A boat and trailer are being pulled along a bumpy road at a velocity

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SUMMARY

The discussion focuses on modeling the dynamics of a boat and trailer being pulled over a bumpy road, approximated as a sine wave with a wavelength of 10 ft and an amplitude of 0.5 in. The system is characterized as an underdamped spring-damper system, with a damping coefficient of 0.05 and a deflection of 1.5 in due to the boat's weight. Key equations involve the motion of the system represented by mx'' + cx' + kx = ky + cy', where the natural frequency is calculated as ωn² = k/m = 257.6. The main questions revolve around determining the speed v for maximum amplitude and the amplitude at a speed of 55 mph.

PREREQUISITES
  • Understanding of underdamped spring-damper systems
  • Familiarity with differential equations in mechanical systems
  • Knowledge of transfer function analysis
  • Basic principles of harmonic motion and resonance
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  • Learn about transfer function analysis in control systems
  • Explore the effects of damping on amplitude and frequency response
  • Investigate the calculation of natural frequency in spring-mass systems
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A boat and trailer are being pulled along a bumpy road at a velocity v. The contour of the road can be approximated by a sine wave with a wavelength l of 10ft and an amplitude y of .5 in. the deflection of the springs in the trailer due to the boats weight is 1.5in. the damping of the system is viscous in nature and has a magnitude of = .05.
What is the speed v at which the amplitude of the boat and trailer will be a maximum?
What is the value of the amplitude at this speed?
What is the amplitude when the boat and trailer are traveling at the speed of 55 mph?

This is supposed to be modeled as an underdamped spring-damper system.


I tried solving this problem by writing the equation of moment which I got to be:

mxdd + cxd + kx = ky + cyd (d denotes derivative)

I don't know if this is right as this is saying that the road is the input in the transfer function analysis but this is what I believe the professor said to do. I got wn ^2 = k/m = 257.6 from mg = kdelta. (delta = deflection) Now I am stuck because i cannot cancel any of the unknown variables.

Thanks in advance
 
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did you ever get the answer to this??
 

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