mass-spring-damper equation

[h4tt3n]

Hello, I'm tampering with a little computer simulation of mass spring damper systems, and somewhere on the net I stumbled across this equation:

 

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

 

F = force

m = mass

g = gavity

k = spring constant

D = string length displacement

Vx = Velocity X

Vy = Velocity Y

 

Maybe someone here can shed a little light on whats going on here and wether this equation is even right? And what does i and j stand for?

[swansont]

i and j are most likely unit vectors in the x and y directions, respectively. Usually they have a little hat (^) on them.

 

It does look like a generic damped oscillator under gravity that can move in two directions: up/down and side-to-side.

[h4tt3n]

Thanks for the reply swansont. It makes sense, but only to a certain degree...

 

If I split the equation up I get:

 

gravity: F = mgj

 

Hooke's law on spring force: F = -k D

 

damping: F -= b (Vx i + Vy j)

 

But what is this for: (sin θ i + cos θ j) ??

 

I've tried implementing the equation, and it works fine, except for the above part. Apparently i and j are just vctors and not unit vectors like you suggsted.

 

Force = -Stiffnes*(Distance-relaxed_distance) - damping*(vel.x*dist.x + vel.y*dist.y)

 

 

By running the attached .exe file you can see the equation in action. Klick-drag the pin, ball or rubberband to interact with the simulation.

 

regards,

Michael

rubberband.zip

[swansont]

"unit" means they have a length of 1

 

The trig factors give you the components if theta is the deviation from vertical.

[OzoneHole]

I understand the equation of a damped mass system (spring plus dashpot) when one end is fixed to a wall as is described in most textbooks. However, I need an equation of the more interesting case where two free floating masses are connected by a single axis spring and a dashpot. How would I model the motion of a seat that is attached to a boat by a damping device? If I can measure accelerations in three axes at the base of the damping device, how do I describe the forced vertical motion of the seat that sits on top of the damper? What is the total mass that I insert into the equation, the mass of the seat plus the mass of the damper or the mass of the seat plus the mass of the damper plus the mass of the boat?

 

Additionally, what is the best way to model the flexibility of the other two axes of the damper system (the almost rigid axes where the damping device bends just a little)? I have heard of the 'hysteretic' model of a beam, but I 1) do not know how to properly force it in this case, and 2) I do not know how to interpret the complex results in a meaningful way. Do I just drop the complex part or does the magnitude represent the deflection? Can I separate each axes of motion or are coupled equations necessary?

 

Thanks for you help!

[swansont]

Sounds like in this setup the boat + damper is one mass, and the seat is the other.

 

You should be able to separate the axes to at least give you the approximation in the case of rigid materials. I imagine the solution where this approximation is not valid is quite ugly.

[Newbies_Kid]

Hello, I'm tampering with a little computer simulation of mass spring damper systems, and somewhere on the net I stumbled across this equation:

 

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

 

F = force

m = mass

g = gavity

k = spring constant

D = string length displacement

Vx = Velocity X

Vy = Velocity Y

 

Maybe someone here can shed a little light on whats going on here and wether this equation is even right? And what does i and j stand for?

 

k = spring constant or a damping coefficient?

D = string length displacement or a symbol for derivation of (t) (d/dt)??

and b should be the spring constant but..

(sin θ i + cos θ j)?? :confused: