Download presentation
Presentation is loading. Please wait.
Published byDulcie Cole Modified over 9 years ago
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.