Simultaneous equation of springs in series

[hallic]

I have a static spring system with 3 springs holding 2 masses in place I have the spring stiffnesses (K1=1N/m k2=3N/m k3=2N/m) and natural lengths (L1=3m L2=1m L3=2m) and the total length of the compressed springs (x1+x2+x3=4m)
I know that I derive simultanious equations and use the iterative or elimination method to solve what the x individual values are but I don't know how to get the equations for 3 springs in series and I can't find a relavent reference in my textbook other than the total of sthe spring constant

 

[HallsofIvy]

I have a static spring system with 3 springs holding 2 masses in place I have the spring stiffnesses (K1=1N/m k2=3N/m k3=2N/m) and natural lengths (L1=3m L2=1m L3=2m) and the total length of the compressed springs (x1+x2+x3=4m)
I know that I derive simultanious equations and use the iterative or elimination method to solve what the x individual values are but I don't know how to get the equations for 3 springs in series and I can't find a relavent reference in my textbook other than the total of sthe spring constant

Your question is rather vague. For one thing, what are you trying to find? The force each spring is exerting? The actual length of each of the compressed springs?

Let x1, x2, and x3 be the compressed length of the three springs, with x1 the length of the spring on the "left", corresponding to K1 and L1, x2 the length of the spring between the two masses, corresponding to K2 and L2, and x3 the length of the spring on the "right", corresponding to K3 and L3. The "compression" on the first spring is L1- x and so the force it is exerting is K1(L1- x1)= 1(3- x1). The compression on the second spring is L2- y and so the force it is exerting is K2(L2- x2)= 3(1- x2). The compression on the third spring is L3- x3 and so the force it is exerting is K3(L3- x3)= 2(2- x3).

The total force on the left mass is 3(1- x2)- 1(3- x1)= x1- 3x2, where I have taken the positive direction to be to the right. The total force on the right mass is 2(2- x3)- 3(1- x2)= -1+ 3x2- 2x3. Because the masses are not moving, those forces must be 0:
x1- 3x2= 0, -1+ 3x2- 2x3= 0, and, of course, x1+x2+ x3= 4.

 

[hallic]

what I was looking for was a set of simultaneous equations to use iterative and elimination methods on, thanks for the help