This document discusses single degree of freedom systems undergoing free vibration. It defines free vibration as oscillation under an initial disturbance with no external forces acting afterwards. It describes degree of freedom as the number of independent coordinates needed to define a system's configuration. For a single degree of freedom system, only one coordinate is required. The document also discusses natural frequency, equivalent spring systems, energy methods, phase plane analysis, and Newton's method for analyzing single degree of freedom vibratory systems.
it is understanding about the undamped free vibration and its equation and function.........its very usefull for vibration system.......
its gives derivation of undamped free vibration.....
this this slideshare presentation we discussed about difference of vibration system for forced damping here this is having with simple definition and for dynamic of machinery without equation and simple method.
it is understanding about the undamped free vibration and its equation and function.........its very usefull for vibration system.......
its gives derivation of undamped free vibration.....
this this slideshare presentation we discussed about difference of vibration system for forced damping here this is having with simple definition and for dynamic of machinery without equation and simple method.
this slide deals with the basic concepts related to mechanical vibrations for more information you can go through any mechanical vibration book available for engineering students
Vibration? Do you know what is vibration and how it is important and unimportant for us. What are the different types of vibration,where vibration effects are desirable and where not. What are the useful effects of the vibration and how it can make useful. How vibration can be eliminated or reduced to some extent. Different terminology or fundamentals of vibration is discussed briefly. So find the easy and simple description about vibration.
this slide deals with the basic concepts related to mechanical vibrations for more information you can go through any mechanical vibration book available for engineering students
Vibration? Do you know what is vibration and how it is important and unimportant for us. What are the different types of vibration,where vibration effects are desirable and where not. What are the useful effects of the vibration and how it can make useful. How vibration can be eliminated or reduced to some extent. Different terminology or fundamentals of vibration is discussed briefly. So find the easy and simple description about vibration.
Introduction to Classical Mechanics:
UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
UNIT-II : Constraints; Definition, Types, cause & effects, Need, Justification for realizing constraints on the system
LECTURE 1 PHY5521 Classical Mechanics Honour to Masters LevelDavidTinarwo1
Classical mechanics, a well-organized introductory lecture. This is easy to follow, and a must-go-through lecture. UNIT-I : Elementary survey of Classical Mechanics: Newtonian mechanics for single particle and system of particles, Types of the forces and the single particle system examples, Limitation of Newton’s program, conservation laws viz Linear momentum, Angular Momentum & Total Energy, work-energy theorem; open systems (with variable mass). Principle of Virtual work, D’Alembert’s principle’ applications.
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Theory of Vibrations: Introduction to the theory of vibrations in multi-degree-of-freedom systems, Normal modes and modal analysis, Nonlinear oscillations and chaos theory.
Canonical Transformations: Properties and classification of canonical transformations, Action-angle variables and their applications in integrable systems, Canonical perturbation theory and perturbation methods.
Poisson's and Lagrange's Brackets: Definitions and properties of Poisson's brackets, Relationship between Poisson's brackets and Hamilton's equations, Lagrange's brackets and their applications in dynamics. UNIT-III : Cyclic coordinates, Integrals of the motion, Concepts of symmetry, homogeneity and isotropy, Invariance under Galilean transformations Hamilton’s equation of motion: Legendre’s dual transformation, Principle of least action; derivation of equations of motion; variation and end points; Hamilton’s principle and characteristic functions; Hamilton-Jacobi equation.
UNIT-IV : Central force fields: Definition and properties, Two-body central force problem, gravitational and electrostatic potentials in central force fields, closure and stability of circular orbits; general analysis of orbits; Kepler’s laws and equation, Classification of orbits, orbital dynamics and celestial mechanics, differential equation of orbit, Virial Theorem.
UNIT-V : Canonical transformation; generating functions; Properties; group property; examples; infinitesimal generators; Poisson bracket; Poisson theorems; angular momentum PBs; Transition from discrete to continuous system, small oscillations (longitudinal oscillations in elastic rod); normal modes and coordinates.
A curious fact on Lorentz transformations for the intrinsic orbital momentum (the orbital momentum measured from its own center of mass and center of innertia).
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Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
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More examples of our work https://www.r-r-consult.dk/en/cases-en/
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1. Single degree of freedom system-
Free vibration
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2. Introduction
A system is said to undergo free vibration when it
oscillates only under an initial disturbance with no
external forces acting after the initial disturbance
3. Degree of Freedom
The number of degrees of freedom of a vibratory system is
the number of independent spatial coordinates necessary
to define its configuration. A is defined as the geometric
location of all the masses of the system. If the inter-
relationship of the masses is such that only one spatial
coordinate is required to define the configuration, the
system is said to possess one degree of freedom. A rigid
body in space requires six coordinates for its complete
identification, namely, three coordinates to define the
rectilinear positions and three to define the angular
rotations. Ordinarily, however, the masses in a system are
constrained to move only in a certain manner. Thus, the
constraints limit the of freedom to a much smaller number.
4. Frequency :-
No. of cycles per unit time or time taken to complete
one cycle.
Natural frequency :-
The frequency of free vibration when no external force
acts on the system after giving it an initial displacement
and body vibrates . These vibrations are called free
vibration and their frequency is called natural
frequency. Units :- Radian/sec or Hertz.
5. Equivalent systems
While we have discussed so far the vibration behavior of a
spring-mass system, in many practical situations we don't
readily find such simple spring-mass systems. Many a time, we
may find several springs and masses vibrating together and
then we will have several second order differential equations
to be solved simultaneously. In some special situations
however, we will be able to simplify the system by considering
equivalent stiffness and inertia. We may then still be able to
model the system as a simple single d.o.f spring-mass case.
When multiple springs are used in an application, they are
mainly found in two basic combinations.
• Series Combination
• Parallel Combination
6. Series Combination
•A typical spring mass system having springs in series
combination is shown above. The two springs can be
replaced by a equivalent spring having equivalent
stiffness equal to k. When springs are in series, they
experience the same force but under go different
deflections.
•For the two systems to be equivalent, the total static
deflection of the original and the equivalent system
must be the same.
7. Parallel combination
For the springs in parallel combination, the equivalent spring stiffness can
be found out as:
Each of the individual spring supports part of the load attached to it but
both the springs undergo same deflection.
Therefore the static deflection of the mass is,
Therefore if the springs are in parallel combination,
the equivalent spring stiffness is sum of individual
stiffnesses of each spring.
8. Newton’s method
Spring mass system in vertical position
Consider a spring mass system constrained to move in a
rectilinear manner along the axis of spring. Spring of constant
stiffness k which is fixed at one end carries a mass m at it’s
free end. The body is displaced from it’s equilibrium position
vertically downwards
9.
10.
11.
12.
13. Energy method
Energy methods are the methods which are based
on the conservation of energy. Assume the system
to be a conservative one. In a conservative system,
the total energy is constant. In a vibratory system
the energy is partly potential and partly kinetic.
The kinetic energy is because of velocity of mass
and potential energy is stored in the spring
because of it’s elastic deformation. According to
conservation law of energy, we know
14.
15. Phase Plane method
The vibratory motion of a spring mass system with initial
conditions and has been obtained earlier give reference to
that section and is reproduced here
_____ (1)
We depict the vibratory motion in the form of a chart showing
displacement, x vs time, t. While this is one common way of
plotting the vibration response, we will now discuss another
very useful method of depicting the response viz., the
phase-plane plot.
16. Radius of the circle is the amplitude of oscillations and centre is at the origin.
17. Time is implicit in this plot and from this diagram, displacement
and velocity of motion are available from single point which
corresponds to a particular time instant. This is called the
phase-plane plot. The horizontal projection of the phase
trajectory on a time base gives the displacement-time plot of
the motion and similarly the vertical projection on time base
gives velocity-time plot of the motion.
The starting point (with finite displacement and velocity at time
t=0) is marked. After seconds, we reach where radians. There
are many other interesting forms of graphical representation of
dynamic response of a system. Since it is an undamped system,
when started with some initial conditions, if continues to move
forever. Staring point P1 is reached after every cycle (time
period).
If the system is damped, then the mass gradually dissipates
away energy and comes to rest.