2011《非线性振动》试题解答

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1、华中科技大学研究生课程考试试卷课程名称：非线性振动与控制课程类别□公共课R专业课考核形式R开卷□闭卷学生类别考试日期2011.12.21学生所在院系学号姓名任课教师题1：（20分）Figure1ParticleonarotatingparabolaxzWOmgConsiderthemotionofaparticleofmassslidingfreelyonawiredescribedbyparabolawhichrotatesaboutthez-axisasshowninFigure1.Weassumethatthewire

2、isweightlessandthatitsangularvelocityischangingwiththepositionofthemassalongthewire.Thereisnooutsideinfluenceactingonthewire.(a)Showthattheequationsofmotionareand(b)Showthatwhereisaconstantofintegration(essentiallythisisastatementofconservationofangularmomentum)and

3、thatthegoverningequationforcanbewrittenintheform(c)Discussthemotionofthemassalongparabola.Showthatthemotionisalwaysboundedinthissystem.(d)Forand,plotthetrajectoriesinthephaseplane.【注：这里为重力加速度，这一值的单位为。】题2：（20分）Determinethesingularpointsandtheirtypesforthesystem-19-S

4、ketchthetrajectoriesandtheseparatricesinthestateplane.题3：（20分）Considerthemotionofasystemgovernedbywhere.(a)Showthatwhere(Notethatmustbepositiveforarealisticsystem.)(b)Determinethestationarymotionsandtheirstabilityasafunctionofthemagnitudesandthesignsofand.题4：（20分）C

5、onsiderthesystemgovernedby(a)Whenisnearunity,showthatforsmallbutfiniteamplitudesoftheresponsewhereHereisameasureoftheamplitudeoftheresponse.Obtainthefrequency-responseequation.Showthat.Howdoesthisvalueofcomparewiththatthecaseoflinearviscousdamping?Plotversusand.Ist

6、hereajumpphenomenon?(b)Whenisnearonethird(superharmonicresponse),showthatwhere-19-Obtainthefrequency-responseequation.Plotversusand.Isthereajumpphenomenon?(c)Whenisnear3(subharmonicresponse),showthatwhereObtainthefrequency-responseequation.Plotversusand.Isthereajum

7、pphenomenon?题5：（20分）ConsiderthesystemshowninFigure5whenthetension.(a)Showthatthegoverningequationis(b)Linearizethegoverningequationtoobtain(c)Determinesecond-orderexpansionsforthetransitioncurvesseparatingstabilityfrominstabilitywhen(d)If,determinetheinfluenceofthe

8、nonlineartermstofirstorderwhen.Figure5ParticleattachedtostretchedstringllmxTTTT-19-注意：所有的题目并没有给出完整的解答，以此作为提供一个解题思路，希望自己推导一遍（使用自己习惯的一套符号），修改和完善其中的