平面运动刚体相对速度瞬心的动量矩定理

() 文章编号 :100025870 20010420064202 平面运动刚体相对速度瞬心的动量矩定理 尹莉 ()石油大学储运不建筑工程学院 , 山东东营 257061 摘要 :建立了刚体作平面运动时的相对速度瞬心动量矩定理 ,证明了在任一瞬时 ,当平面运动刚体的质心不速 度瞬心的距离保持不变时 ,相对速度瞬心的动量矩定理的形式不相对固定点或质心的动量矩定理相同 。用该定理 可以简便地解决相同条件下的平面运动问题 。 关键词 :刚体 ;平面运动 ; 速度瞬心 ; 质心 ; 动量矩 中图分类号 :O 311 . 2 文献标识码 :A n ( )1 Gr × m V , = 1 P问题的提出2 i i ? i = 1 n 理论力学中质点系的动量矩定理是相对于固定 r × m V .)( G2 = 1 i i C? i = 1 点或质心而建立的 ,但在解决平面运动问题时 ,当平 面运动图形的质心不速度瞬心的距离保持不变时 , 若以速度瞬心为矩心 ,建立相同形式的动量矩定理 , 将会简化和运算过程 。文献 1 应用质点系对 速度瞬心动量矩不质点系上所有外力对速度瞬心的 主矩之间的关系来分析问题 ,但却未给出具体的理 论证明 。本文将给出此定理的证明 ,为其实际应用 提供理论依据 。 2 理论证明 如图 1 所示 ,在固定平面坐标系 ox y 中运动的 图 1 刚体平面运动示意图 刚体 , 可由 n 个质点组成的平面运动图形 S 示 。C ( ) 由图 1 可知 , r= r + r , 代入式 1, 并利用式 2 1 点是平面运动图形 S 的质心 , ab 曲线是平面图形的 ( ) 2得速度瞬心在 ox y 固定坐标系下的轨迹 , 即定瞬时中 n 心轨迹 。 ( ) r + r × m V G= 1 i iP? i = 1 在平面图形运动时 , 如果有一动点P 沿着定瞬 n n 时中心轨迹 ab 曲线运动 , 则 P 点始终不瞬时平面图 r × m V +r × m V = i i 1 i i?? i = 1 i = 1 形在该瞬时的速度瞬心相重合 。以 P 为动点 , 建立 r × m V + G= i i Cr 不固定坐标系平行的平动坐标系 Px y 。在任意瞬 1 1 n ( )= G. 3 r × M V +Cr C 时 , 平面运动图形上任意一质点 m 相对动坐标系的 ?i i = 1 位置由矢量 r 表示 , 相对质心的位置由矢量 r 表 2 1 ( ) 将式 3对时间求导数 , 得 示 。平面图形质心 C 在动坐标系的位置由矢量 r 表 d G d G d PC r) ( ( )= r ×M V +. 4 C d t d t d t 示 。 其中 ( ) 平面图形对速度瞬心 即对动点 P和质心 C 点的动量矩表达式分别是 : 收稿日期 :2000210220 ( ) ( ) 作者简介 :尹莉 1965 - ,女 汉族,陕西汉中人 ,讲师 ,硕士 ,从事工程力学教学不研究工作 。 )( d M V d理不相对固定点或质心的动量矩定理有相同的形d r C ( ) ×M V+ r ×= r ×M V = CC d t d t d t 式 , 平面运动刚体对速度瞬心的动量矩对时间的导 n d r e 数等于作用在平面运动刚体上所有外力对速度瞬心 )( ×M V + r × F.5 C i ? d t i = 1 的主矩 。 n )( d M V Ce ( ) = F。 式 5中利用了质心运动定理 :证明完毕 。 i ? d t i = 1 在相对动坐标系 Px y中 , C 点的位置由矢量 r 1 1 3 瞬心动量矩定理的应用 = re 确定 , 如图 2 所示 。e 和 e均为单位矢量 , e 的 θr r r 在处理实际问题时 , 直接利用该结果可简化解 方向为径向 , e不 e 垂直 。 θ r 题步骤 。 d e θr d r d r d r d= e + r= e +( )r .6 eθr r 图 3 给出质量为 M 的均质杆 A B = 2 a , 它在竖 d t d t d t d t d t 直平面内不光滑面接触 。在杆的平面运动中 , 由于质 由于平面运动图形的质心不速度瞬心的距离不 心 C 不速度瞬心 P 的距离保持不变 , 可利用相对速 d r 变 , 即 C P = r = 常数 , 则 = 0 。动坐标轴不固定d t θ 度瞬心动量矩定理比较方便地求出均质杆在任意 θ dd rε瞬时的角加速度。 θ( ) θ坐标轴平行 , 所以 = , 则式 6变为 re = θd t d t 匀质杆 A B 对速度瞬心 P 的动量矩为 :θ= V 。C 点相对动点 P 的速度为 : V = r, 且 V C r Cr Cr2 ω ( ) ωG= J = J + M a , P PC 垂直于 r 。 作用在 A B 杆上所有外力对速度瞬心 P 的力矩 为 : e( ) θM F= M gaco s.P i ? 其中 , J 和 J 是杆 A B 分别对速度瞬心 P 和质心 C P C ω θ 的转动惯量 ;是均质杆在任意瞬时的角速度 。 ω d2 ( ) ( ) θ由式 9得 J + M a= M gaco s, C d t 图 2 相对动坐标系示意图 所以 在任意瞬时 , 由于 P 点是平面图形速度的瞬 ωθθ dM gaco s3co s ε = == g . 2心 , 所以质心 C 相对点 P 的速度必定不质心 C 的速 d t 4 a + M a J C ( ) 度平行 。即 V 平行于 V , 则式 6中 Cr C d r ×M V = V×M V = 0 , C C r C d t ( ) 式 5可写为 n d e ( r × M V = r × F.( )) 7 C i ? d t i = 1 [ 1 ] ( ) 根据相对质心 平移坐标动量矩定理: n d G Cre ( )= r × F,8 1 i ? d t i = 1 ( ) ( ) ( ) 将式 7和式 8代入式 4中得 图 3 待求问题示意图 n n d G Pe e 如果用相对质心动量矩定理求解 , 则须先利用 = r × F+r × Fi 1 i?? d t i = 1 i = 1 质心运动定理求出约束反力 N , N。A B n n ee( )= r +r × F = r× F 1 i 2 i ?? i = 1 1 i = 参考文献 : n e( ) ( )= M F.9 ( ) P i 1 范钦珊. 工程力学教程 ?M . 北京 : 高等教育出版 ? i = 1 社 ,1998 .( ) 由式 9可知 , 当平面运动刚体的质心不速度 () 责任编辑 吕 鹏 瞬心的距离保持不变时 , 相对速度瞬心的动量矩定 Journal of t he U niversity of Pet roleum , China Vol . 25 No . 4 ??? ( )Editio n of Nat ural Science Aug. 2001 rameters o btained f ro m t he field test s of Liao he oilfield co rrespo nd to t he parameters f ro m o ur experiment s. The AVO P field test s in Liao he oilfield got appparent eco no mic p rofit s. Key words : artificial vibratio n technique ; seismic so urce ; visco sity reductio n ; o rt hogo nal design ; test REL ATIO N OF GRO UND STRESS WITH THE P ROPERTIES OF LOW PERM EABL E CO RE/ Z HA N G X i n2 ) ( hon g , et al . Col lege of Pet roleu m En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , Don gyi n g 257061S ( ) hi you D a x ue X uebao 2001 , 25 4:56～57 The gro und st ress is an impo rtant influence f acto r o n fo r matio n parameters of low per meabilit y reservoir , espe2 cially , directly effecting reservoir po ro sit y , per meabilit y , co mp ressibilit y , etc. . An experimental st udy in labo2 rato ry was carried o ut to deter mine t he effect of gro und st ress o n co re po ro sit y , per meabilit y and co mp ressibilit y (by using co nfined p ressure to simulate terra st ress. The per meabilit y2co nfined p ressure curve , t he po ro sit y co m2 ) p ressibilit y2co nfined p ressure curve ,and p ressure gradient t ransfo r matio n curve are o btained. The state equa2 ( ) tio ns amo ng rock po ro sit y co mp ressio n coefficient , per meabilit y and co nfined p ressure are established. The testing result s show t hat t he co re po ro sit y , per meabilit y , co mp ressibilit y of low2per meabilit y co re are sensitive to co nfined p ressure , and t here are expo nential degressio n relatio n amo ng t hem. It is fo und t hat co re p ro duces elas2 tic disto rtio n and plastic disto rtio n wit h t he change of co nfined p ressure . The low2per meabilit y co re has a start2 up p ressure gradient in t he co urse of fluid flow . Key words :low2per meabilit y co re ; gro und st ress ; co nfined p ressure ; per meabilit y ; po ro sit y ; experiment SA ND2CO NSOL IDATIO N EFFECT OF M IDDL E2LOW TEMPERATURE O IL A ND WATER WELLS/ DA I ( ) Cai2l i , et al . Col lege of Pet roleu m En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , Don gyi n g 257061S ( ) hi you D a x ue X uebao 2001 , 25 4:58～60 The sand2co nsolidatio n effectiveness of resino us sand2co nsolidatio n agent s fo r oil and water wells at middle2low temperat ure is evaluated experimentally. The effect s of solidificatio n time and temperat ure , t ypes and amo unt of catalyst , and different grain sizes of quartz sand o n sand2co nsolidatio n effectiveness are discussed. The per me2 abilit y variatio n by injecting sand2co nsolidatio n agent s into sandsto ne co re and critical flow rate of sand p ro duc2 tio n are deter mined. The f acto rs affecting sand2co nsolidatio n effectiveness are analyzed. The p roper sand2co nsol2 idatio n agent s available fo r oil and water wells at middle2low temperat ure can be selected o n t he basis of experi2 ment . Key words : oil2water wells ; sand co nsolidatio n ; resino us sand2co nsolidatio n agent ; experiment EXACT SOL UTIO N OF TRA NSVERSELY FREE VIBRATIO N FO R A RECTA NGUL A R PL ATE AT TWO ( OPPOSITE SIMPL E2S UPPO RTED SID ES/ M EI Fu2l i a n g Col lege of Pet roleu m T ra ns port & S tora ge a n d ) ( ) Ci v i l En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , 257061. S hi you D a x ue X uebao 2001 , 25 4:61～ 63 An analytical solutio n of t ransversely o scillatio n fo r a rectangular plate wit h t wo oppo site simple2suppo rted sides and t he ot her t wo oppo site arbit rary2suppo rted o nes is o btained by viewing elastic point suppo rt s as deter mina2 tio n2awaited exter nal fo rces. U sing t he relatio nship bet ween t he deflectio n and t he deter minatio n2awaited exter2 nal fo rce at an elastic point suppo rt , a f requency equatio n exp ressed in deter minant w ho se ranks equal to t he numbers of elastic point suppo rt s is o btained. A co mp utatio n example is p rovided. Key words : rectangular plate ; elastic point suppo rt ; t ransversely f ree o scillatio n ; exact solutio n D EVELOPM ENT OF MOM ENT OF MOM ENTUM THEO REM TO INSTA NTA NEO US CENTER OF VE2 ( LOCITY FO R RIGID BODY’S PL A NE MOTIO N/ Y I N l i Col lege of Pet roleu m T ra ns port & S tora ge a n d ) Ci v i l En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , Don gyi n g 257061. S hi you D a x ue X uebao 2001 , ( ) 25 4:64～65 A mo ment of mo ment um t heo rem to instantaneo us center of velocity fo r a rigid bo dy in plane motio n is p ro2 po se d. It is p roved t hat t he fo r mula of mo ment of mo ment um t heo rem to an instantaneo us center is same as t hat fo r a fixed point o r moving t he center of mass if t he distance bet ween t he center of mass and t he instantaneo us center of velocit y is co nstant . This mo ment is co nvenient to analyze so me questio n in st udying plane motio n of a rigid bo dy under t he same co nditio ns. Key words : rigid bo dy ; plane motio n ; instantantaneo us center ; center of mass ; mo ment of mo ment um A D ESIGN M ETHOD OF STRAIGHTENING D EVICE IN W HIPSTOC K FO R RAD IAL HO RIZO NTAL ( D RILL ING BY BP NEURAL NETWO RK/ W A N G H ui2yi , et al . Col lege of Pet roleu m En gi nee ri n g i n t he ) ( ) U ni ve rsi t y of Pet roleu m , Chi n a , Don gyi n g 257061S hi you D a x ue X uebao 2001 , 25 4:66～68Acco rding to t he design demand fo r st raightening device in w hip stock of radial ho rizo ntal drilling , t he p rinciples fo r design of t he device are suggested. The po sitio ns of t he w heels in t he st raightening device are calculated by using B P neural net wo r k . The result s of calculatio n co nsist wit h t ho se of experimental well . B P neural net wo r k is used to op timize t he resistance of t he st raightening device . It is p roved t hat t his met ho d can raise t he accuracy of design and experiment remar kably. Key words : radial ho rizo ntal drilling ; w hip stock ; st raightening device ; neural net wo r k INVESTIGATIO N O N PERFO RMA NCE CO NVERSIO N OF CENTRIFUGAL O IL P UMP / X U E D u n2son g , ) ( et al . Col lege of M echa nical a n d Elect ronical En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , 102200S ( ) hi you D a x ue X uebao 2001 , 25 4:69～71 A series of test s o n cent rif ugal oil p ump s wit h different specific speeds were do ne under different co nditio ns of liquid visco sities and p ump rotary speeds. Based o n t hese result s of test s , new co nversio n curves are p resented wit h co nsideratio n of t he effect of rotary speed o n co nversio n coefficient . The result s of test s wit h different met ho ds p ropo sed by t hree co unt ries of Soviet U nio n , America and Ger man because of different p roperties of oil and p ump s used in t hese met ho ds. Key words : oil cent rif ugal p ump ; liquid ; visco sit y co nversio n ; perfo r mance test s ( THIXOTROPY OF WAXY O IL FLOWING THRO UGH PIPEL INE/ Z HA N G Zu2bi n , et al . Col lege of Pet roleu m T ra ns port & S t ora ge a n d Ci v i l En gi nee ri n g i n t he U ni ve rsi t y of Pet roleu m , Chi n a , Don gyi n g ) ( ) 257061S hi you D a x ue X uebao 2001 , 25 4:72～74 An experimental inst rument fo r researching pipe2flow is used to st udy t hixot rop y of waxy oil flowing in pipeline . The result s show t hat t he effect s of histo ries , including shear p rocess befo re shut2up of pipeline and t he co nt ract of crude oil in t he shut2up co urse , o n t he recovery of t he waxy st ruct ure are uncertain . The uncertainty has lit tle influence o n equilibrium shear st ress af ter restart2up of oil flowing. Key words : waxy crude oil ; pipeline t ranspo rtatio n ; t hixot rop y ; waxy st ruct ure ; uncertaint y CO RROSIVITY OF MA RINE ATMOSP HERE AT C HENGDAO REGIO N A ND AFFECTING FACTO RS OF ( CO RROSIO N/ W EN G Y on g2j i , et al . Col lege of M echa nical a n d Elect ronical En gi nee ri n g i n t he U ni ve rsi t y ) ( ) of Pet roleu m , Chi n a , B ei j i n g 102200S hi you D a x ue X uebao 2001 , 25 4:75～78 Weightlessness test s o n t he marine platfo r m at Chengdao regio n were carried o ut . The at mo sp heric co rro sivit y of marine at mo sp here was classified acco rding to ISO 9223 standard. The o ut doo r co rro sio n rates are in t he highest ( ) co rro sio n degree . The influences of temperat ure , relative humidity R Hand t he salt depo sit o n at mo sp heric