材料力学-第3章课件

1、HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG

2、 UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECMM HUAZHONG UNIVERSI

3、TY OF SCI & TECWPtM stM260nM功率功时间力偶矩 角位移角速度每分钟的转数60 ()2( / min)P KWMn r9549()PN mnHUAZHONG UNIVERSITY OF SCI & TEC06595499549137.9()4500PMN mnHUAZHONG UNIVERSITY OF SCI & TEC由平衡方程由平衡方程 0MMTTM0M取左边部分假想切面外力偶外力偶 外力偶外力偶 内力偶内力偶 平衡平衡0M0MHUAZHONG UNIVERSITY OF SCI & TECTMTMTM正负HUAZHONG UNIV

4、ERSITY OF SCI & TECmkN 20mkN 10mkN 10mkN 10TMmkN 20TMmkNMT10mkNMT 20mkNMT/HUAZHONG UNIVERSITY OF SCI & TEC9549AANMn9549BBCNMMn9549DDNMn40095495460700N m12095491640700N m16095492180700N mHUAZHONG UNIVERSITY OF SCI & TEC546016402180ABCDMN mMMN mMN mBC段段11640TMN m 23280TMN m AD段段32180TMN m最

5、大扭矩发生在最大扭矩发生在AC段各横截面上段各横截面上3280TMN mHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TEC HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC 根据圆轴的轴对称性质C、D两点必须具有相同的位移，因而二者必须位于同一圆周上。 根据

6、反对称要求，C、D两点不可能有轴向位移，因而必须仍然位于原来所在的圆周上。HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC正应变是单位长度的线变

7、形量？xu+duxxxuxdddxxu HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC 若将圆轴用同轴柱面分割成许多半径不等的圆柱，根据上述结论，在dx长度上，虽然所有圆柱的两端面均转过相同的角度d，但半径不等的圆柱上产生的切应变各不相同，半径越小者切应变越小。 HUAZHONG UNIVERSITY OF SCI & TEC xdd 设到轴线任意远处的切应变为（），则从图中可得到如下几何关系： HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UN

8、IVERSITY OF SCI & TECGHUAZHONG UNIVERSITY OF SCI & TECxGGdd xddGHUAZHONG UNIVERSITY OF SCI & TECxGGddHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECAdAxMHUAZHONG UNIVERSITY OF SCI & TECAAId2PxGGddPddGIMxxAxMAdHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVE

9、RSITY OF SCI & TECxGGddPddGIMxx PIMxHUAZHONG UNIVERSITY OF SCI & TEC PIMxHUAZHONG UNIVERSITY OF SCI & TECPPmaxmaxWMIMxxmaxPPIW WP 抗扭截面模量HUAZHONG UNIVERSITY OF SCI & TEC43PP,3216ddIW4434PP11,3216DDIW232P00P002,2pIIRdAR tWR tRHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF S

10、CI & TEC7 595499549716 2N m100.xPMTnmax13P111640MPaxxMMWd31616 716 20 045m=45mm40 10d.HUAZHONG UNIVERSITY OF SCI & TECd20.5D223 mmmax234221640MPa1xxPMMWD324616 716 20 046m=46mm 1-40 10D.HUAZHONG UNIVERSITY OF SCI & TEC解：确定实心轴与空心轴的重量之比空心轴取D246 mm d223 mm实心轴取d1=45 mm长度相同的情形下，二轴的重量之比即为横截面面积

11、之比：28. 15 . 01110461045122332222121DdAAHUAZHONG UNIVERSITY OF SCI & TEC3HUAZHONG UNIVERSITY OF SCI & TECP1=14kW, P2= P3= P1/2=7 kWn1=n2= 120r/min360r/minr/min12361203113zznn3HUAZHONG UNIVERSITY OF SCI & TECMx1=T1=1114 N.mMx2=T2=557 N.mMx3=T3=185.7 N.m3HUAZHONG UNIVERSITY OF SCI & TEC

12、 1max3-9P116 1114EPa16.54MPa 7010 xMW 2max3-9P216 557HPa22.69MPa 5010 xMW 3max3-9P316 185.7CPa21.98MPa 3510 xMW3HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECABCDABCD HUAZHONG UNIVERSITY OF SCI & TECxyzdxdydz HUAZHONG UNIVERSITY OF SCI & TECdxdydzxyz = HUAZHONG UNIV

13、ERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECPddGIMxxxGIMxddP 单位长度扭转角ddxHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC微段变形累加的结果xGIMxddPlxABxGIM0PdHUAZHONG UNIVERSITY OF SCI & TEC返回

14、返回HUAZHONG UNIVERSITY OF SCI & TECx y xxy yx xy yx xyHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC塑性材料脆性材料HUAZHONG UNIVERSITY OF SCI & TEC受扭圆轴失效的标志：受扭圆轴失效的标志：塑性材料：屈服，扭转屈服应力： ；s脆性材料：断裂，扭转强度应

15、力： ；b统称扭转极限应力：u因此圆轴的强度条件为因此圆轴的强度条件为maxmaxP TW式中 un（许用切应力， n为安全系数）HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC maxllmax lHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECmax 或maxmax180 (/)PTmGI单位：的数值按照对机器的要求决定：精密机器的轴： (0.25 0.5) / m一般传动轴： (0.5 1) / m

16、精度要求不高的轴： (1 2.5) / mHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC解：解：34P(1)16WD3421() 16DDtD6329.24 10m1DHUAZHONG UNIVERSITY OF SCI & TECmaxmaxPTW6150051.329.24 10MPa 60MPamaxmaxP151.3TMPaW3P1116WDmax316160.05351.3 10TDmHUAZHONG UNIVERSITY OF SCI & TEC2114AD22205m

17、m2222()6874ADdmm221168731%2205GAGA因此在承载能力相同的条件下，使用空心轴因此在承载能力相同的条件下，使用空心轴比较节约材料、比较经济。比较节约材料、比较经济。HUAZHONG UNIVERSITY OF SCI & TEC例例 4 某某传动轴的转速为传动轴的转速为 n= =183.5r/ /min，输出功率为，输出功率为PA=0.756kW，PC=2.98kW，材料的材料的 G=80GPa， =40MPa， =1.5 / /m。试设试设计轴的直径计轴的直径 d。解：由MeAACBMeBMeCe0.756955039 N m183.5AMe2.98955

18、0155 N m183.5CMeee194 N mBACMMM，得到0 xM HUAZHONG UNIVERSITY OF SCI & TEC由扭矩图可知由和39 N mT+.155 N m.-max155 N mT3.按强度条件求 dmaxmaxP TW得到3P16dWmax3361616 155 m0.0272 m 40 10Td MeAACBMeBMeCHUAZHONG UNIVERSITY OF SCI & TEC4. .按刚度条件求按刚度条件求 d由和39 N mT+.155 N m.-得到maxmaxP180 TGI4P32dImax4232180 TdG49232

19、155180 m0.0297 m80101.5 取取 d =29.7mm。 可见：可见：此轴的直径是由刚度条件控制的此轴的直径是由刚度条件控制的MeAACBMeBMeCHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TEC扭转超静定问题：仅由平衡方程不能确定支反力偶矩或扭矩的扭转问题 1. .平衡方程；平衡方程；解扭转超静定问题需要从三方面考虑，即： 2. .变形协调方程；变形协调方程； 3. .物理方程。物理方程。HUAZHONG UNIVERSITY OF SCI & TEC例例 5

20、两端固定的阶梯形圆轴两端固定的阶梯形圆轴AB，在，在C处作用一外力处作用一外力偶矩。已知偶矩。已知CB段段轴的抗扭刚度为轴的抗扭刚度为AC段的二倍，段的二倍， 试求轴两端的支反力偶矩和试求轴两端的支反力偶矩和C截面的截面的扭转角扭转角 C。解解：为一次超静定问题为一次超静定问题。1. .求求支反力偶矩支反力偶矩( (1) )平衡方程平衡方程ACBl2_l2_Me( (2) )变形协调方程变形协调方程( (3) ) 物理方程物理方程e0ABMMMCBCAC pP22, ABCACBCBACllMMGIGI( (1) )( (2) )( (3) )MCACBxMAMBMAMeACBx CMBHUA

21、ZHONG UNIVERSITY OF SCI & TEC 将将( (3) )式代入式代入( (2) )式，并考虑到式，并考虑到由由( (1) )和和( (4) )式求得式求得可得可得 ACCBGIGIpp2 ABMM2 ( (4) )3eMMA 32eMMB HUAZHONG UNIVERSITY OF SCI & TEC2. .求求C截面的扭转角截面的扭转角ePP26ACCAACAClMM lGIGIMAMeACBx CMBHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECHUA