One applied math question talks about an elastic bar with a displacement equation $du^2/dx^2 = 1$ for a uniform load and fixed on both ends. I solved this using the Toeplitz matrix K and u comes out to be a parabola. I am not however able to visualize this uniform load in my head: Teacher says imagine small round balls connected to eachother by threads, hanging vertically. But then if both ends are fixed, arent these balls piled up on eachother? The result being a parabola suggests that there is more displacement in the middle. I also cannot visualize this. A math book had this:
I guess this is not a fixed-fixed situation but it shows the displacement $u(x)$ at least. Should I think about this bar horizontally instead of vertically? Maybe it would make more sense that way.
Any ideas?
Addition: I simulated the system using KineticsKits and VPython
http://kineticskit.sourceforge.net
And using this code
from KineticsKit import * from visual import vector system = System(timestep=0.04, gravity=1) ## generate some masses mass1 = Mass(m=0.1, pos=(0.0, 0.0, 0.0), fixed=1) mass2 = Mass(m=0.1, pos=(0.0, 0.5, 0.0)) mass3 = Mass(m=0.1, pos=(0.0, 1.0, 0.0)) mass4 = Mass(m=0.1, pos=(0.0, 1.5, 0.0)) mass5 = Mass(m=0.1, pos=(0.0, 2.0, 0.0)) mass6 = Mass(m=0.1, pos=(0.0, 2.5, 0.0)) mass7 = Mass(m=0.1, pos=(0.0, 3.0, 0.0), fixed=1) ## insert them into the system system.insertMass(mass1) system.insertMass(mass2) system.insertMass(mass3) system.insertMass(mass4) system.insertMass(mass5) system.insertMass(mass6) system.insertMass(mass7) spring1 = SingleHelixSpring(m0=mass1, m1=mass2, k=1, damping=0.5) system.insertSpring(spring1) spring2 = SingleHelixSpring(m0=mass2, m1=mass3, k=1, damping=0.5) system.insertSpring(spring2) spring3 = SingleHelixSpring(m0=mass3, m1=mass4, k=1, damping=0.5) system.insertSpring(spring3) spring4 = SingleHelixSpring(m0=mass4, m1=mass5, k=1, damping=0.5) system.insertSpring(spring4) spring5 = SingleHelixSpring(m0=mass5, m1=mass6, k=1, damping=0.5) system.insertSpring(spring5) spring5 = SingleHelixSpring(m0=mass6, m1=mass7, k=1, damping=0.5) system.insertSpring(spring5) count = 0 loc_1 = [mass2.sphere.pos.y, mass3.sphere.pos.y, mass4.sphere.pos.y, mass5.sphere.pos.y, mass6.sphere.pos.y] while 1: system.step() count += 1 if count == 100: break loc_2 = [mass2.sphere.pos.y, mass3.sphere.pos.y, mass4.sphere.pos.y, mass5.sphere.pos.y, mass6.sphere.pos.y] from itertools import izip for x,y in izip(loc_1, loc_2): print x-y
I do see the output as
0.237135416331 0.377720360592 0.42427376458 0.377720360592 0.237135416331
which shows more displacement in the middle. And starting from equal positions, after gravity effects are complete, the balls look like this (before left, after right)
How geeky is that? :)