Equation of Motion for 2 DOF spring damper system

[ufone317]

Please guide me towards the "differential equation of motion" for the following 2 DOF Spring-damper system.

And furthermore, if above system is in a uniform speed rotating frame, then what can be the effect on this system?


Thank you very much.

 

[Astronuc]

Please write the equation for one dof.

Does gravity g apply to y-direction?


What force must be considered in a rotating body?

 

[ufone317]

Thank you for your reply.

m*(d²x/dt²)+c*(dx/dt)+k*x = 0 is the equation for one axis.

No effect of gravity is considered here.

The whole frame is rotating at an uniform angular velocity.

If you can provide equation for static (i-e. not rotating) case, that's also fine.

 

[Barkan]

Correct me if I'm wrong.

Here are the motion equations;

mx'' + c1x' + k1x' = 0 (Not forced)
my'' + c2y' + k2y' = 0 (Not forced)

Suppose the frame is rotated thru angle q,

In this case, the mass center position with respect to the frame F is

Xnew = x*cos(q) - y*sin(q)
Ynew = y*cos(q) + x*sin(q)

You can simply differentiate Xnew and Ynew. Once and twice, then replace in motion equations

Xnew' = cos(q)*(x'-yq*') - sin(q)*(y'+xq*')
Xnew'' = x''*cos(q) - 2*x'*q'*sin(q) - x*q''*sin(q) - x*q'*q'*cos(q)
-y''*sin(q) - 2*y'*q'*cos(q) - y*q''*cos(q) + y*q'*q'*sin(q)

Note that q''=0. Eliminate some of the terms above and do the same thing for y-axis

 

[paranormal]

The equation of motion for a 2 DOF spring-damper system can be expressed as a set of coupled differential equations. Let x1 and x2 be the displacements of the two masses, m1 and m2, respectively. The forces acting on the masses are the spring forces, k1x1 and k2(x2-x1), and the damping forces, c1(dx1/dt) and c2(dx2/dt). The equation of motion can be written as:

m1(d^2x1/dt^2) = -k1x1 - k2(x2-x1) - c1(dx1/dt)

m2(d^2x2/dt^2) = k2(x2-x1) - c2(dx2/dt)

This is a set of second-order ordinary differential equations, which can be solved using numerical methods or analytical techniques, such as the method of undetermined coefficients or Laplace transforms.

In a uniform speed rotating frame, the system experiences an additional pseudo force due to the acceleration of the frame. This force can be included in the equations of motion by adding the term -m1ω^2x1 and -m2ω^2x2, where ω is the angular velocity of the frame. This will result in a modified set of equations of motion:

m1(d^2x1/dt^2) = -k1x1 - k2(x2-x1) - c1(dx1/dt) - m1ω^2x1

m2(d^2x2/dt^2) = k2(x2-x1) - c2(dx2/dt) - m2ω^2x2

The effect of the rotating frame on the system will depend on the magnitude and direction of the angular velocity. If the angular velocity is small, the effect may be negligible. However, if the angular velocity is large, it can significantly impact the motion of the masses, potentially leading to unstable or chaotic behavior. It is important to carefully consider the effects of a rotating frame when analyzing the behavior of a 2 DOF spring-damper system.