min-291 chapter 2 (engineering analsysis)(1)
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Chapter 2
ENGINEERING ANALYSIS
Introduction Analysis is the breaking down of an object into its basic
elements to get to its essence.
Studying the nature and identifying its essential features and their relationships.
Tools of analysis are based upon logic and the application of logical systems (e.g. mathematics, physics & mechanics).
The role of analysis in design is a critical one and can be considered the internal guidance system of a project.
A project without analysis is like a sports team without a coaching staff.
The Role of Analysis
The traits of engineers and their relationship with analysis.
What comes first: analysis or experience?
At what point in the evolution of a design should the guidance from theory be given.
priority? Guidance from experiment?
The seamless interplay between hands-on and theoretical components.
Will the application of the tools of engineering remain static?
The Role of Analysis (Cont….)
What happens when theory and experiment do not agree? How is it known that an analysis is flawed? How is a design analyzed? What are the types and levels of analysis? Where does analysis begin? Does a project ever begin based solely upon analysis?
Engineer Traits & Analysis
Universities are developing educational programs to encourage ambidextrous thinking, or both the left & right hemispheres of the brain.
Linear logical process and verbal abilities derive primarily from left side of brain.
Visual spatial properties, intuition and qualitative assessment skills derive primarily from the right side.
An eventual goal as an engineers skill evolves is to move towards “ whole brain” thinking.
Engineer Traits & Analysis (Cont..)
1st year students have significant analytical or creative skills, but lacks broad set of abilities required in engineering.
University education is focused to nurture existing as well as broaden range of talent.
The Role of Analysis in Design Process (Complementary Roles)
Design requires different abilities & perspectives at different stages.
Initially either or both perspectives can derive the choice of project.
Creative brainstorming suggests path investigated.
Next stage involves critical assessment of the possibilities & first level of analysis : resulted in a prioritized list of choices and rational.
After first analysis, followed with iterative & review to include additional avenues.
Creative review the ways of
constructing, testing & implementing the design usually leads to prototypes.
Elements of final design, recommendations for production, use of marketing are key product of the review.
Always an opportunity for another set of iteration and fine tuning.
Iterative Process
Radcliffe [Ref] presents an alternative view of the design process.
Emphasized the iterative nature of design.
New information can be introduced at any design stage, which also necessitates to return to the previous point in the process.
The Design Spiral : Submarine
Designing a submarine is a challenging task.
Design constraints in designing a submarine: Size and weight
Environmental challenges (depth & pressure)
Critical life support needs
Hull design
Mission requirements
Propulsion and energy requirements
Design of each of these parameters effect the other parameters as well.
Different design parameters are methodically integrated with the help of design spiral.
The Design Spiral: Submarine
The Design Spiral (Cont…)
The strong interactions between subcomponents are accounted in all stages.
On moving from outside to inside, each of the systems is revisited in an interactive way, moving toward the final design.
Design of aircrafts, space vehicles, defense vehicles also offer similar challenges.
Analysis can be viewed as an umbrella that protects the whole system-ensuring a minor change in one sector won’t produce disaster changes in another sector.
Interplay Between Theory & Experiments
Design should be a seamless transition between theory and experiments.
An effective engineer respects & understands the relative roles of analysis and practice.
Engineer should be comfortable in handling theory experiments as well as transition between both.
With experience, the choices and application of engineering skills should become a reflex.
Theoretical & Experimental Developments
Evaluation Areas for Theory & Experiments
Several important questions are raised when theory and experiments are not in agreement.
Keys area to evaluate in theory and experiments are tabulated as:
Critical Role of Analysis in Engineering Projects
It is essential for engineers to learn from success & failure.
Success and failure helps to derive valuable lessons by analysis of a great range of designs.
Radio Detection and Ranging
Required an existing base of electronic capabilities to be feasible.
Supporting technology grew to a
threshold to made radar possible. This development eventually led
to parallel remote sensing devices using light and sound.
A successful design exploiting one
area of technology may have fruitful derivatives applications in other areas as well.
Stay on Tabs: Al Cans An example of the value of
being sensitive to a societal need.
Replaced many throw-away
taps and curved pollution. Recovered and recycled
cans along with tabs helps in saving tons of aluminum.
Simple concept with
significant impact.
Boing 777
40% conventional materials are replaced with advanced materials.
Advanced computers and
software's. Networking permitted engineers
world wide to work effectively on the same design.
An example of paperless design
and concurrent engineering. Considered as the most advanced
passenger planes.
Global Positioning System An example of an existing
base of infrastructure and technical capabilities making a concept practical.
Satellite platforms and
electronics enabled the execution of the concept, permitting accurate location world wide.
Example of an area that is
under dynamic expansion with different set of applications.
Tacoma Narrows Bridge Bridge collapsed under
modest wind in 1940. Modest wind exciting a
resonance. An example of extrapolated
engineering, effects of winds were not properly considered.
Engineering design failure
should encourage caution when extending past, seemingly successful, design.
Walkways Regency Hotel
The walkways failed in 1981, resulting in many deaths.
Seemingly non-critical
design change to save time and cost resulted in a weak design of suspension.
Unimportant design
element: NO SUCH THING
Challenger Space Shuttle Exploded on January 28,
1986. Design of O-ring seals failed
at low temperature launch. Seals were critical elements
for separating different stages of rocket.
Decision to use multi-stage
rocket was politically motivated.
This decision made the
design more complicated than necessary, which eventually led to failure.
Three Mile Island Nuclear Plant
Simple component can cause major problems.
Indicates the importance of
working out foolproof displays of system status.
Valve failure led to to
overheating problem. Visual display did not
indicate the actual overheated status of the valve.
Reliability The reliability method of design is one in which we obtain
the distribution of stresses and the distribution of strengths and then relate these two in order to achieve an acceptable success rate.
The reliability R can be expressed by a number having the
range 0 ≤ R ≤ 1 In the reliability method of design, the designer’s task is
to make a judicious selection of materials, processes, and geometry (size) so as to achieve a specific reliability goal.
It is important to note that good statistical data and
estimates are essential to perform an acceptable reliability analysis.
Safety & Liability The strict liability concept of product liability generally prevails
in USA.
It ensures the liability of manufacturer for any damage or harm that results from any defect.
Liability of manufacturer will not be eased if unknown to defects or defective design.
Best approach to the prevention of product liability are: Good engineering in analysis and design.
Quality control.
Comprehensive testing procedure.
Warranties and sales literature should be reviewed carefully.
Statistical Considerations
Introduction
Statistics in mechanical design provides a method of dealing with characteristics whose values are variable.
Products manufactured in large quantities have a life that is variable. One automobile may have so many defects that it must be repaired
repeatedly during the first few months of operation while another may operate satisfactorily for years, requiring only minor maintenance.
The variability inherent in limits and fits, in stress and strength, in
bearing clearances, and in a multitude of other characteristics must be described numerically for proper control.
Evidence gathered from nature by measurement is a mixture of
systematic and random effects. It is the role of statistics to separate these, and, through the sensitive use of data, illuminate the obscure.
Random Variables
Outcome when two dices were tossed:
A Probability Distribution:
A Cumulative Probability Distribution:
Random Variables The strength determined by random experiment
is called a random, or a stochastic, variable.
A probability distribution shows all possible values of a random variable and with the corresponding probabilities.
The probability function p = f (x ) , a function of x, is often called the frequency function or, sometimes, the probability density function (PDF).
A cumulative probability distribution describes the probability that x is less than or equal to a certain value xi.
Arithmetic Mean, Variance , and Standard Deviation
The total number of elements, called the population, may in some cases be quite large.
A small part of the group, called a sample is generally selected for measurement.
Sample mean :
Sample variance :
Sample standard deviation :
Gaussian ( Normal ) Distribution
The Gaussian, or normal, distribution is expressed in terms of its mean μx and its standard deviation σx as
The normally distributed variate x can be expressed as
where N represents the normal distribution function.
To avoid the need for many tables for different values of μ and σ , the deviation from the mean is expressed in units of standard deviation by the transform
Gaussian ( Normal ) Distribution
Probability Distribution Function Cumulative Distribution Function
Lognormal Distribution
The lognormal distribution is one in which the logarithms of the variate have a normal distribution.
Use the transformation
y has a normal distribution
The lognormal distribution has the following two characteristics:
The distribution is asymmetrical about the mean.
The variables have only positive values.
Lognormal Distribution
Probability Distribution Function Cumulative Distribution Function
Uniform Distribution
The uniform distribution is a closed-interval distribution that arises when the chance of an observation is the same as the chance for any other observation.
The probability density function (PDF) for the uniform distribution is
where a is the lower bound and b is the upper bound.
The cumulative density function (CDF) is linear in the range a ≤ x ≤ b given by
The mean and standard deviation are given by
Uniform Distribution
Probability Distribution Function Cumulative Distribution Function
Weibull Distribution
Most reliability information comes from laboratory and field service data, and because of its flexibility, the Weibull distribution is widely used.
The probability density function, for Weibull, is
where x0 = minimum, guaranteed, value of x
θ = a characteristic or scale value (θ ≥ x0)
b = a shape parameter (b > 0)
Weibull Distribution
Probability Distribution Function Cumulative Distribution Function
Propagation of Error
Suppose we wish to add the two variates x and y to form a third variate z.
The mean is given as
The standard deviation
Linear Regression
Statisticians use a process of analysis called regression to obtain a curve that best fits a set of data points.
The process is called linear regression when the best-fitting straight line is to be found.
The standard equation of a straight line is
A correlation coefficient r calculates how well x and y correlate with each other.
Computer Aided Design &
Computer Aided Manufacturing
Introduction to CAD/CAM
• CAD/CAM is commonly used in engineering : Drafting
Design
Simulation and analysis
Manufacturing
• CAD/CAM includes four major areas: Geometric modeling
Computer graphics
Design applications
Manufacturing applications
Product Life Cycle (PLC) (Design Process)
Design Need
Design definitions, specifications
Collecting design information, feasibility study
Design conceptualization
Design modeling & Simulation
Design analysis
Design optimization
Design evaluation
Design communication & documentation
To manufacturing process
Synthesis
Analysis
CAD Process
Product Life Cycle (PLC) (Manufacturing Process)
Process Planning
Production Quality Control
Packaging Design & Procurement of new tools
Order Material
NC,CNC,DNC programming
Shipping Marketing
Production Planning
From design process
CAM Process
Design need (Design Process)
Major Components of PLC Design
Synthesis: Philosophy, functionality and uniqueness is determined. Design takes the form of sketches and layout drawings.
Analysis: Conceptual design in the engineering sciences to evaluate
the performance of the expected product. To finalize the best drawing.
Manufacturing
Process planning: determines the most efficient sequence to manufacture the product. Outcome is production plan, tools procurement, material order and machine programming.
CAD Disciplines
Material properties
Finite element analysis
Dimensioning & tolerances
Assembly modeling
Documentation and drafting
CAD
Geometric modeling
Computer Graphics
Design
CAM Disciplines
Computer aided process planning (CAPP)
NC programming (Numerical modeling) Design of injection molds Coordinate measuring machines
verification (CMM) Inspection Assembly with robots packaging
CAM
CAD Automation
Manufacturing
CAD/CAM Modules
Geometric module
Core of system
Developing, editing and manipulation of geometry
Drafting and documentation
Applications module
Geometry is mean to achieve the goal.
Mass property calculations
Assembly and tolerances analysis
Finite element modeling and analysis
Mechanism analysis
Simulation and analysis of plastic injection molding
CAD/CAM Modules
Programming module
Allows to customize the system
Adapt system with certain design and manufacturing tasks
Communication module
To achieve integration between CAD & CAM, other computer systems and manufacturing facilities
Collaborative module
To established real time connection between design teams working in different geographical locations
Numerical Control
Numerical control (NC) is a form of flexible (programmable) automation in which the process is controlled by numbers, letters, and symbols.
The electronic industries association (EIA) defined NC as
“A system in which actions are controlled by the direct insertion of numerical data at some point. The system must automatically interpret at least some portion of this data.”
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Basic Components
An NC system consists of the machine tools, the part-program, and the machine control unit (MCU).
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Machine Tools
The machine tools perform the useful work.
A machine tool consists of.
– A worktable,
– One or more spindles, motors and controls,
– Cutting tools,
– Work fixtures, and.
– Other auxiliary equipment needed in the machining operation.
55
The Part-Program
The part-program is a collection of all data required to produce the part. It is arranged in the form of blocks of information.
Each block contains the numerical data required for processing a segment of the work piece.
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The Machine Control Unit
The machine control unit consists of the data processing unit (DPU) and the control loop unit (CLU).
The DPU decodes the information contained in the part-program, process it, and provides instructions to the CLU.
The CLU operates the drives attached to the machine lead screws and feedback signals on the actual position and velocity of each one of the axes. The drive units are actuated by voltage pulses.
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Computer Numerical Control (CNC)
The EIA definition of computer numerical control (CNC). – “A numerical control system wherein a dedicated, stored
program computer is used to perform some or all of the basic numerical control functions in accordance with control programs stored in the read-write memory of the computer.”
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The CNC uses a dedicated microprocessor to perform the MCU functions.
CNC supports programming features not available in conventional NC systems:
– Subroutine macros which can be stored in memory and called by the part-program to execute frequently-used cutting sequence.
– Inch-metric conversions, sophisticated interpolation functions (such as cubic interpolation) can be easily accomplished in CNC.
– Absolute or incremental positioning (the coordinate systems used in locating the tool relative to the work piece).
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– The part-program can be edited (correction or optimization of tool path, speeds, and feeds) at the machine site during tape tryout.
– Tool and fixture offsets can be computed and stored.
– Tool path can be verified using graphic display.
– Diagnostics are available to assist maintenance and repair.
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Direct Numerical Control (DNC)
– “A system connecting a set of numerically controlled machines to a
common memory for part program or machine program storage with provision for on-demand distribution of data to machines.”
– In DNC, several NC machines are directly controlled by a computer, eliminating substantial hardware from the individual controller of each machine tool. The part-program is downloaded to the machines directly (thus omitting the tape reader) from the computer memory.
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Terminology of Solid Models
Coordinate Systems
Model(Master) Coordinate System (MCS)
MCS is the reference space of model with respect to which all the model geometries data is stored. In a CAD system MCS is generally shown by displaying X,Y,Z axis.
Working Coordinate System (WCS)
Portable coordinate system often employed when desired plane of sketching is not easily defined as one of the MCS planes.
Sketching Planes
Are the orthogonal planes
created by the axis of MCS or
WCS. Creating or selecting a
sketch plane is the very first
step toward creating a CAD
model.
Three Modeling Approaches
Primitive Approach
Views a solid model as a combination of simple generic, and standard shapes that can be combined. Primitives include, block (box), cylinder, sphere, cone & tores. These primitives are combined with Boolean operations.
Steps :
i. Create the block using block primitive.
ii. Create a cylinder in the desired location/orientation.
iii. Subtract the cylinder from the block.
Feature Approach
Similar to primitive approach, it replaces primitives with features and embeds Boolean operation in the features definition.
Steps:
i. Create the block using block feature.
ii. Create the hole in the block by creating a hole feature.
Block
Hole
Sketching Approach Sketching
Similar to features approach, with one change . Instead of using predefined shapes only, it allows designers to create much more elaborate & more general features starting from a sketch.
2-D Sketch Solid Model
Modeling 3D Operations
Extrusion
Revolving
Sweep
Loft
Types of geometric Model
21/2 D
Have uniform cross-section and thickness in the direction perpendicular to the plane cross sections. Axisymmetric model also falls in this category. Models made up of many 21/2 D features are called composite 21/2D model.
21/2 D 21/2 D Composite Axisymmetric
3 D Model
Are the ones that do not have a uniform cross section and/or not have constant thickness. Require more than one sketch in different sketch planes.
Visualization
Once the model is created, CAD system allows to view those models in many different ways. Viewing operation in a CAD system can be classified into three groups.
i. View orientation
ii. View modes
iii. View manipulation
View Orientation
Includes standard views such as front, top, right and isometric.
View Modes
Allow us to change the display of the model to different types such as wireframe, hidden & shaded
Wire Frame Hidden Dotted Frame Shaded
View Manipulation Allow us to dynamically rotate, pan and zoom the model to gain better control over its viewing.
Software’s
CATIA
Pro-Engineer
Solid works
ANSYS
Abaqus
I-DEAS
LS-DYNA
Introduction to
Finite Element Method
Introduction to FEM Without numerical techniques, it would be almost
impossible to solve practical engineering problems. Finite Element Method (FEM) is a numerical method
for solving engineering problems. The finite element method has been employed in:
(i) structural analysis (ii) fluid flow (iii) heat transfer
Application of FEM
APPLICATIONS
AEROSPACE
AUTOMOTIVE
BIOMECHANICS
MULTIPHYSICS
FEM in Design : Discretization
FEM in Piping
FEM in Safety
FEM in Crashworthiness
Discretization of Continuum Numerical techniques in continuum mechanics are
based on the principle that a continuum can be divided into an equivalent system of smaller bodies.
These bodies are connected at points (nodes) common
to the sub-regions (smaller bodies called elements). As the size of these small bodies gets smaller, the
numerical solution becomes more accurate. The cost of computation time may become prohibitive.
Elements & Nodes
Advantages of FEM
Accurate representation of complex geometry
Inclusion of dissimilar material properties
Easy representation of the total solution
Capture of local effects.
Element Geometries
The Finite-Element Method
Since the finite-element method is a numerical technique that discretizes the domain of a continuous structure, errors are inevitable.
– Computational errors : due to round-off errors from the computer floating-point calculations
and the formulations of the numerical integration schemes that are employed.
– Discretization errors : The geometry and the displacement distribution of a true structure continuously vary. Using a finite number of elements to model the structure introduces errors in matching geometry and the displacement distribution due to the inherent mathematical limitations of the elements.
Mesh Generation
There are three basic ways to generate an element mesh. – Manual mesh generation : This is how the element mesh was created in
the early days of the finite-element method.
– Semiautomatic mesh generation : this method enable the modeler to automatically mesh regions of the structure that he or she has divided up, using well-defined boundaries.
– Fully automated mesh generation. Many software vendors have concentrated their efforts on developing fully automatic mesh generation,
and in some instances, automatic self-adaptive mesh refinement.
The network of elements and nodes that discretize a region is referred to as a mesh.
Results generally improve when the mesh density is increased in areas of high stress gradients and/or when geometric transition zones are meshed smoothly.
Element Mesh
Mesh in Biomechanics
Meshing Curves
Mesh
MGWS
Overview of the Finite Element Method
Strong
form
Weak
form
Galerkin
approx.
Matrix
form
Axial deformation of a bar subjected to a uniform load
(1-D Poisson equation)
Sample Problem
0p=xp
0
00
2
=dx
duEA
=u
p=dx
udEA
Lx
02
L
L=x 0,
u = axial displacement
E=Young’s modulus = 1
A=Cross-sectional area = 1
Strong Form
The set of governing PDE’s, with boundary conditions, is
called the “strong form” of the problem.
Hence, our strong form is (Poisson equation in 1-D):
0
00
2
=dx
du
=u
p=dx
ud
Lx
02
We now reformulate the problem into the weak form.
The weak form is a variational statement of the problem in
which we integrate against a test function. The choice of test
function is up to us.
This has the effect of relaxing the problem; instead of finding
an exact solution everywhere, we are finding a solution that
satisfies the strong form on average over the domain.
Weak Form
Weak Form
0
0
0
0
2
0
2
2
=vdxpdx
ud
=pdx
ud
p=dx
ud
L
2
2
02
Strong Form
Residual R=0
Weak Form
v is our test function
We will choose the test function later.
Why is it “weak”?
It is a weaker statement of the problem.
A solution of the strong form will also satisfy the weak form,
but not vice versa.
Weak Form
Weak Form
Choosing the test function:
We can choose any v we want, so let's choose v such that it
satisfies homogeneous boundary conditions .
Steps Involved in FEM
• Divide continuum into a collection of pre-selected elements
of simple geometries (triangles, rectangles and quadrilateral
elements)
• Derive element equation for all types of elements involved in
the mesh such that
– equilibrium and compatibility are enforced
– assumed displacement within each element is dependent
upon nodal values
– equivalent nodal loads are established using principle of virtual work
[K]e[u]e = [F]e
• Assemble element equations to obtain the equilibrium equation of the whole problem [k]g{u}g = {F}g
• Impose boundary conditions • Solve the equilibrium equations for the nodal displacement • Calculate stresses and strains and post-process results
Fundamentals of FEM
(i) Idealization of structure
simplify the geometrical features of the structure
(ii) Discretization of structures
subdivide the structure into a system of finite elements. The size and number of elements are dictated by the geometrical features of the structure, applied load and restrains, accuracy and size of computer.
(iii) Choice of interpolation function
assume a trial function for the displacement (e.g. polynomial)
Fundamentals (Contd…) (iv) Derivation of the element stiffness matrix Derive the element stiffness matrix using the principle of minimum of potential energy (equilibrium equation). The derived stiffness relates the nodal displacements to the applied nodal forces. The stiffness matrix is a function of the material and geometric properties of an element. (v) Assembly of global stiffness matrix Assemble the global stiffness matrix from the element stiffness matrices
Fundamentals (Contd…)
(vi) Solution for the unknown nodal displacement
apply boundary conditions
solve the global equilibrium equations that can be
described as [K]{u}={F}
(vii) Computation of element & nodal strains and stresses
calculate the element strains and stresses using the appropriate solid mechanics relations
Features of FEM
have no limitations with regards to geometry, physical composition of domain and nature of loading.
involves a systematic procedure that can be automated for use with digital computers, and
yields approximate analysis by assuming a displacement field (or a stress field)
Pre Processing
General features, nodes, elements,
topology, Co-ordinate axes etc.
Material properties, yield strength,
density, coeff. of thermal expansion
Boundary conditions imposed, mechanical & thermal restraints
Applied loads
Preprocessor Appropriate Input Data File
General purpose FE software
Post Processing
Displacement
General purpose FE software
Strain
Stress
Temperature
Velocity
Post Processing
Result Files
Basic FE Algorithm • Mesh geometry • Element type • Boundary condition • Applied load • Symmetry
Input Data
E.S.M.G.
Assembler
Reducer
Solver
Output data
Apply B.C.
Solve for [ug]
• u • σij
• εij
The Finite-Element Solution Process A truss element is a bar loaded in tension or compression and is of
constant cross-sectional area A, length l, and elastic modulus E.
A truss element can be modeled as a simple linear spring with a spring rate
Assuming all forces f and displacements u directed toward the right as positive, the forces at each node can be written as
Consider a two-spring system, the total force at each node is the external force.
FEM in Design: Free Body Diagram
Example
• We combine the two stiffness matrices into the global matrix.
Now that the displacement at u2 has been obtained, the end forces and stress values can be obtained by reverting back to the individual element stiffness matrices
For the stress, you only need to look at the individual node of the stiffness equation
Reactions
Element Forces
Element Stresses
Software’s
ANSYS
Abaqus
I-DEAS
LS-DYNA
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