Here is a short document for IBM 3584/TS3500 firmware upgrade.

 

Get the firmware package

See this Instruction to download IBM 3584/TS3500 firmware package

 

Upgrade firmware via Library web interface

Choose

Service -- > Firmware update

Click Launch Firmware update Wizard

A new window will pop up and let you 'Select Firmware Type', at here you can choose to upgrade 'Library firmware' or a list of type of drives you want to do fw update.

In case you choose a type of Drive, then select either a single drive, or all drives

Browse the fw file, then there you go.

 

Other ways

Traditionally, you would like to use the ways above to perform drive firmware update, but sometime they may fail due to slot connectivity, or other reasons. If that happens, then you can take a try  to Use IBM Tape Diagnostics tool(ITDT) to update drive firmware

Or use tapeutil, the previous tool to ITDT.