Scaling and dimensional analysis - uliege.beCedric J. Gommes What is scaling? Not all physical and...
Transcript of Scaling and dimensional analysis - uliege.beCedric J. Gommes What is scaling? Not all physical and...
Cedric J. Gommes
What is scaling?
Not all physical and chemical phenomena are affected in the same way by a change in length scale. When the size is changed, the structure of system has also to be changed to serve a given function
Cedric J. Gommes
Walking is a complex problem
l At least 6 degrees of freedom for each leg
2 for the ankle 1 for the knee 3 for the hip
l About 29 important muscles in
each leg
Cedric J. Gommes
Dimensional analysis of walking
l The velocity v [v] = L T l The length of the leg l [l] = L l The animal weight m [m] = M l Gravity g [g] = L T-2
l The stride length s [s] = L l …
What are the main variables relevant to walking?
Cedric J. Gommes
Buckingham’s Π theorem
n = 5 parameters k = 3 dimensions
n-k = 2 dimensionless groups
The analysis of walking takes the simple form Y = f(X)
Cedric J. Gommes
Dimensional analysis of animal locomotion
Taking v and s as core variables, the two dimensionless numbers are found to be
Π1 = s/l Π2 = v²/lg
William Froude (1810-1879)
Froude number
Cedric J. Gommes
Real animals do obey dynamical scaling
R. McNeill Alexander, Walking and Running, The mathematical gazette, 80 (1996) 262
1 quadruped =
2 bipeds
Cedric J. Gommes
Dinosaurs ichnites
Chirotherium 250 Ma ago
Π1 = s/l is known for many extinct species
Brontosaur (30 tonnes) 1 m/s Tyrannosaur (5 tonnes) 2 m/s « Smaller ones » (0.5 tonnes) 12 m/s
R. McNeill Alexander, Walking and Running, The mathematical gazette, 80 (1996) 262
Cedric J. Gommes
Laetoli footprints (3.6 Ma ago)
Fossil footprints discovered in Tanzania in 1976. Assuming that they were left by Australopithecus afarensis, the speed was estimated to be 1 m/s
Cedric J. Gommes
Dimensionless groups are ratios of orders of magnitude
EnergynalGravitatioEnergyKinetic
mglmvFroude ==
2
It is physically impossible to walk at Froude > 1
walk trot gallop
Cedric J. Gommes
Why is it difficult to walk on the moon?
For the same v and l, the Froude is 6 times larger on the moon It is hard not to be in the running range
Cedric J. Gommes
There are as many different scaling laws as there are physical phenomena
l Dynamic equilibrium of the body l Static equilibrium of the body l Metabolism l …
This is what we just talked
about
The problem that is analyzed is determined by the choice of the dimensional variables
Cedric J. Gommes
Dimensional analysis of leg diameter
l The diameter of the leg d [d] = L l The animal weight m [m] = M l Gravity g [g] = L T-2
l The maximum stress that the leg can withstand σ [σ] = M L-1 T-2
What are the main variables relevant to leg diameter?
Cedric J. Gommes
Scaling of the diameter of a leg
constant2
==Πmgdσ
n – k = 1 A single dimensionless number
σmgd ≈
Galileo Galilei (1564-1642)
Cedric J. Gommes
Progress in King Kong movies
Merian Cooper, Ernest Schoedsack (1933) Peter Jackson (2005)
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Cedric J. Gommes
Qualitative aspects of scaling: the respiratory system
The respiratory system of insects is (mostly) passive; the size of insects is therefore limited by oxygen diffusion.
spiracles
Cedric J. Gommes
Dimensional analysis of insect breathing
l The size of the insect d [d] = L l Diffusion coeff of O2 D [D] = M2T-1
l O2 concentration C [C] = Mol L-3
l Insect metabolism rV [rV] = Mol L-3 T-1
What are the main variables relevant to insect breathing?
Cedric J. Gommes
Buckingham’s Π theorem
n = 4 variables k = 3 dimensions
n-k = 1 dimensionless group
rVd2
DC0
This is the insect’s Thiele modulus. Insects can live only if that number is small
Cedric J. Gommes
Qualitative aspects of scaling: the respiratory system
Meganeura (300 Ma ago)
75 cm
The giant insects of the Carboniferous could exist only because the oxygen concentration in the atmosphere was about 30-35%.
Polar gigantism dictated by oxygen availability, Nature 399 (1999) 114–115
Cedric J. Gommes
Summing up…
1. Understand the physics
2. Data reduction with dimensionless groups (Buckingham’s Π)
3. Dimensionless groups are ratios of orders of magnitude
With some experience, you can start the analysis dirtectly at this point, and jump from 1 to 3.
Identify dimensionless groups. Similarity is achieved when dimensionless groups take the same value
This is a key step: what are the relevant variables?
Cedric J. Gommes
Further reading
l A. Sonin, The Physical Basis of Dimensional Analysis (2001) http://web.mit.edu/2.25/www/pdf/DA_unified.pdf Very thorough: Π theorem, its justification and applications.
l J.B.S. Haldane, On being the right size http://irl.cs.ucla.edu/papers/right-size.pdf Beautiful essay about why animals have different sizes
l R. McNeill Alexander, Walking and Running, The mathematical gazette, 80 (1996) 262 http://www.jstor.org/pss/3619558 A very clear and entertaining paper by the biologist who estimated first the walking speeds of dinosaurs.