材料力學-第3章課件

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功率功時間力偶矩 角位移角速度每分鐘的轉數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取左邊部分假想切面外力偶外力偶 外力偶外力偶 內力偶內力偶 平衡平衡0M0MHUAZHONG UNIVERSITY OF SCI & TECTMTMTM正負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、大扭矩發(fā)生在最大扭矩發(fā)生在AC段各橫截面上段各橫截面上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 根據圓軸的軸對稱性質C、D兩點必須具有相同的位移，因而二者必須位于同一圓周上。 根據

6、反對稱要求，C、D兩點不可能有軸向位移，因而必須仍然位于原來所在的圓周上。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正應變是單位長度的線變

7、形量？xu+duxxxuxdddxxu HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC 若將圓軸用同軸柱面分割成許多半徑不等的圓柱，根據上述結論，在dx長度上，雖然所有圓柱的兩端面均轉過相同的角度d，但半徑不等的圓柱上產生的切應變各不相同，半徑越小者切應變越小。 HUAZHONG UNIVERSITY OF SCI & TEC xdd 設到軸線任意遠處的切應變?yōu)椋ǎ?，則從圖中可得到如下幾何關系： 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：確定實心軸與空心軸的重量之比空心軸取D246 mm d223 mm實心軸取d1=45 mm長度相同的情形下，二軸的重量之比即為橫截面面積

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 單位長度扭轉角ddxHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC微段變形累加的結果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受扭圓軸失效的標志：受扭圓軸失效的標志：塑性材料：屈服，扭轉屈服應力： ；s脆性材料：斷裂，扭轉強度應

15、力： ；b統(tǒng)稱扭轉極限應力：u因此圓軸的強度條件為因此圓軸的強度條件為maxmaxP TW式中 un（許用切應力， n為安全系數）HUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TEC maxllmax lHUAZHONG UNIVERSITY OF SCI & TECHUAZHONG UNIVERSITY OF SCI & TECmax 或maxmax180 (/)PTmGI單位：的數值按照對機器的要求決定：精密機器的軸： (0.25 0.5) / m一般傳動軸： (0.5 1) / m

16、精度要求不高的軸： (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因此在承載能力相同的條件下，使用空心軸因此在承載能力相同的條件下，使用空心軸比較節(jié)約材料、比較經濟。比較節(jié)約材料、比較經濟。HUAZHONG UNIVERSITY OF SCI & TEC例例 4 某某傳動軸的轉速為傳動軸的轉速為 n= =183.5r/ /min，輸出功率為，輸出功率為PA=0.756kW，PC=2.98kW，材料的材料的 G=80GPa， =40MPa， =1.5 / /m。試設試設計軸的直徑計軸的直徑 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由扭矩圖可知由和39 N mT+.155 N m.-max155 N mT3.按強度條件求 dmaxmaxP TW得到3P16dWmax3361616 155 m0.0272 m 40 10Td MeAACBMeBMeCHUAZHONG UNIVERSITY OF SCI & TEC4. .按剛度條件求按剛度條件求 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。 可見：可見：此軸的直徑是由剛度條件控制的此軸的直徑是由剛度條件控制的MeAACBMeBMeCHUAZHONG UNIVERSITY OF SCI & TEC返回返回HUAZHONG UNIVERSITY OF SCI & TEC扭轉超靜定問題：僅由平衡方程不能確定支反力偶矩或扭矩的扭轉問題 1. .平衡方程；平衡方程；解扭轉超靜定問題需要從三方面考慮，即： 2. .變形協(xié)調方程；變形協(xié)調方程； 3. .物理方程。物理方程。HUAZHONG UNIVERSITY OF SCI & TEC例例 5

20、兩端固定的階梯形圓軸兩端固定的階梯形圓軸AB，在，在C處作用一外力處作用一外力偶矩。已知偶矩。已知CB段段軸的抗扭剛度為軸的抗扭剛度為AC段的二倍，段的二倍， 試求軸兩端的支反力偶矩和試求軸兩端的支反力偶矩和C截面的截面的扭轉角扭轉角 C。解解：為一次超靜定問題為一次超靜定問題。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) )變形協(xié)調方程變形協(xié)調方程( (3) ) 物理方程物理方程e0ABMMMCBCAC pP22, ABCACBCBACllMMGIGI( (1) )( (2) )( (3) )MCACBxMAMBMAMeACBx CMBHUA

21、ZHONG UNIVERSITY OF SCI & TEC 將將( (3) )式代入式代入( (2) )式，并考慮到式，并考慮到由由( (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截面的扭轉角截面的扭轉角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