group5ppt-240508085108-ab6d0112202d.pptx

The uses of mathematics in Mechatronics

1. Que.1 The uses of mathematics in Mechat ronics

2. What is Mechatronics? Mechatronics is an interdisciplinary field that combines , mechanical engineering, electrical, electronics, computer science, and control engineering. It focuses on the design and p r o d u c ti o n o f i n t e llig e n t s y s t e m s a n d products. U ses of mathematics in mechatronics Industry 4.o Lab(Robotics) Automative Mechatronics lab Fab Lab

3. 1.Industry 4.o Lab(Robotics) Mathematics plays a crucial role in robotics in various ways: Kinematics: Mathematics helps in determining the position, velocity, and acceleration of robot manipulators and joints. This involves forward and inverse kinematics calculations Dynamics: Understanding the forces and torques acting on robotic systems requires mathematical modeling, which helps in designing controllers to ensure stability and efficient movement. Trajectory Planning: Mathematics is used to plan optimal paths and trajectories for robots to reach their target positions while avoiding obstacles, often using techniques from optimization and control theory.

4. 2.Automotive Mechatronics Mathematics plays a fundamental role in automotive mechatronics, which integrates mechanical, electrical, and computer engineering principles in automotive systems. Here's how mathematics is used in this field: Control System C o n t r o l s y s te m s in m e c h a tr o n ics r e ly o n m a t h e m a t ica l a lg o r it h m s f o r s e n s i n g , p r o c e ss i n g , a n d a c t u a ti n g . T h e y e n s u r e precision and efficiency in mechatronic devices and processes. Type of math control using systems linear algebra is the branch of mathematics that deals with vectors, matrices, and linear transformations. It is fundamental for control systems design because many control problems can be formulated as linear systems of equations, which can be solved using matrix operations.

5. 3.Fab lab Mathematics is extensively used in laser cutting and 3D printing processes to optimize cutting paths, control beam parameters, and ensure precision. Geometric Modeling: Mathematical representations such as CAD (Computer-Aided Design) models use mathematical equations to define the geometry of 3D objects. These models employ geometric primitives (such as points, lines, curves, and surfaces) and mathematical transformations to create complex shapes. Path Planning: Mathematical algorithms determine the optimal toolpath for the laser cutting beam to follow. Material Deposition: Mathematics is used to control the deposition of material during printing accurately. Material Interaction: Mathematics is used to model the interaction between the laser beam and the material being cut.

6. Directional Derivative 2) Write a computer programming for computing directional derivative. The rate of change of φ at any position along given direction is called directional derivative. D.D = φ .a^ It gives D.D of φ along direction of a’ Find directional derivative of φ =x^2+y+z at point(1,-1,1) towards the point ( 2,1,-1)

7. OUTPUT

8. APPLICATION The directional derivative is used in various fields, including physics, engineering, and economics. In physics, it helps calculate the rate of change of a physical quantity in a specific direction, like the rate of temperature change along a path. In engineering, it's used in fields such as fluid dynamics to determine how a fluid flows in a particular direction. In economics, it can be used to analyse how a function changes with respect to changes in multiple variables. A real-life example: Imagine you're hiking up a mountain, and you want to know the steepest path to reach the summit. The directional derivative can help you determine the direction in which the slope is steepest at any given point on the mountain. This information can be useful for optimizing your hiking route to reach the summit more efficiently.

9. Give uses of Fourier transform in your own branch with particular example Data compression : Fourier transform is utilized in data compression techniques like JPEG compression for images and MP3 compression for audio. In JPEG compression, Fourier transform is applied to convert image data from spatial domain to frequency domain, where compression algorithms can discard less important frequency components. Data compression used to convert image data into frequency components for compression Eg.JPEG compression, Discrete Cosine Transform (DCT) 2) Image processing: Fourier transform is employed in tasks such as image filtering, enhancement, and feature extraction. For example, in student attendance system imaging, Fourier transform can be used to enhance certain features in an image, making it easier for teachers to diagnose conditions.

10. Give uses of Fourier transform in your own branch with particular example 3) AUDIO COMPRESSION : During a Google Meet session, your voice is captured by a microphone, converted into an electrical signal, and then digitized. Before transmitting this audio data over the network, it undergoes compression to reduce the amount of data that needs to be transmitted. Compression algorithms, such as those used in VoIP (Voice over Internet Protocol) systems like Google Meet, often employ Fourier transform techniques to convert audio signals from the time domain to the frequency domain. This conversion helps identify frequency components that can be discarded or compressed without significant loss of audio quality. Eg. Google meet,Zoom , Microsoft Team

11. Control Systems: - System analysis in control systems is about understanding how a system behaves and how to control it effectively -Understanding the system’s behavior : This involves analyzing how the system responds to different inputs, such as changes in voltage or pressure, to predict its performance under various conditions. - Control Design: Designing control strategies: Once the system's behavior is understood, engineers can design control strategies to achieve desired outcomes, such as stability, accuracy, or efficiency, by adjusting input signals or feedback mechanisms. For example, consider a mechatronic system consisting of a robotic arm used in industrial automation. Fourier transforms can be employed to analyse the vibrations experienced by the robotic arm during operation. By performing spectral analysis of the vibration signals using Fourier transforms, engineers can identify the dominant frequencies corresponding to different modes of vibration, such as joint resonances or structural vibrations. This information can then be used to optimize the control algorithms and mechanical design of the robotic arm to minimize vibrations and improve precision and reliability in industrial tasks.

12. REFERANCES Mechatronics: Principles and Applications" by Godfrey C. Onwubolu Mechatronics: Electronic Control Systems in Mechanical and Electrical Engineering by W. Bolton Principles and Applications by Godfrey C. Onwubolu Electronic Control Systems in Mechanical and Electrical Engineering by W. Bolton Advanced Engineering Mathematics by Erwin Kreyszig Vector Calculus" by Jerrold E. Marsden and Anthony J. Tromba