dashpot Rotational Inertia Damper

Mechanical System Building Blocks Basic building block: spring, dashpots, and masses. Springs represent the stiffness of a system Dashpots/Damper represent the forces opposing motion, for example frictional or damping effects. Masses represent the inertia or resistance to acceleration. Mechanical systems does not have to be really made up of springs, dashpots, and masses but have the properties of stiffness, damping, and inertia. All these building blocks may be considered to have a force as an input and displacement as an output.

Mass The mass exhibits the property that the bigger the mass the greater the force required to give it a specific acceleration. The relationship between the force F and acceleration a is Newton’s second law as shown below. Energy is needed to stretch the spring, accelerate the mass and move the piston in the dashpot. In the case of spring and mass we can get the energy back but with the dashpot we cannot. Force Acceleration Mass

Translational Spring, k (N) x(t) Fa(t)

Dashpot/Damper The dashpot block represents the types of forces experienced when pushing an object through a fluid or move an object against frictional forces. The faster the object is pushed the greater becomes the opposing forces. The dashpot which represents these damping forces that slow down moving objects consists of a piston moving in a closed cylinder. Movement of the piston requires the fluid on one side of the piston to flow through or past the piston. This flow produces a resistive force. The damping or resistive force is proportional to the velocity v of the piston: F = cv or F = c dv/dt.

Rotational Damper, Bm (N-m-sec/rad) Fa(t) Bm

Rotational Spring, ks (N-m-sec/rad) Fa(t) ks

Mechanical Building Blocks Equation Energy representation Translational Spring F = kx E = 0.5 F2/k Damper F = c dx/dt P = cv2 Mass F = m d2x/dt2 E = 0.5 mv2 Rotational T = k E = 0.5 T2/k T = c d/dt P = c2 Moment of inertia T = J d2/dt2 P = 0.5 J2

Building Mechanical Blocks Output, displacement Mass Mathematical model of a machine mounted on the ground Ground Input, force

Building Mechanical Blocks Moment of inertia Mathematical model of a rotating a mass Torque Torsional resistance Block model Shaft Physical situation

Example 1 Consider the following system Free Body Diagram M

Example 1 Differential equation of the system is: Taking the Laplace Transform of both sides and ignoring Initial conditions we get

Example 1 if

Example 2 Find the transfer function of the mechanical translational system given in Figure-1. Free Body Diagram M Figure-1

Automobile Suspension

Automobile Suspension Taking Laplace Transform of the equation (2)

Train Suspension

Train Suspension

Electrical System Building Blocks The basic building blocks of electrical systems are resistance, inductance and capacitance.

Resistance, R (ohm) i(t) v(t) R

Inductance, L (H) i(t) v(t) L

Capacitance, C (F) i(t) v(t) C

For a series RLC circuit, find the characteristic equation and define the analytical relationships between the characteristic roots and circuitry parameters.

The Transfer Function of Linear Systems

Mixed Systems Most systems in mechatronics are of the mixed type, e.g., electromechanical, hydro mechanical, etc Each subsystem within a mixed system can be modeled as single discipline system first Power transformation among various subsystems are used to integrate them into the entire system Overall mathematical model may be assembled into a system of equations, or a transfer function

Electro-Mechanical Example Power Transformation: Ra La B Torque-Current: Voltage-Speed: ia u dc  where Kt: torque constant, Kb: velocity constant For an ideal motor Combing previous equations results in the following mathematical model:

Transfer Function of Electromechanical Example Taking Laplace transform of the system’s differential equations with zero initial conditions gives: u ia Kt Ra La  B Eliminating Ia yields the input-output transfer function

Reference Modern Control Theory by Katsuhiko Ogata, Pearson Edu. Int. Other content are taken from http://www.google.co.in