附录(截面图形的几何性质)

附录截面图形的几何性质截面图形的一次矩、二次矩是如何定义的？几种典型截面的二次矩各是多少？形心在计算中起着怎样的作用？如何计算复杂图形的一次矩和二次矩？一、几何图形的一次矩单位：m3或mm3重要结论坐标轴通过形心，则相应的静矩为零。cxcycASAyAyxAcd1AxSAyScycx面积矩（静矩）AySAxdAxSAyd(firstmomentofarea)dAyxxyOAxAxAcd1ASy形心(centerofanarea)公式例求如图半径为R的四分之一圆的形心位置。xyrrRddsin0220241RA34RycAySAxd组合图形的形心公式为组合图形的面积矩组合图形的面积iiciiyiyAxSSiiAAiiiicicAAxx34Rxc同理组合图形ddsin200rrrR331RdArθO例求如图截面的形心位置。232aAayc25二、几何图形的二次矩AyIAxd2AxIAyd2惯性矩(momentofinertia)ArAyxIAApd)d222(极惯性矩(polarmomentofinertia)惯性积(productofinertia)AxyIAxyd3aaa3a7a/23a/27a/23a/25a/2aaSx2732aa2332xy315a3121bhddsind220022rrrAyIRAxrrRddsin03202例求如图半径为R的四分之一圆的惯性矩和极惯性矩。求如图矩形的惯性矩与惯性积。yxpyxIIIIIAyIAxd2yxyhhbbdd222223121hbIyAxyIAxyd0dd2222yyxxhhbb4161RIy481RIp4161RxydArθhbyx动脑又动笔Dxy4161RIIyxDxy4425612161DD4641DIIyx4321DIp44641dDIIyx)1(64144DDd441321DIpDxyd动脑又动笔求图形的惯性矩与惯性积。实心圆空心圆重要结论坐标轴是图形的对称轴，则惯性积为零。轴的惯性矩为。重要数据矩形截面对通过形心且平行于底边的坐标Ix=bh—1213重要数据实心圆截面对通过圆心的坐标轴的惯性矩，极惯性矩为。为Ip=πD—3214Ix=πD—6414，极惯性矩空心圆截面的惯性矩为，α为内径与外径之比。Ix=πD—6414(1–α)4(1–α)4Ip=πD—3214为三、平行移轴定理(parallel-axistheorem)AayIAxd2)(AaAyaAyAAAdd2d220dAyAAaIIxx2abAIIyxxy同理AbIIyy2AaIIxx2AbIIyy2重要公式abAIIyxxydAyxdAyxccbax’y’y’x’例求如图的截面对形心轴的惯性矩。cccxxxIII213aaa3axaa7a/23a/25a/2xc433253121)3(121aaaaaIyy223133121aaaaIcx)(223133121aaaaIcx)(4217aIcx223233121aaaaIcx)(223233121aaaaIcx)(4413a4421a3241bhIK动脑又动笔求三角形对于过形心的C轴的惯性矩。bhCKhKC6133361181bhbhIC例求如图的截面对x和y轴的惯性矩。半圆对K轴的惯性矩4481264121aaIx)(已知半圆对x轴的惯性矩为故图形对x轴的惯性矩为44435520436481244121aaaaaIx.)()(4481264121aaIK)(aaxyaaaaaaaaxyaaaaaaaaxyaaaaaaK半圆对y轴的惯性矩为42448924181aaaaIx故半圆对y轴的惯性矩为故原图形对y轴的惯性矩为4222438817342219881aaaaIy444331383438038817244121aaaaaIy.))((a342y轴与C间的距离为aaxyaaaaaaK4a/3πCK半圆对C轴的惯性矩224342412181aaaIc)(?42249881342412181aaaaIc)(yxOKQPx’y’αOKP’Q’RSPSORPRORPOxPRSKPSSKKPyOPxOQysincosPKOPsincosyxsincosOPPKcossinyx1.两种坐标的转换x’y’αxy四、转轴定理(rotation-axistheorem)POxQOysincosyxxcossinyxy2sin2cos2121xyxyxyxIIIIII)()(2cos2sin21xyxyyxIIII)(2sin2cos2121xyxyxyyIIIIII)()(2.转轴公式的推导AxyyxAxIAAydsincos2sincosd2222AxyAyAxAAAdsincos2dsindcos2222sincos2sincos22xyxyIIIsin221sincoscos2121sincos2121cos22x’y’αxy3.惯性主轴(principalaxesofinertia)yx48127327313194131141141386400001400yxyxyx若对某轴的惯性积为零，则称该轴为惯性主轴，如果惯性主轴通过形心，则称之为形心惯性主轴。6004004001200判断形心惯性主轴重要结论坐标轴是图形的对称轴，则惯性积为零；此时坐标轴一定是惯性主轴。故正方形中任意通过形心的轴都是形心惯性主轴。xy如图，对于平行于底边的形心坐标系，正方形的惯性矩和惯性积为对于其它任意的形心坐标系，其惯性积为例证明正方形中任意通过形心的轴都是形心惯性主轴。x’y’02cos2sin21xyxyyxIIII)(02cos2sin21xyxyyxIIII)(01214xyyxIaII01214xyyxIaII零次矩一次矩二次矩惯性矩惯性积极惯性矩定义符号恒正可正可负恒正可正可负恒正单位轴过形心不为零等于零不为零轴为对称轴时才为零不为零关于形心计算AAAdApAyxId22)(AxyIAxydAyIAxd2AxIAyd2AySAxdAxSAyd2mabAIIcxyxy3m4mAbIIcyy2AaIIcxx2AxScyAyScx本章内容小结静矩形心的计算方法组合图形静矩的计算常用图形的惯性矩平行移轴公式转轴公式AaIIxx2AbIIyy2abAIIyxxy惯性矩AyIAxd2AxIAyd2惯性积AxyIAxyd极惯性矩ArAyxIAApd)d222(