Fem l1(a)
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The document provides an overview of the finite element method (FEM) including its history, basic steps, and direct formulation approach. It describes how FEM was developed in the 1950s for aerospace applications and became widely used after the 1960s. The key steps are discretizing the domain, deriving element equations, assembling to obtain whole problem equations, applying boundary conditions, solving, and post-processing results. Direct formulation relates applied forces to displacements using element stiffness matrices which are assembled into a global matrix to solve for unknown displacements. Examples show applying this to simple structural problems.
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11 - 3 Experiment 11 Simple Harmonic Motio.docx
11 - 3
Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
equations? Again, how do you compare two numbers, which have errors?
Note: This week all students must write a very brief lab report during the lab period. It is
due at the end of the period. The explanation of the equations used, the introduction and the
conclusion are not necessary this week. The discussion section can be as little as three sentences
commenting on whether the two measurements of the spring constant are equivalent given the
propagated errors. This mini-lab report will be graded out of 50 points
Concept
When an object (of mass m) is suspended from the end of a spring, the spring will stretch
a distance x and the mass will come to equilibrium when the tension F in the spring balances the
weight of the body, when F = - kx = mg. This is known as Hooke's Law. k is the force constant
of the spring, and its units are Newtons / meter. This is the basis for Part 1.
In Part 2 the object hanging from the spring is allowed to oscillate after being displaced
down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
for the acceleration of the mass:
Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
negligible, the mass will continue to oscillate with amplitude A. Now, does this guess actually
solve the (differential) equation? A second time-derivative gives:
Comparing this equation to the original differential equation, the correct solution was
chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
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11 - 3 Experiment 11 Simple Harmonic Motio.docx
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Experiment 11
Simple Harmonic Motion
Questions
How are swinging pendulums and masses on springs related? Why are these types of
problems so important in Physics? What is a spring’s force constant and how can you measure
it? What is linear regression? How do you use graphs to ascertain physical meaning from
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Note: This week all students must write a very brief lab report during the lab period. It is
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When an object (of mass m) is suspended from the end of a spring, the spring will stretch
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of the spring, and its units are Newtons / meter. This is the basis for Part 1.
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down from its equilibrium position a distance -x. In this situation, Newton's Second Law gives
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Fnet = m a or
The force of gravity can be omitted from this analysis because it only serves to move the
equilibrium position and doesn’t affect the oscillations. Acceleration is the second time-
derivative of x, so this last equation is a differential equation.
To solve: we make an educated guess:
Here A and w are constants yet to be determined. At t = 0 this solution gives x(t=0) = A,
which indicates that A is the initial distance the spring stretches before it oscillates. If friction is
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solve the (differential) equation? A second time-derivative gives:
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chosen if w2 = k / m. To understand w, consider the first derivative of the solution:
−kx = ma
a = −
k
m
⎛
⎝
⎜⎜⎜⎜
⎞
⎠
⎟⎟⎟⎟
x
d 2x
dt 2
= −
k
m
x x(t) = A cos(ωt)
d 2x(t)
dt 2
= −Aω2 cos(ωt) = −ω2x(t)
James Gering
Florida Institute of Technology
11 - 4
Integrating gives
We assume the object completes one oscillation in a certain period of time, T. This helps
set the limits of integration. Initially, we pull the object a distance A from equilibrium and
release it. So at t = 0 and x = A. (one.
Lec5 total potential_energy_method
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chapter1and2.ppt
1. Statics deals with bodies at rest or in uniform motion, focusing on force and equilibrium. It is a branch of mechanics.
2. A particle is considered to have mass but negligible size, allowing simplified analysis of forces acting at a single point. Forces on a particle can be determined and resolved using vector operations like addition, subtraction and resolution into rectangular components.
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The document discusses the derivation of the stiffness matrix for a bar element in finite element analysis. It begins by outlining the learning objectives, which include deriving the stiffness matrix for a bar element and demonstrating how to solve truss problems using the direct stiffness method. It then describes the derivation process, which involves defining displacement functions, strain-displacement relationships, stress-strain relationships, and assembling the element equations and applying boundary conditions to solve for displacements and forces. An example problem is provided to demonstrate applying the method to a three-bar truss system. Guidelines for selecting appropriate displacement functions are also outlined.
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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.
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Fem l1(a)
- 1. FINITE ELEMENT METHOD ( BDA 4033 ) Lecture #01
- 17. Direct Formulation (Contd..)- Stiffness Matrix Let us write the stiffness matrix for the first element with CSA, A 1 as and For the second element with CSA, A 2 as where k 1 =A 1 E 1 /L 1 and k 2 =A 2 E 2 /L 2 Then the force equation can be written as for element 1 and for element 2 F 1 2 1 2 3
- 18. Direct Formulation (Contd..)- Assembly Physical Assembly: Step 0: Nodes Matrix assembly : Global Matrix Order = No. of nodes X No. of displacements= 3 X 3 1 2 3
- 19. Direct Formulation (Contd..)- Assembly Step 1: Physical Assembly: Matrix assembly : Step 2: Matrix assembly : Physical Assembly: 1 2 3 1 1 2 3 1 2
- 20. Direct Formulation (Contd..)- Apply BCs In the given beam, u 1 =F 1 =F 2 =0 Substituting the values in the assembled equation Solution: Solving the above equation, the unknown displacements u 2 and u 3 can be found. F 1 2 1 2 3
- 21. Direct Formulation (Contd..)- Post Processing From the previous solution, the unknown displacements u 2 and u3 a re found. To find the element stresses: For the element 1, the strain ε 1 = ( u 2 - u 1 )/L 1 Applying Hooke’s law, 1 =E 1 ε 1 =E 1 (u 2 - u 1 )/L 1 Similarly for the element 2, 2 =E 2 ε 2 =E 2 (u 3 - u 2 )/L 2
- 25. Questions ?