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This document discusses different approaches for constructing mathematical models from physical systems: 1) Newtonian mechanics uses Newton's second law to directly obtain equations of motion for lumped mass systems. 2) D'Alembert's principle allows inertia forces to be included in equilibrium diagrams, making it useful for continuous systems. 3) The principle of virtual work equates the total virtual work done by internal and external forces during virtual displacements to zero, providing another approach for developing equations of motion. Examples are provided to illustrate Newtonian mechanics and the principle of virtual work.

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ModelingZillDiffEQ9_LecturePPTs_05_01.pptx

Modeling with high order

DOMV No 7 MATH MODELLING Lagrange Equations.pdf

The document discusses mathematical modeling using Lagrange's equations. It begins by introducing Newtonian mechanics, the principle of virtual work, and Lagrange's equations as three approaches. It then focuses on Lagrange's equations, explaining that they describe the dynamics of systems with N degrees of freedom in terms of energy and generalized coordinates. The document provides details on Lagrange's equations, including examples of their use for conservative and dissipative systems. It also discusses how generalized forces are established and the equations of motion for linear multi-degree-of-freedom systems.

DOMV No 4 PHYSICAL DYNAMIC MODEL TYPES (1).pdf

This document discusses three physical modeling techniques for dynamic analysis of structures:
1. The lumped-mass procedure simplifies structures by concentrating their mass at discrete points and defining displacements only at those points.
2. The generalized displacement model expresses the deflected shape of a structure as the sum of specified displacement patterns defined by shape functions.
3. The finite-element concept divides structures into elements and expresses displacements in terms of the displacements of nodal points where elements connect, using interpolation functions within each element. All three techniques aim to create a system of differential equations relating mass, damping, stiffness, and external forces.

Advanced vibrations

This chapter discusses vibration dynamics and methods for deriving equations of motion. The Newton-Euler and Lagrange methods are commonly used to derive equations of motion for vibrating systems. The Newton-Euler method is well-suited for discrete, lumped parameter models with a low degree of freedom. It involves drawing free body diagrams and applying Newton's second law to each mass to obtain the equations of motion. Having symmetric coefficient matrices is the main advantage of using the Lagrange method for mechanical vibrations.

Small amplitude oscillations

This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.

dicover the new book of halloween

The document provides an overview of vibration dynamics and modeling. It discusses:
1) Vibrations are oscillations about an equilibrium position, which can be desirable or undesirable depending on the application.
2) Analyzing the dynamics of vibrating systems is important for all vibration control approaches, including passive, semi-active, and active controls.
3) Mathematical models of vibrations are introduced, starting with the simple case of free vibration of an undamped single-degree-of-freedom system and gradually increasing the complexity by adding damping and multiple degrees of freedom.

Basic vibration dynamics

The document provides an overview of vibration dynamics and modeling. It discusses:
1) Vibrations are oscillations about an equilibrium position, which can be desirable or undesirable depending on the application.
2) Analyzing the dynamics of vibrating systems is important for all vibration control approaches, including passive, semi-active, and active controls.
3) Mathematical models of vibrations are introduced, starting with the simple single degree of freedom mass-spring-damper system and building in complexity to multi-degree of freedom systems and continuous structures.

REPORT SUMMARYVibration refers to a mechanical.docx

REPORT SUMMARY
Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.
Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.
As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).
In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.
We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differ ...

ModelingZillDiffEQ9_LecturePPTs_05_01.pptx

Modeling with high order

DOMV No 7 MATH MODELLING Lagrange Equations.pdf

The document discusses mathematical modeling using Lagrange's equations. It begins by introducing Newtonian mechanics, the principle of virtual work, and Lagrange's equations as three approaches. It then focuses on Lagrange's equations, explaining that they describe the dynamics of systems with N degrees of freedom in terms of energy and generalized coordinates. The document provides details on Lagrange's equations, including examples of their use for conservative and dissipative systems. It also discusses how generalized forces are established and the equations of motion for linear multi-degree-of-freedom systems.

DOMV No 4 PHYSICAL DYNAMIC MODEL TYPES (1).pdf

This document discusses three physical modeling techniques for dynamic analysis of structures:
1. The lumped-mass procedure simplifies structures by concentrating their mass at discrete points and defining displacements only at those points.
2. The generalized displacement model expresses the deflected shape of a structure as the sum of specified displacement patterns defined by shape functions.
3. The finite-element concept divides structures into elements and expresses displacements in terms of the displacements of nodal points where elements connect, using interpolation functions within each element. All three techniques aim to create a system of differential equations relating mass, damping, stiffness, and external forces.

Advanced vibrations

This chapter discusses vibration dynamics and methods for deriving equations of motion. The Newton-Euler and Lagrange methods are commonly used to derive equations of motion for vibrating systems. The Newton-Euler method is well-suited for discrete, lumped parameter models with a low degree of freedom. It involves drawing free body diagrams and applying Newton's second law to each mass to obtain the equations of motion. Having symmetric coefficient matrices is the main advantage of using the Lagrange method for mechanical vibrations.

Small amplitude oscillations

This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.

dicover the new book of halloween

The document provides an overview of vibration dynamics and modeling. It discusses:
1) Vibrations are oscillations about an equilibrium position, which can be desirable or undesirable depending on the application.
2) Analyzing the dynamics of vibrating systems is important for all vibration control approaches, including passive, semi-active, and active controls.
3) Mathematical models of vibrations are introduced, starting with the simple case of free vibration of an undamped single-degree-of-freedom system and gradually increasing the complexity by adding damping and multiple degrees of freedom.

Basic vibration dynamics

The document provides an overview of vibration dynamics and modeling. It discusses:
1) Vibrations are oscillations about an equilibrium position, which can be desirable or undesirable depending on the application.
2) Analyzing the dynamics of vibrating systems is important for all vibration control approaches, including passive, semi-active, and active controls.
3) Mathematical models of vibrations are introduced, starting with the simple single degree of freedom mass-spring-damper system and building in complexity to multi-degree of freedom systems and continuous structures.

REPORT SUMMARYVibration refers to a mechanical.docx

REPORT SUMMARY
Vibration refers to a mechanical phenomenon involving oscillations about a point. These oscillations can be of any imaginable range of amplitudes and frequencies, with each combination having its own effect. These effects can be positive and purposefully induced, but they can also be unintentional and catastrophic. It's therefore imperative to understand how to classify and model vibration.
Within the classroom portion of ME 345, we discussed damped and undamped vibrations, appropriate models, and several of their properties. The purpose of Lab 3 is to give us the corresponding "hands-on" experience to cement our understanding of the theory.
As it turns out, vibration can be modeled with a simple spring-mass system (spring-mass-damper system for damped vibration). In order to create a mathematical model for our simple spring-mass system, we apply Newton's second law and sum the forces about the mass. After applying some of our knowledge of differential equations, the result is a second order linear differential equation (in vector form). This can easily be converted to the scalar version, from which it's easy to glean various properties of the vibration (i.e. natural frequency, period, etc.).
In the lab, we were provided with a PASCO motion sensor, USB link, ramp, and accompanying software. All of the aforementioned equipment was already assembled and connected. The ramp was set up at an angle with a stop on the elevated end and the motion sensor on the lower end. The sensor was connected to the USB link, which was in turn connected to the computer. We chose to use the Xplorer GLX software to interface with the sensor and record our data. After receiving our equipment, we gathered data on our spring's extension with a known load to derive a spring constant. We were provided with a small cart to which we attached weights to increase its mass. In order to model free vibration, we placed the cart on the track and attached it to the stop at the top of the ramp with a spring. After displacing the cart a certain distance from its equilibrium point, the cart was released and was allowed to oscillate on the track while we recorded its distance from the sensor. This was done with displacements of -20cm, -10cm, +10cm, and +20cm from the system's equilibrium point. After gathering this data for the "free" case, a magnet was attached to the front of the car, spaced as far from the track as possible. As the track is magnetic, this caused a slight damping effect, basically converting our spring-mass system to an underdamped spring-mass-damper system. After repeating the procedure for the "free" case, we moved the magnets as close to the track as possible (causing the system to become overdamped) and again repeated the procedure for the "free" case.
We were finally able to determine the period, phase angle, damping coefficients, and circular and cyclical frequencies for the three systems. There were similarities and differ ...

Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...

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

Mdof

This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.

Bazzucchi-Campolmi-Zatti

This document summarizes a mechanical engineering student project analyzing the kinematics and dynamics of a forging manipulator. It includes:
1) Modeling the hydraulic actuators as spring-damper systems and computing kinematics using vector equations of the degrees of freedom.
2) Computing static preloads on the actuators to achieve equilibrium.
3) Linearizing the equations of motion around the equilibrium position to determine natural frequencies and mode shapes.
4) Calculating frequency response functions by solving the linearized equations with an external forcing function.

Computational Physics - modelling the two-dimensional gravitational problem b...

The document describes a C++ program that models the two-dimensional gravitational interaction between two point masses. The program uses Newton's laws of motion and gravitation to calculate the forces, accelerations, velocities, and positions of the two masses at each time step. It assumes the masses follow elliptical orbits and uses the orbital period and energy to determine the time step for its calculations. The program outputs the positions of the masses over time to a file that can be used to plot their orbital paths.

Vibtraion notes

This document summarizes key terms and concepts related to dynamics of machines including:
1. Basic terms like time period, frequency, angular frequency, and phase of vibration.
2. Classifications of vibration such as free vs forced, damped vs undamped, linear vs non-linear, and deterministic vs random vibration.
3. Components of vibrating systems including springs, masses, and dampers. Equations of motion and natural frequency are derived using various methods.
4. Types of damping and classifications of damped systems based on damping ratio are discussed.

Statics week01

This document provides definitions and concepts related to mechanics, forces, and statics. It introduces coordinate systems, units of measurement, and numerical accuracy. Newton's laws of motion are defined. Vectors are described including operations like addition, subtraction, and dot and cross products. Forces are classified as concentrated or distributed. Statics deals with forces acting on bodies at rest.

Damped and undamped motion differential equations.pptx

Damped and undamped motion, ordinary differential equations

Statics of particle

This document discusses key concepts in engineering mechanics including units, dimensions, Newton's laws of motion, and vector representations of forces. It covers topics like the parallelogram law, triangle law, vector operations of addition, subtraction, dot and cross products. It also discusses concepts like coplanar forces, rectangular components, particle equilibrium, equivalent force systems, and the principle of transmissibility.

lec02.pdf

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.

Basic Principles of Statics

The document discusses basic principles of statics and structural design. It covers:
1) Statics deals with forces on bodies at rest, while dynamics deals with moving bodies. Statics is used to analyze structural systems and ensure strength, stiffness, and stability.
2) Structural design involves preliminary design stages using experience and intuition, followed by detailed analysis and load estimations based on statics principles.
3) Static equilibrium equations must be satisfied for coplanar forces. Systems can be determinate, allowing determination of specific unknowns, or indeterminate.

Simulation of Double Pendulum

ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.

Two

Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.

Chapter26

This chapter discusses molecular dynamics (MD) simulations, which allow modeling the behavior of atomic and molecular systems by numerically solving Newton's equations of motion. It describes the Verlet algorithm and its variants commonly used to integrate the equations of motion in MD simulations. Analysis of the trajectory data generated by MD simulations can provide information on system properties like pressure, diffusion, and the radial distribution function.

DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf

This document discusses free vibration analysis of linear multi-degree-of-freedom (MDOF) systems. It introduces Rayleigh's method, an approximate technique to determine natural frequencies of MDOF systems by assuming harmonic motion. Rayleigh's method equates maximum kinetic energy to maximum potential energy to derive an expression for natural frequencies in terms of mass and stiffness matrices and an assumed mode shape. The document also discusses exact calculation of natural frequencies and mode shapes by solving the eigenvalue problem of the dynamic matrix. It states that natural frequencies and mode shapes, known as normal modes, are important for qualitative analysis and solving forced vibration problems of MDOF systems.

Lit review-Thomas Acton

1) The document is an introduction to Supersymmetric Quantum Mechanics (SUSY QM) aimed at undergraduate students with a basic understanding of quantum mechanics.
2) It explains how SUSY QM involves pairs of partner Hamiltonians that are closely related through factorization methods. The energy eigenstates of the Hamiltonians are related, with one Hamiltonian's excited states corresponding to the other's eigenstates.
3) An example using the Morse potential is worked through to demonstrate how SUSY QM allows all energy eigenstates and wavefunctions to be algebraically determined using the "shape invariance" condition.

4 forced vibration of damped

The document discusses forced vibrations of damped, single degree of freedom linear spring mass systems. It derives the equations of motion for three types of forcing - external forcing, base excitation, and rotor excitation. It presents the steady state solutions and discusses key features, including that the response frequency matches the forcing frequency. The maximum response occurs at resonance when the forcing frequency matches the natural frequency. Engineering applications include designing systems to minimize vibrations by increasing stiffness/natural frequency and damping.

Coordinate systems

1) The document provides an overview of classical mechanics, including definitions of key concepts like space, time, mass, and force. It summarizes Newton's three laws of motion and how they relate to concepts like momentum and inertia.
2) Key principles of classical mechanics are explained, such as reference frames, Newton's laws, and conservation of momentum. Vector operations and products are also defined.
3) Examples are given to illustrate fundamental principles, like Newton's third law and how it relates to conservation of momentum in systems with multiple objects. Coordinate systems are briefly introduced.

Chapter 2 lecture 1 mechanical vibration

This document summarizes key concepts in vibration of single-degree-of-freedom (SDOF) systems. It discusses the generalized model of SDOF systems and provides examples. It then covers the differential equations of motion for SDOF systems using Newton's law and the energy method in the time domain. Specific examples are given for mass-spring, simple pendulum, and cantilever beam systems. Considerations for equivalent mass and stiffness of springs are also addressed.

Lagrange

This document discusses Lagrangian dynamics and Hamilton's principle. It begins by introducing important notation conventions used in the chapter. It then provides an overview of Hamilton's principle and how it can be used to derive Lagrange's equations of motion. This allows problems to be solved in a general manner even when forces are difficult to express or some constraints exist. Examples are provided, including deriving the equation of motion for a simple pendulum using both Cartesian and cylindrical coordinates. The concept of generalized coordinates is also introduced to represent the degrees of freedom of a system.

unit-4 wave optics new_unit 5 physic.pdf

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Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...

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

Mdof

This document discusses multi-degree-of-freedom (MDOF) systems and their analysis. It introduces concepts such as flexibility and stiffness matrices, natural frequencies and mode shapes, orthogonality of modes, and equations of motion. Methods for analyzing free and forced vibration of MDOF systems in the time domain are presented, including modal superposition and direct integration. An example 3DOF system is analyzed to illustrate the concepts.

Bazzucchi-Campolmi-Zatti

This document summarizes a mechanical engineering student project analyzing the kinematics and dynamics of a forging manipulator. It includes:
1) Modeling the hydraulic actuators as spring-damper systems and computing kinematics using vector equations of the degrees of freedom.
2) Computing static preloads on the actuators to achieve equilibrium.
3) Linearizing the equations of motion around the equilibrium position to determine natural frequencies and mode shapes.
4) Calculating frequency response functions by solving the linearized equations with an external forcing function.

Computational Physics - modelling the two-dimensional gravitational problem b...

The document describes a C++ program that models the two-dimensional gravitational interaction between two point masses. The program uses Newton's laws of motion and gravitation to calculate the forces, accelerations, velocities, and positions of the two masses at each time step. It assumes the masses follow elliptical orbits and uses the orbital period and energy to determine the time step for its calculations. The program outputs the positions of the masses over time to a file that can be used to plot their orbital paths.

Vibtraion notes

This document summarizes key terms and concepts related to dynamics of machines including:
1. Basic terms like time period, frequency, angular frequency, and phase of vibration.
2. Classifications of vibration such as free vs forced, damped vs undamped, linear vs non-linear, and deterministic vs random vibration.
3. Components of vibrating systems including springs, masses, and dampers. Equations of motion and natural frequency are derived using various methods.
4. Types of damping and classifications of damped systems based on damping ratio are discussed.

Statics week01

This document provides definitions and concepts related to mechanics, forces, and statics. It introduces coordinate systems, units of measurement, and numerical accuracy. Newton's laws of motion are defined. Vectors are described including operations like addition, subtraction, and dot and cross products. Forces are classified as concentrated or distributed. Statics deals with forces acting on bodies at rest.

Damped and undamped motion differential equations.pptx

Damped and undamped motion, ordinary differential equations

Statics of particle

This document discusses key concepts in engineering mechanics including units, dimensions, Newton's laws of motion, and vector representations of forces. It covers topics like the parallelogram law, triangle law, vector operations of addition, subtraction, dot and cross products. It also discusses concepts like coplanar forces, rectangular components, particle equilibrium, equivalent force systems, and the principle of transmissibility.

lec02.pdf

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.

Basic Principles of Statics

The document discusses basic principles of statics and structural design. It covers:
1) Statics deals with forces on bodies at rest, while dynamics deals with moving bodies. Statics is used to analyze structural systems and ensure strength, stiffness, and stability.
2) Structural design involves preliminary design stages using experience and intuition, followed by detailed analysis and load estimations based on statics principles.
3) Static equilibrium equations must be satisfied for coplanar forces. Systems can be determinate, allowing determination of specific unknowns, or indeterminate.

Simulation of Double Pendulum

ABSTRACT : In this paper, the simulation of a double pendulum with numerical solutions are discussed. The double pendulums are arranged in such a way that in the static equilibrium, one of the pendulum takes the vertical position, while the second pendulum is in a horizontal position and rests on the pad. Characteristic positions and angular velocities of both pendulums, as well as their energies at each instant of time are presented. Obtained results proved to be in accordance with the motion of the real physical system. The differentiation of the double pendulum result in four first order equations mapping the movement of the system.

Two

Richard Feynman's high school physics teacher introduced him to the principle of least action, one of the most profound concepts in physics. The principle states that among all possible paths a physical system can take between two configurations, the actual path taken will be the one that minimizes the action. The action is defined as the time integral of the Lagrangian over the path, where the Lagrangian is the difference between the system's kinetic and potential energies. This principle allows physics to be formulated in terms of variational calculus and is the foundation for classical mechanics, electromagnetism, general relativity, and other physical theories.

Chapter26

This chapter discusses molecular dynamics (MD) simulations, which allow modeling the behavior of atomic and molecular systems by numerically solving Newton's equations of motion. It describes the Verlet algorithm and its variants commonly used to integrate the equations of motion in MD simulations. Analysis of the trajectory data generated by MD simulations can provide information on system properties like pressure, diffusion, and the radial distribution function.

DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf

This document discusses free vibration analysis of linear multi-degree-of-freedom (MDOF) systems. It introduces Rayleigh's method, an approximate technique to determine natural frequencies of MDOF systems by assuming harmonic motion. Rayleigh's method equates maximum kinetic energy to maximum potential energy to derive an expression for natural frequencies in terms of mass and stiffness matrices and an assumed mode shape. The document also discusses exact calculation of natural frequencies and mode shapes by solving the eigenvalue problem of the dynamic matrix. It states that natural frequencies and mode shapes, known as normal modes, are important for qualitative analysis and solving forced vibration problems of MDOF systems.

Lit review-Thomas Acton

1) The document is an introduction to Supersymmetric Quantum Mechanics (SUSY QM) aimed at undergraduate students with a basic understanding of quantum mechanics.
2) It explains how SUSY QM involves pairs of partner Hamiltonians that are closely related through factorization methods. The energy eigenstates of the Hamiltonians are related, with one Hamiltonian's excited states corresponding to the other's eigenstates.
3) An example using the Morse potential is worked through to demonstrate how SUSY QM allows all energy eigenstates and wavefunctions to be algebraically determined using the "shape invariance" condition.

4 forced vibration of damped

The document discusses forced vibrations of damped, single degree of freedom linear spring mass systems. It derives the equations of motion for three types of forcing - external forcing, base excitation, and rotor excitation. It presents the steady state solutions and discusses key features, including that the response frequency matches the forcing frequency. The maximum response occurs at resonance when the forcing frequency matches the natural frequency. Engineering applications include designing systems to minimize vibrations by increasing stiffness/natural frequency and damping.

Coordinate systems

1) The document provides an overview of classical mechanics, including definitions of key concepts like space, time, mass, and force. It summarizes Newton's three laws of motion and how they relate to concepts like momentum and inertia.
2) Key principles of classical mechanics are explained, such as reference frames, Newton's laws, and conservation of momentum. Vector operations and products are also defined.
3) Examples are given to illustrate fundamental principles, like Newton's third law and how it relates to conservation of momentum in systems with multiple objects. Coordinate systems are briefly introduced.

Chapter 2 lecture 1 mechanical vibration

This document summarizes key concepts in vibration of single-degree-of-freedom (SDOF) systems. It discusses the generalized model of SDOF systems and provides examples. It then covers the differential equations of motion for SDOF systems using Newton's law and the energy method in the time domain. Specific examples are given for mass-spring, simple pendulum, and cantilever beam systems. Considerations for equivalent mass and stiffness of springs are also addressed.

Lagrange

This document discusses Lagrangian dynamics and Hamilton's principle. It begins by introducing important notation conventions used in the chapter. It then provides an overview of Hamilton's principle and how it can be used to derive Lagrange's equations of motion. This allows problems to be solved in a general manner even when forces are difficult to express or some constraints exist. Examples are provided, including deriving the equation of motion for a simple pendulum using both Cartesian and cylindrical coordinates. The concept of generalized coordinates is also introduced to represent the degrees of freedom of a system.

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Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...

Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...

Mdof

Mdof

Bazzucchi-Campolmi-Zatti

Bazzucchi-Campolmi-Zatti

Computational Physics - modelling the two-dimensional gravitational problem b...

Computational Physics - modelling the two-dimensional gravitational problem b...

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Vibtraion notes

Statics week01

Statics week01

Damped and undamped motion differential equations.pptx

Damped and undamped motion differential equations.pptx

Statics of particle

Statics of particle

lec02.pdf

lec02.pdf

Basic Principles of Statics

Basic Principles of Statics

Simulation of Double Pendulum

Simulation of Double Pendulum

Two

Two

Chapter26

Chapter26

DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf

DOMV No 8 MDOF LINEAR SYSTEMS - RAYLEIGH'S METHOD - FREE VIBRATION.pdf

Lit review-Thomas Acton

Lit review-Thomas Acton

4 forced vibration of damped

4 forced vibration of damped

Coordinate systems

Coordinate systems

Chapter 2 lecture 1 mechanical vibration

Chapter 2 lecture 1 mechanical vibration

Lagrange

Lagrange

unit-4 wave optics new_unit 5 physic.pdf

unit-4 wave optics new_unit 5 physic.pdf

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3:国家专业人才认证中心颁发入库证书
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- 1. Mathematical Modelling We have now seen the process of creating a physical discretisation of a real structure or machine, with distributed mass (and therefore, a potentially large number of degrees of freedom (i.e. using the Lumped-Mass Procedure, the Generalised Displacement model, and the Finite-Element Concept). The next step is to construct equations of motion, which involves application of physical laws.
- 2. Mathematical Modelling To construct the equations of motion, three different (but very commonly used) approaches are described , based on different physical principles. The choice of approach depends on how easy it is to use, but the resulting mathematical model should be similar regardless of which physical principle is used. These approaches do NOT solve the equations – that involves analysis – later!
- 3. Mathematical Modelling Physical Principle 1) Newtonian Mechanics (conservation of Momentum) in direct form, or using 'Equilibrium' Concepts based on d'Alembert’s principle. Comments Involves application of Newton's second law, therefore requires vector operations (mainly useful for lumped mass models). 2) The Principle of Virtual Work using virtual displacements (an energy principle using d'Alembert’s principle). Work terms are obtained through vector dot products but they may be added algebraically. 3) Lagrange Equations (an energy- based Variational method - a corollary of Hamilton's Principle). This approach is developed entirely using energy (i.e. scalar quantities) which can therefore be added algebraically.
- 4. Mathematical Modelling Newtonian Methods Application of Newton’s 2nd law to a discrete mass m, which has an applied force f(t), gives rise to the statement: 𝑓 𝑡 = 𝑑 𝑑𝑡 𝑚 ሶ 𝑥 = rate of change of momentum where ሶ 𝑥 is the absolute velocity of the mass (i.e. vector differential of position). If the mass is constant, i.e. ሶ 𝑚=0, then: 𝑓(𝑡) = 𝑚 ሷ 𝑥 This equation states that the total external force 𝑓 𝑡 is equal to the mass times the acceleration. This can be used to directly construct the equations of motion for a discrete dynamic system.
- 5. Mathematical Modelling Example: Consider a 2DOF system (undamped Lumped Mass model): k1 k2 x1 x2 F2(t) F1(t) X1 and X2 are displacements from the equilibrium position. First, assume X2 > X1 (in general, assume XN > XN-1 > ... > X1). Then draw free body diagrams for each mass, and apply Newton’s 2nd law to each mass.
- 6. Mathematical Modelling F1(t) k2(x2 – x1) k1 x1 Free body diagrams FBD for Mass 1: FBD for Mass 2:
- 7. Mathematical Modelling Application of Newton’s 2nd Law to the two masses: σ 𝑓 = 𝑚 ሷ 𝑥: 𝑓1 𝑡 + 𝑘2 𝑋2 − 𝑋1 − 𝐾1𝑋1 = 𝑚1 ሷ 𝑋1 σ 𝑓 = 𝑚 ሷ 𝑥: 𝑓2 𝑡 − 𝑘2 𝑋2 − 𝑋1 = 𝑚2 ሷ 𝑋2 and 𝑚1 ሷ 𝑋1 + (𝑘1 + 𝑘2)𝑋1 − 𝐾2𝑋2 = 𝑓1 𝑡 𝑚2 ሷ 𝑋2 + 𝑘2𝑋2 − 𝐾2𝑋1 = 𝑓2 𝑡 A coupled system of linear differential equations and
- 8. Mathematical Modelling The coupled system model can be put into matrix form i.e.: ൯ 𝑀 ሷ 𝑋 + 𝐾 𝑋 = 𝑓(𝑡 where the mass matrix is: 𝑀 = 𝑚1 0 0 𝑚2 and the stiffness mass matrix is: 𝐾 = 𝐾1 + 𝐾2 −𝐾2 −𝐾2 𝐾2
- 9. Mathematical Modelling d'Alembert's Principle Note that Newton's 2nd law is written: 𝑓 𝑡 = 𝑚 ሷ 𝑥 but can be rearranged in the form: 𝑓 𝑡 − 𝑚 ሷ 𝑥 = 0 So the term 𝑚 ሷ 𝑥 can be thought of as an 'inertia' force which, when included on an 'equilibrium diagram’ (rather than a free-body diagram), reduces the problem to one of 'equilibrium'.
- 10. Mathematical Modelling The concept of introducing an inertia force on a mass which is proportional to its acceleration, and which opposes the motion, is called d'Alembert's Principle, and can be very useful in modelling continuous systems. The inertial force is of course fictitious (it doesn't really exist) but it is helpful (for modelling purposes) to think of the system as being in ‘equilibrium’ where the 'inertia force' is included. d'Alembert's Principle
- 11. Mathematical Modelling An example: a SDOF problem. k1 x1 f1 k1x1 f1 𝐹𝐼 = 𝑚 ሷ 𝑥1 Equilibrium diagram using d'Alembert's principle: d'Alembert's principle: 𝑓1−𝑘1𝑥1 − 𝑚 ሷ 𝑥1 = 0 𝑚 ሷ 𝑥1 + 𝑘1𝑥1 = 𝑓1(t) And therefore: No advantage of using d'Alembert's principle, on Lumped-Mass systems since Newton's 2nd law can be applied directly. The real advantage is derived when we use Virtual Work principles.
- 12. Mathematical Modelling The Principle of Virtual Work Again, the focus is on constructing a discrete model of the form: 𝑚 ሷ 𝑍 + 𝑐 ሶ 𝑍 + 𝑘 𝑍 = 𝑝(𝑡). The Principle states that when a system is in ‘equilibrium’ (in the sense of d'Alembert) under the action of external forces, and is forced to move through a virtual displacement, without violating the system constraints, and without the passage of time, at the same time as adhering to a sign convention, then the total virtual work done is zero i.e.: 𝛿𝑤𝑖 = 0
- 13. Mathematical Modelling The Sign Convention: Forces acting in the direction of a Virtual Displacement do –ve (negative) Virtual Work. Strain energy put into a system is always deemed to be positive.
- 14. Mathematical Modelling Virtual Displacements and Virtual Work Consider a system with N degrees-of-freedom, with corresponding coordinates (X1, X2, ..., XN) used to specify the position. Assume forces F1, F2, ...,FN are applied at each coordinate in the (+ve) direction of each coordinate.
- 15. Mathematical Modelling Now if we imagine the system is given an arbitrary set of small displacements 𝛿𝑋1, 𝛿𝑋2, … , 𝛿𝑋𝑁 then the magnitude of the work done by these applied forces will be: 𝛿𝑤 = − 𝑗 𝑁 𝐹𝑗𝛿𝑋𝑗 The small displacements are imaginary and are therefore virtual because they occur without the passage of time, and are different from small changes dx which occur in time dt (i.e. real ones). The virtual displacements conform to the kinematic constraints which apply. Virtual Displacements and Virtual Work
- 16. Mathematical Modelling In general, forces will occur in arbitrary directions and thus the work done is expressed as a dot or vector product i.e. 𝛿𝑤 = − σ𝑗 𝑁 𝐹𝑗 ∙ 𝛿𝜏𝑗 where 𝛿𝜏𝑗 are the small changes in position vectors. Virtual Work Sign Convention and it’s impact on the sign of the Virtual Work Adherence to the Sign convention will always produce the correct sign for all the Virtual terms. Some text books also define Virtual Work as being Internal (suffix I) or external (suffix E). The Principle of Virtual Work can then be stated as: 𝝨(𝛿𝑤𝐸 + 𝛿𝑤𝐼) = 0
- 17. Example: SDOF Linear Oscillator k x f(t) m c Next Lecture! A simple example of applying the Principle of Virtual Work.