1 Point An Intersting Thing Happens When Springs Systems Have No Attachments To The Outside Consider The Following Fr 1 (31.32 KiB) Viewed 88 times
1 Point An Intersting Thing Happens When Springs Systems Have No Attachments To The Outside Consider The Following Fr 2 (32.61 KiB) Viewed 88 times
(1 point) An intersting thing happens when springs systems have no attachments to the outside. Consider the following free system. m Clay To with spring constants c= Assume down is the positive direction. Write the elongation matrix -1 1 0 A= 0 -1 1 Free Displacements. Because the system is unattached, it is possible to displace the masses without causing any internal force. We will find a (nonzero) displacement vector u so that Ku = ATCAU -0. (1) Solve A'w O (forward substitution). 0 -- 0 (2) Solve Ce=w (forward substitution). 0 e = 0 (3) Solve Au e (back substitution) We say that u is the null space of the matrix K. All unattached systems have the same u (which you just computed) in their null space, which corresponds to displacing the entire spring system all together. Since it doesn't stretch any springs (e = 0), this displacement doesn't cause any force.
Balanced Forces. Just as there are certain displacements which cause no force for this system, many external forces cannot be balanced by displacement. We can compute which forces can be balanced by investigating when there is a solution to Ku = ATCAu = f. (1) Write the LU decomposition for AT 1 0 0 AT = -1 1 0 0 -1 1 (2) Using variables for the components off, divide by L (the left matrix above) using forward substitution. fi f2 = (Write f1 for fi, and 12 for f2, and f3 for f3-) a (3) Since the bottom row of U is all O, there will only be a solution to Uu= b if 0 (4) Did you really understand this? Write a nonzero force vector which is balanced by displacement. IT If you apply this force to the spring system, the masses won't just move to a new equilibrium position... instead the whole spring system will fly away.
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