Bill,
A shimmy or wobble at various speeds is something I experience on almost every motorcycle or scooter at some time. On an older bike, it can indicate worn steering head bearings or improper tire pressures or worn tires. But in my experience, the wobbles I've encountered are due to hanging weight (trunks) off the rear of the bike or certain fairings which can upset the bike's balance. For example, on my Super 9, a small scooter, I can take my hand off of the bars at almost any speed (top speed 45-48mph) and it will track straight. But if I attach my medium sized Givi trunk without even loading it, the extra rearward weight can cause the scoot can start to wobble if I take my hands off the bars. This is behavior I observed even when the scooter was new. The wobble is not uncontrollable, so my easy solution is to not take my hands off the bars!
Motorcycles have an advantage in that they are much heavier vehicles and inherently more stable, but shimmy and wobble problems more dangerous (higher speeds) and can literally throw you off the bike at speed. Often, if there is no obvious problems detected, the only solution is a steering damper which will counteract the wobble effect.
Wil B.
Broadview, IL
'71 Guzzi Ambassador
'91 Honda Nighthawk
'04 Kymco Super 9
'10 Kymco Yager 200gt