intro to numerical methods in mechanical engineeringmwr/lecture 2004-08-31/2004-08-31_slides.pdf ·...
TRANSCRIPT
Outlines
Intro to Numerical Methods in MechanicalEngineering
Mike Renfro
August 31, 2004
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
OutlinesPart I: Course Information, AdministriviaPart II: Overview of Numerical MethodsPart III: Types of Problems Solved with Numerical Methods
Course Information, Administrivia
Course InformationCourse Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
OutlinesPart I: Course Information, AdministriviaPart II: Overview of Numerical MethodsPart III: Types of Problems Solved with Numerical Methods
Overview of Numerical MethodsRelevance of Numerical Methods
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types and Sources of ErrorMathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Significant Digits
Rounding
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
OutlinesPart I: Course Information, AdministriviaPart II: Overview of Numerical MethodsPart III: Types of Problems Solved with Numerical Methods
Types of Problems Solved with Numerical Methods
Natural Frequencies of a Vibrating Bar
Static Analysis of a Scaffolding
Critical Loads for Buckling a Column
Realistic Design Properties of Materials
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Part I
Course Information, Administrivia
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 1)
Introductory material (1 day)
Solution of nonlinear equations (2 days)
MATLAB programming (3 days)
Matrix algebra, solution of simultaneous linear algebraicequations (4 days)
Test 1 on October 5
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 1)
Introductory material (1 day)
Solution of nonlinear equations (2 days)
MATLAB programming (3 days)
Matrix algebra, solution of simultaneous linear algebraicequations (4 days)
Test 1 on October 5
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 1)
Introductory material (1 day)
Solution of nonlinear equations (2 days)
MATLAB programming (3 days)
Matrix algebra, solution of simultaneous linear algebraicequations (4 days)
Test 1 on October 5
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 1)
Introductory material (1 day)
Solution of nonlinear equations (2 days)
MATLAB programming (3 days)
Matrix algebra, solution of simultaneous linear algebraicequations (4 days)
Test 1 on October 5
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 1)
Introductory material (1 day)
Solution of nonlinear equations (2 days)
MATLAB programming (3 days)
Matrix algebra, solution of simultaneous linear algebraicequations (4 days)
Test 1 on October 5
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 2)
Solution of matrix eigenvalue problems (2 days)
Curve fitting and interpolation (2 days)
Technical writing (2 days)
Statistics (2 days)
Test 2 on November 9
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 2)
Solution of matrix eigenvalue problems (2 days)
Curve fitting and interpolation (2 days)
Technical writing (2 days)
Statistics (2 days)
Test 2 on November 9
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 2)
Solution of matrix eigenvalue problems (2 days)
Curve fitting and interpolation (2 days)
Technical writing (2 days)
Statistics (2 days)
Test 2 on November 9
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 2)
Solution of matrix eigenvalue problems (2 days)
Curve fitting and interpolation (2 days)
Technical writing (2 days)
Statistics (2 days)
Test 2 on November 9
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 2)
Solution of matrix eigenvalue problems (2 days)
Curve fitting and interpolation (2 days)
Technical writing (2 days)
Statistics (2 days)
Test 2 on November 9
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 3)
Numerical differentiation (2 days)
Numerical integration (2 days)
Solutions to ordinary differential equations (initial valueproblems) (4 days)
Final exam on December 13 (10:30-12:30)
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 3)
Numerical differentiation (2 days)
Numerical integration (2 days)
Solutions to ordinary differential equations (initial valueproblems) (4 days)
Final exam on December 13 (10:30-12:30)
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 3)
Numerical differentiation (2 days)
Numerical integration (2 days)
Solutions to ordinary differential equations (initial valueproblems) (4 days)
Final exam on December 13 (10:30-12:30)
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Course Topics (Part 3)
Numerical differentiation (2 days)
Numerical integration (2 days)
Solutions to ordinary differential equations (initial valueproblems) (4 days)
Final exam on December 13 (10:30-12:30)
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Grading Percentages
20% Homework, roughly one homework grade per week
20% Test 1: October 5
20% Test 2: November 9
20% Projects: given throughout the semester
20% Final: December 13
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Grading Percentages
20% Homework, roughly one homework grade per week
20% Test 1: October 5
20% Test 2: November 9
20% Projects: given throughout the semester
20% Final: December 13
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Grading Percentages
20% Homework, roughly one homework grade per week
20% Test 1: October 5
20% Test 2: November 9
20% Projects: given throughout the semester
20% Final: December 13
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Grading Percentages
20% Homework, roughly one homework grade per week
20% Test 1: October 5
20% Test 2: November 9
20% Projects: given throughout the semester
20% Final: December 13
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Grading Percentages
20% Homework, roughly one homework grade per week
20% Test 1: October 5
20% Test 2: November 9
20% Projects: given throughout the semester
20% Final: December 13
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Mike Renfro ([email protected])Office: Clement Hall 314
Phone: 372-3601Office Hours: 2-5 PM Monday/Wednesday, or by appointment
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Bring These Items To Class Each Day
Textbook
Scientific calculator, preferably one that can perform basicmatrix algebra
Pencil
Engineering graph paper
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Bring These Items To Class Each Day
Textbook
Scientific calculator, preferably one that can perform basicmatrix algebra
Pencil
Engineering graph paper
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Bring These Items To Class Each Day
Textbook
Scientific calculator, preferably one that can perform basicmatrix algebra
Pencil
Engineering graph paper
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Bring These Items To Class Each Day
Textbook
Scientific calculator, preferably one that can perform basicmatrix algebra
Pencil
Engineering graph paper
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Homework Style Requirements
Attached to your course syllabus is an example of the homeworkstyle required for the class, including:
Use engineering graph paper, not notebook paper or regulargraph paper
Use a pencil, not ink
Lay out the problem statement and solution legibly, includingany supporting figures or sketches
Show enough work to leave a clear trail, including unitconversions
Highlight your final answers either with double-underlines,boxes, or something similar.
I reserve the right to not grade homework that doesn’t conform tothe above style.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Programming Requirements
Regardless of the language you write a program in, follow theseguidelines:
Include all code you wrote to solve the problem
Attach any input files you used and all output files created
Attach a log of the program as it was run, regardless of ifanything was read from the keyboard or printed to the screen
Attach any plots or other non-text output generated by theprogram, if any were made
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Programming Requirements
Regardless of the language you write a program in, follow theseguidelines:
Include all code you wrote to solve the problem
Attach any input files you used and all output files created
Attach a log of the program as it was run, regardless of ifanything was read from the keyboard or printed to the screen
Attach any plots or other non-text output generated by theprogram, if any were made
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Programming Requirements
Regardless of the language you write a program in, follow theseguidelines:
Include all code you wrote to solve the problem
Attach any input files you used and all output files created
Attach a log of the program as it was run, regardless of ifanything was read from the keyboard or printed to the screen
Attach any plots or other non-text output generated by theprogram, if any were made
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Course Information
Course Topics and CalendarGradingContact InformationClass, Homework, and Program Requirements
Programming Requirements
Regardless of the language you write a program in, follow theseguidelines:
Include all code you wrote to solve the problem
Attach any input files you used and all output files created
Attach a log of the program as it was run, regardless of ifanything was read from the keyboard or printed to the screen
Attach any plots or other non-text output generated by theprogram, if any were made
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Part II
Overview of Numerical Methods
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
Equilibrium
Newton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motion
Conservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
4 Steps in Engineering Analysis
1 Development of a mathematical model representing allimportant characteristics of the physical system
2 Derivation of governing equations of the model by applyingphysical laws
EquilibriumNewton’s laws of motionConservation of mass and energy
3 Solution of governing equations
4 Interpretation of the solution
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Types of Governing Equations
Linear algebraic
Nonlinear algebraic
Transcendental
Ordinary differential equations
Partial differential equations
Homogeneous differential equations
Equation involving integrals or derivatives
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Benefits
Solutions are exact.
Driving variables are easily visible, leading to easy parametricstudies on the effects of changing those variables.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Benefits
Solutions are exact.
Driving variables are easily visible, leading to easy parametricstudies on the effects of changing those variables.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Limitations
The vast majority of real-world problems do not have ananalytical solution of exacting detail.
Finding out whether or not there is an analytical solution to anew type of problem is tedious and time-consuming. Plus, theresult often is “nope, no analytical solution here — I think.”
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Limitations
The vast majority of real-world problems do not have ananalytical solution of exacting detail.
Finding out whether or not there is an analytical solution to anew type of problem is tedious and time-consuming. Plus, theresult often is “nope, no analytical solution here — I think.”
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Example
Consider the integral
I1 =
∫ b
axe−x2
dx .
Through calculus, you can calculate this integral exactly as
I1 =
(−1
2e−x2
)∣∣∣∣ba
= −1
2e−b2
+1
2e−a2
=1
2
(e−a2 − e−b2
).
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Example
Consider the integral
I1 =
∫ b
axe−x2
dx .
Through calculus, you can calculate this integral exactly as
I1 =
(−1
2e−x2
)∣∣∣∣ba
= −1
2e−b2
+1
2e−a2
=1
2
(e−a2 − e−b2
).
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Example
Consider the integral
I1 =
∫ b
axe−x2
dx .
Through calculus, you can calculate this integral exactly as
I1 =
(−1
2e−x2
)∣∣∣∣ba
= −1
2e−b2
+1
2e−a2
=1
2
(e−a2 − e−b2
).
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Analytical Solution Example
Consider the integral
I1 =
∫ b
axe−x2
dx .
Through calculus, you can calculate this integral exactly as
I1 =
(−1
2e−x2
)∣∣∣∣ba
= −1
2e−b2
+1
2e−a2
=1
2
(e−a2 − e−b2
).
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Benefits
A larger number of problems have numerical solutions thananalytical ones.
Assuming the problem is set up correctly, a correct numericalsolution is mostly a matter of patience.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Benefits
A larger number of problems have numerical solutions thananalytical ones.
Assuming the problem is set up correctly, a correct numericalsolution is mostly a matter of patience.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Limitations
Solutions are not exact, but can be close enough to do the job.
Parametric studies are more difficult, since driving variablesare hidden.
The speed of finding a numerical solution depends on boththe computing tools on hand, your ability to use them, andyour ability to interpret the results for accuracy.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Limitations
Solutions are not exact, but can be close enough to do the job.
Parametric studies are more difficult, since driving variablesare hidden.
The speed of finding a numerical solution depends on boththe computing tools on hand, your ability to use them, andyour ability to interpret the results for accuracy.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Limitations
Solutions are not exact, but can be close enough to do the job.
Parametric studies are more difficult, since driving variablesare hidden.
The speed of finding a numerical solution depends on boththe computing tools on hand, your ability to use them, andyour ability to interpret the results for accuracy.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Example
Consider the integral
I2 =
∫ b
af (x)dx =
∫ b
ae−x2
dx .
There is no closed-form analytic solution to this integral. How canwe evaluate it?
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Example
Recall that a the value of a definiteintegral is identical to the areabeneath the curve of the functionbeing integrated.
We can approximate that area byfinding the area of lots of smallrectangles whose height is determinedby the values of f (x) at differentplaces.
The accuracy of this solution is neverperfect, but increases as we increaseamount of computational time.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Example
Recall that a the value of a definiteintegral is identical to the areabeneath the curve of the functionbeing integrated.
We can approximate that area byfinding the area of lots of smallrectangles whose height is determinedby the values of f (x) at differentplaces.
The accuracy of this solution is neverperfect, but increases as we increaseamount of computational time.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
4 Steps in Engineering AnalysisEquation TypesAnalytical Solutions: Benefits and LimitationsNumerical Solutions: Benefits and Limitations
Numerical Solution Example
Recall that a the value of a definiteintegral is identical to the areabeneath the curve of the functionbeing integrated.
We can approximate that area byfinding the area of lots of smallrectangles whose height is determinedby the values of f (x) at differentplaces.
The accuracy of this solution is neverperfect, but increases as we increaseamount of computational time.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is gravity really a constant 9.808m/s2?
Not necessarily. It varies with latitude and altitude, and evenchanges depending on the density of the Earth’s soil belowyou.
Why would you care? The ability to detect tiny localvariations in gravity provides a non-invasive method formapping out the area’s subsurface geology, including buriednatural resources, fault lines, etc.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is gravity really a constant 9.808m/s2?
Not necessarily. It varies with latitude and altitude, and evenchanges depending on the density of the Earth’s soil belowyou.
Why would you care? The ability to detect tiny localvariations in gravity provides a non-invasive method formapping out the area’s subsurface geology, including buriednatural resources, fault lines, etc.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is gravity really a constant 9.808m/s2?
Not necessarily. It varies with latitude and altitude, and evenchanges depending on the density of the Earth’s soil belowyou.
Why would you care? The ability to detect tiny localvariations in gravity provides a non-invasive method formapping out the area’s subsurface geology, including buriednatural resources, fault lines, etc.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is Newton’s second law of motion F = ma?
Not as it was originally written. Originally, it was shown asF = d(mv)
dt , which does evaluate down to F = ma as long asmass is constant.
Why would you care? F = ma works fine until you get aproblem where mass changes considerably during the analysis,like a rocket launch. The Space Shuttle’s total weight dropsby 30% while its solid rocket boosters are firing during thefirst two minutes after liftoff.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is Newton’s second law of motion F = ma?
Not as it was originally written. Originally, it was shown asF = d(mv)
dt , which does evaluate down to F = ma as long asmass is constant.
Why would you care? F = ma works fine until you get aproblem where mass changes considerably during the analysis,like a rocket launch. The Space Shuttle’s total weight dropsby 30% while its solid rocket boosters are firing during thefirst two minutes after liftoff.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Assumptions and Simplifications Change YourResults?
Is Newton’s second law of motion F = ma?
Not as it was originally written. Originally, it was shown asF = d(mv)
dt , which does evaluate down to F = ma as long asmass is constant.
Why would you care? F = ma works fine until you get aproblem where mass changes considerably during the analysis,like a rocket launch. The Space Shuttle’s total weight dropsby 30% while its solid rocket boosters are firing during thefirst two minutes after liftoff.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Mariner 1 Launch: One Character Can Screw It All Up
Mariner 1 was to be the first spacecraft to fly by Venus. It wasdestroyed during its launch due to the following chain of events:
A hardware failure disabled the rocket’s antenna, breaking itscommunication with ground-based guidance and controlsystems.
An onboard computer took over the guidance and controltasks.
One author of the onboard program mis-transcribed an
averaged derivative of radius Rn as Rn, which was aninstantaneous derivative value.
Since the instantaneous derivative value varied much morequickly than the averaged value, another portion of theprogram thought the rocket had gone out of control anddestroyed it.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Mariner 1 Launch: One Character Can Screw It All Up
Mariner 1 was to be the first spacecraft to fly by Venus. It wasdestroyed during its launch due to the following chain of events:
A hardware failure disabled the rocket’s antenna, breaking itscommunication with ground-based guidance and controlsystems.
An onboard computer took over the guidance and controltasks.
One author of the onboard program mis-transcribed an
averaged derivative of radius Rn as Rn, which was aninstantaneous derivative value.
Since the instantaneous derivative value varied much morequickly than the averaged value, another portion of theprogram thought the rocket had gone out of control anddestroyed it.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Mariner 1 Launch: One Character Can Screw It All Up
Mariner 1 was to be the first spacecraft to fly by Venus. It wasdestroyed during its launch due to the following chain of events:
A hardware failure disabled the rocket’s antenna, breaking itscommunication with ground-based guidance and controlsystems.
An onboard computer took over the guidance and controltasks.
One author of the onboard program mis-transcribed an
averaged derivative of radius Rn as Rn, which was aninstantaneous derivative value.
Since the instantaneous derivative value varied much morequickly than the averaged value, another portion of theprogram thought the rocket had gone out of control anddestroyed it.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Mariner 1 Launch: One Character Can Screw It All Up
Mariner 1 was to be the first spacecraft to fly by Venus. It wasdestroyed during its launch due to the following chain of events:
A hardware failure disabled the rocket’s antenna, breaking itscommunication with ground-based guidance and controlsystems.
An onboard computer took over the guidance and controltasks.
One author of the onboard program mis-transcribed an
averaged derivative of radius Rn as Rn, which was aninstantaneous derivative value.
Since the instantaneous derivative value varied much morequickly than the averaged value, another portion of theprogram thought the rocket had gone out of control anddestroyed it.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Errors in data transfer
Uncertainties in measurements
Incorrect units
One thermodynamics problem involved calculating the speed ofwater flowing through a pipe using both conservation of energyand conservation of mass relationships. Three students all used theexact same method, but differed in their values for the density ofwater:
One student used a density of 2000 kg/m3, and calculated awater speed of 2 m/s.
The second student used a density of 2 kg/m3, and calculateda water speed of 2000 m/s.
The third student used a density of 0.002 kg/m3, andcalculated a water speed of 2000000 m/s.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Numbers in a Computer Are Almost Never Exact
What should the following C code do?
#i n c l u d e <s t d i o . h>
i n t main ( vo i d ) {f l o a t y , z ;y =838861.2;z =1.3 ;p r i n t f (” y : %8.1 f \n” , y ) ;p r i n t f (” z : %8.1 f \n” , z ) ;p r i n t f (” y : %18.11 e\n” , y ) ;p r i n t f (” z : %18.11 e\n” , z ) ;r e t u r n 0 ;
}
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Numbers in a Computer Are Almost Never Exact
You expected:
y : 838861.2z : 1 . 3y : 8 .38861200000 e+05z : 1 .30000000000 e+00
The actual program output:
y : 838861.2z : 1 . 3y : 8 .38861187500 e+05z : 1 .29999995232 e+00
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Any Way to Avoid This?
Changing languages won’t cure it
Changing compilers won’t, either
Nor will changing computer architectures
It can be mitigated by making each number stored occupymore memory. MATLAB stores real numbers using 64 bits ofdata, compared to the 32 used by default in C or FORTRAN.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Any Way to Avoid This?
Changing languages won’t cure it
Changing compilers won’t, either
Nor will changing computer architectures
It can be mitigated by making each number stored occupymore memory. MATLAB stores real numbers using 64 bits ofdata, compared to the 32 used by default in C or FORTRAN.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Any Way to Avoid This?
Changing languages won’t cure it
Changing compilers won’t, either
Nor will changing computer architectures
It can be mitigated by making each number stored occupymore memory. MATLAB stores real numbers using 64 bits ofdata, compared to the 32 used by default in C or FORTRAN.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Any Way to Avoid This?
Changing languages won’t cure it
Changing compilers won’t, either
Nor will changing computer architectures
It can be mitigated by making each number stored occupymore memory. MATLAB stores real numbers using 64 bits ofdata, compared to the 32 used by default in C or FORTRAN.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Causes It?
The problem stems from the way computers store numbers assums of various powers of 2. For example:
1.3 =1 +1
4+
1
32+
1
64+
1
512+
1
1024+ · · ·
=1(20) + 0(2−1) + 1(2−2) + 0(2−3) + 0(2−4) + 1(2−5)+
1(2−6) + 0(2−7) + 0(2−8) + 1(2−9) + 1(2−10) + · · ·
There is no finite sum of powers of 2 that adds up exactly to 1.3,therefore there will always be a small amount of error in anycomputer’s storage of that number using that method.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Causes It?
The problem stems from the way computers store numbers assums of various powers of 2. For example:
1.3 =1 +1
4+
1
32+
1
64+
1
512+
1
1024+ · · ·
=1(20) + 0(2−1) + 1(2−2) + 0(2−3) + 0(2−4) + 1(2−5)+
1(2−6) + 0(2−7) + 0(2−8) + 1(2−9) + 1(2−10) + · · ·
There is no finite sum of powers of 2 that adds up exactly to 1.3,therefore there will always be a small amount of error in anycomputer’s storage of that number using that method.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
What Causes It?
The problem stems from the way computers store numbers assums of various powers of 2. For example:
1.3 =1 +1
4+
1
32+
1
64+
1
512+
1
1024+ · · ·
=1(20) + 0(2−1) + 1(2−2) + 0(2−3) + 0(2−4) + 1(2−5)+
1(2−6) + 0(2−7) + 0(2−8) + 1(2−9) + 1(2−10) + · · ·
There is no finite sum of powers of 2 that adds up exactly to 1.3,therefore there will always be a small amount of error in anycomputer’s storage of that number using that method.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Finite Representations of Infinity Are Not Exact
Anytime ∞ shows up in an equation we evaluate numerically, we’llhave errors. Practically, ∞ is approximated as “the largest numberwe get to before we run out of time or motivation”. This shows upboth in infinite series expansions and in open-ended integrals. Forexample:
y(x) = ln (1 + x) =∞∑i=1
(−1)i+1
ix i
= x − 1
2x2 +
1
3x3 − 1
4x4 +
1
5x5 − 1
6x6 + · · ·
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Mathematical ModelingBlundersInput ErrorsMachine ErrorsTruncation
Effects of Truncation Error
If you take just the first four terms of the approximation toevaluate ln(1 + 1):
ln(1 + 1) = 0.69315
≈ 1− 1
2+
1
3− 1
4= 0.58333
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Why Significant Digits Matter
Experimental measurements are never exact. Instruments mayvary their readouts as temperatures change, or as they wearout.
Whatever number a measurement has, it is not followed by aninfinite number of zeros.
If you measure something with a millimeter scale, yourprecision may only be ± 0.5 mm.
If you measure two results with different precisions andcombine them mathematically into a final result, the precisionof that result is determined by the measurement with thelower precision.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Why Significant Digits Matter
Experimental measurements are never exact. Instruments mayvary their readouts as temperatures change, or as they wearout.
Whatever number a measurement has, it is not followed by aninfinite number of zeros.
If you measure something with a millimeter scale, yourprecision may only be ± 0.5 mm.
If you measure two results with different precisions andcombine them mathematically into a final result, the precisionof that result is determined by the measurement with thelower precision.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Why Significant Digits Matter
Experimental measurements are never exact. Instruments mayvary their readouts as temperatures change, or as they wearout.
Whatever number a measurement has, it is not followed by aninfinite number of zeros.
If you measure something with a millimeter scale, yourprecision may only be ± 0.5 mm.
If you measure two results with different precisions andcombine them mathematically into a final result, the precisionof that result is determined by the measurement with thelower precision.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Why Significant Digits Matter
Experimental measurements are never exact. Instruments mayvary their readouts as temperatures change, or as they wearout.
Whatever number a measurement has, it is not followed by aninfinite number of zeros.
If you measure something with a millimeter scale, yourprecision may only be ± 0.5 mm.
If you measure two results with different precisions andcombine them mathematically into a final result, the precisionof that result is determined by the measurement with thelower precision.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Working with Significant Digits
If the number is in scientific notation, just count the digits,including any trailing zeros: 2.99792458× 108 m/s → 9significant digits.
If the number is not in scientific notation, converting it beforecounting the significant digits helps.
Trailing zeros can be a problem.
A 10k race route should have a length of 10,000 meters.Realistically, it will not have a length of exactly 10,000,000millimeters, and certainly not 10,000,000,000 microns.
Generally, don’t do any unit conversions before countingsignificant digits. If you do any conversions afterwards, makesure you don’t add any digits that aren’t really there.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Working with Significant Digits
If the number is in scientific notation, just count the digits,including any trailing zeros: 2.99792458× 108 m/s → 9significant digits.
If the number is not in scientific notation, converting it beforecounting the significant digits helps.
Trailing zeros can be a problem.
A 10k race route should have a length of 10,000 meters.Realistically, it will not have a length of exactly 10,000,000millimeters, and certainly not 10,000,000,000 microns.
Generally, don’t do any unit conversions before countingsignificant digits. If you do any conversions afterwards, makesure you don’t add any digits that aren’t really there.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Working with Significant Digits
If the number is in scientific notation, just count the digits,including any trailing zeros: 2.99792458× 108 m/s → 9significant digits.
If the number is not in scientific notation, converting it beforecounting the significant digits helps.
Trailing zeros can be a problem.
A 10k race route should have a length of 10,000 meters.Realistically, it will not have a length of exactly 10,000,000millimeters, and certainly not 10,000,000,000 microns.
Generally, don’t do any unit conversions before countingsignificant digits. If you do any conversions afterwards, makesure you don’t add any digits that aren’t really there.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Working with Significant Digits
If the number is in scientific notation, just count the digits,including any trailing zeros: 2.99792458× 108 m/s → 9significant digits.
If the number is not in scientific notation, converting it beforecounting the significant digits helps.
Trailing zeros can be a problem.
A 10k race route should have a length of 10,000 meters.Realistically, it will not have a length of exactly 10,000,000millimeters, and certainly not 10,000,000,000 microns.
Generally, don’t do any unit conversions before countingsignificant digits. If you do any conversions afterwards, makesure you don’t add any digits that aren’t really there.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.
1This is just an example; Newtonian mechanics are not accurate on problemswith speeds this high.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.
1This is just an example; Newtonian mechanics are not accurate on problemswith speeds this high.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808
= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.
1This is just an example; Newtonian mechanics are not accurate on problemswith speeds this high.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.
1This is just an example; Newtonian mechanics are not accurate on problemswith speeds this high.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.
1This is just an example; Newtonian mechanics are not accurate on problemswith speeds this high.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
Significant Digits Example
The speed of light in a vacuum is 2.99792458× 108 m/s. Aspacecraft initially at rest accelerates at a rate of 9.808 m/s2.How much time elapses until the spacecraft reaches lightspeed1?
Solve for tf as follows:
tf =vf
a=
2.99792458× 108
9.808= 3.056611521207178× 107 s (says the calculator)
= 3.057× 107 s (says the engineer)
Notice that the final result has only 4 significant digits.1This is just an example; Newtonian mechanics are not accurate on problems
with speeds this high.Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Relevance of Numerical MethodsTypes and Sources of Error
Significant DigitsRounding
So You Have a Result With Too Many Significant Digits?
How do you chop off digits that you can’t really back up withdata? Rao overcomplicates things on p.20. Simpler rules:
Round the last retained digit up if your first dropped digit is a6 or higher.
Keep the last retained digit as-is if your first dropped digit is a4 or lower.
If the first dropped digit is a 5, round the last retained digit tothe nearest even number.
Numbers rounded to four significant digits:
9.46932 → 9.469
201.72 → 201.7
200.550 → 200.6
200.650 → 200.6
2.013501 → 2.014Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Part III
Types of Problems Solved with Numerical Methods
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Nonlinear Equations: Natural Frequencies of aVibrating Bar
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations
Natural frequencies ω of axial vibration of a bar, fixed at one endand carrying a mass M at the other end, satisfy the equation
cot
(ωl√E/ρ
)=
M
ρAl
ωl√E/ρ
where l is the bar’s length, E is the bar’s elastic modulus, ρ is thebar’s density, and A is the bar’s cross-sectional area.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Governing Equations
0 1 2 3 4 5 6 7 8 9 10−2
−1
0
1
2
3
4
5
6
7
ω L / c
Frequency Equation Solution −− Intersections Represent Solutions
M/m=0.5cot(ω L / c)
Intersections of the red and bluelines represent values of ωl√
E/ρ
that solve the previous equation:
ω1l√E/ρ
= 1.076
ω2l√E/ρ
= 3.642
ω3l√E/ρ
= 6.579
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Simultaneous Linear Algebraic Equations:Static Analysis of a Scaffolding
3 bars supported by 6 cablesform a simple scaffolding. Giventhe positions and magnitudes for3 loads applied to the bars, findthe tension in each cable.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 1
Force equilibrium ∑Fy = 0
TA + TB − TC − TD − TF − P1 = 0
Moment equilibrium ∑M = 0
−9TB + TC + 4TD + 7TF + 5P1 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 1
Force equilibrium ∑Fy = 0
TA + TB − TC − TD − TF − P1 = 0
Moment equilibrium ∑M = 0
−9TB + TC + 4TD + 7TF + 5P1 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 2
Force equilibrium ∑Fy = 0
TC + TD − TE − P2 = 0
Moment equilibrium∑M = 0
−3TD + 2TE + P2 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 2
Force equilibrium ∑Fy = 0
TC + TD − TE − P2 = 0
Moment equilibrium∑M = 0
−3TD + 2TE + P2 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 3
Force equilibrium∑Fy = 0
TE + TF − P3 = 0
Moment equilibrium∑M = 0
−4TF + P3 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Bar 3
Force equilibrium∑Fy = 0
TE + TF − P3 = 0
Moment equilibrium∑M = 0
−4TF + P3 = 0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Assembling Equations
At this point, we have six independent equations (two for eachbar), and six unknowns (cable tensions). Reformat the sixequilibrium equations to isolate the unknown tensions on theleft-hand side of the equations. Make sure the tension variables arein the same order in each equation:
TA +TB −TC −TD −TF = P1
−9TB +TC +4TD +7TF = −5P1
TC +TD −TE = P2
−3TD +2TE = −P2
TE +TF = P3
−4TF = −P3
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Governing Equations
1 1 −1 −1 0 −10 −9 1 4 0 70 0 1 1 −1 00 0 0 −3 2 00 0 0 0 1 10 0 0 0 0 −4
TA
TB
TC
TD
TE
TF
=
P1
−5P1
P2
−P2
P3
−P3
If P1 = 2000 lb, P2 = 1000 lb, P3 = 500 lb, various solutionmethods detailed in Chapter 3 can solve for TA · · ·TF :
TA = 1944.45 lb TB = 1555.55 lbTC = 791.67 lb TD = 583.33 lbTE = 375 lb TF = 125 lb
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Eigenvalue Problems: Critical Loads for Buckling a Column
A long column with elasticmodulus E and cross-sectionalmoment of inertia I is subjectedto an axial load P. If there is asmall deformity in the columndue to misalignment duringconstruction or some otherreason, its strength isconsiderably reduced. Thedeformity will cause the columnto buckle long before a shortercolumn would have been crushed.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Governing Equations for Discretized Column
The continuous differentialequation of deflection
d2y
dx2+
P
EIy = 0
can be discretized with thefollowing substitution:
d2y
dx2≈ yi+1 − 2yi + yi−1
(∆x)2
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Discretized Equations
At any given point i , the governing equation evaluates to
yi+1 − 2yi + yi+1
(∆x)2+ λyi = 0
where λ = P/(EI ). Dividing the column into 4 segments (a totalof 5 points), evaluating the equation at points 2, 3, and 4 yields:
y1 −(
2− λL2
16
)y2 + y3 =0
y2 −(
2− λL2
16
)y3 + y4 =0
y3 −(
2− λL2
16
)y4 + y5 =0
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Solution of Discretized Equations
Since the column is pinned on both ends, we assume that thedeflections y1 = y5 = 0. We can then convert the previous threeequations into the matrix form
(2− λL2
16
)1 0
1(2− λL2
16
)1
0 1(2− λL2
16
)
y2
y3
y4
=
000
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Statistics: Realistic Design Properties of Materials
A shipment of AISI 1020 hot-rolled steel your company bought hasa textbook yield strength of 29700 psi. Upon testing 50 samples ofthe material, you notice that almost no samples measured a yieldstrength of 29700 psi. Assuming these samples are typical, whatstrength should your designers assume as a minimum, so that 95%of the time, the material they use will meet or exceed thatminimum?
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Yield Strength Data (in ksi)
# Sy # Sy # Sy # Sy # Sy
1 29.4 11 29.3 21 28.9 31 31.3 41 29.72 30.5 12 29.8 22 31.2 32 30.4 42 29.23 30.5 13 30.3 23 29.3 33 31.9 43 27.84 28.3 14 28.1 24 28.8 34 31.2 44 31.75 33.0 15 30.7 25 31.2 35 27.6 45 30.66 28.2 16 32.8 26 32.1 36 29.5 46 29.17 31.4 17 29.4 27 30.1 37 28.4 47 30.28 29.7 18 31.6 28 32.2 38 31.3 48 29.49 29.9 19 30.8 29 29.3 39 32.3 49 30.310 30.9 20 29.8 30 30.1 40 29.9 50 27.2
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Statistical Characteristics of Samples
One basic statistical characteristic is the mean or average,indicating the central tendency of the data. Add up all the yieldstrength measurements and divide by the number of samples tocalculate it:
X =1
n
n∑i=1
xi = 30.1
Another characteristic is the sample standard deviation, indicatingthe predictability of the data. Small standard deviations come fromdata that is predominantly clustered around the mean; largestandard deviations come from data that is more scattered.
s =
√√√√ 1
n − 1
n∑i=1
(xi − X
)2= 1.36
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Statistical Characteristics of Samples
One basic statistical characteristic is the mean or average,indicating the central tendency of the data. Add up all the yieldstrength measurements and divide by the number of samples tocalculate it:
X =1
n
n∑i=1
xi = 30.1
Another characteristic is the sample standard deviation, indicatingthe predictability of the data. Small standard deviations come fromdata that is predominantly clustered around the mean; largestandard deviations come from data that is more scattered.
s =
√√√√ 1
n − 1
n∑i=1
(xi − X
)2= 1.36
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Predictions From a Normal Distribution
We assume that the processes controlling the steel’s yield strengthare random, even if they’re well controlled. Many randomphenomena in engineering follow a normal or Gaussian probabilitydistribution.
Standard tables exist that allow us to predict probabilities offinding a particular range of results from a set of randommeasurements, or to take a particular probability and convert itback to a range of results.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Predictions From a Normal Distribution
We assume that the processes controlling the steel’s yield strengthare random, even if they’re well controlled. Many randomphenomena in engineering follow a normal or Gaussian probabilitydistribution.Standard tables exist that allow us to predict probabilities offinding a particular range of results from a set of randommeasurements, or to take a particular probability and convert itback to a range of results.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Using Statistical Tables
The usual set of statistical tables show relationships between avariable z and a definite integral Φ(z). Without going into toomuch detail, z is a function of the measured variable (x in thegeneral case, Sy in ours) and the variable’s mean and standarddeviation.
Our 95% success requirement corresponds to a Φ(z) value of 0.05,which is attached to a z value of -1.64. If z = −1.64, then the Sy
corresponding to that z is 27.9 ksi.If your designers work off an expected yield strength of 27.9 ksi,your supplier will be able to meet that requirement 95% of thetime.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Using Statistical Tables
The usual set of statistical tables show relationships between avariable z and a definite integral Φ(z). Without going into toomuch detail, z is a function of the measured variable (x in thegeneral case, Sy in ours) and the variable’s mean and standarddeviation.Our 95% success requirement corresponds to a Φ(z) value of 0.05,which is attached to a z value of -1.64. If z = −1.64, then the Sy
corresponding to that z is 27.9 ksi.
If your designers work off an expected yield strength of 27.9 ksi,your supplier will be able to meet that requirement 95% of thetime.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Using Statistical Tables
The usual set of statistical tables show relationships between avariable z and a definite integral Φ(z). Without going into toomuch detail, z is a function of the measured variable (x in thegeneral case, Sy in ours) and the variable’s mean and standarddeviation.Our 95% success requirement corresponds to a Φ(z) value of 0.05,which is attached to a z value of -1.64. If z = −1.64, then the Sy
corresponding to that z is 27.9 ksi.If your designers work off an expected yield strength of 27.9 ksi,your supplier will be able to meet that requirement 95% of thetime.
Mike Renfro Intro to Numerical Methods in Mechanical Engineering
Natural Frequencies of a Vibrating BarStatic Analysis of a Scaffolding
Critical Loads for Buckling a ColumnRealistic Design Properties of Materials
Homework
Read Chapter 1, complete problems 1.4, 1.5, and 1.9
Write a short C or FORTRAN program to do the following:
s e t the r e a l−va l u ed v a r i a b l e ’ r e s u l t ’ to 9 .0l oop 100000 t imes :
add ( 1 . 0 / 3 . 0 ) to ’ r e s u l t ’l oop 100000 t imes :
s u b t r a c t ( 1 . 0 / 3 . 0 ) from ’ r e s u l t ’p r i n t the v a l u e o f ’ r e s u l t ’
What do you predict the output of the program will be?
What is the actual output?
Mike Renfro Intro to Numerical Methods in Mechanical Engineering