Obtaining mathematical model for the kinetic system

[ssulun]

Homework Statement

See attachment.

f(t) is the input force and b1 and b2 are kinetic friction constants. There is no static friction.

Homework Equations

$$ƩF = m \ddot{x}$$
$$F_s=kx$$
$$F_f=b\dot{x}$$

Ff is the force from friction and Fs is the force from spring.

The Attempt at a Solution

$$m_1\ddot{x_1}=-b_1\dot{x_1}-k_1x_1-k_1x_2$$
$$m_2\ddot{x_2}=-b_2\dot{x_2}-k_1x_1-k_1x_2-k_2x_2$$
$$m_3\ddot{x_3}=f$$

I have two questions:

1) Should I include the friction between m1 and m3 and m2 and m3 in the equation for x3 and why?

2) When I imagine this system, I think that f(t) should definitely affect x1 and x2, but in my equations it doesn't. Where am I doing wrong?

Thanks in advance.

 

[azizlwl]

Let me try, hope somebody will correct me.
Taking springs as massless.

m₁a=k₁x₁+b₁m₁g
m₂a=k₂x₂+b₂m₂g-k₁x₁
(m₁+m₂+m₃)a=f(t)

 

[ssulun]

b1 and b2 are kinetic friction constants, and k1 spring is squeezed from both x1 and x2 so I can change those parts, but your work gave me new ideas, thank you.

 

[ssulun]

I have modified it as:

$$m_1\ddot{x_1}=-b_1\dot{x_1}-k_1x_1-k_1x_2$$
$$m_2\ddot{x_2}=-b_2\dot{x_2}-k_1x_1-k_1x_2-k_2x_2$$
$$(m_1+m_2+m_3)\ddot{x_3}=f$$

But still, x1 and x2 don't depend on f and that bothers me.

 

[ssulun]

I should have written the effect of k1 in the equation 1 as k1(x1+x2), not k1x1.

Also, the k2 spring will pull m3 with k2x2 (to balance the forces on the k2 spring). So should I write the equation for x3 as:
$$m_3\ddot{x_3}=f-k_2x_2$$
or should I write an overall system with (m1+m2+m3) and include the unbalanced forces from b1 and b2?

I am very confused.