論文：偏微分方程在弦振動(dòng)問題中的應(yīng)用

論文：偏微分方程在弦振動(dòng)問題中的應(yīng)用
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中文摘要

偏微分方程是反映有關(guān)未知變量關(guān)于時(shí)間和空間變量導(dǎo)數(shù)之間制約關(guān)系的等式，在物理學(xué)等領(lǐng)域應(yīng)用廣泛。弦振動(dòng)是一種機(jī)械運(yùn)動(dòng)，但是弦并不是質(zhì)點(diǎn)，質(zhì)點(diǎn)力學(xué)定律并不適用在弦振動(dòng)的研究上，可以用弦振動(dòng)方程來表示。弦振動(dòng)方程實(shí)際上是一個(gè)典型的偏微分方程模型，建立過程相當(dāng)復(fù)雜，主導(dǎo)思想是抓住主要因素，忽略次要因素。為了得到弦振動(dòng)問題的唯一解，必須在弦的兩端給出邊界條件，也就是考慮研究對象所處邊界上的物理狀況。
本文針對弦振動(dòng)的邊值問題，研究了弦振動(dòng)問題在實(shí)際中的一些應(yīng)用：首先我們對弦振動(dòng)問題進(jìn)行適當(dāng)?shù)募僭O(shè)，分別導(dǎo)出了弦不受外力作用時(shí)的弦振動(dòng)方程和弦受N#I-力作用下的弦的強(qiáng)迫橫振動(dòng)方程，應(yīng)用分離變量法研究弦振動(dòng)方程在不同條件下定解問題的存在性，然后研
……（新文秘網(wǎng)http://m.jey722.cn省略731字，正式會(huì)員可完整閱讀）……　
itsdeducedprocessisfairly complicated，thusthe dominantfactors must be considered and thesecondaryfactorsmaybe ignored．Theboundaryconditions must beknown，namelyconsideringitsPh3，sicalconditions of the rescarchobject toobtain theuniquesolutionofstringvibration．Severalactualapplicationsofstringvibration forboundaryvalueproblemwerestudied in thispaper．Atfirst，stringvibrationequationwithout outside force andstringtransverse vibrationequationwith outsideforce wererespectivelydeduced，basedonsuitableassumptionofstringvibration．Method ofseparationof variables Was used tostudythe e*istence of theproblemoffi*ingsolution ofstringvibrationequationunderdifferent conditions．Afterwardsignificance of thecorresponding problemoffi*ingsolution Was studied and itsapplication equationsinrodvibration，etcwere deduced．Secondly,cabletensionofcable—stayed bridgeWas studiedat thepointofbasictheoryofstringvibration and theequationbetweennaturalfrequencyof cable vibration and cabletension was obtainedthroughcalculatingthestringvibrationequation．Understandingandcontrollingits tension hasimportant engineering significance toinsure run-timesafetyofcable-stayed bridge(estimationofloadingchargeandrunning condition)andcablecal"(tension controlling)，etc．Finally,problemofcop applicationWas alsoanalyzedand discussed．Key words：String vibration；Partial differentia equation；method ofseparationofvariables；Cable tension；problem offi*ingsolution；naturalfrequency；windingballoon1

引言

1．1

偏微分方程概述

偏微分方程是反映有關(guān)未知變量關(guān)于時(shí)問和空間變量導(dǎo)數(shù)之間制約關(guān)系的等式。微積分方程這門學(xué)科產(chǎn)生于十八世紀(jì)，歐拉在他的著作中最早提出了弦振動(dòng)的二階方程，隨后不久，法國數(shù)學(xué)家達(dá)朗貝爾也在《論動(dòng)力學(xué)》中提出了特殊 ……（未完，全文共4062字，當(dāng)前僅顯示2051字，請閱讀下面提示信息。收藏《論文：偏微分方程在弦振動(dòng)問題中的應(yīng)用》）