How Do You Solve a Two-Mass, Two-Spring System with External Harmonic Force?

[qwertybob]

Homework Statement

I have a two spring 2 mass system, with masses m1 and m2, spring constants k1 and k2. There is also an external harmonic force acting on m2, $$F sin(\omega\ t )$$

I have to obtain the free ocsillation frequencies $$\omega_{1}$$ and $$\omega_{2}$$

Homework Equations

The Attempt at a Solution

i obtained the following eqns by applying Newtons 2nd law to each of the masses

$$m_{1}x_{1}'' = -k_{1}x_{1}+k_{2}(x_{2}-x_{1})$$

$$m_{2}x_{2}'' = -k_{2}(x_{2}-x_{1})+F sin(\omega\ t)$$

and this is where i am stuck, the notes I have suck tbh and i have no idea what to do next, would be very greatfull for some help.

 

[tiny-tim]

Welcome to PF!

I have a two spring 2 mass system, with masses m1 and m2, spring constants k1 and k2. There is also an external harmonic force acting on m2, $$F sin(\omega\ t )$$

uiop qwertybob! Welcome to PF!

I don't get it …

where are the masses and the springs in relation to each other? ​

 

[qwertybob]

sorry i guess i should have mentioned that, hopefully this makes it clear

/////[m1]//////[m2]

k1 connects m1 to a fixed point, k2 connects m1 and m2. displacement of m1 = x1, displacement of m2 = x2.

ty for the kind welcome

 

[tiny-tim]

uiop qwertybob!

(have an omega: ω )

k1 connects m1 to a fixed point, k2 connects m1 and m2. displacement of m1 = x1, displacement of m2 = x2.

ah! … a fixed point! … all is clear!

ok … multiply each equation by a constant, and add, so that takes the form (ax₁ + bx₂)'' = c(ax₁ + bx₂) + Gsinωt.