两端固定的弦,原先静止不动,单位长度所受横向外力为[tex=9.143x1.357]/taNyHRbAQf8C6bdTD0uNLQWCi+tkQzlaqXTbOfvx0UuSreDdARdRleytizk4Dtu[/tex],求解弦的振动,研究共振的可能性,并求共振时的解.

2022-05-27

两端固定的弦,原先静止不动,单位长度所受横向外力为[tex=9.143x1.357]/taNyHRbAQf8C6bdTD0uNLQWCi+tkQzlaqXTbOfvx0UuSreDdARdRleytizk4Dtu[/tex],求解弦的振动,研究共振的可能性,并求共振时的解.

答案：

解:由题意得定解问题为[tex=11.429x6.357]fnpmC2J6JmQBLyo5NmGAzw0o8Ahogbjk6XUs7uDsJ5wq3d7DuNZLZBlqjoMB8Gr0P1pS7QyQYh5bAUJiA4Jck2y4S7JqXuKsvo7qweuRY1Ihg17TRWnzHSD2Or55P5P6Y4qb0VZzCc/ak0WnoEUpVlc1EQugeGkZyMGqZB8fPZFjHqgnkLZ8mBXwwmXQRXF07H9gzqN0V6dpJxBmEDtShFIJRiMjDpSeMQn3Skw3wV05FA9ELy4t75G7JI14Fwqm[/tex]先将问题转化为求解[tex=3.429x1.357]m7jcD5Wqu62KDVcwLKclTg==[/tex]由第一类边界条件,可知其本征函数为[tex=3.429x2.143]tm6VRRbM/Df8Qog4Yg0Ze+b2vwZQCj/JkRymi2hEadI=[/tex]设问题的解[tex=27.143x3.286]umi4fll1fykhnurJA/2kV+GMNHwL7Y7IFAc5/UAbRHzkueK46nxQn3GkSCcE8YptyqcxR8LuZ9/kalbtHGlj6vzIJ3nRvN8A8d91cn10OmRijIw2yXSDbH5xtHgs3DHC+RXNdPYLq1AlhzlUy3lrQ3uoDixHzjv6YJPD4/FoHJ+VWh2wTCVqCwpQZQLQNIP7gLzo4kYkiCb2LOHGVBst9Q==[/tex]代入初始条件可得[tex=17.071x6.929]fnpmC2J6JmQBLyo5NmGAzxhqh9h0rI4WEuIgGZeIM4McBualruV1uZLbp56HHYakBxsj30+ByFS1gFmQ/Ab9wtJcfofmn5QpRxPTz/naGJbtKACR47iUp1cS9Hf0QTqKp3lFsfvNFpquzRiGRPDoYaV3nQKDpQg3ArJwSzBZYgYi2je0X4KfoqcRqC9V7tfTpCUue8NQmk/SfE635eNwdrFwcHGDekN80NnWzIXmF+L+VQuk45/guq3h2OONcnvCjdNpz3ufXcCijgJvVhLZQA==[/tex][tex=38.357x12.071]c8gX0O6CKBpyqTBZ2fB4DvGcZB7r2xDSc7NsQMg6lafkle2m88OZyti/lxPog2xJ4q8KZIPsCl2+3Tj+l7ZDW0pi+mJyr3ldb7vrTr5TetM9RXlKx9ehlePQVHN3uBopkJ9tamPtJCKHkWCQ+ddS7E1qQXrqRtINjNXnIGfLI6VkyH0m5k0I0hoUQ1Ox28iNj5PhJycOAray5oTBBh3DY1Mw+ZZQfrz7H5ZLrwJ78CJwxXr9jT6GUMiSIceY4wB/2b+3ijKcUA3TcGApXErbHA6xQ1AFDaGxy8Iai9BLHtfJ5fw/H0Ka5+Jvbs09Rs1am3wvKXvPQc7l5vgJQDLsaSAJXxwac2y41SQRsBjIv/GnA8pbpbEWKv2Fy7C6IPWFh0av48DP1U9sE/zpxOKJm6+XBJMROggpUCG1wwNZTph3rGJZ3xphIXwRrgK9CX/CutN465I8YoIEsOkoNyeKrf9JF7CeOHaVS3n1sVjOWFhrC9y1s/X4SYIK/Qw8hTRif1Y3iUZwEIZJhFg2DxBuXsQW3UAc8QAemKcJtYiL+CpciEtCg4fqaPIOmIQvsbzCHJUkcwaQ8TFefqW1/h8960u8sXoLYhkje/W/N0tW/W/gPEW4C0r4T9jU3PZ84Ox2uURkO6RqP1azhlHZrEvhucFpKYPxgLsFBtcKNZLFbc6xoKFOmSxVv3d4fLJvoiaYI4V1VYmynFGYMwoRFuY4MUyFYt4Q71S/NECT3J+BJQfQmpUEb4nfNSmkBINDXE2uS9eCbgmr1jyhPqZlYm7o49uptttAeTJl58VvILjgTz8aa47PjQvtwxriQxOMxhKe[/tex][tex=29.143x4.143]oHet23Ib6WAIr7wg8IHQhRqEbrqH8fJb8eNYWOlXAo86hOC9MtxFhfB97KcTTnD4pOfdfHjrNEPS8XDfoHc1JLkEthSftohKbVHUK62CTbRBkPBzreW3vDleKvkNcB3bmV8QFwE1715YVcRyxG+AgXhAsFrZ3AYU4KSNXNHrNHIO9yqGztHq+/QAiIqLmv6vDphUH+8+j+Pkan6mMmEyKRS+jfa/8flrMGyGoPYWHF8KiMNi5v2BMQycB517zX0PCFdqy6bIXJk2M1SBpdi+FApUWLjbycDblxw9F4IMKzQTyC7gxY+BZ3ZXt4CYhpKcz9coyLLQODiCdQIAmYv5doRkXI3yAPbtbrIcSdbSLW8=[/tex]

举一反三

内容

0
设弦的两端固定于[tex=2.357x1.286]F20DA9b5PZyvxJH27l4LOQ==[/tex]及[tex=2.143x1.286]ZWTEsSiqGBIAl5zcGo6d6A==[/tex]，弦的初始位移如图所示，初速度为零，又没有外力作用，求弦作横向振动时的位移函数[tex=2.643x1.286]niNGv9OCFuWOuAaoq5X6eQ==[/tex].[img=488x297]178afb494906caf.png[/img]
1
【单选题】我们已知弦的振动方程: , c为波速,考虑弦在横向(位移u的正向)上还受到外力的作用,设单位长度所受的外力为f ,此时弦的振动方程为()。 A. B. ,非齐次项是单位质量所受的外力 C.
2
长为[tex=0.357x1.0]5vVfAZliYwqMw8JaLE+iEA==[/tex]的弦，两端固定，弦中张力为[tex=0.643x1.0]awBC2UvU2WxG45VihksPuw==[/tex]，在距一段为[tex=0.929x1.0]XQ8c0totc8uufRPOvpPxwQ==[/tex]的一点以力[tex=1.0x1.214]kLwCQn0d73TWOUUlnBVGBA==[/tex]把弦拉开，然后突然撤除这力，求解弦的振动[img=291x147]178efbe78e53a93.png[/img]
3
我们已知弦的振动方程:, c为波速,考虑弦在横向(位移u的正向)上还受到外力的作用,设单位长度所受的外力为f ,此时弦的振动方程为()。[img=87x48]180653ffaeeeb29.png[/img]
4
半无限长弦的初始位移和初始速度都为0,端点振动规律为 [tex=7.143x1.357]ovdf6hJlCCEUr1UrG9g96LyVrvg4qPokP/F3UTedPMU=[/tex]求解半无界弦的振动规律.