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LIU Jiemin @SJZU 2008CH.4 TORSION
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
Shaft ( 轴 ) : Members subjected mainly to torsion
Torsion: 外力的合力为一力偶,其作用面与轴线垂直 , 变形的特点:两横截面发生绕轴线的相对转动。Circular torsion:
① Torque calculation
② Torsional shear stress and strength
③ Torsional deformation and stiffness
M
F
F
4.1 Introduction
LIU Jiemin @SJZU 2008CH.4 TORSION
4.1 Introduction
LIU Jiemin @SJZU 2008CH.4 TORSION
B
A
剪应变 :直角的改变量。
M
M
扭转角 :任意两截面绕轴线转动而发生的角位移。
O
x
4.1 Introduction
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
4.2.1 Twisting couple exerted to transmission shaft
Power P inputing to a transmission shaft =Twistingcouple M times angle velosityω MP
n
P9549
n
P
2
1060M
m/sN60
n2Mm/sN10P
m/sN60
n2MkWP
r/min
kW3
mN
3
}{
}{}{
min/r
HFmN n
P7024M
m/sN60
n2Mm/s735NP
m/sN60
n2MHFP
}{
}{}{
4.2 External moment
Torque&Torque Diagram
LIU Jiemin @SJZU 2008CH.4 TORSION
2. Symble of T : Torque is povitive when the torque vector leaves the cross-section.
4.2.2 Torque and torque diagram
MM
M T
MT
MT
mx
0
0
x
3. Determination of T by (simple) section method
1. Torque : Moment on cross-sction , T
n
n
x( )x iT M F
4.2 External moment,Torque&Torque Diagram
LIU Jiemin @SJZU 2008CH.4 TORSION
4. Torqur diagram : Curve showing torque veriation with cross-section position
Usage Determine critical section and strength caculation easily
x
T
2T
2T TT
T
4.2 External moment,Torque&Torque Diagram
LIU Jiemin @SJZU 2008CH.4 TORSION
Ex.1 The transmission shaaft. Knowing : n =300r/min , active wheel PA = 500 kW , Passive PB=150kW , PC=150kW , PD=200kW , Plot the torque diagram.
nB C A D
PB PC PA PD
①Twisting couples
m)15.9(kN 300
5009.55
n
P.M A
A 559
m)(kN .4300
1509.55
n
P.MM B
CB 78559
m)(kN .300
2009.55
n
P.M D
D 376559
Solution :
4.2 External moment,Torque&Torque Diagram
LIU Jiemin @SJZU 2008CH.4 TORSION
②Find Torques
mkN.MT
MT , m
B
BC1
784
00
1
1
mkN...
MMT CB
5697847842
mkN.MT D3 376
1
1
2
2n
B C A D
MB MC MA MD
3
3
By the same token :
③Torque diagram x
T (kNm)
4.78
9.56
6.37
––
mkN .Tmax
569
4.2 External moment,Torque&Torque Diagram
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
1. Cross-section remains plane and undistorted, only rotates differential amounts about x-axle ;
2. Non axial deformation
4.3.1 Observation of experiment
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
4.3.2 Shearing stress on cross-section:
1. Geometry relation:
xd
d
xd
GGtan
1
xd
d
(R)
2. Physical relation:
xd
dGG
GHooke’law :
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
3. Static mechanics relation:
O
dA
Adxd
dG
Adxd
dG
AdT
A
A
A
2
2
AdI Ap2set
xd
dGI T p
pGI
T
xd
d
xd
dG
pI
T
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
① Only valid for Isotropy, linear elasticity materials under small deformation condition
② T-Torque on cross-section , obtained from section method
-Distance of the point where shearing stress τis calculated from the center of the circular section.
Ip-Polar inertia moment , a geometry amount
pI
T
4. Discussion:
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
R/I
T
I
TR
ppmax
4.3.3 Maximum torsional stress
R/IW pp
pmax W
T
Section modulus in torsion
AdI Ap2
Popar inertia moment
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Unit : mm4 , m4.
44
2
0
2
2
1032
2
D.D
d
AdID
Ap
For circular section :
D
d
O
dD
D/d
DI p
)( 4
4
132
For circular ring section :
16
DW
3
p
)( 41
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
④ Stress distribution
( Circular section ) ( Circular ring section)
Using circular ring section:
enhancing strength, save material, light in weight
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Thin-walled cylinder:
010
1R ( R0 : Average radius )
dA
R0
τdistributes uniformly
TR R
TRAd A
00
0
2
A
T
R
T
020 22
A0 : Area corresponding to R
0
4.3.4 Torsional stress in thin-walled cylinder
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
a
c d
b
´
´
dA
R0
)(
12
EG
薄壁圆筒扭转试验可建立剪切胡克定律
G
4.3 Torsional stresses in a circular shaft
Strength condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
4.4.1 Ultimate stress in in torsion
Mild steel:τs
Cast iron : fracture along about 45 with axle :τb
Maximum shearing stress when yielding or fracture
45
u
4.4 Strength condition and rational design
LIU Jiemin @SJZU 2008CH.4 TORSION
4.4.2 Strength calculation of circular shaft
Strength condition :
Circular shaft with equal section :
][)( maxP
max W
T
][W
T
p
max
Three types of strength condition :
① Check :
② Design section :
③ Find allowable load :
][ p
maxmax W
T
][max
p
TW
][pmax WT
nu ][Allowable stress
4.4 Strength condition and rational design
LIU Jiemin @SJZU 2008CH.4 TORSION
Ex.4.3 The variable shaft with th same average radias R0, length l=1m, subjected to uniform moment m=3500Nm/m. Chech the strength of the shaft. []=50MPa,knowing that:
m/mN 3500 3500m m/mN
l/2 l/2
x
CA
B
0 AB BCR =50mm, δ =5mm,δ =4mm
4.4 Strength condition and rational design
LIU Jiemin @SJZU 2008CH.4 TORSION
For critical section A
② Check strength
][
6MPa.445502
10003500
R2
ml
W
T
2
AB20p
max1
m/mN 3500 3500m m/mN
l/2 l/2
x
CA
B
T
x
ml
( )AM = ml, T = m l - x
Solution :① Analysis of torque
For critical section B
B2 2
p 0 BC
27 9MPa [ ]W 2 R
T ml /.
2
4.4 Strength condition and rational design
LIU Jiemin @SJZU 2008CH.4 TORSION
例 4.2 功率为 150kW ,转速为 15.4 转 / 秒的电动机转子轴如图,许用切应力 []=30M Pa, 试校核其强度。
nNmTBC 2
103
m)(kN551
m)(N4151432
10150 3
.
..
Tm
解:①求扭矩及扭矩图
② 计算并校核剪应力强度
③ 此轴满足强度要求。
D3 =135 D2=75 D1=70A B C
m m
x
][MPa2316070
105513
3
max
.
.WT
t
4.4 Strength condition and rational design
LIU Jiemin @SJZU 2008CH.4 TORSION
Chapter4 Torsion Chapter4 Torsion
4.1 Introduction
4.2 Twisting couples and torque
4.3 Torsional stresses in a circular shaft
4.4 Torsional strength of circular shaft
4.5 Torsional deformation of circular shaft, Torsional stiffne
ss condition
4.6 Torsion of noncircular shafts
4.7 Torsion of thin-walled shafts
LIU Jiemin @SJZU 2008CH.4 TORSION
4.5.1 Torsional deformation
From:
pGIT
x
dd
Knowing : twisting angle between two cross-sections with length l:
l
p
xdGI
T d
0
pGI
Tl
IG
lT
ipi
ii
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
4.5.2 Unit torsional angle (rad/m) dd
pGIT
x
/m)( 180 dd
pGIT
x
or
4.5.3 Stiffness condition (rad/m) max pGI
T
/m)( 180 max pGI
T or
GIp: Measure capability of the section resisting torsional
deformation, called as Torsion stiffness of section
[ ]: Allowable unit torsional angle 。
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Three types of stiffness calculation :
① Strength check:
② Design section :
③ Fing allowable load:
max
] [
max
GT
I p
] [ max
pGIT
Sometimes , select materials by the condition
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Ex.4 The transmission shaft. n = 500 r / min , Inputting
P1 = 400kW, Outputting P2 = 150kW, P3 = 250kW.
Knowing : G=80GPa , [ ]=70M Pa , [ ]=1º/m ,Determine : ① d1 for portion AB and d2 for portion BC ? ②If the diameter of shaft is same , how to select d ? ③How to arrange rationally active wheel and passive wheel ?
Solution :⑴ Plot torque D 500 400
P
1
P3P2
A CB
T x
7.644.775
(kNm)
11
22
32
P=7.64kN m
P=2.865kN m
P=4.775kN m
m .n
m .n
m .n
9 55
9 55
9 55
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
][ Td
W p 16
3
3
16 4775000=70.3mm
3.14 70
Td
32
16
] [ 32 4
GTdI p
3
2
16 7640000N mm=82.2mm
3.14 70N/mm
Td
31
16
⑶ From Stiff.C :
⑵ From S.C. :
6
4 2 32 3
32 7.64 10 N mm 18086.4mm
N 1 [ ] 3.14 80 10mm 10 mm
Td
G
41
32
6
42
32 4.775 10 180=76.68mm
[ ] 3.14 80 1
Td
G
42
32
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
轴上的绝对值最大的扭矩越小越合理,所以, 1 轮和 2 轮应该换位。换位后 , 轴的扭矩如图所示 , 此时 , 轴的最大直径才为 75mm 。
T
x
4.21
(kNm)2.814
1 2 mm mmd , d 87 77
② 全轴选同一直径时 mmd d 1 87
①
T x
7.0244.21
(kNm)
Result:
③
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
例 5 长为 L=2m 的圆杆受均布力偶 m=20Nm/m 的作用,如图,若杆的内外径之比为 =0.8 ,外径 D=0.0226m , G=80GPa ,试求固端反力偶。
解:①杆的受力图如图示, 这是一次超静定问题。 平衡方程为:
02 BA mmm
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
② 几何方程——变形协调方程 0BA
③ 综合物理方程与几何方程,得补充方程:
0402202
00
P
A
P
AL
PBA GI
mdx
GIxm
dxGIT
mN 20 Am
④ 由平衡方程和补充方程得:
另 : 此题可由对称性直接求得结果。
mN 20 Bm
LIU Jiemin @SJZU 2008CH.4 TORSION
(rad/m) dd
pGIT
x
l/2
l/2
l/2m/mN 3500 3500m m/mN
l/2 l/2
x
CA
B
LIU Jiemin @SJZU 2008CH.4 TORSION
例 3.4 组合机床主轴箱内第四轴。动力由 5 轴经Ⅲ齿轮输入到 4 轴,齿轮Ⅱ和 Ⅳ带动轴 1 、 2 和 3 。轴 1 、 2 钻孔,共消耗功 率 0.755kW; 轴 3 扩孔,消耗功率 2.98kW 。若轴 4转速为 183.5r/min,G=80GPa,[τ]=40 MPa,[θ]=1.5°/m 。试设计轴的直径。
4 ⅣⅡ
5
1 , 2 3
Ⅲ
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
4 ⅣⅡ
5
Ⅲ
1 , 2 3mⅢ
(a)
mⅡmⅣ(b)
39.3Nm
155 Nm
-
+
扭矩图(c)
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
(1) Calculation of externall torque: m Ⅱ=9549N Ⅱ /n=9549×0.735/183.5=39.3 Nm m Ⅳ=9549N Ⅳ /n=9549×2.98/183.5=155 Nm m Ⅲ= m Ⅱ+ m Ⅳ=39.3+155=194.3 Nm(2)Design of the shaft diameter Tmax = 155 Nm From strength condition: τ max =Tmax /Wp=16Tmax/(πD3 ) ≤[τ] D1≥0.0272m
39.3Nm
155 Nm
-
+
扭矩图
4.5 Torsional deformation and stiffness condition
Solution:
LIU Jiemin @SJZU 2008CH.4 TORSION
From stiffness condition:
θ max =Tmax ×180/(GIp×π) ≤[θ]
D2 ≥0.0297m
D>D2 =30mm
Finally, The diameter of the shaft is determined as 30 mm. It can be seen that the design of the shaft is controlled by stiffness condition.
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
φ
m
轴受扭而转
薄壁筒不受扭
例 3 。 5 ( 107 页)轴受扭矩产生扭转角 φ 。这时把薄壁筒与轴 的凸缘焊接在一起,然后解除扭矩 m 。设轴和筒的抗扭刚度 分别是 G1Ip1 和 G2Ip2 试求轴内和筒内的扭矩。
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
φ1
φ2
φ
T2
T1
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
Solution: (1) Equilibrium equation: T1-T2=0 (1) (2) Compliant condition: φ1 + φ2 = φ (2)
(3) Physical relationship: φ =ml/(G1Ip1) φ1 =T1l/(G1Ip1) (3) φ2 =T2l/(G2Ip2)
From Eqs.(1)-(3),
T1=T2=mG2Ip2 /(G1Ip1+ G2Ip2)
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
1. 点 M 的应力单元体如图 (b) :
(a)
M
(b)
´́ ´́
(c)
2. 斜截面上的应力;
取分离体如图 (d)
:
(d)
´́
x
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
(d)
´́
x
n
t
转角规定:
轴正向转至截面外法线逆时针:为“ +”
顺时针:为“–”由平衡方程:
0)cossind()sincosd(d ; 0 AAAFn
0)sinsind()coscosd(d ; 0 AAAFt
解得: 2cos ; 2sin
4.5 Torsional deformation and stiffness condition
LIU Jiemin @SJZU 2008CH.4 TORSION
2cos ; 2sin 分析:
当 = 0° 时,
max00 , 0
当 = 45° 时,
0 , 45min45
当 = – 45° 时,
0 , 45max45
当 = 90° 时,
max9090 , 0
´́
45°
由此可见:圆轴扭转时,在横截面和纵截面上的剪应力为最大值;在方向角 = 45 的斜截面上作用有最大压应力和最大拉应力。根据这一结论,就可解释前述的破坏现象。
4.5 Torsional deformation and stiffness condition