#7: Spring Equilibrium
Fall 2020
Problem
Given
free objects P1 , P2 , · · · , Pn
fixed objects F1 , F2 , · · · , F`
springs S1 , · · · , Sm with elastic coefficients(탄성계수) w1 , · · · , wm
I each spring connect free/free objects or free/fixed objects
find equilibrium points of each free objects
Simplifying Assumption
Assumption: Initial length of each spring is 0
Then, the forces Fx and Fy are represented simply:
F = w ·∆` (since the initial length is 0)
Fx = F ·cos θ = w ·∆`·cos θ = w ·∆x
Fy = F ·sin θ = w ·∆`·sin θ = w ·∆y
Observations
Let (xi , yi ) be the position of free object Pi in equilibrium
I Thus, there are 2n variables x1 , · · · , xn , y1 , · · · , yn
A pair of linear equations can be derived for each Pi
I x-direction: w4 ·(x1 −x3 ) + w5 ·(x2 −x3 ) + w7 ·(x4 −x3 ) = 0
I y-direction: w4 ·(y1 −y3 ) + w5 ·(y2 −y3 ) + w7 ·(y4 −y3 ) = 0
With these 2n linear equations, we can find the value of 2n
variables x1 , · · · , xn , y1 , · · · , yn
Deriving a pair of equations of each free object
For each free object Pi , let
Ai = the set of indices of adjacent free objects (A1 = {2, 3})
Bi = the set of indices of adjacent fixed objects (B1 = {1})
wik = elastic coefficient of the spring that connects Pi and Pk
wik0 = elastic coefficient of the spring that connects Pi and Fk
Linear equations for each Pi
P 0
wik ·(xk0 −xi ) = 0
P
wik ·(xk −xi ) + (xk0 : x-coord of Fk )
k∈A i k∈B
P 0i
wik ·(yk0 −yi ) = 0
P
wik ·(yk −yi ) + (yk0 : y-coord of Fk )
k∈Ai k∈Bi
Collecting into a system of linear equations
By symmetry, it suffices to consider equations from x-direction only
P 0
wik ·(xk0 −xi ) = 0
P
wik ·(xk −xi ) +
k∈Ai k∈Bi
With some manipulation,
!
P P 0 P P 0 0
wik + wik ·xi + −wik · xk = wik ·xk
k∈Ai k∈Bi k∈Ai k∈Bi
a11 x1 + a12 x2 + . . . + a1n xn = b1 ···
··· ai1 x1 + . . . + aii xi + . . . + ain xn = bi
an1 x1 + an2 x2 + . . . + ann xn = bn ···
What are the coefficients of the above system of linear equation?
P
0
P
k∈Ai wik + k∈Bi wik if i = j
aij = −wij elif Pi and Pj are connected
0 else
0 0
P
bi = k∈Bi wik ·xk (= 0 if Pi is not adjacent to a fixed object)
How to efficiently construct the system of linear equations
P
0
P
k∈Ai wik + k∈Bi wik
if i = j
aij = −wij elif Pi and Pj are connected
0 else
0 0
P
bi = k∈Bi wik ·xk (= 0 if Pi is not adjacent to a fixed object)
Input is given as a list of informations of each spring
I elastic coefficient, free object ID, free object ID or
I elastic coefficient, free object ID, x/y-coord of fixed object
Initially, set all aij and bi to 0
For each spring that connects Pi /Pj , accumulate aii , ajj , aij , aji
I aii += wij , ajj += wij , aij -= wij , aji -= wij
For each spring that connects Pi /Fj , accumulate aii , bi
I aii += wij0 , bi += wij0 ·xj0
Do similarly for a system of equations from y-direction
Finally, run Gaussian Elimination twice to obtain (xi , yi )’s!
The system of linear equations for the example input
wij = wji
0 (x 0 −x ) = 0
P1 : w12 (x2 −x1 ) + w13 (x3 −x1 ) + w11 1 1
0 0 0
I (w
12 +w13 +w11 )x1 −w12 x2 −w13 x3 = w11 x1
0 (x 0 −x ) + w 0 (x 0 −x ) = 0
P2 : w21 (x1 −x2 ) + w23 (x3 −x2 ) + w22 2 2 23 3 2
I −w 0 0 0 0 0 0
21 x1 + (w21+w23+w22+w23 )x2 −w23 x3 = w22 x2 + w23 x3
P3 : w31 (x1 −x3 ) + w32 (x2 −x3 ) + w34 (x4 −x3 ) = 0
I −w
31 x1 −w32 x2 + (w31 +w32 +w34 )x3 −w34 x4 = 0
0 (x 0 −x ) + w 0 (x 0 −x ) = 0
P4 : w43 (x3 −x4 ) + w44 4 4 45 5 4
I −w 0 0 0 0 0 0
43 x3 + (w43 +w44 +w45 )x4 = w44 x4 + w45 x5
0 0 0
w12 +w13 +w11 −w12 −w13 0 w11 x1
0 0 0 0 0 0
−w21 w21 +w23 +w22 +w23 −w23 0
x =
w22 x2 +w23
x3
−w31 −w32 w31 +w32 +w34 −w34 0
0 0 0 0 0 0
0 0 −w43 w43 +w44 +w45 w44 x4 +w45 x5
To which aij ’s does w13 (= w31 ) contribute?
