# Multiple mass-spring-damper system

1. Oct 30, 2004

### Jamesss

I posted this in the general forum but it probably belongs here.
This is a tricky one!

I'm unsure about the forces on the ground level.

The problem is a 2-spring-damper system with three levels. A ground level, level 1 and level 2. Then a force is applied to the ground level simulating an earth quake (eg the force could be sinusoidal). See the attachment for a diagram.

These are the forces I have resolved:

-level 2 mass:
gravitational force, m0*g (down)
Spring Force2, K1*(x1-x0) (up)
Damper Force2, b1*(x1'-x2') (up)
Inertial Force, m*x0'' (down)

-level 1 mass:
gravitational force, m*g (down)
Spring Force1, K1*(x2-x1) (up)
Damper Force1, b1*(x2'-x1')(up)
Inertial Force, m*x1'' (down)
Spring Force2, K1*(x1-x0) (down)
Damper Force2, b1*(x1'-x2') (down)

-level ground mass:
driving force (earthquake) = some function eg sine
g-force, mg (down)
Spring Force1, K1*(x2-x1) (down)
Damper Force1, b1*(x2'-x1')(down)
Inertial Force = m*x2'' (down)

I have tried to solve this using simulink, but for the positions of each level they fall towards negative oblivion. I am thinking I either have forgotten some force added too many.

Have I left anything out? *thinking*
James

#### Attached Files:

• ###### mass.JPG
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2. Oct 30, 2004

### marlon

Hi, this looks a bit like active suspension control...

i get following equations :

m0x0" = -k1(x0-x1)-b1(x0'-x1') + U
m1x1" = k1(x0-x1) + b1(x0'-x1') - U - k1(x1-x2) - b1(x1'-x2')
m2x2" = k1(x1-x2) + b1(x1'-x2') - U

Ofcourse you need to add the overall gravitational force which you implemented correctly in my opinion.

x : position
x' : velocity
x" : acceleration

just for clarity...

regards
marlon

3. Oct 30, 2004

### marlon

m0 is level two
m1 is level 1
m2 is groundlevel

Just to make sure, ok ???

regards
marlon

4. Oct 30, 2004