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The document discusses the derivation of position, velocity, and acceleration vectors for a particle moving in a plane when described using a rotating reference frame. It shows that the position vector in the rotating frame is simply the particle's radius vector. The velocity vector has components of radial velocity and tangential velocity due to rotation. Similarly, the acceleration vector has radial and tangential acceleration components as well as a centrifugal acceleration term. These relationships are obtained through rotation of axes transformations.

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Cartesian coordinates

This document discusses velocity and acceleration in different coordinate systems, including Cartesian, spherical, and cylindrical coordinates. It provides the equations to calculate velocity and acceleration vectors in each system. In spherical coordinates, it defines the radial, angular, and azimuthal positions and shows how to write the position, velocity, and acceleration vectors in terms of these variables. Similarly, it defines the radial, angular, and vertical positions in cylindrical coordinates and gives the corresponding equations for velocity and acceleration.

Presentation on components of angular velocity vector along the body set of a...

This document presents the components of an angular velocity vector along the body set of axes. It discusses angular velocity, finding the components of angular velocity, and the components of an angular velocity vector along the body set of axes. Equations are provided to calculate the components of angular velocity vectors ωφ, ωθ, and ωψ along the x', y', and z' axes of the body frame. The components are added vectorially to obtain the full angular velocity vector ω with respect to the body axes.

Chapter 5 - Oscillation.pptx

MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 5
OSCILLATIONS
CONTENT
Introduction
Periodic and oscillatory
motions
Simple harmonic motion
Simple harmonic motion
and uniform circular
motion
Velocity and acceleration
in simple harmonic motion
Force law for simple
harmonic motion
Energy in simple harmonic
motion
Some systems executing
simple harmonic motion
Damped simple harmonic
motion
Forced oscillations and
resonance

Coordinatetransformation 130405095156-phpapp01-converted

This document discusses different types of coordinate transformations, including translation, rotation, scaling, and reflection. Translation moves all points the same distance in the same direction. Rotation turns the coordinate system around a fixed point. Scaling changes the units of measurement along the axes. Reflection mirrors the coordinate system across an axis. Each transformation has a corresponding inverse that undoes the original transformation.

Chapter 11_0 velocity, acceleration.pdf

This problem involves analyzing the motion of a ball thrown vertically upwards in an elevator shaft, and an open-platform elevator moving upwards at a constant velocity.
The key steps are:
1) Use kinematic equations to find the velocity and position of the ball as a function of time, assuming constant downward acceleration due to gravity.
2) Determine the velocity and position of the elevator as a constant upward velocity.
3) Express the relative motion of the ball with respect to the elevator to determine when they meet.
By setting the position of the ball equal to the position of the elevator and solving for time, we can determine when the ball and elevator meet at 26.4 seconds after the ball is thrown

Servo systems

This document describes the design of a servo system using state feedback and integral control. It defines the plant state and output equations, and shows the block diagram of the servo system. The state equation of the augmented system is derived, combining the plant states and integrator states. The gains K1 and K2 are selected using pole placement so that the closed-loop poles of the combined system are located at the desired locations. An example is provided to illustrate the design process.

2D-transformation-1.pdf

This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.

2D transformation (Computer Graphics)

with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.

Cartesian coordinates

This document discusses velocity and acceleration in different coordinate systems, including Cartesian, spherical, and cylindrical coordinates. It provides the equations to calculate velocity and acceleration vectors in each system. In spherical coordinates, it defines the radial, angular, and azimuthal positions and shows how to write the position, velocity, and acceleration vectors in terms of these variables. Similarly, it defines the radial, angular, and vertical positions in cylindrical coordinates and gives the corresponding equations for velocity and acceleration.

Presentation on components of angular velocity vector along the body set of a...

This document presents the components of an angular velocity vector along the body set of axes. It discusses angular velocity, finding the components of angular velocity, and the components of an angular velocity vector along the body set of axes. Equations are provided to calculate the components of angular velocity vectors ωφ, ωθ, and ωψ along the x', y', and z' axes of the body frame. The components are added vectorially to obtain the full angular velocity vector ω with respect to the body axes.

Chapter 5 - Oscillation.pptx

MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 5
OSCILLATIONS
CONTENT
Introduction
Periodic and oscillatory
motions
Simple harmonic motion
Simple harmonic motion
and uniform circular
motion
Velocity and acceleration
in simple harmonic motion
Force law for simple
harmonic motion
Energy in simple harmonic
motion
Some systems executing
simple harmonic motion
Damped simple harmonic
motion
Forced oscillations and
resonance

Coordinatetransformation 130405095156-phpapp01-converted

This document discusses different types of coordinate transformations, including translation, rotation, scaling, and reflection. Translation moves all points the same distance in the same direction. Rotation turns the coordinate system around a fixed point. Scaling changes the units of measurement along the axes. Reflection mirrors the coordinate system across an axis. Each transformation has a corresponding inverse that undoes the original transformation.

Chapter 11_0 velocity, acceleration.pdf

This problem involves analyzing the motion of a ball thrown vertically upwards in an elevator shaft, and an open-platform elevator moving upwards at a constant velocity.
The key steps are:
1) Use kinematic equations to find the velocity and position of the ball as a function of time, assuming constant downward acceleration due to gravity.
2) Determine the velocity and position of the elevator as a constant upward velocity.
3) Express the relative motion of the ball with respect to the elevator to determine when they meet.
By setting the position of the ball equal to the position of the elevator and solving for time, we can determine when the ball and elevator meet at 26.4 seconds after the ball is thrown

Servo systems

This document describes the design of a servo system using state feedback and integral control. It defines the plant state and output equations, and shows the block diagram of the servo system. The state equation of the augmented system is derived, combining the plant states and integrator states. The gains K1 and K2 are selected using pole placement so that the closed-loop poles of the combined system are located at the desired locations. An example is provided to illustrate the design process.

2D-transformation-1.pdf

This document discusses various 2D transformations in computer graphics including translation, rotation, and scaling. Translation moves an object by adding offsets to the x and y coordinates. Rotation uses trigonometric functions and rotation matrices to reposition objects around a central point. Scaling enlarges or shrinks objects by multiplying their coordinates by scaling factors. Homogeneous coordinates generalize these transformations into matrix operations.

2D transformation (Computer Graphics)

with today's advanced technology like photoshop, paint etc. we need to understand some basic concepts like how they are cropping the image , tilt the image etc.
In our presentation you will find basic introduction of 2D transformation.

Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/

Simple harmonic oscillator - Classical Mechanics

A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.

MT102 Лекц 6

1. The document discusses methods for calculating the area of regions bounded by curves using integral calculus.
2. Six methods are presented for computing the area of regions bounded above and below by curves including the use of polar coordinates.
3. One example calculates the area between the curves y=x2 and y=2x from x=0 to x=2 as 8π/15 using the integral of the difference of the two curves.

Gravitational field and potential, escape velocity, universal gravitational l...

What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples

Two dimentional transform

This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.

Parallel tansport sssqrd

This document provides a review of equations for parallel transport of vectors in Schwarzschild spacetime. It presents the general equation for parallel transport and specifies it for Schwarzschild metric. Several examples are worked through, parallel transporting vectors along timelike and spacelike paths. Consistency of the solutions is checked by taking derivatives and substituting known relations.

CLASS XII - CHAPTER 5: OSCILLATION (PHYSICS - MAHARASHTRA STATE BOARD)

This document provides information about oscillations and simple harmonic motion (SHM). It defines oscillation as periodic motion that repeats after a definite time interval. SHM is described as oscillatory motion where the restoring force is directly proportional to displacement from the equilibrium position. The key characteristics of SHM include:
- The differential equation relating displacement, velocity, and acceleration.
- Expressions for displacement, velocity, and acceleration as functions of time and constants.
- Definitions and calculations of important terms like amplitude, period, frequency, phase.
- Conditions required for motion to be considered SHM.
- Examples of SHM and calculations related to restoring force and period.

2D transformations

This document discusses 2D transformations in computer graphics, including rotation, reflection, and shearing. It explains rotation using trigonometric equations to express transformed coordinates in terms of an angle, and represents rotation using a rotation matrix. Reflection is described as rotating an object 180 degrees about an axis, and reflection about the x-axis is represented using a matrix. Shearing is defined as a transformation that changes an object's shape by sliding its layers, and shearing matrices for the x and y directions are provided.

Derivational Error of Albert Einstein

This document summarizes a paper that points out a major error in Albert Einstein's 1905 paper on special relativity. Specifically, it shows that Einstein's assumption that the time coordinate of a moving clock (τ2) can be expressed as a function of the time (t) and spatial (x) coordinates of a stationary system is incorrect. An alternative derivation is presented that expresses τ2 in terms of t, x, the velocity (v) of the moving system, and other variables. This challenges one of the foundational assumptions of Einstein's original formulation of special relativity.

GATE Engineering Maths : Vector Calculus

1. This document covers key concepts in vector calculus including vector basics, vector differentiation, and vector integration. It defines concepts like position vectors, gradients, divergence, curl, line integrals, and surface integrals.
2. Formulas are provided for calculating directional derivatives, divergence, curl, line integrals, surface integrals, and theorems like Green's theorem and Gauss's divergence theorem.
3. Vector operations like dot products, cross products, and triple products are defined along with their geometric interpretations and formulas for calculation.

2 d transformations by amit kumar (maimt)

Transformations are operations that change the position, orientation, or size of an object in computer graphics. The main 2D transformations are translation, rotation, scaling, reflection, shear, and combinations of these. Transformations allow objects to be manipulated and displayed in modified forms without needing to redraw them from scratch.

Introduction to mechanics

This document discusses concepts in mechanics including kinematics, dynamics, and statics. It defines key terms like reference frames, position vectors, displacement, average speed, average velocity, and instantaneous acceleration. It also provides examples of determining trajectory, displacement, velocity, and center of mass for systems of particles.

Section 2 part 1 coordinate transformation

This document discusses coordinate transformations, including translating and rotating coordinate frames. It provides examples of how to calculate the coordinates of a point in a new frame after a translation or rotation from the original frame. Specifically, it shows how to calculate the new coordinates of point P if the original frame is translated to a new origin or rotated by 30 degrees. The key steps are to first translate the frame if needed, then apply the rotation matrix to calculate the new x' and y' coordinates of the point in the rotated frame.

COORDINATE SYSTEM.pdf

Coordinate systems
orthogonal coordinate system
Rectangular or Cartesian coordinate system
Cylindrical or circular coordinate system
Spherical coordinate system
Relationship between various coordinate system
Transformation Matrix
DIFFERENTIAL VECTOR
Curvilinear, Cartesian, Cylindrical, Spherical table

COORDINATE SYSTEM.pdf

COORDINATE SYSTEM & TRANSFORMATION,Transformation Matrix, DIFFERENTIAL VECTOR,Curvilinear, Cartesian, Cylindrical, Spherical

Chapter2powerpoint 090816163937-phpapp02

This document provides an overview of kinematics concepts including displacement, speed, velocity, acceleration, and equations of motion. Key points covered include:
- Kinematics deals with describing motion without considering causes of motion like forces.
- Displacement, speed, velocity, and acceleration are defined. Equations of motion that relate these variables for constant acceleration are presented.
- Position-time and velocity-time graphs are introduced as ways to represent motion. The slope and area under graphs relate to velocity and displacement.
- Free fall near the Earth's surface provides a specific example where acceleration due to gravity is constant.
- Graphical analysis techniques are described for determining acceleration from velocity-time graphs.

06.Transformation.ppt

This document discusses 2D and 3D geometric transformations. It describes two types of transformations: geometric transformations that alter the object itself, and coordinate transformations that alter the coordinate system. Several common 2D geometric transformations are covered, including translation, rotation, scaling, reflection and shear. Matrix representations are introduced to combine multiple transformations into a single operation. The concept of homogeneous coordinates is explained for representing 2D transformations with 3x3 matrices. Finally, a general method for 2D rotation around a pivot point is described.

Lecture Dynamics Kinetics of Particles.pdf

The document discusses kinematics of particles, including rectilinear and curvilinear motion. It defines key concepts like displacement, velocity, and acceleration. It presents equations for calculating these values for rectilinear motion under different conditions of acceleration, such as constant acceleration, acceleration as a function of time, velocity, or displacement. Graphical interpretations are also described. An example problem is worked through to demonstrate finding velocity, acceleration, and displacement at different times for a particle moving in a straight line.

Motion in a plane

This document discusses vectors and their properties. It provides examples of vector addition and multiplication. Some key points:
- Vectors have both magnitude and direction, while scalars only have magnitude. Vector addition follows the triangle and parallelogram laws.
- There are two types of vector multiplication: the dot product, which results in a scalar, and the cross product, which results in another vector.
- The dot product of two vectors is equal to their magnitudes multiplied by the cosine of the angle between them. It is used to calculate quantities like work and power.
- Vectors can be resolved into rectangular components using a set of base vectors like the i, j, k unit vectors. The magnitude

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...National Cheng Kung University

The document discusses rotation matrix (DCM) and quaternions. It provides the definitions and equations for representing 3D rotations using DCM and quaternions. It then gives an example of calculating the DCM, quaternion elements, and rotated axes given the Euler angles of 45.827° for roll, 12.346° for pitch, and -198.542° for yaw in a 1-2-3 rotation sequence (roll-pitch-yaw). It also provides the inverse calculation of determining the Euler angles given a quaternion of [-0.425 -0.0537 -0.1950.782].Sensors_2020.pptx

This document discusses various types of robotic sensors. It begins by explaining why robots need sensors to provide awareness of their surroundings, allow interaction with the environment, and enable goal-seeking behaviors. The document then describes different things that can be sensed by robotic sensors, such as light, sound, heat, chemicals, and object proximity. Several common types of robotic sensors are outlined, including feelers, photoelectric, infrared, ultrasonic, visual, and chemical sensors. The characteristics and functions of proximity, inductive, capacitive, and optical proximity sensors are explained in more detail. The document aims to provide an overview of the role and functionality of different robotic sensors.

DOMV No 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

This document discusses the response of linear single degree of freedom (SDOF) systems to general loading through the use of superposition. It introduces the mass-spring-damper model and defines two special free response functions: the unit amplitude free decay function and the unit velocity free decay function. It explains that the general solution to the forced response of a SDOF system can be constructed by taking a superposition of responses to these two base functions using the initial conditions and applied force over time.

Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/

Simple harmonic oscillator - Classical Mechanics

A brief and easy concept of Simple harmonic oscillator. How we can get simple harmonic motion equation from Lagrange's equation of motion. How can we obtain this from Lagrange's equation of motion.

MT102 Лекц 6

1. The document discusses methods for calculating the area of regions bounded by curves using integral calculus.
2. Six methods are presented for computing the area of regions bounded above and below by curves including the use of polar coordinates.
3. One example calculates the area between the curves y=x2 and y=2x from x=0 to x=2 as 8π/15 using the integral of the difference of the two curves.

Gravitational field and potential, escape velocity, universal gravitational l...

What is Escape Velocity-its derivation-examples-applications
Universal Gravitational Law-Derivation and Examples
Gravitational Field And Gravitational Potential-Derivation, Realation and numericals
Radial Velocity and acceleration-derivation and examples
Transverse Velocity and acceleration and examples

Two dimentional transform

This document discusses 2D geometric transformations including translation, rotation, scaling, and composite transformations. It provides definitions and formulas for each type of transformation. Translation moves objects by adding offsets to coordinates without deformation. Rotation rotates objects around an origin by a certain angle. Scaling enlarges or shrinks objects by multiplying coordinates by scaling factors. Composite transformations apply multiple transformations sequentially by multiplying their matrices. Homogeneous coordinates are also introduced to represent transformations in matrix form.

Parallel tansport sssqrd

This document provides a review of equations for parallel transport of vectors in Schwarzschild spacetime. It presents the general equation for parallel transport and specifies it for Schwarzschild metric. Several examples are worked through, parallel transporting vectors along timelike and spacelike paths. Consistency of the solutions is checked by taking derivatives and substituting known relations.

CLASS XII - CHAPTER 5: OSCILLATION (PHYSICS - MAHARASHTRA STATE BOARD)

This document provides information about oscillations and simple harmonic motion (SHM). It defines oscillation as periodic motion that repeats after a definite time interval. SHM is described as oscillatory motion where the restoring force is directly proportional to displacement from the equilibrium position. The key characteristics of SHM include:
- The differential equation relating displacement, velocity, and acceleration.
- Expressions for displacement, velocity, and acceleration as functions of time and constants.
- Definitions and calculations of important terms like amplitude, period, frequency, phase.
- Conditions required for motion to be considered SHM.
- Examples of SHM and calculations related to restoring force and period.

2D transformations

This document discusses 2D transformations in computer graphics, including rotation, reflection, and shearing. It explains rotation using trigonometric equations to express transformed coordinates in terms of an angle, and represents rotation using a rotation matrix. Reflection is described as rotating an object 180 degrees about an axis, and reflection about the x-axis is represented using a matrix. Shearing is defined as a transformation that changes an object's shape by sliding its layers, and shearing matrices for the x and y directions are provided.

Derivational Error of Albert Einstein

This document summarizes a paper that points out a major error in Albert Einstein's 1905 paper on special relativity. Specifically, it shows that Einstein's assumption that the time coordinate of a moving clock (τ2) can be expressed as a function of the time (t) and spatial (x) coordinates of a stationary system is incorrect. An alternative derivation is presented that expresses τ2 in terms of t, x, the velocity (v) of the moving system, and other variables. This challenges one of the foundational assumptions of Einstein's original formulation of special relativity.

GATE Engineering Maths : Vector Calculus

1. This document covers key concepts in vector calculus including vector basics, vector differentiation, and vector integration. It defines concepts like position vectors, gradients, divergence, curl, line integrals, and surface integrals.
2. Formulas are provided for calculating directional derivatives, divergence, curl, line integrals, surface integrals, and theorems like Green's theorem and Gauss's divergence theorem.
3. Vector operations like dot products, cross products, and triple products are defined along with their geometric interpretations and formulas for calculation.

2 d transformations by amit kumar (maimt)

Transformations are operations that change the position, orientation, or size of an object in computer graphics. The main 2D transformations are translation, rotation, scaling, reflection, shear, and combinations of these. Transformations allow objects to be manipulated and displayed in modified forms without needing to redraw them from scratch.

Introduction to mechanics

This document discusses concepts in mechanics including kinematics, dynamics, and statics. It defines key terms like reference frames, position vectors, displacement, average speed, average velocity, and instantaneous acceleration. It also provides examples of determining trajectory, displacement, velocity, and center of mass for systems of particles.

Section 2 part 1 coordinate transformation

This document discusses coordinate transformations, including translating and rotating coordinate frames. It provides examples of how to calculate the coordinates of a point in a new frame after a translation or rotation from the original frame. Specifically, it shows how to calculate the new coordinates of point P if the original frame is translated to a new origin or rotated by 30 degrees. The key steps are to first translate the frame if needed, then apply the rotation matrix to calculate the new x' and y' coordinates of the point in the rotated frame.

COORDINATE SYSTEM.pdf

Coordinate systems
orthogonal coordinate system
Rectangular or Cartesian coordinate system
Cylindrical or circular coordinate system
Spherical coordinate system
Relationship between various coordinate system
Transformation Matrix
DIFFERENTIAL VECTOR
Curvilinear, Cartesian, Cylindrical, Spherical table

COORDINATE SYSTEM.pdf

COORDINATE SYSTEM & TRANSFORMATION,Transformation Matrix, DIFFERENTIAL VECTOR,Curvilinear, Cartesian, Cylindrical, Spherical

Chapter2powerpoint 090816163937-phpapp02

This document provides an overview of kinematics concepts including displacement, speed, velocity, acceleration, and equations of motion. Key points covered include:
- Kinematics deals with describing motion without considering causes of motion like forces.
- Displacement, speed, velocity, and acceleration are defined. Equations of motion that relate these variables for constant acceleration are presented.
- Position-time and velocity-time graphs are introduced as ways to represent motion. The slope and area under graphs relate to velocity and displacement.
- Free fall near the Earth's surface provides a specific example where acceleration due to gravity is constant.
- Graphical analysis techniques are described for determining acceleration from velocity-time graphs.

06.Transformation.ppt

This document discusses 2D and 3D geometric transformations. It describes two types of transformations: geometric transformations that alter the object itself, and coordinate transformations that alter the coordinate system. Several common 2D geometric transformations are covered, including translation, rotation, scaling, reflection and shear. Matrix representations are introduced to combine multiple transformations into a single operation. The concept of homogeneous coordinates is explained for representing 2D transformations with 3x3 matrices. Finally, a general method for 2D rotation around a pivot point is described.

Lecture Dynamics Kinetics of Particles.pdf

The document discusses kinematics of particles, including rectilinear and curvilinear motion. It defines key concepts like displacement, velocity, and acceleration. It presents equations for calculating these values for rectilinear motion under different conditions of acceleration, such as constant acceleration, acceleration as a function of time, velocity, or displacement. Graphical interpretations are also described. An example problem is worked through to demonstrate finding velocity, acceleration, and displacement at different times for a particle moving in a straight line.

Motion in a plane

This document discusses vectors and their properties. It provides examples of vector addition and multiplication. Some key points:
- Vectors have both magnitude and direction, while scalars only have magnitude. Vector addition follows the triangle and parallelogram laws.
- There are two types of vector multiplication: the dot product, which results in a scalar, and the cross product, which results in another vector.
- The dot product of two vectors is equal to their magnitudes multiplied by the cosine of the angle between them. It is used to calculate quantities like work and power.
- Vectors can be resolved into rectangular components using a set of base vectors like the i, j, k unit vectors. The magnitude

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...National Cheng Kung University

The document discusses rotation matrix (DCM) and quaternions. It provides the definitions and equations for representing 3D rotations using DCM and quaternions. It then gives an example of calculating the DCM, quaternion elements, and rotated axes given the Euler angles of 45.827° for roll, 12.346° for pitch, and -198.542° for yaw in a 1-2-3 rotation sequence (roll-pitch-yaw). It also provides the inverse calculation of determining the Euler angles given a quaternion of [-0.425 -0.0537 -0.1950.782].Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

Rotation in 3d Space: Euler Angles, Quaternions, Marix Descriptions

Simple harmonic oscillator - Classical Mechanics

Simple harmonic oscillator - Classical Mechanics

MT102 Лекц 6

MT102 Лекц 6

Gravitational field and potential, escape velocity, universal gravitational l...

Gravitational field and potential, escape velocity, universal gravitational l...

Two dimentional transform

Two dimentional transform

Parallel tansport sssqrd

Parallel tansport sssqrd

CLASS XII - CHAPTER 5: OSCILLATION (PHYSICS - MAHARASHTRA STATE BOARD)

CLASS XII - CHAPTER 5: OSCILLATION (PHYSICS - MAHARASHTRA STATE BOARD)

2D transformations

2D transformations

Derivational Error of Albert Einstein

Derivational Error of Albert Einstein

GATE Engineering Maths : Vector Calculus

GATE Engineering Maths : Vector Calculus

2 d transformations by amit kumar (maimt)

2 d transformations by amit kumar (maimt)

Introduction to mechanics

Introduction to mechanics

Section 2 part 1 coordinate transformation

Section 2 part 1 coordinate transformation

COORDINATE SYSTEM.pdf

COORDINATE SYSTEM.pdf

COORDINATE SYSTEM.pdf

COORDINATE SYSTEM.pdf

Chapter2powerpoint 090816163937-phpapp02

Chapter2powerpoint 090816163937-phpapp02

06.Transformation.ppt

06.Transformation.ppt

Lecture Dynamics Kinetics of Particles.pdf

Lecture Dynamics Kinetics of Particles.pdf

Motion in a plane

Motion in a plane

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...

The rotation matrix (DCM) and quaternion in Inertial Survey and Navigation Sy...

Sensors_2020.pptx

This document discusses various types of robotic sensors. It begins by explaining why robots need sensors to provide awareness of their surroundings, allow interaction with the environment, and enable goal-seeking behaviors. The document then describes different things that can be sensed by robotic sensors, such as light, sound, heat, chemicals, and object proximity. Several common types of robotic sensors are outlined, including feelers, photoelectric, infrared, ultrasonic, visual, and chemical sensors. The characteristics and functions of proximity, inductive, capacitive, and optical proximity sensors are explained in more detail. The document aims to provide an overview of the role and functionality of different robotic sensors.

DOMV No 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

This document discusses the response of linear single degree of freedom (SDOF) systems to general loading through the use of superposition. It introduces the mass-spring-damper model and defines two special free response functions: the unit amplitude free decay function and the unit velocity free decay function. It explains that the general solution to the forced response of a SDOF system can be constructed by taking a superposition of responses to these two base functions using the initial conditions and applied force over time.

DOMV No 5 MATH MODELLING Newtonian d'Alembert Virtual Work (1).pdf

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.

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.

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.

DOMV No 3 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (1).pdf

There are two unique functions needed to generate the general response of a single-degree-of-freedom (SDOF) system to arbitrary forcing: the unit amplitude free decay function and the unit velocity free decay function. The impulse response function is identical to the unit velocity free decay function. The lecture will consider four scenarios involving the impulse response function to build up the solution to general forcing. This will demonstrate that only the impulse response function is needed to determine the response of an SDOF system to any input, from any initial conditions.

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 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

This document discusses the response of linear single degree of freedom (SDOF) systems to general loading through the use of superposition. It introduces the mass-spring-damper model and defines two special free response functions: the unit amplitude free decay function and the unit velocity free decay function. It explains that the general solution to the forced response of a SDOF system can be constructed by taking a superposition of responses to these two base functions using the initial conditions and applied force over time.

Sensors_2020.pptx

Sensors_2020.pptx

DOMV No 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

DOMV No 2 RESPONSE OF LINEAR SDOF SYSTEMS TO GENERAL LOADING (2).pdf

DOMV No 5 MATH MODELLING Newtonian d'Alembert Virtual Work (1).pdf

DOMV No 5 MATH MODELLING Newtonian d'Alembert Virtual Work (1).pdf

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

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

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

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

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2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
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- 1. We saw in the previous lecture that the components of a fixed vector with respect to a frame that has been rotated, are related to the components of the original system as follows: 𝑥𝑥 𝑦𝑦 𝑧𝑧 = 𝐴𝐴 𝑋𝑋 𝑌𝑌 𝑍𝑍 where 𝐴𝐴 is the ‘Matrix of Direction Cosines’: 𝐴𝐴 = Cos(𝑥𝑥𝑥𝑥𝑥𝑥) Cos(𝑥𝑥𝑥𝑥𝑥𝑥) Cos(𝑥𝑥𝑥𝑥𝑥𝑥) Cos(𝑦𝑦𝑦𝑦𝑦𝑦) Cos(𝑦𝑦𝑦𝑦𝑦𝑦) Cos(𝑦𝑦𝑦𝑦𝑦𝑦) Cos(𝑧𝑧𝑧𝑧𝑧𝑧) Cos(𝑧𝑧𝑧𝑧𝑧𝑧) Cos(𝑧𝑧𝑧𝑧𝑧𝑧) where for 𝑥𝑥𝑥𝑥𝑥𝑥 is the angle between the x and the X axes, 𝑥𝑥𝑥𝑥𝑥𝑥 is the angle between the x and the Y axes etc. Rotation of Axes Advanced Kinematic Analysis
- 2. We will now prove this from geometry. X Y y x A 2D Rotation We also saw that the transformation for a 2D rotation about the z axis simplifies to: 𝑥𝑥 𝑦𝑦 𝑧𝑧 = Cos(𝑥𝑥𝑥𝑥𝑥𝑥) Cos(𝑥𝑥𝑥𝑥𝑥𝑥) 0 Cos(𝑦𝑦𝑦𝑦𝑦𝑦) Cos(𝑦𝑦𝑦𝑦𝑦𝑦) 0 0 0 1 𝑋𝑋 𝑌𝑌 𝑍𝑍 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 0 −𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 0 0 0 1 𝑋𝑋 𝑌𝑌 𝑍𝑍 i.e.: 𝑥𝑥 𝑦𝑦 = 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 −𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 𝑋𝑋 𝑌𝑌 Advanced Kinematic Analysis
- 3. Advanced Kinematic Analysis 2D Transformation - proof from geometry. A 2D Rotation Y X y Y X y x And from the figure (using similar triangles) it is therefore evident that: 𝑥𝑥 𝑦𝑦 = cos 𝜃𝜃 sin 𝜃𝜃 − sin 𝜃𝜃 cos 𝜃𝜃 𝑋𝑋 𝑌𝑌 end of proof x
- 4. Advanced Kinematic Analysis 3D rotation of axes achieved by 3 successive 2D rotations In general, we can always achieve any 3D rotation by 3 successive 2D rotations (about the appropriate axes using the appropriate (3 x 3) 2D rotation matrix of Direction Cosines) i.e. i.e. 𝑥𝑥′ = 𝐴𝐴𝑋𝑋 ⟹ 𝑥𝑥′′ = 𝐴𝐴′𝑥𝑥′ ⟹ 𝑥𝑥′′′ = 𝐴𝐴′′𝑥𝑥′′ where the direction cosine matrices in each case (𝐴𝐴, 𝐴𝐴′ , and 𝐴𝐴′′ ) are 2D rotations about corresponding axes. Orthogonality of matrix A 𝐴𝐴𝑇𝑇 = 𝐴𝐴−1 𝑖𝑖. 𝑒𝑒. 𝐴𝐴𝑇𝑇𝐴𝐴 = 𝐼𝐼 (the unit matrix)
- 5. Advanced Kinematic Analysis A Physical rotation A physical rotation can be obtained by keeping the axes fixed but rotating a vector. Consider a point P on a disc. If the disc is rotated through angle θ, the new position vector P* can be obtained by multiplying vector P by 𝐴𝐴−1 e.g.: Y X P P* i.e. 𝑃𝑃∗ = 𝐴𝐴−1𝑃𝑃 = cos 𝜃𝜃 − sin 𝜃𝜃 sin 𝜃𝜃 cos 𝜃𝜃 𝑃𝑃𝑥𝑥 𝑃𝑃𝑦𝑦
- 6. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Here we return to the original task, namely the development of tools that enable us to obtain the derivatives of vectors (particularly velocity and acceleration) when the position vector is described in terms of a frame of reference that is moving (i.e. a rotating frame). To do this, we initially approach the problem in a ‘sledge-hammer’ way by rotation of axes (which, from the previous section, we now know how to do).
- 7. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Consider a particle P, with position vector r, that is moving arbitrarily in the (fixed) XY plane as described in the following figure where P’ is a new position. Here the particle is ‘tracked’ by a frame of reference xy such that the x axis always points straight at the particle. The xy axes are therefore moving polar coordinates. The question is: what are the absolute velocity and acceleration vector for particle P? We will answer this question using a rotation of axes. P X Y y P’ x r P moves anyway in the plane P’ is a new position Particle P moving arbitrarily in the XY Plane (where the XY frame is fixed). In addition, a (polar) coordinate system xy is chosen as a special case to track particle P - the xy frame is therefore moving.
- 8. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The position vector Note the position vector 𝑟𝑟 of P is: 𝑟𝑟 = 𝑋𝑋 𝑡𝑡 𝐼𝐼 + 𝑌𝑌 𝑡𝑡 𝐽𝐽 = 𝑟𝑟 𝑡𝑡 cos 𝜃𝜃 𝑡𝑡 𝐼𝐼 + 𝑟𝑟 𝑡𝑡 sin 𝜃𝜃(𝑡𝑡)𝐽𝐽 𝐼𝐼 𝑎𝑎𝑎𝑎𝑎𝑎 𝐽𝐽 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 The velocity vector The velocity vector can be obtained by differentiation of the position vector with respect to the fixed frame of reference, i.e.: 𝑉𝑉 = ̇ 𝑟𝑟 = 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑟𝑟 cos 𝜃𝜃 𝐼𝐼 + 𝑟𝑟 sin 𝜃𝜃 𝐽𝐽 = ̇ 𝑟𝑟 cos 𝜃𝜃 − 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 𝐼𝐼 + ̇ 𝑟𝑟 sin 𝜃𝜃 + 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 𝐽𝐽 (i.e. in the fixed system)
- 9. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The acceleration vector The acceleration vector can be obtained again by differentiation of the velocity vector with respect to the fixed frame of reference, i.e.: 𝑎𝑎 = ̈ 𝑟𝑟 = ̈ 𝑋𝑋𝐼𝐼 + ̈ 𝑌𝑌𝐽𝐽 = ̈ 𝑟𝑟 cos 𝜃𝜃 − ̇ 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 − ̇ 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 − 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃2 − 𝑟𝑟 sin 𝜃𝜃 ̈ 𝜃𝜃 𝐼𝐼 + ̈ 𝑟𝑟 sin 𝜃𝜃 + ̇ 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 + ̇ 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 + 𝑟𝑟 cos 𝜃𝜃 ̈ 𝜃𝜃 − 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃2 𝐽𝐽 (i.e. again in the fixed system)
- 10. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES The position, velocity, and acceleration vectors in the moving system The components of the position vector 𝑟𝑟 in the moving (polar) system can be obtained by a 2D rotation matrix i.e.: 𝑥𝑥 𝑦𝑦 = cos 𝜃𝜃 sin 𝜃𝜃 − sin 𝜃𝜃 cos 𝜃𝜃 𝑋𝑋(𝑡𝑡) 𝑌𝑌(𝑡𝑡) i.e. since 𝑟𝑟 = 𝑋𝑋 𝑡𝑡 𝐼𝐼 + 𝑌𝑌 𝑡𝑡 𝐽𝐽 = 𝑟𝑟 𝑡𝑡 cos 𝜃𝜃 𝑡𝑡 𝐼𝐼 + 𝑟𝑟 𝑡𝑡 sin 𝜃𝜃(𝑡𝑡)𝐽𝐽 : 𝑥𝑥 𝑦𝑦 = cos 𝜃𝜃 sin 𝜃𝜃 − sin 𝜃𝜃 cos 𝜃𝜃 𝑟𝑟 cos 𝜃𝜃 𝑟𝑟 sin 𝜃𝜃 And by noting that 𝑐𝑐𝑐𝑐𝑐𝑐2𝜃𝜃 + 𝑠𝑠𝑠𝑠𝑠𝑠2𝜃𝜃 = 1, we get: 𝑥𝑥 𝑦𝑦 = 𝑟𝑟 0 i.e. 𝑟𝑟 = 𝑟𝑟𝑖𝑖 (where 𝑖𝑖 is moving with angular velocity ̇ 𝜃𝜃). This result is obvious because the x axis always points straight at the particle so the frame of reference xy (polar coordinates) is defined precisely to ‘track’ the particle.
- 11. Advanced Kinematic Analysis The Velocity vector 𝑉𝑉 in the moving system The components of the velocity vector obtained by a 2D rotation matrix i.e.: 𝑉𝑉 = ̇ 𝑋𝑋𝐼𝐼 + ̇ 𝑌𝑌𝐽𝐽 = ̇ 𝑟𝑟 cos 𝜃𝜃 − 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 𝐼𝐼 + ̇ 𝑟𝑟 sin 𝜃𝜃 + 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 𝐽𝐽 ̇ 𝑥𝑥 ̇ 𝑦𝑦 = cos 𝜃𝜃 sin 𝜃𝜃 − sin 𝜃𝜃 cos 𝜃𝜃 ̇ 𝑋𝑋 ̇ 𝑌𝑌 = ̇ 𝑟𝑟 𝑟𝑟 ̇ 𝜃𝜃 i.e. the velocity vector in the moving system is: ̇ 𝑟𝑟 = ̇ 𝑟𝑟𝑖𝑖 + 𝑟𝑟 ̇ 𝜃𝜃𝑗𝑗 KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES
- 12. Advanced Kinematic Analysis KINEMATICS OF A PARTICLE OBTAINED BY ROTATION OF AXES Acceleration vector 𝒂𝒂 in the moving system The components of the acceleration vector also obtained by a 2D rotation matrix are: 𝑎𝑎 = ̈ 𝑋𝑋𝐼𝐼 + ̈ 𝑌𝑌𝐽𝐽 = ̈ 𝑟𝑟 cos 𝜃𝜃 − ̇ 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 − ̇ 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃 − 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃2 − 𝑟𝑟 sin 𝜃𝜃 ̈ 𝜃𝜃 𝐼𝐼 + ̈ 𝑟𝑟 sin 𝜃𝜃 + ̇ 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 + ̇ 𝑟𝑟 cos 𝜃𝜃 ̇ 𝜃𝜃 + 𝑟𝑟 cos 𝜃𝜃 ̈ 𝜃𝜃 − 𝑟𝑟 sin 𝜃𝜃 ̇ 𝜃𝜃2 𝐽𝐽 And in terms of the xy frame: ̈ 𝑥𝑥 ̈ 𝑦𝑦 = cos 𝜃𝜃 sin 𝜃𝜃 − sin 𝜃𝜃 cos 𝜃𝜃 ̈ 𝑋𝑋 ̈ 𝑌𝑌 And after some manipulation we get: = ̈ 𝑟𝑟 − 𝑟𝑟 ̇ 𝜃𝜃2 2 ̇ 𝑟𝑟 ̇ 𝜃𝜃 + 𝑟𝑟 ̈ 𝜃𝜃 i.e. the acceleration vector in the moving system is: 𝑎𝑎 = ( ̈ 𝑟𝑟 − 𝑟𝑟 ̇ 𝜃𝜃2)𝑖𝑖 + (2 ̇ 𝑟𝑟 ̇ 𝜃𝜃 + 𝑟𝑟 ̈ 𝜃𝜃)𝑗𝑗
- 13. Advanced Kinematic Analysis Physical Interpretation of Acceleration Terms The components of the acceleration vector are now shown in the figure below where the unit vectors 𝑖𝑖 , 𝑗𝑗 are moving. P X Y y x r=ri r ̈ 𝑟𝑟: is the radial acceleration. −𝑟𝑟 ̇ 𝜃𝜃2: is the centripetal acceleration. 𝑟𝑟 ̈ 𝜃𝜃: is the tangential acceleration. 2 ̇ 𝑟𝑟 ̇ 𝜃𝜃: is the coriolis component.
- 14. Advanced Kinematic Analysis The Coriolis acceleration stems from the combined radial and angular motion. Imagine moving radially outwards on a spinning disc (e.g. a carousel or roundabout) with constant angular velocity ω. At radius r1, the tangential velocity is v1= ωr1. At radius r2, the tangential velocity is v2= ωr2. Since r2 > r1 the tangential velocity must increase, representing an acceleration component in the tangential direction. ωr1 𝑟𝑟2 > 𝑟𝑟1 ̇ 𝜃𝜃 = 𝜔𝜔 ̇ 𝜔𝜔 = 0 ωr2 r1 r2