This document discusses analytical methods for determining velocity and acceleration of points in mechanisms. It presents:
1. An approximate method for determining velocity and acceleration of the slider in a slider-crank mechanism using a binomial expansion.
2. A general method for velocity and acceleration using polar coordinates and defining radial and tangential components.
3. Methods for determining relative velocity and acceleration between two moving points and for analyzing the plane motion of links with examples of equivalent motions comprising translation and rotation.
This document discusses double patterning lithography, which is a technique used to print integrated circuit designs smaller than the wavelength of light allows. It involves splitting the mask into two exposures to print features half as dense. The document outlines the challenges in splitting layouts and describes techniques used, including a novel polygon fracturing algorithm, dynamic priority search trees for plane sweeping, converting the problem to a graph coloring one. It shows how to decompose the graph to solve conflicts and presents results on test cases.
The document discusses solving locus problems by eliminating parameters from coordinate points. It provides 3 types of problems: 1) no parameters in coordinates, 2) obvious relationship between coordinates allowing one parameter to be eliminated, 3) no obvious relationship requiring use of a separate proven relationship between parameters. Examples of each type are given along with multi-part word problems involving parabolas, tangents and loci of intersection points.
This document provides an introduction to kinematics and the analysis of mechanisms using velocity and acceleration diagrams. It discusses:
1. Key concepts in mechanisms including different types of motion transformations and common mechanism components like four-bar linkages.
2. How to determine the displacement, velocity, and acceleration of points within a mechanism using either mathematical equations or graphical methods using velocity and acceleration diagrams.
3. How to construct velocity diagrams by determining the absolute and relative velocities of points and drawing them as vectors. This allows solving for unknown velocities.
4. How to extend the method to acceleration diagrams to determine centripetal and other accelerations which are important for calculating inertia forces.
The document provides examples
This document provides an overview of kinematics concepts related to velocity and acceleration diagrams for mechanisms. It describes:
1) How mechanisms can convert different types of motion and forces.
2) The concepts of absolute and relative velocity and acceleration, as well as radial and tangential components.
3) How to construct velocity diagrams by calculating tangential velocities and combining relative velocities to find absolute velocities.
The document discusses static and dynamic force analysis of mechanisms. It defines key terms like static equilibrium, inertia force, inertia torque, and D'Alembert's principle. It explains the conditions for a body to be in equilibrium under different force configurations. Dynamic force analysis considers inertia forces to determine input torque required. Equivalent masses and Klein's construction diagram are discussed for dynamic analysis of reciprocating engines. Correction couple and torque are also summarized.
the presentation consists of various important terms that are generally linked with the analysis of a common four bar mechanism which are as follows - coupler curves, toggle positions, transmission angles, mechanical advantage, acc analysis and coriolis component.
This project report describes the modeling, simulation, and kinematic analysis of a radial engine using CATIA V5 software. The report includes an introduction to radial engines and their basic components. It then discusses modeling the engine components in CATIA and performing kinematic analysis using the software. The results of the velocity and acceleration calculations from CATIA are then compared to approximate analytical methods. The report concludes that CATIA allows understanding the complete working of a radial engine before actual design and manufacturing.
This document discusses double patterning lithography, which is a technique used to print integrated circuit designs smaller than the wavelength of light allows. It involves splitting the mask into two exposures to print features half as dense. The document outlines the challenges in splitting layouts and describes techniques used, including a novel polygon fracturing algorithm, dynamic priority search trees for plane sweeping, converting the problem to a graph coloring one. It shows how to decompose the graph to solve conflicts and presents results on test cases.
The document discusses solving locus problems by eliminating parameters from coordinate points. It provides 3 types of problems: 1) no parameters in coordinates, 2) obvious relationship between coordinates allowing one parameter to be eliminated, 3) no obvious relationship requiring use of a separate proven relationship between parameters. Examples of each type are given along with multi-part word problems involving parabolas, tangents and loci of intersection points.
This document provides an introduction to kinematics and the analysis of mechanisms using velocity and acceleration diagrams. It discusses:
1. Key concepts in mechanisms including different types of motion transformations and common mechanism components like four-bar linkages.
2. How to determine the displacement, velocity, and acceleration of points within a mechanism using either mathematical equations or graphical methods using velocity and acceleration diagrams.
3. How to construct velocity diagrams by determining the absolute and relative velocities of points and drawing them as vectors. This allows solving for unknown velocities.
4. How to extend the method to acceleration diagrams to determine centripetal and other accelerations which are important for calculating inertia forces.
The document provides examples
This document provides an overview of kinematics concepts related to velocity and acceleration diagrams for mechanisms. It describes:
1) How mechanisms can convert different types of motion and forces.
2) The concepts of absolute and relative velocity and acceleration, as well as radial and tangential components.
3) How to construct velocity diagrams by calculating tangential velocities and combining relative velocities to find absolute velocities.
The document discusses static and dynamic force analysis of mechanisms. It defines key terms like static equilibrium, inertia force, inertia torque, and D'Alembert's principle. It explains the conditions for a body to be in equilibrium under different force configurations. Dynamic force analysis considers inertia forces to determine input torque required. Equivalent masses and Klein's construction diagram are discussed for dynamic analysis of reciprocating engines. Correction couple and torque are also summarized.
the presentation consists of various important terms that are generally linked with the analysis of a common four bar mechanism which are as follows - coupler curves, toggle positions, transmission angles, mechanical advantage, acc analysis and coriolis component.
This project report describes the modeling, simulation, and kinematic analysis of a radial engine using CATIA V5 software. The report includes an introduction to radial engines and their basic components. It then discusses modeling the engine components in CATIA and performing kinematic analysis using the software. The results of the velocity and acceleration calculations from CATIA are then compared to approximate analytical methods. The report concludes that CATIA allows understanding the complete working of a radial engine before actual design and manufacturing.
1) To analyze accelerations, positions must first be found to calculate velocities by differentiation and accelerations by further differentiation.
2) Acceleration has two components - tangential and centripetal. For uniform motion only centripetal acceleration exists, and for straight-line motion only tangential acceleration exists.
3) The Coriolis component arises for points moving on rotating links and is perpendicular to the link and proportional to the product of linear and angular velocities.
This document discusses kinematic analysis and various methods for velocity analysis of mechanisms. It covers graphical methods, the relative velocity method, instantaneous center method, and the vector loop method. The instantaneous center method is described in detail, including locating instantaneous centers, Kennedy's theorem on three centers in a line, and examples of applying this method to determine velocities and angular velocities in different mechanisms.
Design, Fabrication and Analysis of Crank and Slotted Lever Quick Return Mech...Mohammed Naseeruddin Shah
In industry it is always desirable to increase the productivity or reduce the time loss. A beautiful mechanism in which, for the constant input rotation, the forward stroke takes larger time(cutting stroke) than the return stroke(idle stroke).
Unit 1 – Basics of Mechanics
Topics to be covered – unit1
Basic kinematic concepts and definitions
Degree of freedom & Mobility
Kutzbach criterion & Gruebler’s criterion
Grashof’s Law
Kinematic inversions of four-bar-chain and slider crank chains – Limit positions
Mechanical advantage – Transmission Angle
Classification of mechanisms
Description of some common mechanisms
Unit 1-introduction to Mechanisms, Kinematics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
The document discusses the history and development of chocolate over centuries. It details how cocoa beans were first used by Mesoamerican cultures before being introduced to Europe, where it became popular in drinks and confections. The document also notes that modern chocolate production methods were established in the 19th century to allow chocolate to be consumed on a larger scale.
Here are the steps to evaluate (8)2 in either order:
1) Evaluate the exponent first:
(8)2 = 8 * 8 = 64
2) Evaluate the base first:
8 = 23
(23)2 = (2 * 2 * 2 * 2 * 2 * 2 * 2 * 2)2 = 25 * 25 = 625
So evaluating the exponent first yields 64, while evaluating the base first yields 625.
In general, the order of operations matters for expressions involving exponents. Exponents should be evaluated before other operations like multiplication or division.
Rational exponents can be written in three different forms. To evaluate or simplify expressions with rational exponents:
- Use properties of exponents like power-to-a-power, product-to-a-power, and quotient-to-a-power laws
- Simplify to remove negative exponents, fractional exponents in the denominator, or complex fractions
- Write expressions with rational exponents as radicals and simplify if possible in 1-2 sentences
The document describes various techniques for implementing a Petri net state space search:
1. It discusses how transitions are fired and states are evaluated by marking changed places and checking enabled transitions.
2. State predicates are stored in negation-free normal form to efficiently check state properties.
3. The state space is managed by representing states as bit vectors and organizing them in a decision tree for fast lookup and insertion.
4. Search organization involves firing transitions, finding/inserting states, and backtracking with a search stack and write-only memory approach.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller overlapping subproblems and storing the solutions to subproblems to avoid recomputing them. It involves building up a solution using previously found subsolutions and working in a bottom-up fashion to solve larger subproblems from solutions to smaller subproblems. Examples where dynamic programming has been applied include computing Fibonacci numbers, finding the shortest paths in a graph, and solving optimization problems like the knapsack problem.
This document summarizes acceleration analysis techniques for mechanisms:
1. It defines linear and angular acceleration and outlines graphical and algebraic methods.
2. Key steps are formulating loop equations relating link positions, velocities, and accelerations, then differentiating to solve for unknowns.
3. Coriolis acceleration, which depends on both rotational and tangential motion, is also explained.
4. Rolling acceleration analysis is presented for both fixed and inverse curvature with and without angular acceleration.
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
1. The document discusses acceleration analysis of mechanisms using both graphical and algebraic methods.
2. Key steps in the graphical method include drawing acceleration vectors on links and solving the equation relating accelerations of connected points.
3. The algebraic method involves writing loop closure equations relating positions, velocities, and accelerations of links, then differentiating the equations to solve for unknown accelerations.
The document discusses analytical kinematics for planar mechanisms. It provides details on:
1) Performing analytical kinematic analysis by projecting the vector loop equation onto a Cartesian frame to obtain algebraic equations relating the position, velocity, and acceleration of links.
2) Working through an example problem of the kinematics of a slider-crank mechanism by obtaining position, velocity, and acceleration equations and solving for unknown values.
3) Noting observations about the analytical kinematic process for slider-crank and four-bar mechanisms.
This document contains formulas for calculus, trigonometric functions, and their inverses. It includes formulas for:
- Differentiation and integration of polynomials using the power, product, and quotient rules.
- Trigonometric identities for sums and differences of sines, cosines, tangents, cotangents, secants and cosecants.
- Graphs of trigonometric functions and their inverses over their domains.
- Absolute value definitions.
- Sum and difference formulas for trigonometric functions of angles.
Pascal's triangle is a triangular array of the binomial coefficients that arises in many areas of mathematics. It is constructed by taking 1 at the top and then each entry is the sum of the two entries directly above it. The entries also represent the coefficients in the binomial expansion of (1+x)^n as well as the combinatorial coefficients that give the number of combinations of choosing r objects from a set of n. The relationships between these interpretations are proven through examples like interpreting the coefficients as counting the number of ways to choose subsets containing a special object.
This document provides an overview of advanced power system protection topics including definitions, objectives of power system protection, relay characteristics, methods of discrimination, distance protection, current balance protection, phase comparison protection, fault detection techniques using zero sequence systems, sequence filters, general relay equations, fuses, and time-current characteristics. It discusses concepts such as discrimination, stability, sensitivity, repeatability and provides examples to illustrate relay equations and characteristics.
The document reviews trigonometry concepts including the unit circle and finding trigonometric functions of special angles. It provides examples of the unit circle with coordinates marked around it and homework problems involving finding the trigonometric functions of various angles in radians and degrees. The review is intended to help remember things learned in trigonometry class.
The document discusses the double modal transformation technique for analyzing the dynamic response of linear structures subjected to stochastic loading. It proposes a method called double modal transformation that simultaneously transforms the equations of motion and the loading process. This allows the structural response to be obtained through a double series expansion where structural and loading modal contributions are superimposed. The effectiveness of this technique is illustrated through two classic wind engineering problems: alongwind response and vortex-induced crosswind response of slender structures.
The document provides information and examples on linear laws and linear relationships. It discusses:
- Drawing lines of best fit by inspection of data points.
- Writing equations for lines of best fit in the form of y = mx + c.
- Determining values of variables from lines of best fit and equations.
- Reducing non-linear relationships to linear form by rearranging variables.
- Finding values of constants in non-linear relationships by plotting graphs of best fit lines and determining the gradient and y-intercept.
Worked examples are provided to illustrate key concepts like identifying dependent and independent variables, determining the gradient and y-intercept, and using these to solve for constants in non-
This document discusses algorithms for solving the point in polygon problem for arbitrary polygons. It presents two main concepts: the even-odd rule and the winding number rule. It shows that both concepts are closely related and can be based on determining the winding number. The document derives an incremental angle algorithm for computing the winding number and modifies it to accelerate the computation and handle special cases. It compares the resulting winding number algorithm to those found in literature.
1) To analyze accelerations, positions must first be found to calculate velocities by differentiation and accelerations by further differentiation.
2) Acceleration has two components - tangential and centripetal. For uniform motion only centripetal acceleration exists, and for straight-line motion only tangential acceleration exists.
3) The Coriolis component arises for points moving on rotating links and is perpendicular to the link and proportional to the product of linear and angular velocities.
This document discusses kinematic analysis and various methods for velocity analysis of mechanisms. It covers graphical methods, the relative velocity method, instantaneous center method, and the vector loop method. The instantaneous center method is described in detail, including locating instantaneous centers, Kennedy's theorem on three centers in a line, and examples of applying this method to determine velocities and angular velocities in different mechanisms.
Design, Fabrication and Analysis of Crank and Slotted Lever Quick Return Mech...Mohammed Naseeruddin Shah
In industry it is always desirable to increase the productivity or reduce the time loss. A beautiful mechanism in which, for the constant input rotation, the forward stroke takes larger time(cutting stroke) than the return stroke(idle stroke).
Unit 1 – Basics of Mechanics
Topics to be covered – unit1
Basic kinematic concepts and definitions
Degree of freedom & Mobility
Kutzbach criterion & Gruebler’s criterion
Grashof’s Law
Kinematic inversions of four-bar-chain and slider crank chains – Limit positions
Mechanical advantage – Transmission Angle
Classification of mechanisms
Description of some common mechanisms
Unit 1-introduction to Mechanisms, Kinematics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
The document discusses the history and development of chocolate over centuries. It details how cocoa beans were first used by Mesoamerican cultures before being introduced to Europe, where it became popular in drinks and confections. The document also notes that modern chocolate production methods were established in the 19th century to allow chocolate to be consumed on a larger scale.
Here are the steps to evaluate (8)2 in either order:
1) Evaluate the exponent first:
(8)2 = 8 * 8 = 64
2) Evaluate the base first:
8 = 23
(23)2 = (2 * 2 * 2 * 2 * 2 * 2 * 2 * 2)2 = 25 * 25 = 625
So evaluating the exponent first yields 64, while evaluating the base first yields 625.
In general, the order of operations matters for expressions involving exponents. Exponents should be evaluated before other operations like multiplication or division.
Rational exponents can be written in three different forms. To evaluate or simplify expressions with rational exponents:
- Use properties of exponents like power-to-a-power, product-to-a-power, and quotient-to-a-power laws
- Simplify to remove negative exponents, fractional exponents in the denominator, or complex fractions
- Write expressions with rational exponents as radicals and simplify if possible in 1-2 sentences
The document describes various techniques for implementing a Petri net state space search:
1. It discusses how transitions are fired and states are evaluated by marking changed places and checking enabled transitions.
2. State predicates are stored in negation-free normal form to efficiently check state properties.
3. The state space is managed by representing states as bit vectors and organizing them in a decision tree for fast lookup and insertion.
4. Search organization involves firing transitions, finding/inserting states, and backtracking with a search stack and write-only memory approach.
Dynamic programming is an algorithm design technique that solves problems by breaking them down into smaller overlapping subproblems and storing the solutions to subproblems to avoid recomputing them. It involves building up a solution using previously found subsolutions and working in a bottom-up fashion to solve larger subproblems from solutions to smaller subproblems. Examples where dynamic programming has been applied include computing Fibonacci numbers, finding the shortest paths in a graph, and solving optimization problems like the knapsack problem.
This document summarizes acceleration analysis techniques for mechanisms:
1. It defines linear and angular acceleration and outlines graphical and algebraic methods.
2. Key steps are formulating loop equations relating link positions, velocities, and accelerations, then differentiating to solve for unknowns.
3. Coriolis acceleration, which depends on both rotational and tangential motion, is also explained.
4. Rolling acceleration analysis is presented for both fixed and inverse curvature with and without angular acceleration.
This presentation explains how the differentiation is applied to identify increasing and decreasing functions,identifying the nature of stationary points and also finding maximum or minimum values.
1. The document discusses acceleration analysis of mechanisms using both graphical and algebraic methods.
2. Key steps in the graphical method include drawing acceleration vectors on links and solving the equation relating accelerations of connected points.
3. The algebraic method involves writing loop closure equations relating positions, velocities, and accelerations of links, then differentiating the equations to solve for unknown accelerations.
The document discusses analytical kinematics for planar mechanisms. It provides details on:
1) Performing analytical kinematic analysis by projecting the vector loop equation onto a Cartesian frame to obtain algebraic equations relating the position, velocity, and acceleration of links.
2) Working through an example problem of the kinematics of a slider-crank mechanism by obtaining position, velocity, and acceleration equations and solving for unknown values.
3) Noting observations about the analytical kinematic process for slider-crank and four-bar mechanisms.
This document contains formulas for calculus, trigonometric functions, and their inverses. It includes formulas for:
- Differentiation and integration of polynomials using the power, product, and quotient rules.
- Trigonometric identities for sums and differences of sines, cosines, tangents, cotangents, secants and cosecants.
- Graphs of trigonometric functions and their inverses over their domains.
- Absolute value definitions.
- Sum and difference formulas for trigonometric functions of angles.
Pascal's triangle is a triangular array of the binomial coefficients that arises in many areas of mathematics. It is constructed by taking 1 at the top and then each entry is the sum of the two entries directly above it. The entries also represent the coefficients in the binomial expansion of (1+x)^n as well as the combinatorial coefficients that give the number of combinations of choosing r objects from a set of n. The relationships between these interpretations are proven through examples like interpreting the coefficients as counting the number of ways to choose subsets containing a special object.
This document provides an overview of advanced power system protection topics including definitions, objectives of power system protection, relay characteristics, methods of discrimination, distance protection, current balance protection, phase comparison protection, fault detection techniques using zero sequence systems, sequence filters, general relay equations, fuses, and time-current characteristics. It discusses concepts such as discrimination, stability, sensitivity, repeatability and provides examples to illustrate relay equations and characteristics.
The document reviews trigonometry concepts including the unit circle and finding trigonometric functions of special angles. It provides examples of the unit circle with coordinates marked around it and homework problems involving finding the trigonometric functions of various angles in radians and degrees. The review is intended to help remember things learned in trigonometry class.
The document discusses the double modal transformation technique for analyzing the dynamic response of linear structures subjected to stochastic loading. It proposes a method called double modal transformation that simultaneously transforms the equations of motion and the loading process. This allows the structural response to be obtained through a double series expansion where structural and loading modal contributions are superimposed. The effectiveness of this technique is illustrated through two classic wind engineering problems: alongwind response and vortex-induced crosswind response of slender structures.
The document provides information and examples on linear laws and linear relationships. It discusses:
- Drawing lines of best fit by inspection of data points.
- Writing equations for lines of best fit in the form of y = mx + c.
- Determining values of variables from lines of best fit and equations.
- Reducing non-linear relationships to linear form by rearranging variables.
- Finding values of constants in non-linear relationships by plotting graphs of best fit lines and determining the gradient and y-intercept.
Worked examples are provided to illustrate key concepts like identifying dependent and independent variables, determining the gradient and y-intercept, and using these to solve for constants in non-
This document discusses algorithms for solving the point in polygon problem for arbitrary polygons. It presents two main concepts: the even-odd rule and the winding number rule. It shows that both concepts are closely related and can be based on determining the winding number. The document derives an incremental angle algorithm for computing the winding number and modifies it to accelerate the computation and handle special cases. It compares the resulting winding number algorithm to those found in literature.
This document discusses methods for finding the roots of polynomial equations, including Muller's method, Bairstow's method, and the Bairstow algorithm. It provides details on how each method works, such as deriving the coefficients of a parabola through three points for Muller's method or using synthetic division and solving simultaneous equations to estimate changes in values for Bairstow's method. An example is also shown applying Bairstow's method to find the roots of a 5th order polynomial equation.
This document discusses different coordinate systems used to describe points in two-dimensional and three-dimensional spaces, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems, and gives examples of performing these conversions as well as writing equations of basic geometric shapes in different coordinate systems.
This document discusses different coordinate systems used to describe points in 2D and 3D space, including polar, cylindrical, and spherical coordinates. It provides the key formulas for converting between Cartesian and these other coordinate systems. Examples are given of converting points and equations between the different coordinate systems. The key points are that polar coordinates use an angle and distance to specify a 2D point, cylindrical coordinates extend this to 3D using a z-value, and spherical coordinates specify a 3D point using a distance from the origin, an angle, and an azimuthal angle.
This document provides an overview of MPI (Message Passing Interface), which is a standard for parallel programming using message passing. The key points covered include:
- MPI allows programs to run across multiple computers in a distributed memory environment. It has functions for point-to-point and collective communication.
- Common MPI functions introduced are MPI_Send, MPI_Recv for point-to-point communication, and MPI_Bcast, MPI_Gather for collective operations.
- More advanced topics like derived data types and examples of Poisson equation and FFT solvers are also briefly discussed.
This document describes a student project to build a washing machine prototype. It includes a summary, instruction manual, state diagrams, schematics and code for the washing machine. The prototype has two modes - manual, where the user selects the function, or automatic, where it runs through the cycle automatically. It was built using multiplexers, Moore state machines, registers and other components on a Xilinx Basys 2 board. The code provided includes modules for functions like filling, washing, draining and spinning in manual mode.
La tabla lista los puertos renombrados y servicios comunes como FTP (puerto 20), SSH (puerto 22), e IMAP (puerto 143) que utiliza Red Hat Enterprise Linux. También incluye puertos específicos de UNIX como los utilizados para correo electrónico (puerto 512), inicio de sesión (puerto 513), e impresión (puerto 515). Finalmente, lista puertos suministrados a la IANA para su registro formal.
Este documento presenta los resultados de 5 circuitos analizados como parte de una práctica de laboratorio sobre funciones de transferencia. Describe brevemente cada circuito, incluyendo las señales de entrada y salida medidas. El objetivo general era obtener las funciones de transferencia de los circuitos y comparar los datos experimentales con cálculos teóricos.
La práctica describe el uso de un sistema de control de presión que consta de los módulos G35 y TY35. El objetivo es familiarizarse con cada elemento del módulo controlador G35 y alambrarlo correctamente, así como conocer los transductores de presión y sus características. Se conectan y calibran los módulos, midiendo valores de presión y voltaje. Las gráficas resultantes muestran una relación lineal entre la presión, el voltaje de entrada y el voltaje de salida.
Este documento presenta los resultados de una práctica de control de velocidad utilizando transductores. Se describen las características de un transductor de velocidad y uno de posición, y se muestran tablas con datos de velocidad y voltaje de salida para diferentes voltajes de entrada, así como datos de posición angular y voltaje. Adicionalmente, se grafican las relaciones entre las variables medidas y se calculan las pendientes de las curvas de ajuste por mínimos cuadrados.
El documento describe el puerto paralelo de una computadora, incluyendo su conector DB-25, sus modos de operación definidos por el estándar IEEE 1284, y sus registros de datos, estado y control. También explica cómo determinar la dirección base del puerto paralelo usando el programa Debug.exe y cómo grabar y leer información de un arreglo de memoria usando los puertos paralelos y programas diseñados.
Este documento describe una serie de experimentos realizados para determinar las constantes de tiempo de las acciones proporcional, integral y derivativa de un controlador PID. Se midió la respuesta del sistema al variar las señales de entrada y las constantes del controlador en lazo abierto y cerrado. Finalmente, se analizó el efecto de cada acción del PID en el control automático de la velocidad de un motor.
Este documento describe una práctica de laboratorio sobre el control de temperatura. Se explican tres tipos de transductores de temperatura: de semiconductor, de termo-resistencia y de termopar. Se muestran tablas con datos de temperatura vs voltaje para cada transductor y gráficas de sus curvas características. El objetivo de la práctica es familiarizar a los estudiantes con estos dispositivos y su funcionamiento.
Este documento describe el puerto paralelo de una computadora. Explica que un conector DB25 puede conectarse a uno Centronix y muestra la equivalencia de pines. También indica que el puerto paralelo se usa para comunicación con dispositivos de E/S de forma sincrónica y asincrónica, y para controlar dispositivos como motores. Finalmente, presenta un diagrama de flujo para controlar el movimiento de un motor de pasos a través del puerto paralelo.
El documento describe dos programas para usar el puerto paralelo de una computadora. El primer programa recibe un número del 0 al 255 desde el teclado, lo convierte a binario y lo muestra en pantalla y LEDs. El segundo programa recibe letras del abecedario, las codifica para mostrar en un display de 7 segmentos conectado al puerto paralelo. Ambos programas usan funciones como scanf, outportb y delay.
El documento describe los pasos para implementar memorias de mayor capacidad, como definir la organización, los circuitos integrados disponibles, las restricciones de materiales, y el número de circuitos necesarios. También explica las diferencias entre RAM estática y dinámica, como la necesidad de refresco periódico en RAM dinámica. Además, indica que la celda de almacenamiento de una SRAM consiste en flip-flops. Por último, pide implementar varios arreglos de memoria usando un CI SRAM específico.
Este documento discute el uso y manejo de registros de corrimiento como memorias secuenciales. Explica que los flip-flops D pueden operar a frecuencias máximas de 15 MHz para el 7474 y 30 MHz para el 74273. También enumera algunas aplicaciones de los registros de corrimiento como generadores pseudoaleatorios, multiplicadores en serie y retardar bits. Finalmente, compara las ventajas y desventajas de la memoria serial frente a la de acceso aleatorio.
Una memoria CAM (Content Addressable Memory) funciona de manera opuesta a las memorias convencionales como SRAM y ROM. En lugar de proporcionar una dirección de memoria para recuperar datos, una CAM busca datos específicos en toda la memoria y devuelve las direcciones donde se encuentran los datos coincidentes. Esto hace que la búsqueda sea más rápida que en una SRAM convencional. Las memorias CAM se utilizan comúnmente en switches de red y cachés de procesador donde se requieren tiempos de búsqueda muy rápidos.
Este documento explica las diferencias entre memorias de acceso secuencial (SAM) y aleatorio (RAM), así como entre memorias RAM estáticas (SRAM) y dinámicas (DRAM). También describe cómo se implementa un bus común bidireccional en las memorias RAM y por qué las memorias DRAM son de lectura destructiva debido a que los capacitores se descargan y borran los bits almacenados si no reciben energía. Finalmente, pide dibujar un habilitador general para una SRAM 2x2.
Este documento contiene las preguntas y respuestas de un examen previo sobre memorias de solo lectura semiconductoras (ROM). Explica que las ROM tienen acceso aleatorio y se usan para almacenar datos que solo se leerán, como código de computadora. No son útiles como memoria de trabajo porque rara vez se pueden modificar los datos. También define términos clave relacionados con las memorias como celda de almacenamiento, registro de almacenamiento, unidad de almacenamiento, palabra de memoria y dirección de memoria.
El documento describe el protocolo estándar RS-232 para la comunicación serie en puertos de computadoras. Explica que usa una señal serial bipolar entre +10 y -10 voltios para transmitir bits de forma asíncrona. También describe las velocidades de transmisión estándar, las ventajas e desventajas de puertos serie frente a paralelos, y pide construir un cable serial para uso en el laboratorio.
Este documento trata sobre las memorias RAM y sus componentes básicos. Explica que la decodificación convierte un código binario de entrada en líneas de salida activadas, y que los dispositivos de memoria almacenan datos de forma permanente o temporal para su lectura y escritura. Identifica los cuatro bloques principales de una memoria RAM: almacenamiento, direccionamiento, control e I/O, y explica que el bloque de direccionamiento le da acceso aleatorio mediante decodificadores. Finalmente, menciona que las salidas de colector
Este documento presenta una antología sobre el desarrollo de software seguro dividida en 8 unidades. La unidad I introduce conceptos clave como software, casos reales de fallas, fuentes de información sobre vulnerabilidades y metas de seguridad. Las unidades II a VII cubren temas como administración de riesgos, código abierto vs cerrado, principios de software seguro, auditoría, código seguro, pruebas y derechos de autor. La antología provee una guía completa sobre los fundamentos y mejores prácticas para el desarrollo de software
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LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
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Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) CurriculumMJDuyan
(𝐓𝐋𝐄 𝟏𝟎𝟎) (𝐋𝐞𝐬𝐬𝐨𝐧 𝟏)-𝐏𝐫𝐞𝐥𝐢𝐦𝐬
𝐃𝐢𝐬𝐜𝐮𝐬𝐬 𝐭𝐡𝐞 𝐄𝐏𝐏 𝐂𝐮𝐫𝐫𝐢𝐜𝐮𝐥𝐮𝐦 𝐢𝐧 𝐭𝐡𝐞 𝐏𝐡𝐢𝐥𝐢𝐩𝐩𝐢𝐧𝐞𝐬:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
𝐄𝐱𝐩𝐥𝐚𝐢𝐧 𝐭𝐡𝐞 𝐍𝐚𝐭𝐮𝐫𝐞 𝐚𝐧𝐝 𝐒𝐜𝐨𝐩𝐞 𝐨𝐟 𝐚𝐧 𝐄𝐧𝐭𝐫𝐞𝐩𝐫𝐞𝐧𝐞𝐮𝐫:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
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How to Setup Warehouse & Location in Odoo 17 InventoryCeline George
In this slide, we'll explore how to set up warehouses and locations in Odoo 17 Inventory. This will help us manage our stock effectively, track inventory levels, and streamline warehouse operations.
1. Analysis by
UNIT 5 ANALYSIS BY ANALYTICAL Analytical Methods
METHODS
Structure
5.1 Introduction
Objectives
5.2 Analytical Method for Velocity and Acceleration
5.2.1 Approximate Method for Slider Crank Mechanism
5.2.2 General Method for Velocity and Acceleration
5.3 Motion Referred to Motion Frames of Reference
5.3.1 Relative Motion of Two Points
5.3.2 Plane Motion of a Link
5.4 Method of Relative Velocity and Acceleration
5.5 Alternative Method of Determining Coriolis’ Component of Acceleration
5.6 Klein’s Construction for Determining Velocity and Acceleration of Slider
Crank Mechanism
5.7 Summary
5.8 Key Words
5.9 Answers to SAQs
5.1 INTRODUCTION
In unit 4, you have studied instantaneous centre method for determining velocity of any
point in a mechanism. This method is good for determining velocity and cannot be used
determining acceleration. Also, if lines drawn for determining instantaneous centre are
parallel, they cannot locate instantaneous centre. Even if these lines are nearly parallel,
they will meet at a large distance to locate instantaneous centre within the limited size of
the paper. In this unit, you will be explained relative velocity method and analytical
method and how they can be used to determine velocity and then acceleration of a point
in a mechanism. Acceleration is required for determining inertia forces which are
required in dynamic analysis of a mechanism.
Objectives
After going through this unit you should be able to
• analyse plane motion,
• analyse plane motion with moving frames of references,
• determine magnitude of Corioli’s acceleration and its direction,
• determine velocity and acceleration of any point in a mechanism, and
• apply Klein’s construction for determination of acceleration of slider in
slider crank mechanism.
21
2. Motion Analysis of
Planar Mechanism and 5.2 ANALYTICAL METHOD FOR VELOCITY AND
Synthesis
ACCELERATION
The analytical method can be used for determining velocity and acceleration of any point
in a mechanism and angular acceleration of any link.
5.2.1 Approximate Method for Slider Crank Mechanism
The slider crank mechanism OAB is shown in Figure 5.1. Let l and r be lengths of the
connecting rod AB and crank OA respectively. Let x be the distance of the piston pin
from its farthest position B, (for this point OA and AB are aligned).
A
l
r 2
3 B
θ C φ
O B1
4
1 x
Figure 5.1
x = (r + l ) − (OC + CB ) = (r + l ) − (r cos θ + l cos φ)
⎧ l ⎫
or x = r (1 − cos θ) + l (1 − cos φ) = r ⎨(1 − cos θ) + (1 − cos φ) ⎬
⎩ r ⎭
Also AC = r sin θ = l sin φ
r
∴ sin φ = sin θ
l
r2
Also cos 2 φ = 1 − sin 2 φ = 1 − sin 2 θ
l2
1
⎡ r2 ⎤2
∴ cos φ = ⎢1 − 2 sin 2 θ ⎥
⎢
⎣ l ⎥
⎦
By Binomial theorem
2 4 6
1 ⎛r⎞ 1 ⎛r⎞ 1 ⎛r⎞
cos φ = 1 − ⎜ ⎟ sin θ − ⎜ ⎟ sin θ −
2 4
⎜ ⎟ sin θ
6
2 ⎝l⎠ 8⎝l⎠ 16 ⎝ l⎠
2 4 6
1 ⎛r⎞ 1 ⎛r⎞ 1 ⎛r⎞
or (1 − cos φ) = ⎜ ⎟ sin 2 θ + ⎜ ⎟ sin 4 θ + ⎜ ⎟ sin θ + . . .
6
2⎝l⎠ 8⎝l⎠ 16 ⎝ l ⎠
Substituting (1 – cos φ) in expression of x you will get
⎡ 1 ⎛r⎞ 1 ⎛r⎞
3 ⎤
x = r ⎢(1 − cos θ) + ⎜ ⎟ sin 2 θ + ⎜ ⎟ sin 4 θ + . . .⎥
⎢
⎣ 2⎝l⎠ 8⎝l⎠ ⎥
⎦
r ⎛r⎞
In this mechanism, is approximately 0.25 and higher powers of ⎜ ⎟ shall have
l ⎝l⎠
negligible value.
⎡ 1 ⎛r⎞ ⎤
Therefore, x ≈ r ⎢(1 − cos θ) + ⎜ ⎟ sin 2 θ ⎥
⎣ 2⎝l⎠ ⎦
22
3. dx dx d θ Analysis by
Velocity of slider, V = = × Analytical Methods
dt d θ dt
dθ
=ω (angular speed of crank)
dt
dx
Therefore, V =ω
dθ
⎡ 1 ⎛r⎞ ⎤
or V = r ω ⎢sin θ + ⎜ ⎟ 2 sin θ cos θ ⎥
⎣ 2⎝l⎠ ⎦
⎛ 1 ⎛r⎞ ⎞
or V = r ω ⎜ sin θ + ⎜ ⎟ sin 2 θ ⎟
⎝ 2⎝l⎠ ⎠
Let crank rotate at the constant angular velocity.
dV dV d θ
Acceleration of slider, a= = ×
dt d θ dt
⎡ r ⎤
or a = r ω2 ⎢cos θ + cos 2 θ⎥
⎣ l ⎦
5.2.2 General Method for Velocity and Acceleration
We start with plane curvilinear motion using polar coordinates to denote the position
vector ‘r’ and its angular coordinate ‘θ’ measured from a fixed reference axis as shown in
Figure 5.2. Let er and eθ unit vectors in radial and transverse directions respectively.
These unit vectors have positive directions in increasing r and θ. The vectors er and eθ are
parallel to the positive senses of Vr and Vθ. Both these unit vectors will swing through the
angle in the time dt. As shown in figure, their tips move through the distances | d er | = 1
× dθ = dθ in the direction of eθ, and through d eθ = 1 × d θ = d θ in the direction opposite
to that of er. Therefore,
d er d er d θ dθ
= × = θ eθ , where →θ
dt dθ dt dt
d eθ d eθ d θ
and = × = θ ( − er ) = − θ er
dt dθ dt
V
Vθ Vr
P
deθ
der
eθ
er Y
r
dθ dθ
er
eθ
Z
θ
θ
X
Figure 5.2
The velocity of a particle P which has position vector ‘r’ is given by
dr d dr d er
V = = (r er ) = er + r
dt dt dt dt
= r er + r θ eθ = Vt er + Vθ eθ 23
4. Motion Analysis of where Vr and Vθ are the components of velocity in radial direction and transverse
Planar Mechanism and direction. The acceleration of particle P is given by
Synthesis
dV d
a= = (r er + r θ eθ )
dt dt
d d
= ( r er ) + (r θ eθ )
dt dt
dr d er dr d
= er + r + (θ eθ ) + r (θ eθ )
dt dt dt dt
⎛ de ⎞
= r er + r (θ eθ ) + r θ eθ + r ⎜ θ eθ + θ θ ⎟
⎝ dt ⎠
= r er + 2 r θ eθ + r θ eθ − r θ2 er
= ( r − r θ2 ) er + ( r θ + 2 r θ) eθ
= ar er + aθ eθ
where ar represents radial component and aθ represents transverse component of
acceleration.
ar = r − r θ2 and aα = r θ + 2 r θ
Here, r → Acceleration of slider along slotted link,
r θ2 → Centripetal acceleration,
r θ → Tangential component of acceleration, and
2r θ → Coriolis’ component of acceleration.
SAQ 1
Explain Corioli’s component of acceleration and what is its magnitude?
Example 5.1
The length of the crank of quick return crank and slotted lever mechanism is
15 cm and it rotates at 10 rod/s in counter clockwise sense. For the configuration
shown in Figure 5.3 determine angular acceleration of link BD.
D
A A
15 13
0
ω 60
B 0 θ
60 B
ω = 10 rad/sec 32.5 cm E
O
(a) (b)
0
42 D
0
48 A
eθ
er
θ′′
θ′
ω 0
B 60
18
0 C
O
(c)
Figure 5.3
24
5. Solution Analysis by
Analytical Methods
AE 13 13
In Figure 5.3(b), tan θ = = = or θ = 18o
BE 32.5 + 7.5 40
13 AB
Also, = = AB or AB = 42.06 cm
sin θ sin 90o
The motion of point A is common to both OA and BD. The velocity and
acceleration of A are given by
15
VA = OA × ω = × 10 = 1.5 m/s
100
15
a A = OA × ω2 = × 10 = 15 m/s 2
100
The components of VA along BD are perpendicular to it are given by
Vr = r = − 1.5 cos 48o = − 1.0 m/s
1.5 sin 48o
Vθ = r θ = 1.5 sin 48o ∴ θ =
r
1.5 sin 48o
or θ= = 2.65 rad/s
42.06
100
The transverse component of acceleration of A is give by
aθ = − a A sin 42o = − 15 sin 42o = − 10.037 m/s 2
Also aθ = (r θ) + 2 (r θ)
42.06
or − 10.037 = θ + 2 (− 1.0) (2.65)
100
5.30 − 10.037
or θ= = − 11.26 rad/s 2
0.4206
Therefore, angular acceleration of BD is 11.26 rad/s2 in clockwise sense.
5.3 MOTION REFERRED TO MOVING FRAMES OF
REFERENCE
The moving frame may, in general, translate and rotate as well as accelerate linearly or
angularly. The purpose of the following treatment is to arrive at a systematic procedure
for refering motion of a point with respect to a frame of reference if its motion is known
in relation to another moving with respect to the former.
5.3.1 Relative Motion of Two Points
Consier two points P1 and P2 moving with velocities V1 and V2 and accelerations a1
and a2. The velocity of P1 with respect to P2 is given and acceleration of P1 with respect
to P2 is given by
a12 = a1 − a2
V12 = V1 − V2
Z 25
a1
B B
6. Motion Analysis of
Planar Mechanism and
Synthesis
Figure 5.4
Also V21 = V2 − V2 = − V12
and a21 = a2 − a1 = − a12
5.3.2 Plane Motion of a Link
A motion is said to be a plane motion if all the points in the body stay in the same and
parallel planes. The concept of plane motion enables us to consider only one of the
parallel planes and analyse the motion of the points lying in that plane.
The parallel planes may not appear to be identical in shape and size but there is no
difficulty because any plane of the body can be hypothetically extended by a massless
extension of the rigid body for the purpose of kinematic analysis.
The mechanisms use links. Consider a link PQ which is shown displaced in Figure 5.5 to
position P′ Q′ in a general plane motion. The motion may be considered to comprise
translation of an arbitrary point on the link plus a rotation about an axis perpendicular to
the plane and passing through that point. In an example in Figure 5.5 eight combinations
are equivalent.
P P′
θ
Q′
Q Q1
Figure 5.5(a) : Translation from PQ to P′Q1 and Rotation about P′
P P1
P′
θ
Q Q′
Figure 5.5(b) : Translation from PQ to P1 Q′ and Rotation about Q′
P1
P
θ P′
26
C C′
7. Analysis by
Analytical Methods
Figure 5.5(c) : Translation from PQ to P1 Q1 and Rotation about C′
P
θ P′
O O′
θ
Q′
Q Q1
Figure 5.5(d) : Translation from PQ to P1 Q1 and Rotation about R′
P
P′
θ
Q1 Q′
Q
Figure 5.5(e) : Rotation from PQ to PQ1 and Translation to P′ Q′
P P1 P′
θ
Q′
Q
Figure 5.5(f) : Rotation from PQ to P1Q and Translation to P′ Q′
P P1 P′
θ
C C′
Q′
Q1 Q
Figure 5.5(g) : Rotation from PQ to Pi Q1 and Translation to about P′ Q′
P′
P1
P
θ
27
O′
O
8. Motion Analysis of
Planar Mechanism and
Synthesis
Figure 5.5(h) : Rotation from PQ to P1 Q1 and Translation to P′ Q′
The translation and rotation are commutative. In Figures 5.5(e) and (f) show equal
amount of translation and rotation but in Figures 5.5(a) and (b) translation is done first
and rotation later. Similarly, Figures 5.5(a) and (e), (c) and (g), (d) and (h) are
commutative.
It may also be noted that, whatever be the mode of combination, the amount of rotation is
same. The angular velocity of every point on the link is the same. It is, therefore, the
customary to use the term angular velocity of the link rather than of any particular point
on it.
The fact that a general plane motion can be thought of as a superpositon of translation
and rotation is a special case of Chasle’s theorem. The theorem, in general, states that any
general motion of a rigid body can be considered as an appropriate superposition of a
translational motion and a rotational motion.
SAQ 2
Determine velocity of top point of a rolling wheel if centre of the wheel is moving
with velocity ‘v’.
5.4 METHOD OF RELATIVE VELOCITY AND
ACCELERATION
For this purpose, a link of a general shape may be considered to start with. It is shown in
Figure 5.6. Let there by any arbitrary two points A and B moving with velocities VA
and VB. The velocity of B relative to A may be represented by VBA. Let relative velocity
B
VBA makes an angle θ with line joining A and B. VBA can be resolved into two components
VBA cos θ along AB and VBA sin θ perpendicular to AB. Since the link is rigid, distance AB
remains constant. Therefore, component of velocity of B relative to A along AB cannot
exist.
Therefore, VBA cos θ = 0 or cos θ = 0
π
or, θ=
2
This means a rigid link has rotation relative to point A, as well as translation along
velocity of A.
VBA
VB
B
aB
28
aB
B B
o′
b′
a′
9. Analysis by
Analytical Methods
Figure 5.6
Therefore, the direction of VBA is perpendicular to the line joining A and B. If VA is known
in magnitude and direction and for VB only direction is known, the magnitude can be
B
determined by drawing a polygon as stated below :
(a) First the velocity of A, i.e. VA is plotted by assuming a suitable scale from an
arbitrary selected point o which represents fixed reference. Let it be
represented by ‘oa’.
(b) Now, draw a line through o parallel to the direction of velocity of B, i.e. VB. B
(c) Next, draw a line through point ‘a’ which is parallel to the perpendicular to
the line AB which meets the direction of VB at point ‘b’.
B
In this velocity polygon oab, ‘ob’ represents velocity VB in magnitude and direction.
B
After drawing velocity polygon, we can proceed for determination of magnitude of
acceleration of point B. If direction of acceleration of B is known and acceleration of
point A is known in magnitude and direction. The vector equation can be written as
follows :
a B = a A + a BA
Since a BA = a BA + a BA ∴ a B = a A + a BA + a BA
t c c t
t c
Where aBA is acceleration of B relative to A; aBA is tangential component of aBA and aBA
is centripetal component of aBA.
2
VBA
c
aBA =
AB
t
It is directed from B to A along AB whereas aBA has direction perpendicular to AB.
Acceleration polygon can be drawn as follows :
(a) Plot acceleration of A in magnitude and direction by assuming suitable
scale. aA is represented by o′a′.
c
(b) Plot centripetal component a BA by drawing line parallel to AB and represent
c
its magnitude according to the vector equation. a BA is represented by a′a′1.
(c) ′
From a1 draw a line perpendicular to aa1 or parallel to the perpendicular to
t
AB to represent direction of aBA .
(d) Draw a line parallel to the direction of acceleration of B, i.e. aB from fixed
B
t
reference o′ to meet the line representing direction of a BA . They meet at
b′ . o′b′ represents acceleration of B in direction and magnitude.
Example 5.2
29
10. Motion Analysis of In a slider crank mechanism shown in Figure 5.7, the crank OA rotates at 600 rpm.
Planar Mechanism and Determine acceleration of slider B when crank is at 45o. The lengths of crank OA
Synthesis
and connecting rod AB are 7.5 cm and 30 cm respectively.
a
A
aB r
b′ o ω
o′ b
B
0
O 45
aA
a′1
a′
Figure 5.7
Solution
2π × 600
The angular velocity of crank OA, ω = = 6.28 rad/s . The crank OA is a
60
rotating body about fixed centre O. Therefore, the velocity of point A is given by
V A = ω × OA = 6.28 × 7.5 = 47.1 cm/s or 0.471 m/s
Select suitable position of pole o which represents fixed reference. Draw a line oa
perpendicular to OA to represented velocity of A, i.e. VA in magnitude and
direction. From point a, draw a line perpendicular to AB to represent direction of
relative velocity VBA. Now draw another line parallel to the motion of the slider B
from O to represent direction of the velocity of slider VB to meet another line
B
through a at b. Thus ob represents velocity of B in magnitude and direction.
Since crank OA rotates at uniform angular speed, therefore, acceleration of A will
be centripetal acceleration.
2
VA (0.471)2
aA = ac =
A = = 2.96 m/s 2
OA 7.5
100
It is directed from A towards O.
a B = a A + a BA = a A + a BA + a BA
c t
2
VBA
aBC =
c
along AB directed from B to A
AB
(3.425) 2
a BC =
c
= 39.17 m/s
0.3
Select a suitable position of pole O which represents fixed reference. Plot
acceleration of A, i.e. aA by drawing parallel to OA and representing its magnitude
by a suitable scale. This is represented by o′a′. Now, draw a line from ‘a′’ parallel
c
to AB for aBA and represent its magnitude. This is represented by a′a′1. Now from
t
a′1 draw a line perpendicular to a′a′1 or AB to represent direction of aBA . Draw a
line from o′ parallel to the motion of slider B to represent direction of motion of B.
This line from o′ will meet another line from a′1 at b′. The acceleration of slider B
is represented by o′b′ in magnitude and direction and it gives
aB = 215 m/s 2
Example 5.3
30
11. A four bar chain O2 AB O4 is shown in Figure 5.8. The point C is on link AB. The Analysis by
crank O2A rotates in clockwise sense with 100 rad/s and angular acceleration Analytical Methods
4400 rad/s2. The dimensions are shown in Figure 5.8(a). Determine acceleration of
point C and angular acceleration of link O4B.
O′2, O′4
A 3 B
28 mm C
80 mm 37 mm
4 i′,O′
O4
α = 4400 rad/sec
2
75 mm 2
ω = 100 rad/sec 1
0
53
O2 125 mm a′
1 b′ a′
(a) c′
b
b′
O2, O4 c
a b′
(b) (c)
Figure 5.8
Solution
Draw configuration diagram to the scale.
75
The velocity of point A, VA = O2 A × ω = × 100 = 7.5 m/s
1000
Velocity Diagram
For drawing velocity polygon the following steps may be followed :
(a) Assume suitable scale say 1 cm → 5 m/s.
(b) Plot velocity of A, VA perpendicular to O2A. It is represented by
o2a in velocity polygon Figure 5.8(b).
(c) Draw a line from point a perpendicular to AB to represent
direction of VBA.
(d) Draw a line from o2 perpendicular to O4B to represent direction
of VB to meet line representing direction of VBA at point b. o2b
B
represents VB in magnitude and direction.
B
(e) For determining velocity of C, plot point c on ab such that
AC
‘ac’ = × ab .
AB
From velocity polygon VBA = 6.5 m/s
and VB = 9 m/s
Acceleration Diagram
For drawing acceleration polygon, the following steps may be followed :
(a) Assume suitable scale depending on value of centripetal
acceleration of A which is a c = O2 A ω2 . Or
A
31
12. Motion Analysis of 75
Planar Mechanism and ac =
A × (100)2 = 750 m/s 2 . A scale 1 cm → 250 m/s2 may
Synthesis 1000
serve the purpose.
(b) Draw a line parallel to O2A and plot a c which will be
A
represented by 3 cm. It is represented by o2′ a1′ in polygon.
(c) Draw a line perpendicular to O2A from a′1 in the sense of
angular acceleration to represent tangential acceleration which
75
is atA = O2 A × α = × 4400 = 300 m/s 2 . It will be denoted
1000
by a′1 a′ in polygon. o′2 a′ can be joined to get acceleration of
A which is represented by o′2 a′ in magnitude and direction.
(d) Draw a line parallel to AB from a′ and plot centripetal
c
component of acceleration aBA which is given by
2
VBA (6.5)2
aBA =
c
= = 528.15 m/s 2 . Plot magnitude of aBA . It
c
AB 0.08
is represented by a′ b′1.
(e) Draw a line perpendicular to AB from b′1 to represent direction
t
of tangential component of acceleration aBA .
(f) Draw a line parallel to O4B from o′2 to represent centripetal
component of acceleration of B. The magnitude is given by
2
VB 92
aB =
c
= = 2189.189 m/s 2 . It will be denoted by
O4 B 0.037
8.76 cm and it is represented by o′2 b′2.
(g) Draw a line perpendicular to O4B or o′2 b′2 to represent
direction of tangential component of acceleration of B to meet
t
the line representing direction aBA at b′. Join a′b′ which
represents aBA.
Join o′2 to b′ to get acceleration of B, i.e. aB. B
(h) To determine acceleration of C, plot a point c′ on line a′b′ such
AC 28
that a′ c′ = × a ′ b′ = × 72 = 2.52 . Joint o′2 with c′ and
AB 80
o′2 c′ represent acceleration of C in magnitude and direction.
From acceleration polygon Figure 5.8(c), ac = 1400 m/s2.
(i) Angular acceleration of link O4B is given by
t
aB
αO4 B = = 3479.3 rad/s 2
O4 B
The acceleration polygon is shown in Figure 5.8(c).
Example 5.4
Determine acceleration of slider D in a combined four bar chain and slider crank
mechanism shown in Figure 5.9(a). The dimensions are shown in the figure. The
crank OA rotates at 240 rpm in counterclockwise sense.
d′1
c VD
aDB
d O,C
b′ t
aBA t
b′1 aDB
32 VDB
VB VA
aB
B B
t c
aBC aBA
a′
aA
B B
aDB
B B
13. Analysis by
Analytical Methods
(a) (b)
28 mm O
0
A 75
2
44 mm 3 65 mm
B
49 mm
C
D 11
(c)
Figure 5.9
Solution
Plot configuration diagram to the suitable scale.
Velocity Diagram
2π × 240
Velocity of A, VA = O2 A × = 0.7037 m/s
60
It is perpendicular to OA in sense of rotation.
(a) Plot velocity of A, VA by assuming a suitable scale say
1 cm → 0.2 m/s. It is perpendicular to OA and represented by
‘oa’ on velocity polygon.
(b) From a, draw a line perpendicular to AB to represent direction
of VBA.
(c) From o, draw a line perpendicular to BC to represent direction
of velocity of B. Extend it if necessary to meet the line
representing direction of relative velocity VBA. These two lines
join at b.
(d) From b, draw a line perpendicular to BD to represent direction
of VDB.
(e) From o, draw a line parallel to line of motion of slider D to
meet line representing direction of VDB at d.
(f) In velocity polygon od represents velocity of slider D
From velocity polygon VB = 0.5 m/s, VBA = 0.4 m/s,
B
VDB = 0.602 m/s
Velocity polygon is shown in Figure 5.9(b).
Acceleration Diagram
33
14. Motion Analysis of ⎛ 2π × 240 ⎞
Planar Mechanism and Acceleration of A, a A = a c = OA × ⎜
A ⎟ = 17.686 m/s Select a
2
Synthesis ⎝ 60 ⎠
suitable scale, say 1 cm → 5 m/s2.
(a) Draw a line parallel to OA to represent acceleration of A, aA. It
is represented by o′a′.
c
(b) Plot centripetal component of acceleration aBA, i.e. aBA by
drawing line parallel to AB.
2
VB
aBA =
c
= 3.636 m/s 2
AB
It is represented by a′ b′1.
(c) From b′1, draw a line perpendicular to AB to represent direction
of tangential component of acceleration aBA.
(d) From o′ draw a line o′ b′2 parallel to BC to represent centripetal
component of acceleration of B which has magnitude given by
2
VB
aB =
c
= 5.102 m/s 2
BC
(e) From b′2, draw a line perpendicular to BC or o′ b′2 to represent
t
direction of tangential component of acceleration of B, i.e. aB .
t
This line meets direction of aBA at b′.
(f) From b′, draw a line b′ d′1 parallel to BD to represent
c
centripetal component of acceleration of aDB, i.e. aDB .
2
VDB
aDB =
c
= 7.878 m/s 2
BD
(g) From d′1, draw a line perpendicular to b′ d′1 or BD to represent
direction of tangential component of acceleration of aDB, i.e.
t
aDB .
(h) From o′ draw a line parallel to the line of motion of slider D to
t
meet direction of aDB at d′.
o′ d′ represents acceleration of slider in magnitude and
direction. Magnitude of acceleration of slider
aD = 31.5 m/s 2
Example 5.5
Figure 5.10(a) shows Andreau Variable Stroke engine Mechanism in which
links 2 and 7 have pure rolling motion. The dimensions of various links are
indicated in the Figure 5.10. Determine acceleration of slider D if link 2 rotates at
1800 rpm.
Solution
Draw configuration diagram to the scale.
2π × 1800
Velocity of point A, VA = OA × = 3.581 m/s .
60
The velocity of point B will also be equal to VA.
Velocity Diagram
34
15. The vector equations are as follows : Analysis by
Analytical Methods
VC = V A + VCA
and VC = VB + VCB
VD = VC + VDC
C
0
30
76 mm 4 D 5
38 mm 0
15
62 mm
6
D
3
B
7
0
30 Q Q 2 A
38 mm dia
1 Wheel
Wheel
63 mm dia
(a)
d′1
C
c
a
b
a′
o′,q′
c′
b′
c′1
o,q
d d′
(b) (c)
Figure 5.10
Velocity polygon is shown in Figure 5.10(a).
(a) Assume suitable scale say 1 cm → 1 m/s.
(b) Assume suitable position of pole o, and plot VA and VB B
perpendicular to OA and QB respectively. They are represented
by oa and ob.
(c) Draw lines to represent directions of VCA and VCB
perpendicular to AC and BC from points a and b respectively to
meet at point c.
(d) Draw line to represent direction of VDC perpendicular to DC at
point c.
(e) Draw line parallel to motion of slider from o to represent
direction of motion of slider to meet direction of VDC at d.
VCA = 1.28 m/s
VCB = 1.3 m/s
VDC = 5.2 m/s
Acceleration Diagram
35
16. Motion Analysis of Vector equations are as follows :
Planar Mechanism and
Synthesis aC = a A + aCA + aCA
c t
Also aC = a B + aCB + aCB
c t
a D = aC + a DC + a DC
c t
Both the wheels are rotating at uniform angular speed
2
VA
∴ a A = ac =
A = 674.9 m/s 2
OA
2
VB (3.581)2
∴ aB = a B =
c
= = 407.1 m/s
OB 0.0315
2
VCA 1.282
aCA =
c
= = 26.425 m/s 2
AC 0.062
2
VCB 1.32
aCB =
c
= = 22.236 m/s 2
BC 0.076
Assuming scale for acceleration as 1 cm → 200 m/s2.
(a) Assuming suitable position for o′, plot aA and aB parallel to OA
B
and BQ respectively. They are represented in acceleration
polygon by o′ a′ and o′ b′ respectively.
c c
(b) From a′ and b′ plot aCA and aCB parallel to AC and BC. They
are represented by a c″2 and b′ c′1 respectively.
(c) Draw lines perpendicular to AC from c″2 and perpendicular to
t t
BC from c′1 to represent directions of aCA and aCB . These
lines meet at c′.
c
(d) Plot aDC by drawing line parallel to DC from c′. This is
represented by c′ d′1.
(e) Now draw a line from d′1 perpendicular to DC to represent
t
direction of aDC .
(f) From o′, draw a line parallel to motion of slider D to meet
t
direction of aDC at d′.
o′ d′ represents acceleration of slider D in magnitude and
direction.
Acceleration of slider = 1320 m/s2.
5.5 ALTERNATIVE METHOD OF DETERMINING
CORIOLIS’ COMPONENT OF ACCELERATION
In rigid body rotation, distance of a point from the axis of rotation remains fixed. If a link
rotates about a fixed centre and ,at the same time, a point moves over it along the link,
the absolute acceleration of the point is given by the vector sum of
(a) the absolute acceleration of the coincident point relative to which the point,
under consideration, is moving;
(b) acceleration of the point relative to the coincident point; and
36
17. (c) Coriolis’ component of acceleration. Analysis by
Analytical Methods
Figure 5.11 shows a link 2 rotating at constant angular speed say ω2, it moves from
position OC to OC1. During this interval of time, the slider link 3 moves outwards from
position B to B2 with constant velocity VBA (A is point on link 2 shich coincides with B
B
which is on link 3). The motion of the slider 3 from B to B2 may be considered in the
following three stages :
(a) B to A1 due to rotation of link 2,
(b) A1 to B1 due to outward velocity VBA,
B
(c) B1 to B2 due to acceleration perpendicular to the link which is the Coriolis’
B B
component of acceleration.
Arc B1 B2 = Arc DB2 − Arc DB1
= Arc DB2 − Arc AA1
Also Arc B1 B2 = A1 B1 δ θ = VBA δ t ω2 δ t
= VBA ω2 (δ t ) 2
The tangential component of the velocity perpendicular to the link is say Vt
and it is given by
Vt = r ω
In this case ω has been assumed constant and slider moves with constant velocity.
Therefore, tangential velocity of point B on the slider 3 will result in uniform increase in
tangential velocity because of uniform increase in value of r in the above equation.
C
C1
2
B1
D
2ω2,VBA
B2
VBA
B on link 3 δθ ω2
VBA ω2
3
VBA
A1 2ω2,VBA
A on link 2 (b) (c)
δθ
VBA 2ω2,VBA
VBA ω2
ω2
VBA
ω2
2ω2,VBA VBA
O
(d) (e)
1
(a)
Figure 5.11
To result in uniform increase in value of Vt, there has to exist constant acceleration
perpendicular to link 2.
1
Therefore, B1 B2 = a (δ t ) 2 (From Unit 1)
2
where a is the acceleration. 37
18. Motion Analysis of 1
Planar Mechanism and ∴ a (δ t )2 = VBA ω2 (δ t )2
Synthesis 2
a = 2 VBA ω2
cor
This is Coriolis’ component of acceleration and will be denoted by aBA .
Therefore, aBA = 2 ω2 VBA
cor
The directional relationship of VBA and 2 ω2 VBA is shown in Figures 5.11(b), (c), (d) and
(e). If slider moves towards centre of rotation o, its velocity can be transmitted to other
side so that it is directed outwards.
The direction of Coriolis’ component of acceleration is given by the direction of the
relative velocity vector for the two coincident points rotated by 90o in the direction of the
angular velocity of the rotation of the link.
If the angular velocity ω2 and the velocity VBA are varying, they will not affect the
expression of the Coriolis’ component of acceleration but their instant values will be used
in determining the magnitude and this value will be applicable at that instant.
SAQ 3
What are the necessary and sufficient conditions for the Coriolis’ component of
acceleration to exist?
Example 5.6
A cam and follower mechanism is shown in Figure 5.12(a), the dotted line shows
the path of point B (on the follower). The cam rotates at 100 rad/s. Draw the
velocity and acceleration diagram for the mechanism and determine the linear
acceleration of the follower. Minimum radius of cam = 30 mm and maximum
lift = 35 mm.
t b′
aBA
b′1
4 aB
2ω,VBA
A on cam. path
B on folower o′
c
a AO
3 VBA
45 mm
a′
O 30
o
(b)
2 VBA
2ω2 b
1
VBA VB
VA
a o
(a) (c)
Figure 5.12
Solution
38
19. Linear velocity of point A which is on the cam, VA = ω × OA. Analysis by
Analytical Methods
45
or, VA = 100 × = 4.5 m/s
1000
VB = V A + VBA
Velocity Diagram
(a) Assume suitable scale say 1 cm → 1 m/s.
(b) Plot velocity of A by drawing a line perpendicular to OA from pole o.
It is represented by oa.
(c) From o, draw a line parallel to motion of follower to represent
direction of velocity of follower.
(d) Draw a line from ‘a’, parallel to the motion of B which is parallel to
the tangent at cam profile to meet line representing direction of
velocity of follower at b. ob represents velocity of follower in
magnitude and direction
VB = 1.75 m/s and VBA = 4.85 m/s
Velocity polygon is shown in Figure 5.12(b).
Acceleration Diagram
aB = a A + aBA + a cor = a A + aBA + aBA + a cor
c t
(4.5) 2
2
VA
aA = = 45 = 450 m/s 2
OA 1000
2
VBA
aBA =
c
=0
∞
a cor = 2 VBA × ω = 2 × 4.85 × 100 = 970 m/s 2
(a) Select suitable scale say 1 cm → 200 m/s.
(b) Plot aA by drawing a line parallel to OA from o′ and length equal to
2.25 cm.
(c) Now plot acor perpendicular to profile of cam and length equal to
4.85 cm. It is represented by a′ b′1.
t
(d) Now draw a line perpendicular to a′ b′1 to represent direction of aBA .
(e) Draw another line from o′ to represent direction of motion of B to
t
meet direction of aBA at b′.
o′ b′ represents acceleration of B in magnitude and direction.
aB = 3 × 200 = 600 m/s 2
The acceleration polygon is shown in Figure 5.12(c).
Example 5.7
A quick return motion mechanism is shown in Figure 5.13(a). The crank rotates at
20 rad/s. Determine angular acceleration of the slotted link 3.
Solution
The configuration diagram is drawn to the scale according to the given dimensions. 39
20. Motion Analysis of Let point B (on the link 3) coincides point A (on the crank 2). Velocity of point A
Planar Mechanism and is perpendicular to OA and
Synthesis
15
VA = OA × ω = × 20 = 3 m/s
100
V A = VB + V AB
O
1 15 cm
2
4 b
A On Link B On Link 2
2 and 4
35 cm O,C
3
25 cm (b)
a
b′′ C
a′
VAB
cr
a AB
aBA aA (c)
b′
aB
t o′c′
aBC
c
aBC
(a)
b′1
Figure 5.13
Velocity Diagram
(a) Assume suitable scale say 1 cm → 1 m/s.
(b) Plot velocity VA by drawing line from o perpendicular to OA. It is
represented by oa.
(c) Draw a line from o perpendicular to slotted link 3 to represent
direction of velocity VB. B
(d) From a, draw a line parallel to slotted link 3 to represent VAB which
represents movement of slider. This line meets direction of VB at b. B
Here ob represents velocity of B in magnitude and direction and ba
represents velocity of slider VAB in magnitude and direction.
VB = 1.5 m/s ; V AB = 2.6 m/s
The velocity polygon is shown in Figure 5.13(b).
Acceleration Diagram
a A = a B + a AB + a cor = a B + a B + a tAB + a c + a cor
c t
AB
40
21. Analysis by
V2 32
aA = ac
A = A = = 60 m/s 2 Analytical Methods
OA 0.15
2
VB 1.52 V2
aB =
c
= = 9 m/s 2 ; a c B = AB = 0
A
BC 0.25 ∞
VB 1.5
a cor = 2 VAB × = 2 × 2.6 × = 31.2 m/s 2
BC 0.25
(a) Assume suitable scale say 1 cm → 10 m/s2.
(b) Plot aA by drawing line parallel to OA from a suitable point o′. It is
represented by o′ a′.
(c) Plot acor (Coriolis, component of acceleration) as per direction
determined in Figure 5.13(c) and according to the vector equation. It
is represented by b″ a′. It is perpendicular to link 3.
(d) From b″, draw a line perpendicular to b″ a′, i.e. parallel to link 3 to
represent direction of motion of slider ( a tAB ) .
c
(e) Starting from o′, plot aB by drawing line parallel to link 3. It is
represented by o′ b′1.
(f) From b′1, draw a line perpendicular to o′ b′1, i.e. perpendicular to
t
link 3 to represent direction of aB and meet line representing
direction of acceleration a tAB at b′.
Here o′ b′ represents acceleration of B in magnitude and direction and
b′ b″ represents acceleration of slider in magnitude and direction.
Vector represented by b′1 b′ represents tangential component of
acceleration of B
aB = 83.5 m/s 2
t
∴ BC × α BC = 83.5
83.5
or Angular acceleration of link 3 (α BC ) =
0.25
or α BC = 334 rad/s 2
Example 5.8
Figure 5.14(a) shows Whitworth Quick Return Mechanism. The dimensions are
written in the Figure 5.14. Determine velocity and acceleration of slider D when
crank rotates at 120 rpm uniformly in the sense indicated in Figure 5.14.
Solution
2π N 2 π × 120
The velocity of A, VA = OA × = 0.2 × = 2.51 m/s .
60 60
Draw configuration diagram to the scale. The point B is on the slotted link which
coincides with point A on link OA.
Velocity Diagram
V A = VB + V AB
VB
VC = × QC
QB
and VD = VC + VDC 41
22. Motion Analysis of
Planar Mechanism and c
Synthesis C
50 cm
15 cm D VO
Q o,q d
10 cm 26 cm
VB
Q
20 cm
B
A
b
VAB
VQB= ba
a
(a) (b)
b′′
b′
b′1
o′,q′ d′
a AB = 2ωaB VAB
r
ωQB
d′1 c′
(c) (d)
Figure 5.14
(a) Plot VA by selecting proper scale say 1 cm → 0.5 m/s by a line
perpendicular to OA. It is represented by oa.
(b) From o, draw a line perpendicular to OB for direction of VB. B
(c) From a, draw a line parallel to QB to represent direction of VAB, i.e.
sliding velocity of slider to meet direction of velocity of B, i.e. VB at b. B
The vector ob represents VB in magnitude and direction.
B
2.4
VB = 2.4 m/s ∴ VC = × 0.15 = 1.38 m/s
0.26
(d) Plot VC perpendicular to QC from o. It is represented by oc.
(e) Draw a line parallel to motion of slider from o.
(f) Draw a line perpendicular to CD from c to meet direction of velocity
of slider at d.
The velocity of slider in magnitude and direction is given by od.
2.512
VD = 0.75 m/s, VAB = = 31.5 m/s 2
0.20
Acceleration Diagram
2
VA 2.512
Acceleration of A, a A = = = 31.5 m/s2
OA 0.20
42
23. Analysis by
a A = a B + a AB + a cor = a B + a B + a c + a tAB + a cor
c t
AB Analytical Methods
2
VB 2.42 V
aB =
c
= = 22.15 m/s 2 ; a c B = AB = 0
A
QB 0.26 ∞
VB 2.4
a cor = 2 VAB × = 2 × 0.9 × = 16.61 m/s 2
QB 0.26
Acceleration of B can be determined as explained in Example 5.7. Assume
suitable scale say 1 cm → 5 m/s2. Extend o′ b′ to the other side of o′ (C is on
other side of B) upto c′ such that
QC
o′ c′ = × o′ b′
QB
a D = aC + a DC = aC + a DC + a DC
c t
2
VDC 0.82
aDC =
c
= = 1.28 m/s 2
CD 0.5
c
(a) Draw a line parallel to CD and plot aDC . It is represented
by c′ d′1.
(b) Draw a line perpendicular to c′ d′1, i.e. perpendicular to CD to
t
represent direction of aDC .
(c) Draw a line from o′ parallel to motion of slider D to meet
t
direction of aDC at d′.
The acceleration of slider is represented by o′ d′ in magnitude
and direction
aD = 6.5 m/s 2
Example 5.9
A swivelling point mechanism is shown in Figure 5.15. If the crank OA rotates at
200 rpm, determine the acceleration of sliding of link DE in the trunion. The
following dimensions are known AB = 18 cm, DE = 10 cm, EF = 10 cm, AD = DB,
BC = 6 cm, OC = 15 cm, OA = 2.5 cm.
Solution
The configuration diagram is plotted to the scale say 1 cm → 4 cm.
Velocity Diagram
VB = V A + VBA ; VS = VD + VSD and VF = VE + VFE
The point S is on the link DE which slides in the swivel block.
2π N 2.5 2 π × 200
VA = × OA = × = 0.52 m/s
60 100 60
Velocity of S related to D, VSD is perpendicular to SD. Velocity of S related
to fixed block is parallel to link DE. Select a suitable scale say
1 cm → 0.2 m/s.
(a) Plot VA. It is represented by oa in velocity polygon
(Figure 5.15(b)).
(b) Draw a line perpendicular to BC to represent direction of VB B
from o.
(c) Draw a line perpendicular to AB from a to represent direction
of VBA to meet direction of VB at b.
B
43
24. Motion Analysis of AD
Planar Mechanism and (d) Plot point d on ab such that ad = ab .
Synthesis AB
DE
(e) Plot de such that de = ds .
DS
(f) From e, draw ef perpendicular EF to represent direction of VFE.
(g) Draw a line parallel to motion of slider to meet direction of VFE
at f.
VSD = VS = 0.32 m/s, VDE = 0.84
E F
S
Q B
12 cm
D
7.5 cm
A
8.5 cm
0
45 O C
(a)
s′′
O′,C′
a
e
s
b′
d b′′
d′
b s′
f
O, C a′
b′
(b) (c)
S′1
Figure 5.15
Acceleration Diagram
2
VA (0.52) 2
Acceleration of A, a A = = = 10.816 m/s 2
OA 2.5
100
a B = a A + a BA + a BA
c t
Also, aB = aB + aB
c t
2
VSD (0.32)2 V2 (0.44) 2
aSD =
C
= ; aBA = BA =
C
= 1.075 m/s 2
SD 0.04 AB 18
100
(0.48) 2
a cor = 2 ωDE × VS ; aB =
C
= 3.84 m/s 2
0.1
VDE (0.84) 2
=2 × VS = 2 × × 0.32
DE 0.1
aS = aSD + a cor = aSD + a SD + a cor
C t
Select suitable scale say 1 cm → 2 m/s2.
44