Can I Make a Support Double after Doubling for Takeout?

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Is West's 2nd double on the auction below a Support Double?

A Support Double is used by opener to show exactly 3-card support for responder's suit.  Only the opener can make a Support Double.  West's second double in the above auction isn't a Support Double.   While West's double on the above auction isn't a Support Double, it typically shows 3-card support for their partner's suit, extra values, and a desire to compete.  If West had a hand that was too strong to overcall, West would show their suit or bid notrump.  West didn't bid a new suit or bid notrump.  So West is expected to have a normal takeout double.  With a minimum takeout double, West would pass.  So West; is expected to have extra values.  West would have raised their partner's suit with 4-card support.  So West has 3-card support for East's hearts.  That explains why West's 2nd double shows 3-card support for their partner's suit, extra values, and a desire to compete.  West might have a hand like ♠Q873 ♥KQ5 ♦9 ♣AKT72.   

Similarly when responder raises opener's suit, a 2nd double by the takeout doubler is takeout oriented,, typically with 3-card support for their partner's suit.  

West's second double in the above auction is takeout oriented.  It typically shows 3-card support for their partner's suit, extra values, and a desire to compete.  West might have a hand like ♠AQ73 ♥A85 ♦94 ♣KQ72. 

If opener bids a new suit and West doubles, the double is more penalty oriented.  Opener is expected to have 3-card support for their partner's suit, extra values, and a good 4+ card holding in opener's 2nd suit.  The takeout doubler's partner uses their judgment as to whether it is better to defend or declare.  If the opponents run, subsequent doubles are penalty oriented.

West might have a hand like ♠AQ73 ♥A85 ♦94 ♣KQ72 on the above auction.