Modelling a spring system with damping force and external forces

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SUMMARY

The discussion focuses on modeling a spring system with damping forces and external forces, specifically addressing critically damped systems. Participants inquire about examples of systems exhibiting damping without external forces and those with external forces. The equation presented, 10y'' + 9y" + 2y' = -2e^(-t/2), raises questions about the distinction between y'' and y", as well as the implications of modifying coefficients in the equation.

PREREQUISITES
  • Understanding of differential equations, specifically second-order linear differential equations.
  • Familiarity with concepts of damping in mechanical systems.
  • Knowledge of external forces in the context of Newtonian mechanics.
  • Basic grasp of exponential functions and their applications in dynamic systems.
NEXT STEPS
  • Research critically damped systems in mechanical engineering.
  • Study the implications of varying coefficients in differential equations.
  • Explore examples of damping in real-world spring systems.
  • Learn about the differences between notation in differential equations, specifically y'' versus y".
USEFUL FOR

Mechanical engineers, physics students, and anyone interested in the dynamics of spring systems and the effects of damping forces.

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Homework Statement
I know for a system with no external forces there are conditions for being underdamping, overdamping and critically damped. Is there also such conditions for systems having external forces acting on them also? Specifically, for the example 10y''+9y"+2y'=-2e^(-t/2) with conditions y(0)=0 and y'(0)=0, is the system critically damped?
Relevant Equations
10y''+9y"+2y'=-2e^(-t/2)
I think its critically damped by looking at the graph of the solution.
 
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Can you give an example of a system with no external forces that exhibits damping and a system with external forces that also exhibits damping? I do not understand your use of "external force" at least not in the Newtonian sense.

Also, if 10y''+9y"+2y'=-2e^(-t/2), why not 19y''+ 2y'=-2e^(-t/2)? Is there a real difference between y'' and y"?
 

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