Equivalent
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The document discusses equivalent dynamical systems used to model the motion of rigid bodies acted on by external forces. It defines an equivalent dynamical system as one where: 1) the sum of the masses equals the total body mass, 2) the center of gravity of the masses matches the body's, and 3) the sum of the mass moments of inertia about the center of gravity equals the body's. It then gives an example of calculating the equivalent dynamical system for a connecting rod suspended and oscillating, determining the radius of gyration and placing one mass at the small end center and the other a distance l2 from the center of gravity.
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Study Unit Ill Engineerin M Part4 an1cs By And.docx
Study Unit
Ill Engineerin M
Part4
an1cs
By
Andrew Pytel, Ph.D.
Associate Professor, Engineering Mechanics
The Pennsylvania State University
When you complete this study unit, you'll be able to
• Calculate the mass moment of inertia
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translation
• Describe centripetal and centrifugal force
• Describe the forces that impact the rotation of a rigid body without translation
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friction forces
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iii
PRELIMINARY EXPLANATIONS PERTAINING TO KINETICS .
FORCE-MASS-ACCELERATION METHOD .....
Translation of Rigid Body
Rotation of Rigid Body without Translation
General Plane Motion of Rigid Body
23
WORK-ENERGY METHOD . . . . . . . . . . . . . . . . . . . . . . . . . 53
Application of Method for Translation
Other Applications of Work-Energy Method
IMPULSE-MOMENTUM METHOD . . . . . . .
Rectilinear Translation of Single Body
Collision of Two Bodies
PRACTICE PROBLEMS ANSWERS
EXAMINATION . . . . . . . . . .
. ........... 77
93
95
Engineering Nlechanics, Part 4
PRELIMINARY EXPLANATIONS PERTAINING
TO KINETICS
Scope of This Text
1
1 • In the preceding texts on engineering mechanics, we have discussed
separately the relations of forces in a system and the conditions of mo-
tion of bodies. In this text, we shall consider the relation between the
motion of a body and the force or forces acting on the body to produce
the motion. The basis for the relationship between motion and force
is Newton's second law of motion. However, there are three different
methods of applying this law. These are commonly called the force-
mass acceleration method, the work-energy method, and the impulse-
momentum method. Each method is most useful for solving certain
types of problems.
Statement of Newton's Second Law of Motion
2 • In Engineering Mechanics, Part 1, Newton's second law of motion was
stated as follows:
If a resultant force acts upon a particle, the particle will be accelerated
in the direction of the force. Furthermore, the magnitude of the accel-
eration will be directly proportional to the magnitude of the resultant
force and inversely proportional to the mass of the particle.
Newton's second law can be expressed mathematically by the following
equation:
F
a=k-
m
in which a = magnitude of the acceleration of a particle
k = a numerical factor
F = magnitude of the force acting upon the particle
m = mass of the particle
(1)
The mass of a particle is a measure of the exact amount of matter in
the particle. Any body is composed of a number of particles, and the
mass of a body is the sum of the masses of all the particles.
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This document provides an overview of modern physics concepts covered in a 1EE modern physics course, including special relativity. It discusses how classical physics concepts of space, time, and velocity were found to be inconsistent with experimental results. The document uses examples such as particle lifetime experiments and velocity addition to show how observations cannot be explained unless space and time are treated as relative and velocities are limited to the speed of light. The principles of relativity were developed by Einstein to resolve these inconsistencies and form the foundations of modern physics.
Kinetic energy and gravitational potential energy
1) The document discusses kinetic energy and gravitational potential energy. Kinetic energy is the energy of an object in motion and depends on the object's mass and speed. Gravitational potential energy is the energy contained in an object above the ground and depends on the object's mass and position.
2) The gravitational field can be compared to an electric field. Both are conservative fields that represent the gradient of potential energy. Calculations of fields and potentials can be adapted from electricity to gravity.
3) The document makes analogies between the Coulomb force law in electricity and the gravitational force law. Both are foundations for their respective fields and phenomena can be described similarly using mathematical formulas.
Small amplitude oscillations
This document summarizes classical dynamics and small amplitude oscillations. It discusses oscillatory motion near equilibrium positions and developing the theory using Lagrange's equations. Normal modes of coupled oscillating systems are explored, where the normal coordinates represent eigenvectors that oscillate at characteristic frequencies. The principles of superposition and matrix representations are used to analyze examples like two coupled pendulums and a system of two masses connected by three springs.
7. Springs In Series And Parallel K 1 K
1. The document discusses simple harmonic motion and elasticity. It provides 10 examples solving problems related to springs, pendulums, simple harmonic oscillators, and periodic motion.
2. Key concepts covered include springs in series and parallel, reduced mass of coupled oscillators, conservation of mechanical energy, and using restoring force to determine angular frequency and time period of oscillations.
3. Various examples solve for time periods of physical pendulums, springs, particles oscillating due to gravity, and coupled oscillators. Restoring force and energy methods are applied.
ALL THE LAGRANGIAN RELATIVE EQUILIBRIA OF THE CURVED 3-BODY PROBLEM HAVE EQUA...
We consider the 3-body problem in 3-dimensional spaces of nonzero
constant Gaussian curvature and study the relationship between the masses of
the Lagrangian relative equilibria, which are orbits that form a rigidly rotating
equilateral triangle at all times. There are three classes of Lagrangian relative equilibria in 3-dimensional spaces of constant nonzero curvature: positive
elliptic and positive elliptic-elliptic, on 3-spheres, and negative elliptic, on hyperbolic 3-spheres. We prove that all these Lagrangian relative equilibria exist
only for equal values of the masses.
Planetary Motion- The simple Physics Behind the heavenly bodies
The document summarizes Kepler's laws of planetary motion and Newton's law of universal gravitation. It explains how Newton was able to derive Kepler's laws from his law of gravitation. It then discusses the central force problem and how to solve it to obtain planetary orbits. Finally, it briefly touches on limitations of the two-body approximation and possibilities like Lagrange points when more than two bodies are involved.
Engineering mechanics system of coplanar forces by
Engineering Mechanics covers various topics in mechanics including:
1. Relativistic mechanics deals with mechanics compatible with special and general relativity.
2. Quantum mechanics provides a mathematical description of energy, matter, and their interactions at nanoscopic scales.
3. Mechanics of deformable bodies deals with how forces are distributed in solid bodies and the resulting stresses and deformations.
Law Of Gravitation PPT For All The Students | With Modern Animations and Info...
Law Of Gravitation PPT For All The Students | With Modern Animations and Infographics
All the Students od Class 1,2,3,4,5,6,7,8,9,10,11,12 and all the students of engineering, medical, CBSE, GSEB, U.P from beginner to Top and high level can get used. All The informtion are gathered to help you all the people.
All colleges and School students can use it.
All the people can reuse it by downloading by giving credits.
Copyright @ Jay Butani 2019
DISCLAIMER :- ALL THE INFORMARION ARE NOT EXACT OR 100% CORRECT THERE MAY BE MISTAKE. WE ARE NOT RESPONSIBLE OVER THAT.
1234567890-Chapter 11b_Dynamic Force Analysis.pptx
1. The document discusses concepts related to inertia forces generated in moving mechanisms due to linear and angular accelerations. It introduces concepts of mass moment of inertia and centroid.
2. Mass moment of inertia measures an object's resistance to changes in rotational motion and depends on its mass distribution. It is calculated through integration of mass elements and use of parallel axis theorem.
3. The document provides equations to calculate mass moment of inertia for various regular shapes, and discusses radius of gyration as an alternative measure related to mass moment of inertia.
PHYSICS CLASS XI Chapter 5 - gravitation
Here are the key steps to calculate the mass of Earth (M) from the given data:
1) Acceleration due to gravity on Earth's surface (g) = 9.81 m/s^2
2) Universal gravitational constant (G) = 6.67x10^-11 Nm^2/kg^2
3) Radius of Earth (R) = 6.37x10^6 m
4) Using the formula for acceleration due to gravity:
g = GM/R^2
5) Rearranging the terms:
M = gR^2/G
6) Substituting the values:
M = (9.81 m/s^2)(
chapter5-gravitationppt-copy-211229151431 (2).pdf
Here are the key steps to calculate the mass of Earth (M) from the given data:
1) Acceleration due to gravity on Earth's surface (g) = 9.81 m/s^2
2) Universal gravitational constant (G) = 6.67x10^-11 Nm^2/kg^2
3) Radius of Earth (R) = 6.37x10^6 m
4) Using the formula for acceleration due to gravity:
g = GM/R^2
5) Rearranging the terms:
M = gR^2/G
6) Substituting the values:
M = (9.81 m/s^2)(
Simple Harmonic Motion
This document discusses simple harmonic motion (SHM) and related concepts like angular velocity, restoring forces, displacement, velocity, acceleration, energy, resonance, and damping. It provides equations for angular velocity, SHM acceleration, displacement, velocity, energy, and the simple pendulum. Examples are given and questions provided for practice calculations involving these equations and concepts.
Ch03 8
The motion of a mass attached to a spring can be modeled by a second order differential equation. For undamped free vibrations with no external force, the solution is a simple harmonic motion with natural frequency ω0 and period T. Small damping causes the motion to be damped oscillations with slightly reduced quasi-frequency μ and increased quasi-period Td. The ratio γ2/4km determines whether damping can be neglected - if small, damping has little effect on frequency and period.
Physics notes revision
This document provides notes on various physics concepts including:
- Units of measurement for length, time, speed, velocity, and acceleration.
- Concepts of mass vs weight, density, forces, Hooke's law, circular motion, moments, and center of mass.
- Forms of energy, work, power, and conservation of energy.
- Renewable and non-renewable energy resources.
The document covers essential high school physics concepts in a concise yet comprehensive manner.
Physics notes
1. The document provides notes on various topics in physics including thermal physics, mechanics, and properties of matter.
2. Key concepts discussed include states of matter, temperature and pressure of gases, work, energy, density, forces, and heat transfer mechanisms.
3. Equations for important quantities like kinetic energy, pressure, density and motion are defined.
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ALL THE LAGRANGIAN RELATIVE EQUILIBRIA OF THE CURVED 3-BODY PROBLEM HAVE EQUA...
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Design of Temporary and Permanent Joints
This document discusses different types of temporary and permanent joints used in structures, including bolted joints, knuckle joints, cotter joints, welded joints, and riveted joints. It specifically focuses on eccentrically loaded riveted joints and provides examples of calculating the size of rivets needed based on the load and permissible stresses.
Balancing of reciprocating masses
The document discusses balancing of reciprocating masses in engines. It describes:
1. The various forces acting on reciprocating parts and how the inertia force is balanced by an opposing force on the crankshaft, leaving an unbalanced force.
2. Methods to partially balance the primary unbalanced force using a balancing mass on the crank, which changes the direction of the maximum unbalanced force.
3. How balancing is applied to two-cylinder locomotives, reducing variation in tractive force, swaying couple, and hammer blow.
Balancing of rotating masses
This document discusses balancing rotating masses on a shaft. It provides examples of balancing a single mass, balancing multiple masses in the same plane, and balancing masses in different planes. It then gives two example problems:
1) A shaft carries four masses rotating at different radii and angles, and asks to find the balancing mass and its position if placed at a given radius.
2) A shaft carries four masses at different radii and angles across three planes, and asks to find the magnitudes and positions of balancing masses placed at a given radius in two other planes.
Design of fasteners and welded joints
The document contains several examples of calculating weld sizes and bolt sizes for different welded and bolted joints. The examples involve calculating shear stresses in welds due to direct loads and bending moments and determining necessary weld sizes. They also involve calculating bolt sizes for different bolted joints based on allowable shear stresses and safety factors. The final example calculates the necessary bolt size for connecting a steam engine cylinder head based on the steam pressure, cylinder dimensions, bolt material properties, and a preload on the bolts.
Design of springs and levers
The document discusses problems related to the design of helical springs, leaf springs, and coned disc springs. It provides examples of spring design problems involving the calculation of spring dimensions, loads, stresses, and deflections given various input parameters such as material properties, load ranges, spring indices, and dimensional constraints. The problems cover topics such as determining wire diameter, number of coils, spring rates, thicknesses of leaf springs, and stresses induced in coned disc springs. Solutions to the problems are presented using relevant spring design equations.
Design of couplings
The document discusses the design of various types of rigid and flexible couplings. It provides steps to design a flange coupling connecting two shafts transmitting 37.5 kW power at 180 rpm. Key details include calculating torque from power, selecting shaft diameter, coupling dimensions based on standards, and checking design of key and bolts for shearing and crushing. The document also provides problems and solutions for designing flange, muff, and clamp couplings for given power and speed conditions.
Design of bearings and flywheel
Here are the steps to design and draw a flywheel for a four stroke four cylinder 133 kW engine running at 375 rpm with a diameter not exceeding 1 m:
Given:
Power of engine, P = 133 kW = 133000 W
Number of cylinders, n = 4
Speed of engine, N = 375 rpm
Maximum diameter, Dmax = 1 m
Step 1) Calculate the mean effective pressure (p):
p = (2*P)/(n*π*D^2*N)
p = (2*133000)/(4*π*(0.5)^2*375) = 7 bar
Step 2) Calculate the mass moment of inertia (I) required:
I
Forced vibrations unit 4
1. The document discusses forced vibrations of mechanical systems subjected to periodic external forces such as harmonic, stepped, or periodic disturbances. It provides examples and discusses the amplitude of forced vibrations.
2. Vibration isolation techniques are introduced to minimize the transmission of vibrations from machines to foundations using springs and dampers. The transmissibility ratio, which is the ratio of transmitted force to applied force, is defined.
3. Several examples are provided to calculate the natural frequency, stiffness, amplitude of vibrations, damping coefficient, and transmissibility ratio of systems undergoing forced vibrations. Resonance conditions and their effects are also considered.
Governors
Here are the key steps to solve this problem:
1) Calculate the centrifugal forces at the minimum and maximum radii using Fc = mω2r
2) Use the lever equation to relate the centrifugal forces to the spring forces:
Fc1/S1 = r1/x
Fc2/S2 = r2/x
3) The initial compression of the spring is r2 - r1
4) Use the relation between spring force and compression to find the spring constant:
ΔS/Δx = s
Where ΔS is the change in spring force (S2 - S1) and Δx is the change in compression (r2 - r1
Gyroscopic couple
1. The document discusses gyroscopic couple, which acts on a spinning object that is rotating about another axis.
2. It provides examples of gyroscopic couple in naval ships, where the spinning of propeller shafts affects steering, pitching, and rolling.
3. The document also examines the gyroscopic couple and centrifugal couple in vehicles like cars and motorcycles taking turns, and how this affects their stability.
flywheel
A punching press punches 38mm holes in 32mm thick plates, requiring 7 N-m of energy per square mm of sheared area. It punches one hole every 10 seconds. The mean flywheel speed is 25 m/s.
To calculate the motor power required, the energy per hole is calculated based on the sheared area of the hole. The mass of the flywheel required to limit speed fluctuations to 3% of the mean is then calculated using the energy variation and flywheel properties.
Unit 3 free vibration
This document covers free and damped vibrations. It defines key terms like natural frequency, damping, and damping ratio. It describes the equations of motion for an undamped single degree of freedom system and how to calculate the natural frequency. It also covers calculating the natural frequency of damped systems and defines types of damping like overdamped, underdamped, and critically damped systems. Formulas are provided for damped vibration frequency, logarithmic decrement, and damping ratio. Examples are given on calculating natural frequency, damping coefficient, and damping ratio from data provided on an oscillating system.
Transverse vibrations
This document provides instructions on how to calculate the natural frequency of vibration of a shaft that is carrying multiple loads at different points. It describes calculating: 1) the moment of inertia of the shaft, 2) the static deflection caused by each point load individually, 3) the total static deflection by summing the individual deflections, and 4) the natural frequency of transverse vibration of the shaft based on the total deflection. It then gives two example problems involving shafts of different sizes, materials, and load configurations to calculate the natural frequency.
Torsional vibrations
This document discusses torsional vibrations in shafts. It provides equations to calculate the natural frequency of torsional vibrations based on the shaft's torsional stiffness, mass moment of inertia, and material properties. As an example, it calculates the natural frequency of a flywheel mounted on a vertical shaft. It then discusses multi-rotor shaft systems and how to determine the location of nodes. Finally, it provides methods to calculate the natural frequency and node locations of stepped shafts with varying diameters connecting multiple flywheels.
Free vibrations
The document provides information on vibrations including definitions of key terms like natural frequency, forced vibrations, and damped vibrations. It also discusses single degree of freedom systems and how to calculate the natural frequency of longitudinal and transverse vibrations in beams and shafts using different methods. An example is provided to show how to calculate the natural frequencies of longitudinal and transverse vibrations for a cantilever shaft with a mass at the free end. Formulas are given for stiffness, static deflection, and natural frequency of beams and shafts.
Damped vibrations
The document discusses different types of damping in vibrating systems, including viscous, Coulomb, and critical damping. It provides equations to calculate damping coefficient, logarithmic decrement, damping ratio, natural frequency, and damped vibration frequency. Examples are given to show how to determine damping coefficient, critical damping coefficient, damping factor, logarithmic decrement, and ratio of damped to undamped frequencies based on given mass, spring constant, amplitude decay between cycles.
Three rotor system
The document describes a three rotor system with rotors A, B, and C connected by a shaft. Rotor A has an inertia of 0.15 kg-m2, rotor B has an inertia of 0.30 kg-m2, and rotor C has an inertia of 0.09 kg-m2. The system can vibrate with nodes forming between rotors C and A or between rotors C and B depending on the direction of rotation. The task is to find the natural frequency of the torsional vibrations using the given inertias, shaft dimensions, and modulus of rigidity.
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This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
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本公司拥有海外各大学样板无数，能完美还原。
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Equivalent
- 3. • In order to determine the motion of a rigid body, under the action of external forces, it is usually convenient to replace the rigid body by two masses placed at a fixed distance apart, in such a way that, 1. the sum of their masses is equal to the total mass of the body ; 2. the centre of gravity of the two masses coincides with that of the body 3. the sum of mass moment of inertia of the masses about their centre of gravity is equal to the mass moment of inertia of the body. • When these three conditions are satisfied, then it is said to be an equivalent dynamical system. Consider a rigid body, having its centre of gravity at G
- 4. Let m = Mass of the body, KG = Radius of gyration about its centre of gravity G, m1 and m2 = Two masses which form a dynamical equivalent system, l1 = Distance of mass m1 from G, l2 = Distance of mass m2 from G,
- 7. A connecting rod is suspended from a point 25 mm above the centre of small end, and 650 mm above its centre of gravity, its mass being 37.5 kg. When permitted to oscil- late, the time period is found to be 1.87 seconds. Find the dynamical equivalent system constituted of two masses, one of which is located at the small end centre. Solution. Given : h = 650 mm = 0.65 m ; l1 = 650–25 = 625 mm = 0.625 m ; m = 37.5 kg ; tp = 1.87 s First of all, let us find the radius of gyration (kG) of the connecting rod (considering it is a compound pendulum), about an axis passing through its centre of gravity, G. We know that for a compound pendulum, time period of oscillation (tp),
- 8. It is given that one of the masses is located at the small end centre. Let the other mass is located at a distance l2 from the centre of gravity G, as shown in Fig. 15.19. We know that, for a dynamicallyequivalentsystem, l1.l2 = (kG)2