How is a 3D Mass-Spring-Damper System Expressed in Mathematical Physics?

[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: $(\sin \theta \cos \phi, \sin \theta \sin phi, \cos \phi)$ where z is "up."

($\theta$ is the polar angle and $\phi$ 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 anybody 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]

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

($\theta$ is the polar angle and $\phi$ is the azimuthal angle in standard spherical coordinates.)

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