1. EULER, CARNOT, TORSION by Evie Antholis A Cluster 3 Production

2. LEONHARD EULER

3. LIFE  1707-1783  Swiss  Majored in philosophy  Taught by Johann Bernoulli  Taught at the Imperial Russian Academy of Sciences and Berlin Academy  Cyclops

4. CONTRIBUTIONS  Mathematical Notation  Power Series  Euler Identity (“beautiful”)  Geometry  Graph theory/ Topology  Applied Mathematics  Physics:  Euler’s Equations  Fluid Dynamics and Incompressible Flow  Beam Theory

5. CONTRIBUTIONS CONTINUED (DUH!) Euler – Bernoulli Beam Theory  Developed with Daniel Bernoulli (aka Classical Beam Theory)  Fourth-order ODE  BVP Euler Equations

6. NICOLAS LÉONARD SADI CARNOT

7. LIFE  1796-1832  French  Tried to advance steam engine technology  Analyzed the connection between fuel consumption, work, and expansion and compression of steam.  Died in a cholera epidemic

8. CARNOT CYCLE

9. TORSION

10. WHAT IS TORSION? & APPLICATIONS  What is torsion?  Applications of torsion  Torsion occurs everywhere!!!!!  It is important to understand torsion to evaluate the durability of shafts and other members > failure  Manufacturing  Machines  Vehicles and Airplanes  Etc.

11. TORSION OF A CIRCULAR SHAFT  Circular cross sections do not deform  Axial strain does not occur  γ = ρ (d ϕ /dx)  T = ( τ max /c) J  Polar Moment of Inertia = J Important Terms to Know: -Shear Strain γ -Angle of twist ϕ -Radius ρ -Shear Stress τ -External/Applied Torque Τ -Polar moment of inertia J

12. Hooke’s Law & Shear Modulus  F = k∆L which can also be written as….  σ =E ε and can be applied to stress and strain like so…  τ =G γ  Aka the modulus of rigidity  G  = shear stress/shear strain  = τ / γ Important Terms to Know: -Shear Strain γ -Angle of twist ϕ -Radius ρ -Shear Stress τ -External/Applied Torque Τ -Polar moment of inertia J -Normal Stress σ -Normal Strain ε

13. Poisson’s Ratio  When a load is applied to a material strains will occur both perpendicular and parallel to the direction of the load  ν = - d ε transverse /d ε axial  ν remains constant for elastic, homogenous, and isotropic materials Connecting E, G, and υ

14. TORSION OF NON-CIRCULAR SHAFTS  Mini History Lesson:  Before 1820, it was believed that shear stress of a member is always proportional to the distance from the longitudinal axis  But alas, it is not the case, for non- circular cross-sections.  A. Duleau  1855 – Saint-Venant published the first correct analysis  Every section will warp and not remain a plane when torsion occurs  Except for circular cross-sections  Ex. Torsion of a shaft with rectangular cross-sections  τ max = Τ /( α a 2 b)  α can be determined from a table  a and b are the sides of the rectangular cross section

15. FIN

16. WORKS CITED  http://www.maverickmath.org/2015/01/26/hed/ http://www.maverickmath.org/2015/01/26/hed/  http://www.wikiwand.com/fr/Sadi_Carnot_(physicien ) http://www.wikiwand.com/fr/Sadi_Carnot_(physicien  http://www.biography.com/people/leonhard-euler-21342391 http://www.biography.com/people/leonhard-euler-21342391  http://www.britannica.com/biography/Sadi-Carnot-French-scientist http://www.britannica.com/biography/Sadi-Carnot-French-scientist  https://www.youtube.com/watch?v=rBsWAfTWOks https://www.youtube.com/watch?v=rBsWAfTWOks  Philpot, Timothy A. Mechanics of Materials: An Integrated Learning System. 2nd ed. Hoboken, NJ: John Wiley, 2011. Print.  https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/PoissonRatio.svg/300px-PoissonRatio.svg.png https://upload.wikimedia.org/wikipedia/commons/thumb/e/ec/PoissonRatio.svg/300px-PoissonRatio.svg.png  http://engineering-references.sbainvent.com/strength_of_materials/pictures/shear_poisson_equation.jpg http://engineering-references.sbainvent.com/strength_of_materials/pictures/shear_poisson_equation.jpg  http://civil.njit.edu/images/torsion1.gif http://civil.njit.edu/images/torsion1.gif  http://img01.ibnlive.in/ibnlive/uploads/2013/04/leonhard-euler-google-doodle-150413.jpg http://img01.ibnlive.in/ibnlive/uploads/2013/04/leonhard-euler-google-doodle-150413.jpg  https://www.grc.nasa.gov/www/k-12/airplane/carnot.html https://www.grc.nasa.gov/www/k-12/airplane/carnot.html  https://www.grc.nasa.gov/www/k-12/airplane/eulereqs.html https://www.grc.nasa.gov/www/k-12/airplane/eulereqs.html  http://www.colorado.edu/engineering/CAS/courses.d/AVMM.d/AVMM.Ch08.d/AVMM.Ch08.pdf http://www.colorado.edu/engineering/CAS/courses.d/AVMM.d/AVMM.Ch08.d/AVMM.Ch08.pdf