I've had to deal with Logitech in regards to manufacturing faults and product failures before. The good news is, once you get involved with their support procedures they give you one-on-one service and will work to resolve the issues at hand. If your product has a genuine fault they will certainly do the right thing by you, in my experience.
For many products we buy the warranty is dealt with at store level. This can actually be more convenient for the customer sometimes than dealing with the manufacturer. Additionally, most product warranties these days are "repair or replacement", meaning they can decide to take the time to repair the original item or supply a new one, at their own discretion.
My advice to you if you believe your product is faulty, is to first check with the store where you purchased your wheel. I know this isn't always possible if the item was a gift or was purchased in a far away location. If the store has stock they may just swap out a new wheel right away. If they do not have stock they may wish to send it away for repair, this may be less desireable for you, but it may also be less desireable for the manufacturer, depending on their service regime in your location.
The alternative to returning the product to the place of purchase would be to contact Logitech support directly. This may not be the fastest path to satisfaction though, so it is probably best to try the store first (may even be a warranty condition in some countries). If you do end up communicating with Logitech support staff, just tell them the facts and be prepared to provide proof of purchase. If your claim is legitimate they will action it quickly (but delivery of replacement goods may take weeks, depending on location and availability).
Logitech have angered and frustrated me many times over the years I've dealt with them and their products, but when it came to the crunch they didn't fail to rectify my problems promptly and professionally.