We refer to this constraint as force balance the

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Unformatted text preview: rium position of a system of springs. We consider a collection of n masses in R2 , with locations (x1 , y1 ), . . . , (xn , yn ), and masses m1 , . . . , mn . (In other words, the vector x ∈ Rn gives the x-coordinates, and y ∈ Rn gives the y-coordinates, of the points.) The masses mi are, of course, positive. For i = 1, . . . , n − 1, mass i is connected to mass i + 1 by a spring. The potential energy in the ith spring is a function of the (Euclidean) distance di = (xi , yi ) − (xi+1 , yi+1 ) 2 between the ith and (i + 1)st masses, given by 0 di < li Ei = 2 d ≥l (ki /2)(di − li ) i i where li ≥ 0 is the rest length, and ki > 0 is the stiffness, of the ith spring. The gravitational potential energy of the ith mass is gmi yi , where g is a positive constant. The total potential energy of the system is therefore n −1 Ei + gmT y. E= i=1 The locations of the first and last mass are fixed. The equilibrium location of the other masses is the one that minimizes E . (a) Show how to find the equilibrium pos...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at University of California, Berkeley.

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