SuperDrive won't read burned DVDs

I have some DVDs that I burned several years ago using my Titanium Powerbook. I know that recordable DVDs don't last for too many years so I want to copy them to disk images. They still play just fine on DVD players, but when I put them into computer drives, they won't read. I've tried the old PowerBook, a MacBook Pro, an iMac, and even an external Sony DVD drive. Sometimes DVDPlayer can start playing the disks. But when I try to make a copy with either Disk Utility or Toast, I get an error saying the disk can't be read.

Does anyone have an idea why a DVD has no trouble but the computer's drive does?

Any suggestions for how I can make the copies I want?

Thanks for the help.