The elements of the “swing-axle” type of independent rear suspension for automobiles are depicted in the figure. The differential D is rigidly attached to the car frame. The half-axles are pivoted at their inboard ends (point O for the half-axle shown) and are rigidly attached to the wheels. Suspension elements not shown constrain the wheel motion to the plane of the figure. The weight of the wheel– tire assembly is W = 100 lb, and its mass moment of inertia about a diametral axis passing through its mass center G is 1 lb-ft-sec². The weight of the half-axle is negligible. The spring rate and shockabsorber damping coefficient are k = 50 lb/in. and c = 200 lb-sec/ft, respectively. If a static tire imbalance is present, as represented by the additional concentrated weight w = 0.5 lb as shown, determine the angular velocity w which results in the suspension system being driven at its undamped natural frequency. What would be the corresponding vehicle speed v? Determine the damping ratio ζ. Assume small angular deflections and neglect gyroscopic effects and any car frame vibration. In order to avoid the complications associated with the varying normal force exerted by the road on the tire, treat the vehicle as being on a lift with the wheels hanging free.

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W r = 14" %3D 4 = 27" l2 = 36" -