Analyzing the free body diagram around the three masses with Newton's second law (Sigma F=ma)
resulted in the following three equations, which form a tridiagonal and symmetrical system.
[[(k_(1)+k_(2))/(m_(1)),-(k_(2))/(m_(1)),0],[-(k_(2))/(m_(2)),(k_(2)+k_(3))/(m_(2)),-(k_(3))/(m_(2))],[0,-(k_(3))/(m_(3)),((k_(3)+k_(4)))/(m_(3))]][[x_(1)],[x_(2)],[x_(3)]]=[[a_(1)],[a_(2)],[a_(3)]]
Find the unknown displacements of each of the masses in the above system (x_(1),x_(2),x_(3)) with:
(a) Thomas algorithm
(b) Cholesky's decomposition
Knowing that for masses 1, 2, and 3:
*k_(1)=k_(4)=20(N)/(m)
*k_(2)=k_(3)=60(N)/(m)
*m_(1)=m_(2)=m_(3)=6kg,
*a_(1)=0.25(m)/(s^(2)),a_(2)=0.7(m)/(s^(2)), and a_(3)=0.5(m)/(s^(2))
3. (25%) Mass and spring systems are often used in civil engineering to analyze structures and equipment that are subject to accelerations and displacements that vary over time. For example, for a system of three masses and four springs:
000
m2
m3
000
Analyzing the free body diagram around the three masses with Newton's second law (F = ma) resulted in the following three equations, which form a tridiagonal and symmetrical system.
k1+k2 k2 0 m ms k2 k2+k3 k3 [x1] X2 m2 m2 m2 [x3] k3 (kz+k4) 0 m3 m3
[a1] ar [a3]
Find the unknown displacements of each of the masses in the above system (x1, X2, x3) with: (a) Thomas algorithm (b) Cholesky's decomposition
Knowing that for masses 1, 2, and 3: k=k4=20 N/m k2=k3=60 N/m m=m=m=6kg a1 = 0.25 m/s2, a2 = 0.7 m/s2, and a3 = 0.5 m/s2