- An Intersting Thing Happens When Springs Systems Have No Attachments To The Outside Consider The Following Free System 1 (47.35 KiB) Viewed 33 times
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 = A'CAU = f. (1) Write the LU decomposition for AT LIE AT (2) Using variables for the components of f, divide by L (the left matrix above) using forward substitution. L h府所 (Write f1 for fi, and f2 for f2, and f3 for f3-) 同 (3) Since the bottom row of U is all O, there will only be a solution to Uu bif 0 (4) Did you really understand this? Write a nonzero force vector which is balanced by displacement. f = 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.
An intersting thing happens when springs systems have no attachments to the outside. Consider the following free system. £ 500004 3000 5 with spring constants c = C2 Assume down is the positive direction. Write the elongation matrix. A = 00 Free Displacements. V
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 Atw = 0 (forward substitution). w 8 (2) Solve Ce = w (forward substitution). e (3) Solve Au = e (back substitution). 1 u = t 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 = A'CAu=f. (1) Write the LU decomposition for AT AT (2) Using variables for the components of f, divide by L (the left matrix above) using forward substitution. - [] Write f1 for fi and 12 for f3, and 43 for fs) (3) Since the bottom row of U is all O, there will only be a solution to 1:1 le Uu If (4) Did you really understand this? Write a nonzero force vector which is balanced by displacement. 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.
Final Comments. Note that both of the parts above (finding motions that did not cause force and forces that were not balanced by motion) relied only on the elongation matrix A rather than all of K. We will see this again later when we look at collapse mechanisms for truss systems. • To find motions that did not cause internal force, we computed the null space of A. • To find forces that could not be balanced by internal displacement (and thus would cause the whole spring system to move without any equilibrium), we computed the column space of AT. The column space of AT is equal to the row space of A which is perpendicular to the null space. This is why your answers to the first and second parts looked so similar. (The motion vector in the first part is the normal vector for the plane of forces in the second part.)