PE Meche I.A. Foundation
Overview
Engineering is like a home. YOU NEED a good foundation! This is where I will add concepts that you should be familiar with as you dive into a journey of exploration.
Table of Contents
Data
Uncertainties
The only answer that is 100% accurate is 42 (Hitchhiker's Guide to the Galaxy). Every other number that we use in science and engineering has an inherit "error" associated with the measurements (systematic & random). According to Webster "error" is defiend as "the difference between an observed or calculated value and the true value." While we can go full depth into statistics, we will get to that at a later time. This section is just the summary of the arithmatics of numbers with "errors" given x is a function of both u and v (note a & b are just constants). Reference: Data Reduction and Error Analysis for the Physical Sciences (3rd ed) Benvington & Robinson (pg 48). If in doubt check orignal reference for details.
Covariance
\( \sigma^2_{uv} = \big \langle \big( u-\bar u \big ) \big ( v-\bar v ) \big \rangle \)
Uncorrelated u and v then: \( \sigma^2_{uv} = 0 \)
Addition
\( x = a \cdot u + b \cdot v \)
\( \sigma^2_x = a^2\cdot \sigma^2_u + b^2\cdot \sigma^2_v + 2ab\cdot \sigma^2_{uv}\)
Multiplication
\( x = a \cdot u\cdot v \)
\( \dfrac{\sigma^2_x }{x^2} = \dfrac{\sigma^2_u }{u^2} + \dfrac{\sigma^2_v }{v^2} + 2 \dfrac{\sigma^2_{uv} }{u \ v} \)
Division
\( x = \dfrac {a \cdot u}{v}\)
\( \dfrac{\sigma^2_x }{x^2} = \dfrac{\sigma^2_u }{u^2} + \dfrac{\sigma^2_v }{v^2} - 2 \dfrac{\sigma^2_{uv} }{u \ v} \)
Exponentials
\( x = a \cdot e^{bu} \)
\( \dfrac{\sigma_x}{x} = b \ \sigma_u \)
\( x = a^{bu}\)
\( \dfrac{\sigma_x }{x} = b \ln{(a)} \cdot \sigma_u \)
\( x = a \ln{(bu)} \)
\( \sigma_x = ab \dfrac{\sigma_u}{u} \)
Powers
\( x = a \cdot u^b\)
\( \dfrac{\sigma_x }{x} = b \ \dfrac {\sigma_u}{u} \)
Trig
\( x = a \cos{(bu)} \)
\( \sigma_x = -\sigma_u \ ab \sin{(bu)} \)
\( x = a \sin{(bu)} \)
\( \sigma_x = \sigma_u \ ab \cos{(bu)} \)
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Schedule
- Overview Meche PE
- Product Dev. Process
- Explore → Hands On
- I - Principles
- → Foundation
- → A - Basic Engineering Practices
- → B - Engineering Science and Mechanics
- → C - Material Properties
- → D - Strength of Materials
- → E - Vibration
- II - Applications
- → A - Mechanical Components
- → B - Joints and Fasteners
- → C - Supportive Knowledge
- III - Holistic Knowledge
- → A - Heat Transfer
- → B - Fluid Dynamics
- → C - Controls
- → D - Thermodynamics
- → E - Electronics
- IIII - Precision Machine
- → A - Error Apportionment
- → B - Metrology