Couple questions:

1. Is figure 1 the initial condition? That is, are all the masses all and all the spring forces are zero?

I would like calculate the new position of m2 and m3 mass after that user move the m4 mass.

2. If you're only interested in calculating the final position of m4, you don't need to consider damping coefficients. Damping is only a function of velocity. When all the masses are at rest there is no damping. The mass positions will all be colinear (all in a line).

This is just a quick derivation, so double check the algebra!

The letter L represents the distance between two masses
and assumes each spring was at it's rest position before m4 was moved

and ALL k's and distances are > 0.

Forces on m2 must sum to 0 since masses are at rest:
k12 * L12 = k24 * L24, so
1) L12 / L24 = k24 / k12

Forces on m3:
k13 * L13 = k34 * L34
2) L13 / L34 = k34 / k13

Distances:
You know L14 from the final position of m4, so:

L12 + L24 = L14
L13 + L34 = L14

L12 = L14 - L24
1) becomes ( L14 - L24 ) / L24 = k24 / k12
or L14 / L24 - 1 = k24 / k12
L14 / L24 = k24 / k12 + 1
L24 = L14 / ( k24 / k12 + 1 ) [everything on the right is known]

Similarly for L13 / L14

FYI: The answer I posted was just the implementation of the formulae in the image you posted, after an infinite amount of time.

The question you originally posted is greatly different from implementing the model described in the pdf! The model is for a dynamic situation, changing over time (between the start and end points,) and, using your original post as an example, would take into consideration contraints and other forces on m2 and m3, possibly from other masses not shown. In your posted example, when m4 is moved, m2 and m3 will oscillate, and continue to oscillate after m4 has come to rest. If they're free to move without other constraint, eventually the damping will "stop" m2 and m3, and they will come to rest as I described. If you're not interested in the behavior of m2 and m3 between the start and end of their movement, the method in the article may not be what you're looking for.

With regard to modeling the system described in the article, you may find someone (other than me) that will work with you on that. I would, however, suggest that you fully understand what you really want to do, and whether implementation of that model is the correct approach.

Whatever you decide, I hope it works for you.