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a b - Duke Universitypeople.duke.edu/~jthomas/papers/papers/transonicairfoilflutter.pdf · (*)#+( ,.-/ /,0 1$2#+,43 5 34 6 7 8 9 ( :,4 ,; =!*- 3? @A)- ,03434 B ,.#C(D EF HGI,?#+

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Page 1: a b - Duke Universitypeople.duke.edu/~jthomas/papers/papers/transonicairfoilflutter.pdf · (*)#+( ,.-/ /,0 1$2#+,43 5 34 6 7 8 9 ( :,4 ,; =!*- 3? @A)- ,03434 B ,.#C(D EF HGI,?#+

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Page 2: a b - Duke Universitypeople.duke.edu/~jthomas/papers/papers/transonicairfoilflutter.pdf · (*)#+( ,.-/ /,0 1$2#+,43 5 34 6 7 8 9 ( :,4 ,; =!*- 3? @A)- ,03434 B ,.#C(D EF HGI,?#+

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0

0.2

0.4

0.6

0.8

1

20 40 60 80 100 120 140

Vf /

õ

Mass Ratio, µ

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9

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0.2

0.4

0.6

0.8

1

1.2

20 40 60 80 100 120 140

ωf /

ωα

Mass Ratio, µ

ω1 / ωα = 1.06

ω2 / ωα = 0.49

pitch dominated

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9+ ' ` 21���S���" ` ����N������."] F ����01(��� ¢� ê

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

20 40 60 80 100 120 140

ϖf

Mass Ratio, µ

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9

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2

4

6

8

10

12

20 40 60 80 100 120 140

|h| f

/ αr

Mass Ratio, µ

|h / α|1 = 0.32

|h / α|2 = 9.32

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9

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Page 9: a b - Duke Universitypeople.duke.edu/~jthomas/papers/papers/transonicairfoilflutter.pdf · (*)#+( ,.-/ /,0 1$2#+,43 5 34 6 7 8 9 ( :,4 ,; =!*- 3? @A)- ,03434 B ,.#C(D EF HGI,?#+

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Ratio of Uncoupled Frequencies, ωh / ωα

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7

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Ratio of Uncoupled Frequencies, ωh / ωα

ω1 / ωα

ω2 / ωα

M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9M = 1.0

+ ' ` 21���S���" ` ����N�������"] F ����01(��� ¢� ê

0

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M = 0.8M = 0.9M = 1.0

�5' ` 21���S�$�� ��N!?�"�"�-BD�����Y\���� ¢� � 0

2

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6

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/ αr

Ratio of Uncoupled Frequencies, ωh / ωα

|h / α|1

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M = 0.0M = 0.3M = 0.5M = 0.6M = 0.7M = 0.8M = 0.9M = 1.0

; ' @�%&!�210 �$������(�* ` 21�N�$�$�" � J@��" �('�"29��:$�$0����A35016���� ) ���Y��(� ���� � � � �������������� ��� @ �!�!���YB8� @ +D;��_��% B8����� �����_� � �I� @ +(�)�!����+ � ������� E �_�!���<� � � +(�N� @ 'P���e;A������� @ ��+(�]���� �� � � � � � � � �»v¹Ï¿�¾ÁÃP¾Ëß0¼Töaä7¼;Ð?ÊcºP¹Ç¹?à ¿~¹?×�H>§_ö ² æ Ú µXÐ0ºP¼\¿PÌ�¹#±~ÂB¾Á ¸M¾�ëÎö � Ð�ö- Ì�¼ÎÂ�ÃPÌ�¼�Äιϼ .�Äž˼ÎÂÏÃ Û t »eÐ0¿Y¿P¼;¿ ÃPÌ�ºY¹0È�ë4ÌÀìμ;ºP¹vÍ4ÉeºYÐ0ÂeÄ Ì�¼;¿¹?×   �é  �ê Ð0¿Y¿�Æǽp»ÖÃP¹0ÃP¾ÀÄÎÐ0ÊÁÊËƺë0¹CÃY¹ ��� ö °XÌe¼�ØÖ¼;»v¼;ÂeØÖ¼;ÂeÄż¹?× Û t ±~¾ÁÃPÌIºY¼;¿P» ¼;ÄÑÃBÃY¹ ä¦ \ ¾À¿B¿�Ìe¹#±~ÂI¾Ë¸M¾�ë_ö � É2ö´¸e¹0º¼ÅÜ�Ð?½p»�Ê˼0ÍǾËÂÔ¸M¾�ëÎö�14Ä7× ¹4º õ Ü Û v è Í Û t ë4¹Ï¼t¿1ÃPÌ�ºY¹0È�ë4ÌBì;¼ÎºY¹Ð.Ã ä¦ \�� Û v � Í,±~Ì�¾ËÄ Ì�¾À¿�±~Ìe¼Î ÃYÌ�¼pÉ�º Ð?ÂeÄ Ì�¼t¿�¹0×   �é  �ê ë4¹�ÃP¹��� ¾ÁÂj¸M¾�ëÎö � ÐÎöV�¹?ÃY¼BÃPÌeÐ?Ã�ÃPÌ�¼ Ø�¾�ø2¼ÎºY¼ÎÂeÄμ É ¼ÅÃ"±a¼Î¼ÎÂ6ÃYÌ�¼jÃ"±a¹�Äι0È�»eÊÁ¼tØÂeÐ.ÃYÈ�º Ð?ÊÏ× ºP¼�§ÏÈ�¼ÎÂeÄξÁ¼t¿ ² ¦)© Ð0ÂeØ ¦ t Ð4¿c¾ËÂeØÖ¾ÀÄÎÐ?ÃP¼tØã¾ËÂd¸M¾�ëÎö�1?É5µ¾Ë¿~ÃPÌ�¼ã¿P½HÐ?ÊËÊÁ¼t¿}Ã7Âe¼;Ð?º   é  ê Ü Ù öX°XÌ�¾À¿~¾Ë¿�ÊÁ¾R¹0¼;ÊÁÆ�ºP¼t¿�» ¹0Âv¿�¾ËÉ�ÊË¼× ¹0ºcÃPÌ�¼1½d¾ËÂ�¾Ë½ãÈ�½U¾ËÂ�ÃYÌ�¼ ¥ È�Ã�ÃP¼;ºM¿P» ¼Î¼;Ø\¾ÁÂeØ�¼ÅÜ�¾ÁÂ\ÃPÌ�¼1ßϾÀÄžËÂhã¾áÃ}Æ�¹?×XÃYÌeÐ.Ãd»v¹4¾ÁÂÏÃN>�ÃPÌe¼�ÄÅÊ˹4¿P¼Îº\ÃPÌ�¼ÔÄŹ0Èe»�ÊÁ¼tØ�ÂeÐ.ÃYÈ�º Ð?Ê�× ºY¼»ã

§ÏÈ�¼ÎÂeÄξÁ¼t¿gÐ?ºY¼s¾ÁÂe¾áÃY¾ËÐ0ÊÁÊËÆ0ÍtÃPÌe¼ ½p¹0ºY¼MºY¼;Ð4ØÖ¾ËÊÁÆ4ÍN±~Ì�¼;Â8¾ÁÂeÄκP¼tÐ0¿P¾ÁÂÎëß0¼;ÊÁ¹ÖÄžÁÃ}Æ0Í 2PÄŹÏÐ?Ê˼;¿YÄż;ÂeÄż)4ã¹4º$2�½p¼Îºjë0¾ËÂ�ëd× ºP¼�§ÏÈ�¼ÎÂeÄÎÆ�4 ¥ ÈÖÃ�ÃY¼ÎºÄÎÐ0Â�¹ÖÄÎÄÎÈ�º ² ¿P¼Î¼8»2ö Ù4Ù æ�¹?×sä~¼Å×�ö�» Ú?¼ µ»öû+Ü Ù;Û4Û "¸e¹0º ûYÜ ÙtÛ0Û ÍgÃPÌ�¼ ¥ ÈÖÃ�ÃY¼Îº�¿P» ¼Î¼;Ø�¾ËÂeØÖ¼ÎÜ ² ¸_¾�ëÎö Ú ÐBÐ?ÂvØÄNµpÐWë4Ð?¾ËÂ"ÌeÐ0¿HÃPÌ�¼BÃY¼ÎÂeØÖ¼;ÂeÄÅƪÃP¹+ÌvÐ ß0¼ Ð�½p¾ÁÂe¾Á½�È�½ Âe¼;Ð?º  é  ê Ü Ù ÍvÐ0ÂeØÔ¼;¿P»v¼tÄžÀÐ?ÊËÊÁÆB¿�¹p× ¹4º�¿PÈ�Ée¿P¹0Â�¾ÀĤù Ð0Ä ÌBÂÇÈ�½�Év¼;ºY¿² ¸M¾�ëÎö Ú Ð�µ»ö1Ò�¿7¾ÁÂBÃYÌ�¼ãÄÎÐ4¿�¼8× ¹4º ûfÜ æ0â�ÍÖÃPÌ�¼ ¥ ÈÖÃ�ÃY¼Îº�¿�» ¼Î¼tؾÁÂvØÖ¼ÅÜC¾À¿dÂ�¼;Ð0ºPÊËÆ+ÄŹ0Âv¿}à Ð?ÂÏÃd±~¾ÁÃPÌ õ × ¹0ºH¿PÈ�Ée¿P¹0Â�¾ÀÄ�Äι0ÂeØ�¾�ã

ÃP¾Ë¹0Âe¿xöªÒaÃHÌ�¾�ë0Ìe¼ÎºpÃPº Ð?Âv¿�¹4Â�¾ËÄpù Ð0Ä Ì6ÂÇÈ�½ãÉ ¼Îº ¿ ² ¸M¾�ëÎö Ú Ä�µÑÍÃPÌ�¼ ¥ È�Ã�ÃP¼;º ¿�» ¼Î¼tØd¾ÁÂeØ�¼ÅÜãÉ ¼;Äι0½p¼;¿Mß0¼;ºPÆ\¿P¼ÎÂv¿�¾ÁÃP¾Ëß0¼aÃP¹�ù Ð4Ä ÌÂÇÈ�½ãÉ ¼ÎºUöf°XÌ�¾À¿\É ¼ÎÌeÐ ßǾ˹0º\¾À¿ãÐ4¿P¿P¹4¿P¾ËÐ?ÃP¼tØ ±~¾ÁÃPÌ�ÃYÌ�¼ 2�ÃPº Ð?Âhã¿�¹4Â�¾ÀÄ ¥ ÈÖÃPÃP¼ÎºBØ־˻*4�öqÒ7ÊÀ¿�¹ªÂ�¹?ÃY¼jÃPÌeÐ?Ã�ÃYÌ�¼ ¥ ÈÖÃ�ÃY¼ÎºB¿�» ¼Î¼tؾÁÂvØÖ¼ÅÜ�Ä;Ð?ÂpÉ ¼7Ð ½�È�ÊÁÃP¾Ëß Ð0ÊÁÈe¼;Ø�× È�ÂeÄÅÃP¾Ë¹0Âp¹?×   é  ê × ¹0º õ Ü Û v ÚÐ?ÂeØ õ Ü Û v è ö*V�¹?ÃY¼ ÃPÌvÐ.à ² ¸M¾�ë_ö Ú É5µ«ÃPÌ�¼ ¥ ÈÖÃPÃP¼Îº_× ºY¼N§ÏÈ�¼ÎÂvÄÅƾ˿dß4¼ÎºYÆ�ÄÎÊÁ¹Ï¿�¼�ÃP¹�ÃYÌ�¼ÔÄι0È�»eÊÁ¼tØ�ÂeÐ?ÃPÈ�º Ð?Êa»�¾ÁÃYÄ Ìª× ºP¼�§ÏÈ�¼ÎÂeÄÎÆ0Í ��  ê Í_× ¹4º õ Ü Û v è ² »�¾ÁÃYÄ Ì�ØÖ¹4½p¾ÁÂeÐ?ÃP¼tØ+½p¹?ÃP¾Ë¹0Â4µyöH°XÌ�¾À¿�¾Ë¿¾ÁÂvØÖ¾ËÄ;Ð.ÃY¾Áß4¼�¹?× 2P¿P¾ÁÂ�ë4ÊÁ¼·ã Ø�¼·ë0ºY¼Î¼·ã ¹0×Çã × ºY¼Î¼tØÖ¹0½$4 ¥ ÈÖÃ�ÃY¼Îº ² ¿�¼;¼\»2öÙ0Ùtë ¹0×sä~¼Î×�ö�» Ú?¼ µ»ö� �_�N� J @ � + ��� � ! ���)�°XÌe¼ Øּλ ¼ÎÂeØ�¼ÎÂeÄμj¹?×�ÃPÌ�¼ ¥ È�Ã�ÃP¼;º�É ¹0È�ÂvØ�Ð?ºYÆC¹0ºù Ð4Ä ÌÂÇÈ�½ãÉ ¼Îº ² ¿�¼;¼p¸M¾�ë4¿xö è Ð?ÂeØ Ù;Û µ�× ¹0º8ÃPÌe¼�ÄÎÐ0¿P¼;¿�¹0× û^Ü æ0âÖÍÙ;Û4Û Ð0ÂeØ   é  ê Ü Û v â�Í Û v Ú ¾À¿~»�ºY¼;¿P¼ÎÂÏÃY¼;Ø�Â�¼ÅÜÇÃxö

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0.2

0.4

0.6

0.8

1

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Mach number, M

µ = 25

µ = 100

�y' ` 21���S���" ��N!?���"��BD�����Y\���� ¢� � 0.4

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ωα

Mach number, M

µ = 25

µ = 100

ω1 / ωα = 1.06

ω2 / ωα = 0.49

+ ' ` 21���S���" ` ����N�������"] F ����01(��� ¢� ê

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Mach number, M

µ = 25

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��' ` 21���$�$�" F �"��� �"�"� ` ����'���"� �Y]�� ��� 0

2

4

6

8

10

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/ αr

Mach number, M

µ = 25

µ = 100

|h / α|1 = 0.32

|h / α|2 =9.32

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