Can gravity be ignored in a spring problem with multiple masses and springs?

[sol66]

So I have a spring problem where I have two masses and three springs hanging down from a ceiling. Each spring has a different force constant and each mass has a different weight.

|.....|
|Spring 1...|Spring 3
Mass 1...|
|.....|
|Spring 2...|
...Mass 2...
That is the diagram. Anyways I'm constructing a matrix to find my normal modes of oscillation being dependent on the displacement of Mass 1 and the displacement of Mass 2. I know that for a spring problem that contains only one spring with a hanging mass that I can ignore gravity and solve for the homogenous complementary solution finding my angular frequencies. Then to get the complete solution and just add the particular solution that takes gravity into account. My question is this ... for this particular problem, when I create my K matrix to solve for my normal modes, normal coordinates, and angular frequencies can I simply leave out gravity? Unless the force of gravity was somehow dependent on x position, which I can't see ... I don't see a reason/ way to add it in my K matrix.

 

[AlephZero]

Start by solving the statics problem to find the extensions of the springs caused by gravity.

Then set up the dynamics equations using the displacement from the static position. If you draw a free body diagram for each mass, it should be obvious that the weight and "statics" forces in the springs will sum to zero if you solved the statics problem correctly, so those forces will not affect the dynamics equations.