3d mass-spring-damper

dduardo

I know a 2d mass-spring-damper is expressed:

F = m g j − k D (sin θ i + cos θ j) − b (Vx i + Vy j)

m = mass
g = gavity
k = spring constant
D = string length displacement
Vx = Velocity X
Vy = Velocity Y

But how would you extend this to three dimensions?

Tide

You'll just need to express the components of the radial vector using polar and azimuth angles: [itex](\sin \theta \cos \phi, \sin \theta \sin phi, \cos \phi)[/itex] where z is "up."

([itex]\theta[/itex] is the polar angle and [itex]\phi[/itex] is the azimuthal angle in standard spherical coordinates.)

dduardo

Ah, spherical coordinate system. I should have realized that. So it should be the following for a case where the spring is anchored from above and the mass is dangling:

m (ax i + ay j + az k) = m g j − k D (sin(θ)cos(phi) i + sin(theta)sin(phi) j + cos(phi) k) − b (Vx i + Vy j + Vz k)

Now let's say a spring-damper is added to each face of a cube. I guess you could consider this new system a spring lattice but the ends of the springs are anchored instead of going to other masses. Could you apply a perpendicular rotation to the equation above for each face and sum up the forces due to each spring?

Just a note: I would eventually want to linearize the system by assuming very small deflections.

Tide

Sure, you could do that. It sounds like you're trying to analyze the motion of an atom in a lattice?

dduardo

No i'm trying to model the motion of the mass inside of a 3d MEMS (Micro electro-mechanical system) accelerometer. The system is basically composed of a cube in the center that is held in place by piezoelectric bridges. When the bridges are compressed they generate a voltage. Based on position of the mass I can figure out what type of votage i'm generating which thus tells me the acceleration.

Prince007

Mr dduardo you did not mentioned what does 'F' mean here ?

Prince007

I am looking for a finite element model (actually a 2D spring mass lattice model which has springs not only at its sides but also 4 sides spring crossings at the center like 'x' or 2 sides spring crossings at the center like '/'), can be extended upto infinite length. I need the equations of motion for frequency and (phase)velocity with pre-stress and stressed conditions.
If any body does know any helping material, paper, book, website or software for this. Then let me know, it would be a nice help for me.Thanks !

ashapi

how would you do the same for an n-dimensional case?