One gig should be enough, though on the low end. SolidWorks seems to work much better with 2 gigs of memory. We have had 2 gigs in all of our engineering computers for several years now. It appears to me that SW 2007 demands a bit more out of the computer hardware then previous version of SolidWorks.
What are the rest of your computer specs? Details. details..... CPU model and speed, what specific video card do you have? Is it a new computer or a few years old? I am assuming you are running Windows XP SP2, do you have the 3Gb switch set in the boot.ini file.
What kind of assembly are you trying to bring this part into. How big and complex?
What else are you running besides SolidWorks? If you turn off everything other program you typically run and just run SolidWorks do you still get an out of memory error?
Can you bring this same part into other similar assemblies?
Do you have your computer set to Adjust For Best Performance? RMB on My Computer > Properties > Advanced > Performance Settings > Adjust for best performance.
Also RMB anywhere not on a shortcut on your screen and set your Windows Theme in Properties to Windows Classic.
All the Windows eye candy consumes system resources.
2 gigs of memory on your computer is going to give you a much better experience in SolidWorks. That is the minimum RAM required on a workstation these days, in my opinion. We spec our regular office pc's with a gig of RAM. I also expect that we will be spec'ing all new workstations with 4 gig minimum of RAM the next go round in about a year for engineering workstations.
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