三．解答题 1. 解：
振形函数w?ysin最大形变势能V??xa
max2222??D?22?w?w??w?????V?max??????w??2?1????2????dxdy 2??2??x?y?x?y?????????D?4b3D?1????2b?代入得V?max? 312a2a2wm2?xEkmax?wdxdy 同理得 w?ysin代入??2amw2ab3有EKmax? 又V?max?12EKmax
D?4b3D?1????2bmw2ab3即 ??312a2a12得最低自然频率?min
?min?2. 解：
?2a26?1???a21??2b2D m?Dw?w?d???qw?d?411
?dw1dw?64C1?4w??2???4?d??a?d?64DC1a??2?32DC1Dw?w?d??1??d?????1?a40?a2?3a24222
q0????8q0a2?qw1?d???a?1?a2?d??105
??22232DC18q0a2q0a4q0a4??2?解得C1? ∴w???1?2? 23a105140D140D?a?q0a4∴wmax?
140D23. 解：
?w?2w?w?m ?mx ?my （1）证明：由w?mxy得
?x?y?y?x22?2w?2w?w?w2??0?2?0 ?w?222?x?y?x?y??2w?2w?Mx??D???x2???y2????Mx?0??2w?2w?My??D???y2???x2??My?0???2w1?Mxy?Myx??D?1??? →Mxy?Myx??mD??x?yFsx?0?Fsx??D?2wFsy?0?x?Fsy??D?2w?y??
?w? OC边：?w?边界条件：OA边：
x?0?0 ?Mx?x?0 ?0 ?My?y?0
y?0??Mxy??Mx?x?a?0 Fsxx?a?? BC边：?Fsx??y??x?a?0
????tt?Myx???y?b?0 BA边：?My?y?b?0 ?Fsy?x?b??Fsy??x??验证可知 w?mxy满足边界条件 （2）根据B点平衡条件 FSB?FSBA?FSBC?2?Mxy???F
FxyF即 -2D?1???m??F→ m? w?
??21??D2D?1???Ftt故内力：Mx?My?0 Fsx?Fsy?0 Mxy?Myx?? Fsx?Fsy?0
2 反力：FSA?FSAO?FSBA??F FSC?FSCO?FSCB??F FSO?FSOA?FSOC??F