Chapter-13; Mathematical Modelling

Introduction:

Welcome back to ThinkTankNotes! In this blog post, we will explore the concept of models and their importance in system analysis. Additionally, we will delve into the classification of physical systems based on their types and discuss the nodal method and analogous systems. Let's dive in!

What is a Model?

A model is an elemental or mathematical representation of a plant or system. It serves as a tool for analyzing the behavior and dynamics of the system. By capturing the system's dynamics, models enable us to study and understand the relationships between system variables.

Types of Mathematical Modelling:

1. Differential Equation Model: This type of model represents a system using differential equations that describe the relationships between system variables and their derivatives.

2. Transfer Function Model: Transfer function models are commonly used in control system analysis. They represent the system's input-output relationship using a ratio of polynomials in the Laplace domain.

3. State Space Model: State space models describe a system's behavior using a set of first-order differential equations. They represent the system's internal states and their evolution over time.

Classification of Physical Systems:

1. Mechanical System: These systems involve the mechanical components that exhibit motion or displacement.

2. Electrical System: Electrical systems are classified based on the type of source used, such as voltage sourced systems or current sourced systems.

3. Electronics System: Electronics systems involve electronic components and circuits for various applications.

4. Hydraulics System: Hydraulics systems utilize fluids and hydraulic components to transmit power and control motion.

5. Thermal System: Thermal systems focus on heat transfer and energy conversion processes.

Electrical System Classification:

1. Voltage Sourced System: These systems utilize a voltage source as the primary input or source of energy.

2. Current Sourced System: Current sourced systems rely on a current source as the primary input or energy source.

Mechanical System Classification:

1. Translational System: This type of mechanical system exhibits linear motion or displacement.

2. Rotational System: Rotational systems involve angular motion about a fixed axis.

Key Components in Mechanical Systems:

1. Mass: Represents the inertial property of the system.

2. Linear Spring: Models the elasticity and stiffness of the system.

3. Damper or Dashpot: Represents the damping effect in the system.

Nodal Method for Drawing Mechanical Networks:

The nodal method is a technique used to draw mechanical networks systematically. The following steps can be followed:

1. Determine the total number of nodes, which equals the total number of displacement elements or mass elements.

2. Establish a reference node in addition to representing the ground line.

3. Connect the mass and inertia elements between the principal nodes and the reference node.

4. Connect the spring and damper elements either between principal nodes or between principal nodes and the reference node, depending on their positions.

5. Obtain the nodal diagram and write the describing differential equations at each node.

Analogous Systems:

Analogous systems refer to different physical systems that can be described by the same mathematical model. The concept of analogous systems is useful due to the following reasons:

1. Solutions derived from one system's set of differential equations can be directly applied to analogous systems of other types.

2. It is easier to build electrical or electronic systems, making them convenient for experimental studies compared to mechanical or hydraulic systems.

Electrical Analogous Systems:

There are four electrical analogous systems commonly used for experimental studies:

1. Force-Voltage Analogous

2. Force-Current Analogous

3. Torque-Voltage Analogous

4. Torque-Current Analogous

In this chapter, we have covered the fundamental concepts of system models, the classification of physical systems, the nodal method for drawing mechanical networks, and the many more.