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A 12 kg spool that is 1 m in radius is pinned to a viscoelastic rod of negligible mass with effective properties k=10 N/m and c=8 Nsec/m. The end of the rod is attached to a rigid support as shown. Determine the natural frequency of the system if the spool rolls without slipping.

Short Answer

Expert verified
Answer: The natural frequency of the system with the spool rolling without slipping is 56 rad/s.

Step by step solution

01

Determine the moment of inertia of the spool

To determine the moment of inertia, we will use the formula for a cylinder, which is I=12mR2, where I is the moment of inertia, m is the mass, and R is the radius. In this case, we have m=12 kg and R=1 m: I=12(12 kg)(1 m)2=6 kgm2
02

Apply the equation of motion

Considering the forces acting on the spool - gravitational force and the forces exerted by the rod, we will apply Newton's second law of motion. For rolling without slipping, the translational and rotational acceleration are related as αR=a, where α is the angular acceleration and a is the linear acceleration. The forces exerted by the rod can be calculated as: Spring force: Fs=kx Damping force: Fd=cv Total force: FT=Fs+Fd=kx+cv For translational motion: FT=ma For rotational motion: FTR=Iα Substituting the rotational equation with the translational one, we get: kx+cv=IaR
03

Find the natural frequency

To find the natural frequency, we use the formula ωn=km, where ωn is the natural frequency, k is the stiffness, and m is the mass. ωn=10 N/m12 kg=56 rad/s The natural frequency of the system with the spool rolling without slipping is 56 rad/s.

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