Equation of Motion for 2 DOF spring damper system

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Homework Help Overview

The discussion revolves around deriving the differential equations of motion for a two degrees of freedom (2 DOF) spring-damper system, particularly in the context of a rotating frame. Participants are exploring the implications of rotation on the system's dynamics and the effects of gravity.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster seeks guidance on the differential equations governing the system and questions the effects of rotation. Other participants discuss the equations for a single degree of freedom and the role of gravity in a rotating frame. There are attempts to derive motion equations and transformations related to the rotating frame.

Discussion Status

The discussion is active, with participants sharing equations and questioning assumptions about gravity and rotation. Some have provided equations for motion in a rotating frame, while others are exploring the implications of these transformations. No consensus has been reached, but various interpretations and approaches are being examined.

Contextual Notes

Participants are working under the assumption that gravity may not apply in the rotating frame, and there is a request for equations in both rotating and static cases. The discussion includes specific mathematical expressions and transformations that are being analyzed.

ufone317
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Please guide me towards the "differential equation of motion" for the following 2 DOF Spring-damper system.

Image.jpg


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.
 
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Please write the equation for one dof.

Does gravity g apply to y-direction?


What force must be considered in a rotating body?
 
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.
 
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
 

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