This document provides an introduction to dynamics and mechanics. It discusses two models used in dynamics - particles and rigid bodies. It describes the motion of each, including their degrees of freedom and parameters. The document then covers equations of motion using vector mechanics and analytical mechanics approaches. It introduces concepts like the center of mass and linear momentum. Key theorems discussed include the linear momentum theorem, conservation of linear momentum, and the center-of-mass motion theorem. Examples are provided to demonstrate applying these theorems to problems involving particles and rigid bodies.
1) The center of mass of a system of particles or solid body represents the point where all the mass can be considered to be concentrated and external forces are applied.
2) Newton's second law can be applied to the center of mass of a system, where the net external force equals the mass times the acceleration of the center of mass.
3) For a closed, isolated system where no external forces act, the total linear momentum of the system remains constant over time.
The document outlines the plan for a lecture on state-space models of systems and linearization. It will begin with a review of control and feedback concepts. The main topic will be introducing state-space models, which provide a general framework for representing different types of systems with differential equations. Examples will be used to illustrate how to derive state-space models from physical systems like masses on springs, electrical circuits, and pendulums. The goal is for students to master the state-space modeling approach in order to enable later analysis and design of systems.
1. Engineering mechanics deals with the behavior of physical bodies under the action of forces or displacements. It examines topics such as statics, dynamics, solid and fluid mechanics using principles from Newtonian physics.
2. Statics analyzes bodies at rest or in constant motion, while dynamics considers bodies in motion and the forces involved in causing and changing that motion. Rigid body mechanics examines both topics for systems that do not deform under loading.
3. Fundamental concepts in mechanics include length, time, mass, and force. Newton's laws of motion describe how forces affect motion. Moments represent the tendency of forces to cause rotation and are important in mechanics of rigid bodies.
This document provides an overview of engineering mechanics as taught in the course ME101:
1) Engineering mechanics deals with the motion and equilibrium of rigid bodies under the action of forces, and includes statics, dynamics, and rigid body mechanics. 2) Rigid body mechanics assumes bodies do not deform under loading and is a prerequisite for more advanced topics. 3) Statics analyzes equilibrium of bodies under constant forces, while dynamics analyzes accelerated motion of bodies.
This document discusses mathematical modeling of mechanical systems. It provides examples of obtaining transfer functions for single-degree-of-freedom and multi-degree-of-freedom translational mechanical systems using Newton's second law and Laplace transforms. Transfer functions are derived for various spring-mass-damper systems, and the effects of adding different mechanical elements like springs, masses, and dampers are explored. Methods for analyzing coupled and uncoupled multi-degree-of-freedom systems are also presented.
Equation of motion of a variable mass system3Solo Hermelin
This is the third of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Lagrangian approach. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation..
This is the more difficult of the three presentations.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
Control system note for 6th sem electricalMustafaNasser9
This document provides an overview of the course EC 6405 – Control System Engineering. It includes 5 units that cover various topics in control systems including:
1) Control system modeling using block diagrams, transfer functions, and modeling different physical systems.
2) Time response analysis using concepts like impulse response, step response, and stability analysis tools.
3) Frequency response analysis using tools like Bode plots, Nyquist plots, and Nichol's chart.
4) Stability analysis using tools like the Routh-Hurwitz criterion and root locus analysis.
5) State variable analysis and digital control systems including discrete-time systems and sampled data control systems.
The document lists textbooks and references for
1) The center of mass of a system of particles or solid body represents the point where all the mass can be considered to be concentrated and external forces are applied.
2) Newton's second law can be applied to the center of mass of a system, where the net external force equals the mass times the acceleration of the center of mass.
3) For a closed, isolated system where no external forces act, the total linear momentum of the system remains constant over time.
The document outlines the plan for a lecture on state-space models of systems and linearization. It will begin with a review of control and feedback concepts. The main topic will be introducing state-space models, which provide a general framework for representing different types of systems with differential equations. Examples will be used to illustrate how to derive state-space models from physical systems like masses on springs, electrical circuits, and pendulums. The goal is for students to master the state-space modeling approach in order to enable later analysis and design of systems.
1. Engineering mechanics deals with the behavior of physical bodies under the action of forces or displacements. It examines topics such as statics, dynamics, solid and fluid mechanics using principles from Newtonian physics.
2. Statics analyzes bodies at rest or in constant motion, while dynamics considers bodies in motion and the forces involved in causing and changing that motion. Rigid body mechanics examines both topics for systems that do not deform under loading.
3. Fundamental concepts in mechanics include length, time, mass, and force. Newton's laws of motion describe how forces affect motion. Moments represent the tendency of forces to cause rotation and are important in mechanics of rigid bodies.
This document provides an overview of engineering mechanics as taught in the course ME101:
1) Engineering mechanics deals with the motion and equilibrium of rigid bodies under the action of forces, and includes statics, dynamics, and rigid body mechanics. 2) Rigid body mechanics assumes bodies do not deform under loading and is a prerequisite for more advanced topics. 3) Statics analyzes equilibrium of bodies under constant forces, while dynamics analyzes accelerated motion of bodies.
This document discusses mathematical modeling of mechanical systems. It provides examples of obtaining transfer functions for single-degree-of-freedom and multi-degree-of-freedom translational mechanical systems using Newton's second law and Laplace transforms. Transfer functions are derived for various spring-mass-damper systems, and the effects of adding different mechanical elements like springs, masses, and dampers are explored. Methods for analyzing coupled and uncoupled multi-degree-of-freedom systems are also presented.
Equation of motion of a variable mass system3Solo Hermelin
This is the third of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Lagrangian approach. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation..
This is the more difficult of the three presentations.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
Control system note for 6th sem electricalMustafaNasser9
This document provides an overview of the course EC 6405 – Control System Engineering. It includes 5 units that cover various topics in control systems including:
1) Control system modeling using block diagrams, transfer functions, and modeling different physical systems.
2) Time response analysis using concepts like impulse response, step response, and stability analysis tools.
3) Frequency response analysis using tools like Bode plots, Nyquist plots, and Nichol's chart.
4) Stability analysis using tools like the Routh-Hurwitz criterion and root locus analysis.
5) State variable analysis and digital control systems including discrete-time systems and sampled data control systems.
The document lists textbooks and references for
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
EGME 431 Term ProjectJake Bailey, CSU FullertonSpring .docxSALU18
EGME 431 Term Project
Jake Bailey, CSU Fullerton
Spring 2016
This document serves to set forth the requirements for your term project, and the criteria
which such project submissions shall be judged. This outline should be the first point of inquiry
for any questions you may have about your project.
The project consists of a thorough investigation, analysis, and set of design improvement sug-
gestions for a simplified automobile suspension model. The dynamics of this model are rather
complex: as such, I have provided a detailed derivation of the equations of motion for this system
to you in a separate document. Your responsibility will be that of the analyst: use the provided dy-
namic models to investigate the system’s response to typical inputs, judge these responses critically,
and suggest improvements to the system.
Your project submissions shall consist of a single analysis and design report. The project
report shall be turned in no later than the final class meeting of the semester, which is May 10,
2016 at 7:00 PM. As always, late assignments will not be accepted. The report shall, at a minimum,
include:
• A description of your analysis methodology
• A summary of the important results from your analyses, including plots and data tables where
appropriate
• A thorough defense of your analysis results, including (but not limited to):
– comparison with analytical approximations
– investigation of typical results of published investigations, and
– discussion and investigation of the approximate errors accrued in your simulations
• A succinct description of the modifications you propose to improve the performance of the
system, including justification of your choices
The dynamic models which have been provided to you include both a fully coupled, non-linear
model and a simplified, linearized version. It is up to you to decide which to use for each portion
of the tasks outlined below. Note, however, that you should, at a minimum, simulate both models
under a common input. This will server as a basis for comparison.
Your specific tasks for this project are as follows:
1. Find the response of the system to a variety of inputs, including steps, impulses, and harmonic
excitation.
2. Determine the Displacement Transmissibility Ratio and Force Transmissibility Ratio of the
system over a range of input frequencies.
1
3. Using judgment, analytical techniques, and/or optimization methods, find a new set of sys-
tem parameters (stiffnesses and damping coefficients) which will improve the response of the
system to the selected inputs.
4. Finally, prepare a report which thoroughly summarizes and defends your methodology and
results.
A final word on collaboration. You are encouraged to discuss your ideas and your solution
approach with your classmates and colleagues. You are, however, expressly forbidden from sharing
simulation data, code, spreadsheets, scripts, or the like with anyone. Two students submitting
substantially similar s ...
The document discusses the concept of center of mass and how to solve center of mass problems. It defines center of mass as the point where the total mass of a system can be considered to be concentrated. The velocity and acceleration of the center of mass can be calculated using the formulas provided. In the absence of external forces, the velocity of the center of mass remains constant. Examples are provided to demonstrate how to calculate the position of the center of mass for a system of particles and how the center of mass will continue along the original path after a rocket explodes into parts due to no change in external forces.
The document provides an overview of applied mechanics, including definitions of mechanics, engineering, applied mechanics, and their various branches and topics. It also covers fundamental concepts such as units, scalars, vectors, and trigonometry functions that are important to mechanics. Examples of static force analysis using vector operations like resolution and resultant are presented.
Charactteristics of forces;
Vector to represent forces;
Classification of forces;
What is force system;
Principles of forces;
Resultant of forces;
Components of forces;
Solved numericals;
examples;
Solved problems;
excercise;
Mechanics is a branch of physics which deals with the state of restmostafaadel290260
This document provides an overview of mechanical engineering and engineering mechanics. It discusses key topics including statics, dynamics, rigid bodies, Newton's laws of motion, and vector analysis. Specifically, it defines statics as the study of bodies at rest or in constant motion, while dynamics examines bodies in motion. It also outlines fundamental concepts such as mass, force, and rigid bodies. Vector quantities like force are defined, and vector operations including component resolution and force combination are demonstrated through examples.
Here are the key steps to solve this problem:
1. Resolve each force into horizontal and vertical components.
2. Take the algebraic sum of the horizontal components to get the horizontal component (Fx) of the resultant.
3. Take the algebraic sum of the vertical components to get the vertical component (Fy) of the resultant.
4. Use the equations:
Resultant (R) = √(Fx)2 + (Fy)2
tan(θ) = Fy/Fx
to find the magnitude and direction of the resultant.
5. Use Varignon's theorem to locate the position of the resultant from point O.
By going through these steps, we find
1. Solid mechanics deals with the behavior of solid bodies subjected to various types of loading. It is subdivided into mechanics of rigid bodies and mechanics of deformable solids.
2. Mechanics of rigid bodies is concerned with static and dynamic behavior under external forces, while mechanics of deformable solids determines internal forces and associated changes in geometry from applied loads.
3. Stresses, strains, and deflections in solids are analyzed using concepts such as stress which defines force intensity, strain which defines deformation, elasticity defined by Hooke's Law, and Poisson's ratio which relates longitudinal and lateral strains.
- Newton's laws of motion describe the relationship between an object and the forces acting upon it.
- The first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
- The second law states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
The 3 conservation laws are:
1) Conservation of energy - the total energy of an isolated system remains constant over time.
2) Conservation of linear momentum - the total momentum of a system remains constant, as long as no external force acts on the system.
3) Conservation of angular momentum - the angular momentum of a system remains constant, as long as no external torque acts on it.
This document discusses mathematical modeling of mechanical systems involving translational and rotational motion. It explains how to form differential equations of motion using Newton's laws and analogies to electrical systems. Models with multiple degrees of freedom are addressed by considering the independent motion of individual points/components and summing the relevant forces for each. Examples of 2 and 3 degree of freedom systems are presented for both translation and rotation.
This document is a chapter from a textbook on engineering mechanics and dynamics. It covers kinetics of particles, including Newton's laws of motion. Key topics summarized include Newton's second law relating force, mass and acceleration; applying this law to rectilinear and curvilinear motion using different coordinate systems; working through examples of applying these concepts; and introducing work, energy, impulse and momentum as other ways to analyze particle motion. Equations of motion are presented for various cases along with explanations of concepts like kinetic energy, potential energy and conservation of energy.
INTRODUCTION_TO_STATICS of rigid bodies.pptxMariyaMariya35
1) The document describes an introduction to statics course, covering topics like forces, equilibrium, moments, distributed forces, centroids, and virtual work.
2) The course objectives are to provide definitions of forces and moments, explain particle and rigid body equilibrium, calculate support reactions, analyze structures, and explain virtual work.
3) Statics is the study of systems at rest or in constant motion, while dynamics considers systems with acceleration. Rigid bodies are assumed to deform negligibly under loads.
The document describes a mechanical system project presented by group members Ali Ahssan, Faysal Shahzad, M. Aaqib, and Nafees Ahmed. It discusses translational and rotational mechanical systems. Translational systems move in a straight line and include mass, spring, and dashpot elements. Rotational systems move about a fixed axis and include moment of inertia, dashpot, and torsional spring elements. The document also provides equations to calculate the opposing forces or torques in each element when a force or torque is applied based on Newton's second law of motion.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
The document discusses modeling of electrical and mechanical systems. It provides examples of modeling an RLC network, DC motor, spring-mass-damper system, and closed-loop position control system using transfer functions derived from equations of motion. Transfer functions are obtained using Laplace transforms of differential equations describing the dynamics of the systems.
The document discusses modeling of electrical and mechanical systems. It provides examples of modeling translational and rotational mechanical systems including springs, masses, dampers, motors, and gears. It derives transfer functions for various systems using equations of motion and Laplace transforms. One example shows modeling a closed-loop position control system with a motor, gears, load, and position feedback.
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
EGME 431 Term ProjectJake Bailey, CSU FullertonSpring .docxSALU18
EGME 431 Term Project
Jake Bailey, CSU Fullerton
Spring 2016
This document serves to set forth the requirements for your term project, and the criteria
which such project submissions shall be judged. This outline should be the first point of inquiry
for any questions you may have about your project.
The project consists of a thorough investigation, analysis, and set of design improvement sug-
gestions for a simplified automobile suspension model. The dynamics of this model are rather
complex: as such, I have provided a detailed derivation of the equations of motion for this system
to you in a separate document. Your responsibility will be that of the analyst: use the provided dy-
namic models to investigate the system’s response to typical inputs, judge these responses critically,
and suggest improvements to the system.
Your project submissions shall consist of a single analysis and design report. The project
report shall be turned in no later than the final class meeting of the semester, which is May 10,
2016 at 7:00 PM. As always, late assignments will not be accepted. The report shall, at a minimum,
include:
• A description of your analysis methodology
• A summary of the important results from your analyses, including plots and data tables where
appropriate
• A thorough defense of your analysis results, including (but not limited to):
– comparison with analytical approximations
– investigation of typical results of published investigations, and
– discussion and investigation of the approximate errors accrued in your simulations
• A succinct description of the modifications you propose to improve the performance of the
system, including justification of your choices
The dynamic models which have been provided to you include both a fully coupled, non-linear
model and a simplified, linearized version. It is up to you to decide which to use for each portion
of the tasks outlined below. Note, however, that you should, at a minimum, simulate both models
under a common input. This will server as a basis for comparison.
Your specific tasks for this project are as follows:
1. Find the response of the system to a variety of inputs, including steps, impulses, and harmonic
excitation.
2. Determine the Displacement Transmissibility Ratio and Force Transmissibility Ratio of the
system over a range of input frequencies.
1
3. Using judgment, analytical techniques, and/or optimization methods, find a new set of sys-
tem parameters (stiffnesses and damping coefficients) which will improve the response of the
system to the selected inputs.
4. Finally, prepare a report which thoroughly summarizes and defends your methodology and
results.
A final word on collaboration. You are encouraged to discuss your ideas and your solution
approach with your classmates and colleagues. You are, however, expressly forbidden from sharing
simulation data, code, spreadsheets, scripts, or the like with anyone. Two students submitting
substantially similar s ...
The document discusses the concept of center of mass and how to solve center of mass problems. It defines center of mass as the point where the total mass of a system can be considered to be concentrated. The velocity and acceleration of the center of mass can be calculated using the formulas provided. In the absence of external forces, the velocity of the center of mass remains constant. Examples are provided to demonstrate how to calculate the position of the center of mass for a system of particles and how the center of mass will continue along the original path after a rocket explodes into parts due to no change in external forces.
The document provides an overview of applied mechanics, including definitions of mechanics, engineering, applied mechanics, and their various branches and topics. It also covers fundamental concepts such as units, scalars, vectors, and trigonometry functions that are important to mechanics. Examples of static force analysis using vector operations like resolution and resultant are presented.
Charactteristics of forces;
Vector to represent forces;
Classification of forces;
What is force system;
Principles of forces;
Resultant of forces;
Components of forces;
Solved numericals;
examples;
Solved problems;
excercise;
Mechanics is a branch of physics which deals with the state of restmostafaadel290260
This document provides an overview of mechanical engineering and engineering mechanics. It discusses key topics including statics, dynamics, rigid bodies, Newton's laws of motion, and vector analysis. Specifically, it defines statics as the study of bodies at rest or in constant motion, while dynamics examines bodies in motion. It also outlines fundamental concepts such as mass, force, and rigid bodies. Vector quantities like force are defined, and vector operations including component resolution and force combination are demonstrated through examples.
Here are the key steps to solve this problem:
1. Resolve each force into horizontal and vertical components.
2. Take the algebraic sum of the horizontal components to get the horizontal component (Fx) of the resultant.
3. Take the algebraic sum of the vertical components to get the vertical component (Fy) of the resultant.
4. Use the equations:
Resultant (R) = √(Fx)2 + (Fy)2
tan(θ) = Fy/Fx
to find the magnitude and direction of the resultant.
5. Use Varignon's theorem to locate the position of the resultant from point O.
By going through these steps, we find
1. Solid mechanics deals with the behavior of solid bodies subjected to various types of loading. It is subdivided into mechanics of rigid bodies and mechanics of deformable solids.
2. Mechanics of rigid bodies is concerned with static and dynamic behavior under external forces, while mechanics of deformable solids determines internal forces and associated changes in geometry from applied loads.
3. Stresses, strains, and deflections in solids are analyzed using concepts such as stress which defines force intensity, strain which defines deformation, elasticity defined by Hooke's Law, and Poisson's ratio which relates longitudinal and lateral strains.
- Newton's laws of motion describe the relationship between an object and the forces acting upon it.
- The first law states that an object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
- The second law states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
The 3 conservation laws are:
1) Conservation of energy - the total energy of an isolated system remains constant over time.
2) Conservation of linear momentum - the total momentum of a system remains constant, as long as no external force acts on the system.
3) Conservation of angular momentum - the angular momentum of a system remains constant, as long as no external torque acts on it.
This document discusses mathematical modeling of mechanical systems involving translational and rotational motion. It explains how to form differential equations of motion using Newton's laws and analogies to electrical systems. Models with multiple degrees of freedom are addressed by considering the independent motion of individual points/components and summing the relevant forces for each. Examples of 2 and 3 degree of freedom systems are presented for both translation and rotation.
This document is a chapter from a textbook on engineering mechanics and dynamics. It covers kinetics of particles, including Newton's laws of motion. Key topics summarized include Newton's second law relating force, mass and acceleration; applying this law to rectilinear and curvilinear motion using different coordinate systems; working through examples of applying these concepts; and introducing work, energy, impulse and momentum as other ways to analyze particle motion. Equations of motion are presented for various cases along with explanations of concepts like kinetic energy, potential energy and conservation of energy.
INTRODUCTION_TO_STATICS of rigid bodies.pptxMariyaMariya35
1) The document describes an introduction to statics course, covering topics like forces, equilibrium, moments, distributed forces, centroids, and virtual work.
2) The course objectives are to provide definitions of forces and moments, explain particle and rigid body equilibrium, calculate support reactions, analyze structures, and explain virtual work.
3) Statics is the study of systems at rest or in constant motion, while dynamics considers systems with acceleration. Rigid bodies are assumed to deform negligibly under loads.
The document describes a mechanical system project presented by group members Ali Ahssan, Faysal Shahzad, M. Aaqib, and Nafees Ahmed. It discusses translational and rotational mechanical systems. Translational systems move in a straight line and include mass, spring, and dashpot elements. Rotational systems move about a fixed axis and include moment of inertia, dashpot, and torsional spring elements. The document also provides equations to calculate the opposing forces or torques in each element when a force or torque is applied based on Newton's second law of motion.
This document contains formulas and equations related to finite element analysis (FEA) for one-dimensional structural and heat transfer problems. It includes formulas for weighted residual methods, Ritz method, beam deflection and stress, springs, one-dimensional bars and frames, and one-dimensional heat transfer through walls and fins. Displacement functions, stiffness matrices, thermal loads, and conduction/convection equations are provided for linear and quadratic elements undergoing static structural and thermal analysis.
The document discusses modeling of electrical and mechanical systems. It provides examples of modeling an RLC network, DC motor, spring-mass-damper system, and closed-loop position control system using transfer functions derived from equations of motion. Transfer functions are obtained using Laplace transforms of differential equations describing the dynamics of the systems.
The document discusses modeling of electrical and mechanical systems. It provides examples of modeling translational and rotational mechanical systems including springs, masses, dampers, motors, and gears. It derives transfer functions for various systems using equations of motion and Laplace transforms. One example shows modeling a closed-loop position control system with a motor, gears, load, and position feedback.
International Conference on NLP, Artificial Intelligence, Machine Learning an...gerogepatton
International Conference on NLP, Artificial Intelligence, Machine Learning and Applications (NLAIM 2024) offers a premier global platform for exchanging insights and findings in the theory, methodology, and applications of NLP, Artificial Intelligence, Machine Learning, and their applications. The conference seeks substantial contributions across all key domains of NLP, Artificial Intelligence, Machine Learning, and their practical applications, aiming to foster both theoretical advancements and real-world implementations. With a focus on facilitating collaboration between researchers and practitioners from academia and industry, the conference serves as a nexus for sharing the latest developments in the field.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
DEEP LEARNING FOR SMART GRID INTRUSION DETECTION: A HYBRID CNN-LSTM-BASED MODELgerogepatton
As digital technology becomes more deeply embedded in power systems, protecting the communication
networks of Smart Grids (SG) has emerged as a critical concern. Distributed Network Protocol 3 (DNP3)
represents a multi-tiered application layer protocol extensively utilized in Supervisory Control and Data
Acquisition (SCADA)-based smart grids to facilitate real-time data gathering and control functionalities.
Robust Intrusion Detection Systems (IDS) are necessary for early threat detection and mitigation because
of the interconnection of these networks, which makes them vulnerable to a variety of cyberattacks. To
solve this issue, this paper develops a hybrid Deep Learning (DL) model specifically designed for intrusion
detection in smart grids. The proposed approach is a combination of the Convolutional Neural Network
(CNN) and the Long-Short-Term Memory algorithms (LSTM). We employed a recent intrusion detection
dataset (DNP3), which focuses on unauthorized commands and Denial of Service (DoS) cyberattacks, to
train and test our model. The results of our experiments show that our CNN-LSTM method is much better
at finding smart grid intrusions than other deep learning algorithms used for classification. In addition,
our proposed approach improves accuracy, precision, recall, and F1 score, achieving a high detection
accuracy rate of 99.50%.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
bank management system in java and mysql report1.pdf
E_Presentation_slides_03_week.pdf
1. Engineering
Mechanics:
Dynamics
• ME2211E
• Prof.Dr. Dinh Van Phong
• Dept.of Applied Mechanics, School of
Mechanical Engineering
• C3/307-308 (C1/224)
• Hanoi University of Science and
Technology
• Email: phong.dinhvan@hust.edu.vn
• 0903200960/ (024) 36230949
2. Chapter 1. Introduction to Dynamics -2-
Department of Applied Mechanics-SME
Two models using in Dynamics: Particle and Rigid body
P
rP
x
z
y
O
A
rA
x
z
y
O
B
Motion of a particle:
Motion of a rigid body:
• Motion in plane, motion in space
• Rectilinear & Curvilinear motion
• Velocity and Acceleration of a
particle
• Maximum 3 position parameters (3
Degree of Freedom)
• Translation
• Rotation about a fixed axis / fixed point
• General spatial motion
• General planar motion
• Helical motion (screw motion)
• Velocity and Acceleration of a point and angular
velocity vector, angular acceleration vector
• Maximum 6 position parameters (6 Degree of
Freedom)
Particle model Rigid body model
3. Equations of
Motion
Model of particles or
rigid bodies
• Force and acceleration
• Work and energy
• Impulse and
momentum
3 methods:
2 Approaches:
-Vector Mechanics
- Analytical Mechanics
4. Applied Mechanics - Department of Mechatronics - SME
Chapter 3.
A Method of Vector Mechanics:
Momentum Method
5. Applied Mechanics - Department of Mechatronics - SME
-5-
Content
I. SOME CONCEPTS
1. Mechanical system, Internal Force, External Force
2. The center of mass of mechanical system
• The center of mass of system of particle
• The center of mass of rigid body, system of rigid bodies
II. LINEAR MOMENTUM THEOREM
1. Linear momentum of mechanical system
2. Linear momentum theorem
3. Conservation Linear momentum
III. THE CENTER-OF-MASS MOTION THEOREM
1. The center-of-mass motion theorem
2. Conservation of the center-of-mass motion
6. Applied Mechanics - Department of Mechatronics - SME
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Content
I. SOME CONCEPTS
1. Mechanical system, Internal Force,
External Force
2. The center of mass of mechanical system
• The center of mass of system of particles
• The center of mass of rigid body, system of
rigid bodies
7. Applied Mechanics - Department of Mechatronics - SME
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1. Mechanical system – Internal, External force
Mechanical system includes particles and rigid bodies that interact to
each other (forces and constraints of motion)
Boundary, Inside and outside of the system.
mk
m1
e
k
F
i
k
F
i
j
F
e
l
F
Boundary
outside
internal
x
y
z
O
k
m
g
j
m
e
j
F
e
l
F
e
k
F
boundary
8. Applied Mechanics - Department of Mechatronics - SME
-8-
1. Mechanical system – Internal, External force
Internal force: forces interact between
particles / rigid bodies in a system.
External force: forces from outside the
system act on the system, such as gravity,
force associated with the environment, ..
mk
m1
e
k
F
i
k
F
i
j
F
e
l
F
Boundary
outside
internal
i i
k O k
k k
F m F O
0, ( ) 0,
= =
e
k
F k
, 1,2,...
i
k
F k
, 1,2,...
Properties of the internal force system
9. Applied Mechanics - Department of Mechatronics - SME
-9-
2. Center of mass of a system of particles
1
1
1
1
,
1
,
1
n
C k k
k
n
C k k
k
n
C k k
k
x m x
m
y m y
m
z m z
m
=
=
=
=
=
=
For a system of n particles with masses mk (k=1,…,n), the
Center of mass is a geometric point C that satisfies:
1
0
n
k k
k
m u
=
=
x
y
z
mk
O
m1
C
k
x
k
r k
y
k
z
C
r
k
u
k
u - vector from C to point Mk
1 1
( ) 0
n n
k k C k k k k C
k k
u r r m u m r r
= =
= − = − =
In Oxyz:
- Position vectors of particles
- Position vector of center of mass C
1 2
, ,..., n
r r r
n n
C k k k
k k
r m r m m
m 1 1
1
,
= =
= =
In case, g = const, center of mass C center of gravity G.
C
r
10. Applied Mechanics - Department of Mechatronics - SME
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2. Center of mass of a rigid body
C C C
x m x dm y m ydm z m z dm
1 1 1
, ,
− − −
= = =
The Center of mass of a rigid body is point C:
x
y
z
O
C
M
x
y
z
dm
r
C
r
u
11. Applied Mechanics - Department of Mechatronics - SME
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2. Center of mass: mechanical system
x
y
z
O
C
Ci
i
C
r
k
r
k
u
Ci
u
C
r
k
m
i
m Cj
1 1 1 1
1 1
1 1
, ,
1
,
p p
n n
C k k i Ci C k k i Ci
k i k i
p
n
C k k i Ci
k i
x m x m x y m y m y
m m
z m z m z
m
= = = =
= =
= + = +
= +
1 1
1 1
1
( ),
p
n
C k k i Ci
k i
p
n
k i
k i
r m r m r
m
m m m
= =
= =
= +
= +
The Center of mass of system including n particles and p rigid bodies
12. Applied Mechanics - Department of Mechatronics - SME
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2. Center of mass: Rigid body and rigid body system
If the system is on or near the earth’s surface, the gravitational acceleration is
constant, the center of mass coincides with the center of gravity. Thus, the
methods used in calculation of the center of gravity can be applied to the
calculation of the center of mass.
The center of mass of the body always exists but the center of gravity exists
only when the system is in the gravity field (of the Earth).
Some remarks:
• If the homogeneous rigid body has a symmetric center (axis, plane), its
center of mass lies on that symmetric center (axis, plane).
• If the rigid body consists of several parts that have center of mass lie on an axis
(plane), then its center of mass lies on that axis (plane)
13. Applied Mechanics - Department of Mechatronics - SME
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Content
II. LINEAR MOMENTUM THEOREM
1. Linear momentum of mechanical system
2. Impulse of a force
3. Linear momentum theorem
4. Conservation Linear momentum
14. Applied Mechanics - Department of Mechatronics - SME
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1. Linear momentum of a mechanical system
• Linear momentum of a particle
• Linear momentum of a system of particles
p mv
= [kg.m/s]
1 1
( )
n n
k k k C
k k
p m v m v
= =
= =
• Linear momentum of a rigid body
• Linear momentum of a system
C C
p vdm rdm mr mv
= = = =
,
k C k
p p Mv M m
= = =
The linear momentum of a system is equal to the
product of the total mass M of the system and the
velocity of the center of mass. x
y
z
O
C
vdm
C
r
u
C
p mv
=
mk
m1
i i
m v
k k
m v
C
p mv
=
15. Applied Mechanics - Department of Mechatronics - SME
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2. Impulse
• Impulse of a force during infinitesimal time dt is defined by:
(elementary impulse)
( )
dS F t dt
=
• Impulse of force F(t) during the length of time t1 to t2
2
1
( )
t
t
S F t dt
=
t
t1
t2
F(t)
S
• The unit of impulse is Newton.second [N.s] ,
(equivalent unit: the kilogram meter per second
[kg⋅m/s]).
16. Applied Mechanics - Department of Mechatronics - SME
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3. Linear momentum theorem
• In derivative form: The time rate of change of the momentum of a
particle is equal to the net force acting on the particle and it is in the
direction of that force.
For a particle
p mv ma F
= = = Newton’s second law. (mass is constant)
• In integral form: The change in the linear momentum of a particle is
equal to the impulse that acts on that particle.
dp
F dp Fdt
dt
= =
1
( )
mv t
2
1
t
t
Fdt
2
( )
mv t
=
+
2
1
2 1
( ) ( ) ( )
t
t
p t p t S F t dt
− = =
17. Applied Mechanics - Department of Mechatronics - SME
-17-
3. Linear momentum theorem
• In derivative form: The time rate of change of the momentum of a system
is equal to the net of all external forces acting on the system:
For a system
e
k k k k k
p m v F
= =
• In integral form: The change in the linear momentum of a system is equal
to the impulse of all external forces acting on the system:
e e
k k k k
dp
F dp F dt
dt
= =
2
1
2 1
( ) ( )
t
e
k k k
t
p t p t Se F dt
− = =
Proof.
, 0
e i i
k k k k k k k
e i
k k k k k k k
p m v F F F
p p m v F F
= = + =
= = = +
18. Applied Mechanics - Department of Mechatronics - SME
-18-
3. Conservation of Linear momentum
0 ,
e
k C
k
If F p const mv const
= = =
• If sum of all external forces act on a system is zero, then the total
momentum of the system is a constant vector.
0 ,
e
kx x C
k
If F p const mx const
= = =
• If the component of the net external force on a closed system is zero
along an axis, then component of the linear momentum along that
axis cannot change:
The linear momentum theorem is valid not only for particle and system of particles,
but also valid for a rigid body and system of rigid bodies. This is one of general
theorems in dynamics of the system in which the 3. Newton law is valid.
19. Applied Mechanics - Department of Mechatronics - SME
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Examples
Force depending on time P(t)
Example 1. A car of mass m starts
from the rest on a straight road by
repulsion force P = P0(1-exp(-a.t)),
in which P0 and a are const. Find
the velocity expression as function
of time.
m
P
0
( ) (0) ( ) , (0) 0
t
mv t mv F t dt v
− = =
Solution.
Force depends on time, apply the
momentum theorem, on axis x.
t t
at at
at
P e dt P t e a
P t e a P a
0 0 0
0
0 0
(1 ) ( / )
( / ) /
− −
−
− = +
= + −
0
0
0
( ) (0)
(0) (1 )
t
x
t
at
mv t mv F dt
mv P e dt
−
= +
= + −
( )
0 0
1
( ) (0) ( / ) /
at
v t v P t e a P a
m
−
= + + −
x
y
20. Applied Mechanics - Department of Mechatronics - SME
-20-
Examples
Example 2. Conservation of momentum. A system
includes body A (mass m1) which is put on an inclined
plane of a prism (mass m2). The angle between the
inclined plane of the prism and the horizontal plane is
. The prism is put on a smooth horizontal plane (as
shown in Figure). At first, the body stays still on the
prism plane relatively, the prism slides to the right with
velocity v0. After that, body A slide on the inclined plane
with relative velocity u=at. Find the velocity of the
prism.
v0
u
A
B
m1
m2
x
y
21. Applied Mechanics - Department of Mechatronics - SME
-21-
Example
v0
u
A
B
m1
m2
m
v v u
m m
1
0
1 2
cos
Solution
External forces of the system: P1, P2, N1, N2
e
k x x x
F p const p
,
0 (0)
x x x
p m v m v
m m v
1 1 2 2
1 2 0
(0) (0) (0)
( )
= +
= +
(1)
(2)
x x x
p t m v t m v t
m v t u m v t
1 1 2 2
1 2
( ) ( ) ( )
( ( ) cos ) ( ) (3)
x
y
P2
P1
N1
N2
( ) (0)
x x
p t p
22. Applied Mechanics - Department of Mechatronics - SME
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III. THE CENTER-OF-MASS MOTION THEOREM
1. The center-of-mass motion theorem
2. Conservation of the center-of-mass motion
23. Applied Mechanics - Department of Mechatronics - SME
-23-
1. The center of mass motion theorem
The center of mass motion theorem
Apply Newton's second law to the particle mk that belongs to system:
e i
k k k k
m a F F k n
, 1,2,...,
Take the sum of the two sides with all particles of the system,
paying attention to the properties of the internal force system
n n n n
e i e
k k k k k
k k k k
m a F F F
1 1 1 1
n n n
k k C k k C k k C
k k k
m r m r m r m r m a m a
1 1 1
n n
e
C k k
k k
m a F m m
1 1
,
Note: The center of mass motion
theorem is valid for both body
and body system.
The sum of all external forces acting on the system is
equal to the total mass of the system multiplied by the
acceleration of the center of mass of the system.
24. Applied Mechanics - Department of Mechatronics - SME
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2. The center of mass motion theorem / conservation case
The center of mass motion theorem
n
e
C k
k
m a F
1
e
C kx
e
C ky
e
C kz
m x F
m y F
m z F
,
,
Some cases of conservation
1
If 0 (0)
(0) 0 (0)
n
e
k C C
k
C C C
F m v const m v
If m v m r const m r
1
If 0 (0)
(0) 0 (0)
n
e
kx C C
k
C C C
F m x const m x
If m x m x const m x
25. Applied Mechanics - Department of Mechatronics - SME
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Examples
Example 1. The electric motor is fixed on the
horizontal ground as shown in the figure. The
fixed part of the motor (stator) has mass m1,
the rotating part (rotor) has mass m0 with the
center of mass at point A and OA = e.
(a)Find the maximum shear force acting on
the bolts if the motor rotates with angular
velocity .
(b)Assume that the motor is freely placed on
a smooth ground (no bolts, no friction), find
the allowable rotation speed so that the
motor is not bounced off the ground.
Solution:
a) We examine the whole motor. External forces act on the system as figure.
With center of mass C, apply the center of mass motion theorem
t
O
A
x
y
G
P1
P0
R
N
n
e
C k C
k
m a F m m a P P N R
0 1 0 1
1
( )
26. Applied Mechanics - Department of Mechatronics - SME
-26-
Examples
2
1 0 0
( ) cos
C
R m m x R m e t
= + = −
A G A
C C A
m x m x m x m
x x x
m m m m m m
0 1 0 0
0 1 0 1 0 1
+
= = =
+ + +
A A
x e t x e t
2
cos cos
= = −
C
m e t
x
m m
2
0
1 0
cos
= −
+
R m e 2
max 0
=
Note: If the rotor rotates at high speed (), the force Rmax will be large even though e is small.
In engineering, it is common to find ways to reduce the eccentricity as small as possible,
ideally, e = 0.
The maximum value of the shear force on bolts
Determine the center of mass acceleration in the x axis
t
O
A
x
y
G
P1
P0
R
N
0 1
,
C C
m x R m y N P P
27. Applied Mechanics - Department of Mechatronics - SME
-27-
Examples
0 1 0
0 1 0 1
A G A
C C
m y m y m y
y y
m m m m
+
= =
+ +
A A
y e t y e t
2
sin sin
= = −
C
y m m m e t
1 2
1 0 0
( ) sin
−
= − +
C
N P P m m y m m g m e t
2
0 1 1 0 1 0 0
( ) ( ) sin
= + + + = + −
m m
N m m g m e g
m e
2 0 1
min 0 1 0 max
0
( ) 0
+
= + − =
b) Where the motor is freely placed on a non-friction ground without shear forces
from bolts.
C C
m y N P P N m y P P
0 1 0 1
The motor does not bounce off the ground, if the maximum allowable angular velocity needs to satisfy:
Calculating yC t
O
A
x
y
G
P1
P0
R
N
28. Applied Mechanics - Department of Mechatronics - SME
-28-
Examples / conservation case
A
B
m
M
x
y
Example 2.
Determine the horizontal displacement
of the ship carrying the crane, when
crane AB is needed to be lifted vertically
from the original angle = 30 as shown
in figure. Givens : M = 20 tons, m = 2
tons, and AB = L = 8 m. Neglect the
water resistance and the mass of AB
crane.
29. Applied Mechanics - Department of Mechatronics - SME
-29-
Examples / conservation case
1 1 2
( ) ( ) ( sin )
C
Mx t M x m x d L
= + + + + −
1
sin
(0) ( ) 0,36 m
C C
mL
Mx Mx t
M m
= = =
+
A
B
m
M
x
y
Solution
By neglecting horizontal resistance, all
external forces are vertical.
Case (= 0): Suppose the ship moves a distance of horizontally to the
right:
1 2
(0) ,
C
Mx Mx mx
= + 2 1
sin
x x d L
= + +
= 30
Determine the center of mass of system
at two cases : = 30 and = 0
( ) 0
=const (0) 0
=const
C
C C
C
M m x
x x
x
+ =
= =
30. Applied Mechanics - Department of Mechatronics - SME
-30-
IV. MOMENT OF INERTIA OF A RIGID BODY
ABOUT AN AXIS
1. Mass moment of inertia
2. Radius of gyration
3. Parallel-axis theorem
4. Moment of Inertia of some simple rigid body
5. Method of composite bodies
31. Applied Mechanics - Department of Mechatronics - SME
-31-
Mass moment of inertia
Definition: Moment of Inertia of rigid body about
z-axis, Iz, is scalar quantity calculated by
x
y
z
O
x
y
z dm
h
u
2 2 2
( )
z
I h dm x y dm
= = +
z
z
M
z
I
Moment of Inertia is a measure of the resistance of a body in rotation
z z z
I I M
= =
m
ma F
=
F
Moment of Inertia of rigid body about an axis
32. Applied Mechanics - Department of Mechatronics - SME
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Mass moment of inertia
Moment of Inertia of rigid body about a point
x
y
z
O
x
y
z dm
u
Definition: Moment of Inertia of rigid body about a point O, IO
O
I u dm x y z dm
2 2 2 2
( )
= = + +
O
I y z dm x z dm x y dm
2 2 2 2 2 2
1
( ) ( ) ( )
2
= + + + + +
O x y z
I I I I
1
( )
2
= + +
2 2
2 2
( ) ,
( )
x
y
I y z dm
I x z dm
= +
= +
Radius of gyration of a body about
an axis of rotation z is defined
2
/
z z z z
I m I m
= =
Similarly, Ix and Iy z
m
z
33. Applied Mechanics - Department of Mechatronics - SME
-33-
Parallel-Axis Theorem
Huygens-Steiner Theorem
z Cz
I I md2
= +
z Cz
I r dm I r dm
2 2
,
= =
2 2 2
2 2 2
2 2 2 2
2 2
, ,
2( )
2( )
C C
C C C C
C C
x x x y y y
r x y
r x y
x y x y x x y y
r d x x y y
= + = +
= +
= +
= + + + + +
= + + +
x y
z
O
C
d
r’
r
C
x
C
y
y
z
x
dm
Proof
C-center of mass
x dm y dm
0, 0
= =
z C C z
I r dm r d x x y y dm I md
2 2 2 2
'
[ 2( )] .
= = + + + = +
C-center of mass
As shown in figure, we have
34. Applied Mechanics - Department of Mechatronics - SME
-34-
Moment of Inertia of homogeneous rigid body
a) Slender rod AB with mass m and length L.
z
I x dm
2
=
m
dm dx
l
=
z
dm
C x
x dx
A
B
z’
With constant cross-section:
l
l
z l l
m
I x m l dx x ml
l
/2
/2
2 3 2
/2 /2
1
( / )
3 12
− −
= = =
l
l
Az
m
I x m l dx x ml
l
2 3 2
' 0 0
1
( / )
3 3
= = =
Cz Az
I ml I ml
2 2
'
1 1
&
12 3
= =
Axis Cz ⊥ AB, CA=CB
Axis Az’ ⊥ AB
35. Applied Mechanics - Department of Mechatronics - SME
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Moment of Inertia of homogeneous rigid body
b) Thin ring (m, r)
2 2
2 2
Cz
O
I dm r dm
r dm mr I
= =
= = =
x y
I I
=
2 2
1
,
2
x y O Cz
I I mr I I mr
= = = =
dm
O=C x
y
m
r
1
( )
2
O x y z z Cz
I I I I I I
= + + = =
Note: IO and Ix = Iy
Similarly, for a thin hollow round
cylinder (m, r)
Cz
I mr2
=
z
36. Applied Mechanics - Department of Mechatronics - SME
-36-
Moment of Inertia of homogeneous rigid body
c) Thin Disk (m, r)
Cz O
dI dm dI
2
= =
2
( / )2
dm m r d
= : 0 r
→
r
r
Cz O
m m
I d mr I
r r
2 4 2
2 2
0 0
2 1
2
2
4
= = = =
dm
O=C x
y
d
Elementary ring (dm, , d). Moment of Inertia
of this element about Cz
x y Cz
I I mr I mr
2 2
1 1
,
4 2
= = =
O x y z z
I I I I I
1
( )
2
= + + =
Note: IO and Ix = Iy
Similarly, for a cylinder (m, r)
Cz
I mr2
1
2
=
z
37. Applied Mechanics - Department of Mechatronics - SME
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Moment of Inertia of homogeneous rigid body
Disk with a hole (m, r, R)
Cz O
dI dm dI
2
= =
2 2
,
( )
mdA
dm dA d d
R r
= =
−
r R
: , : 0 2
→ →
R
R
Cz r
r
m m
I d d
R r R r
m R r m R r
R r
4
2
2
2 2 2 2
0
4 4 2 2
2 2
2
4
( ) ( )
( ) ( )
2
2( )
= =
− −
− +
= =
−
Element of mass (dm, , d, , d). Moment of
Inertia of this element about Cz-axis :
Similarly, for a thick hollow round
cylinder (m, r, R)
Cz
m R r
I
2 2
( )
2
+
=
dm
O=C x
y
d
R
r
d
z
38. Applied Mechanics - Department of Mechatronics - SME
-38-
Moment of Inertia of homogeneous rigid body
Rectangle plate (m, a, b)
To determine Moment of Inertia of plate about Cy-axis:
2
2
/2
2 2
/2
1
( ) ,
12
1
12
1 1
12 12
Cy
Cy
b
Cy
b
m
dI dm a dm dy
b
m
dI a dy
b
m
I a dy ma
b
−
= =
=
= =
Similarly, Moment of Inertia of plate about Cx-axis : 2
1
12
Cx
I mb
=
For thin plate:
Cz C
I I
=
( )
1
2
C Cx Cy Cz
I I I I
= + +
2
1
12
Cy
I ma
=
m
C x
y
a
b
dm dy
y
2 2
1
( )
12
Cz Cx Cy
I I I m a b
= + = +
39. Applied Mechanics - Department of Mechatronics - SME
-39-
Summary
z
C
A
z
L/2 L/2
2
2
1
12
1
3
Cz
Az
I ml
I ml
=
=
m
O=C x
y
r
Huygens-Steiner theorem 2
z Cz
I I md
= + C-center of mass
m
O=C x
y
r
Cz
I mr2
1
2
=
Slender Rod Thin ring Disc
2
Cz
I mr
=
40. Chapter 5 Planar Kinetics of a Rigid Body
-40-
Composite Bodies
If a body is constructed of a number of simple shapes, such as
disks, spheres, or rods, the mass moment of inertia of the body
about any axis can be determined by algebraically adding
together all the mass moments of inertia, found about the same
axis, of the different shapes.
COMPOSITE BODIES
Algorithm:
1. Calculation of the mass moment of inertia for
each shape (about arbitrary axis)
2. Converting all moments of inertia of shapes to
common axis (using Parallel Axis Theorem)
3. Algebraic adding
41. Chapter 5 Planar Kinetics of a Rigid Body
-41-
The pendulum can be divided into a
slender rod (r) and a circular plate (p).
Calculate the moment of inertia of the
composite body. Then, determine the
radius of gyration.
Given: The pendulum consists of a slender
rod with a mass 2 kg and a circular
plate with a mass of 4 kg.
Find: The pendulum’s radius of gyration
about an axis perpendicular to the
screen and passing through point O.
Plan:
EXAMPLE
42. Chapter 5 Planar Kinetics of a Rigid Body
-42-
The center of mass for rod is at point Gr, 1 m from
Point O. The center of mass for circular plate is
at Gp, 2.5 m from point O.
IO = IG + (m) (d) 2
For the rod: IOr = (1/12) (2) (2)2 + 2 (1)2 = 2.667 kg·m2
For the plate: IOp = (1/2) (4) (0.5)2 + 4 (2.5)2 = 25.5 kg·m2
3. Now add the two MMIs about point O.
IO = IOr + IOp = 28.17 kg·m2
1. Calculate MMI for a slender rod and a circular
plate (or using handbook).
2. Using those data and the parallel-axis theorem,
calculate the following:
SOLUTION
R
P
43. Chapter 5 Planar Kinetics of a Rigid Body
-43-
SOLUTION (continued)
R
P
4. Total mass (m) equals 6 kg
Thus the radius of gyration about O:
ρO = 𝐼O/𝑚 = 28.17/6 = 2.17 m
44. Applied Mechanics - Department of Mechatronics - SME
-44-
Moment of inertia of composite bodies
Determine the moment of inertia about an
axis perpendicular to the page and passing
through the pin at O. The thin plate has a
hole in its center. Its thickness is h = 50 mm,
and the material has a density = 50 kg/m3.
O
r
a
a
a
a
C
= + =
(p) 2 2 2
1
( ),
12
C p p
I m a a m a h
For a square plate
For the hole
= =
(h) 2 2
1
,
2
C h h
I m r m r h
= − =
= + − =
(p) (h)
2
...
( ) , 2 / 2
C C C
O C p h
I I I
I I m m d d a
45. Homework
• Read Textbooks and slides
• Do exercises : TEAMS: Force-Acceleration: slides 6-14
• Send your homework to my email
( only in 1 file and in Subject of Email specify:
your name, class, Force_Acceleration_slidenumber_.....)
• See you soon