Clear out old messages in Exchange 2010 journal mailbox after messages are archived to .PST Getting Error on Date

Question

# THIS IS THE FIRST COMMAND THAT I RUN IN A SEPARATE .PS1 FILE. THIS SEEMS TO WORK FINE:
New-MailboxExportRequest JournalMailbox -FilePath '\\dfsdr.dc.pud\PowershellEmailArchive\Journal.pst'
# NEXT I AM TRYING TO CLEAN OUT EVERYTHING ELSE UP UNTIL 1-2 DAYS AGO BUT GETTING THIS ERROR:
[PS] C:\scripts>.\CleanOutOldMsg.ps1
Please adjust the date time. Make sure it is in the correct format.
    + CategoryInfo          : InvalidArgument: (:) [], ParserException
    + FullyQualifiedErrorId : 44F6D37F
# HERE'S WHAT I'VE DONE:
$global:startDate = (Get-Date).AddDays(-31).ToString('MM/dd/yyy')
$global:endDate = (get-date).AddDays(-1).ToString('MM/dd/yyy')
# HAVE TRIE THIS DATE FORMAT TOO:
$endDate = (Get-Date).AddDays(-1).ToShortDateString()
$startDate = (Get-Date).AddDays(-31).ToShortDateString()
# AND ALSO THIS DATE FORMAT:
# $endDate = Get-Date -Day 2 "00:00:00"
# $startDate = $endDate.AddDays(-30)
# $endDate = $endDate.ToShortDateString()
# $startDate = $startDate.ToShortDateString()
# THIS COMMAND GIVES DATE ERRORS:
Search-Mailbox -Identity JournalMailbox -TargetFolder inbox -TargetMailbox jimk -SearchQuery "Received:'${startDate}'..'${endDate}'" -DeleteContent -Force
# THIS COMMAND GIVES DATE ERRORS TOO:
Get-Mailbox -Identity "JournalMailbox" | Search-Mailbox -SearchQuery "Received:'${startDate}'..'${endDate}'" -DeleteContent -Force
# THIS WORKS:
Search-Mailbox -Identity JournalMailbox -TargetFolder inbox -TargetMailbox jimk -SearchQuery "Received:02/07/2014..03/07/2014" -DeleteContent -Force
# I FOUND SOME OF THE INFORMATION HERE:
https://serverfault.com/questions/511610/how-can-i-move-exchange-items-to-a-pst-using-powershell
Thanks in advance! Jim
Sorry if I wasn't as clear as I could have been.  I'm trying to clean out the Journal mailbox after exporting the messages to a .PST file.  I've tried formatting the date different ways and when I check it the format seems OK. For some reason I'm
still getting the same error.  I've tried this which was suggested and I received the same error listed above:
Search-Mailbox -Identity JournalMailbox -TargetFolder inbox -TargetMailbox jimk -SearchQuery "Received:'${$startDate}'..'${$endDate}'" -DeleteContent -Force
For some reason the -SearchQuery is not accepting the variable.
Thanks,  Jim

Answer 1

Thank You Yan Li!   I ran commands both ways and still received errors.  Here's the first command and the error message: 
Search-Mailbox -Identity JournalMailbox -TargetFolder inbox -TargetMailbox jimk -SearchQuery "Received:"$startDate".."$endDate"" -DeleteContent -Force
Error: 
[PS] C:\scripts>.\CleanOutOldMsg.ps1
A positional parameter cannot be found that accepts argument '02/10/2014..03/12/2014'.
    + CategoryInfo          : InvalidArgument: (:) [Search-Mailbox], ParameterBindingException
    + FullyQualifiedErrorId : PositionalParameterNotFound,Search-Mailbox
Here's the result of the following command which is formatted a bit different per your suggestion:
Search-Mailbox -Identity JournalMailbox -TargetFolder inbox -TargetMailbox jimk -searchQuery "Received:>$startDate and Recieved:< $enddate" -DeleteContent -Force
[PS] C:\scripts>.\CleanOutOldMsg.ps1
The property keyword isn't supported.
    + CategoryInfo          : InvalidArgument: (:) [], ParserException
    + FullyQualifiedErrorId : 34A6B2A1
Would you happen to have any other suggestions?  Thanks in advance!  Jim