Research ArticleStiffness Calculation Model of Thread ConnectionConsidering Friction Factors
Shi-kun Lu ,1,2 Deng-xin Hua ,1 Yan Li ,1 Fang-yuan Cui ,1 and Peng-yang Li 1
1Xiβan University of Technology, Xiβan 710048, China2Laiwu Vocational and Technical College, Laiwu 271100, China
Correspondence should be addressed to Deng-xin Hua; [email protected] and Yan Li; [email protected]
Received 9 September 2018; Revised 12 December 2018; Accepted 1 January 2019; Published 23 January 2019
Guest Editor: Yingyot Aue-u-lan
Copyright Β© 2019 Shi-kun Lu et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
In order to design a reasonable thread connection structure, it is necessary to understand the axial force distribution of threadedconnections. For the application of bolted connection in mechanical design, it is necessary to estimate the stiffness of threadedconnections. A calculation model for the distribution of axial force and stiffness considering the friction factor of the threadedconnection is established in this paper. The method regards the thread as a tapered cantilever beam. Under the action of thethread axial force, in the consideration of friction, the two cantilever beams interact and the beam will be deformed, thesedeformations include bending deformation, shear deformation, inclinationdeformation of cantilever beam root, shear deformationof cantilever beam root, radial expansion deformation and radial shrinkage deformation, etc.; calculate each deformation of thethread, respectively, and sum them, that is, the total deformation of the thread. In this paper, on the one hand, the threadedconnection stiffness was measured by experiments; on the other hand, the finite element models were established to calculatethe thread stiffness; the calculation results of the method of this paper, the test results, and the finite element analysis (FEA) resultswere compared, respectively; the results were found to be in a reasonable range; therefore, the validity of the calculation of themethod of this paper is verified.
1. Introduction
The bolts connect the equipment parts into a whole, whichis used to transmit force, moment, torque, or movement. Thebolt connection is widely used in various engineering fields,such as aviation machine tools, precision instruments, etc.The precision of the threaded connection affects the qualityof the equipment. Especially for high-end CNC machinetools, the precision of the thread connection is very high.Therefore, it is important to study the stiffness of the threadedconnection to improve the precision of the device. Manyresearchers have conducted research in this area.
Dongmei Zhang et al. [1], propose a method which cancompute the engaged screw stiffness, and the validity of themethod was verified by FEA and experiments. Maruyama etal. [2] used the point matching method and the FEM, basedon the experimental results of Boenick and the assumptionsmade by Fernlund [3] in calculating the pressure distributionbetween joint plates. Motash [4] assumed that the pressure
distribution on any plane perpendicular to the bolt axis hadzero gradient at r = πβ/2 and r = ππ, where it also vanishes.Theymainly use numericalmethods to calculate the influenceof different parameters on the stiffness of bolted connections.Kenny B, Patterson E A. [5] introduced a method formeasuring thread strains and stresses. Kenny B [6] et al.reviewed the distribution of loads and stresses in fasteningthreads. Miller D L et al. [7] established the spring model ofthread force analysis and, combined with the mathematicaltheory, analyzed the stress of the thread and comparedwith the FEA results and experimental results to verify thecorrectness of the spring mathematical model. Wang Wand Marshek K M. et al. [8] proposed an improved springmodel to analyze the thread load distribution, comparedthe load distributions of elastic threads and yielding threadjoints, and discussed the effect of the yield line on the loaddistribution.Wileman et al. [9] performed a two-dimensional(2D) FEA for members stiffness of joint connection. DeAgostinis M et al. [10] studied the effect of lubrication on
HindawiMathematical Problems in EngineeringVolume 2019, Article ID 8424283, 19 pageshttps://doi.org/10.1155/2019/8424283
2 Mathematical Problems in Engineering
thread friction characteristics or torque. Dario Croccoloet al. [11β13] studied the effect of Engagement Ratio (ER,namely, the thread length over the thread diameter) on thetightening and untightening torque and friction coefficientof threaded joints using medium strength threaded lockingdevices. Zou Q. et al. [14, 15] studied the use of contactmechanics to determine the effective radius of the boltedjoint and also studied the effect of lubrication on frictionand torque-tension relationship in threaded fasteners. NassarS. A. et al. [16, 17] studied the thread friction and threadfriction torque in thread connection. Nassar S. A. et al.[18, 19] also investigated the effects of tightening speed andcoating on the torque-tension relationship and wear patternin threaded fastener applications in order to improve thereliability of the clamping load estimation in bolted joints.Kopfer et al. [20] believe that suitable formulations shouldconsider contact pressure and sliding speed; based on this, thecontribution shows experimental examples for main uncer-tainties of frictional behavior during tightening with differentmaterial combinations (results from assembly test stand).Kenny B et al. [21] reviewed the distribution of load andstress in the threads of fasteners. Shigley et al. [22] presentedan analytical solution for member stiffness, based on thework of Lehnhoff and Wistehuff [23]. Nasser [24], Mustoand Konkle [25], Nawras [26], and Nassar and Abbound[27] also proposed mathematical model for the bolted-jointstiffness. Qin et al. [28] established an analytical model ofbolted disk-drum joints and introduced its application todynamic analysis of joint rotor. Liu et al. [29] conductedexperimental and numerical studies on axially excited boltconnections.
There are also several authors that, starting from thenature of thread stiffness, from the perspective of threaddeformation, established a mathematical model of the calcu-lation of the distribution of thread axial force. The Sopwithmethod [30] and the Yamamoto method [31] received exten-sive recognition. The Sopwith method gave a method for cal-culating the axial force distribution of threaded connections.Yamamoto method can not only calculate the axial force dis-tribution of threads but also calculate the stiffness of threadedconnections. The assumption for Yamamoto method is thatthe load per unit width along the helix direction is uniformlydistributed. In fact, for the three-dimensional (3D) helixthread, the load distribution is not uniform.Therefore, basedon the Yamamoto method, Dongmei Zhang et al. [1] proposea method which can compute the engaged screw stiffnessby considering the load distribution, and the validity of themethod was verified by FEA and experiments. The methodof Zhang Dongmei does not consider the influence of thefriction coefficient of the thread contact surface. In fact,the friction coefficient of the contact surface of the threadconnection has an influence on the distribution of the axialforce of the thread and the stiffness of the thread. Therefore,we propose a new method which can compute the engagedscrew stiffness more accurately by considering the effects offriction and the load distribution.The accuracy of themethodwas verified by the FEA and bolt tensile test. The flow chartof the article is shown in Figure 2.
2. Mathematical Model
2.1. Axial Load Distribution. According to Yamamoto [31],the thread is regarded as a cantilever beam, and the thread isdeformed under axial force and preload. These deformationsinclude the following (shown in Figure 3): thread bendingdeformation, thread shear deformation, thread root incli-nation deformation, thread root shear deformation, radialdirection extended deformation, or radial shrinkage defor-mation.
For the ISO thread, the axial deformation of the threadat π₯ at the axial unit width force π€z is thread bendingdeformation πΏ1, thread shear deformation πΏ2, thread rootinclination deformation πΏ3, thread root shear deformationπΏ4, and radial direction extended deformation πΏ5 (nut) orradial shrinkage deformation πΏ5 (screw), and calculate thesedeformations of the thread, respectively, and then sum them,that is, the total deformation.
2.1.1. Bending Deformation. In the threaded connection,under the action of the load, the contact surface frictioncoefficient isπ, when the sliding force along the inclined planeis greater than the friction force along the inclined plane, therelative sliding occurs between the two inclined planes, andthe axial unit width force (shown in Figure 3) is π€π§; if theinfluence of the lead angle is ignored, the force per unit widthperpendicular to the thread surface can be expressed as
π€ = π€π§π sin πΌ + cos πΌ (1)
The force per unit width perpendicular to the threadsurface can be decomposed into the x-direction componentforce and the y-direction component force, respectively
π€ cos πΌ = π€π§ cos πΌπ sin πΌ + cos πΌ (2)
and
π€ sin πΌ = π€π§ sin πΌπ sin πΌ + cosπΌ (3)
The friction generated along the slope is wπ; i.e.,π€π = π€π§ππ sin πΌ + cos πΌ (4)
The force wπ is also decomposed into x-direction forceand y-direction force, which are π sin πΌ and π cos πΌ, respec-tively.
π€π sin πΌ = π€π§π sin πΌπ sin πΌ + cos πΌ (5)
π€π cosπΌ = π€π§π cos πΌπ sin πΌ + cos πΌ (6)
In the unit width, the thread is regarded as a rectangularvariable-section cantilever beam. Under the action of the
Mathematical Problems in Engineering 3
above-mentioned force, the thread undergoes bending defor-mation, and the virtual work done by the bending moment πΈon the beam ππ¦ section is
πΈππ = πΈ ππ€πΈππΌ (π¦)ππ¦ (7)
According to the principle of virtual work, the deflectionπΏ1 (see Figure 3(a)) of the beam subjected to the load is
πΏ1 = β«π0
πΈππ€πΈππΌ (π¦)ππ¦ (8)
where πΈ is the bending moment of the unit load beam.ππ€ is the bending moment of the beam under the actualload. I(y) is the area moment of inertia of the beam at π¦. πΈπ isYoungβs Modulus of the material. c is the length of the beam.Here, the forces are assumed as acting on the mean diameterof the thread.
As shown in Figure 5, the height β(π¦) of the beam sectionper unit width and the area moment of inertia πΌ(π¦) of thesection can be expressed by using the function interpolation.
β (π¦) = [1 + (π½1 β 1) (π β π¦)π ] β, (0 β€ π¦ β€ π)
πΌ (π¦) = 112πβ3 [1 + (π½1 β 1) (π β π¦)π ]3 , (0 β€ π¦ β€ π)
π½1 = πβ
(9)
where h is the beam end section height; b is the beamsection width; π½1 is the beam root section height and thebeam end section height ratio; see Figure 5.
FromFigure 5, the bendingmoment of the beam is relatedto the y-axis component of π€ and wπ, and these componentscause the beam to bend; therefore, the analytical solutionshows that the bending moment of the unit width beamsubjected to the friction force and the vertical load of thethread surface is
ππ€ = π€π§π sin πΌ + cos πΌ β [cosπΌ β c + π sin πΌ β πβ sin πΌ(π2 β π tan πΌ) + π cos πΌ(π2 β π tan πΌ)]
(10)
Substituting (10) and (9) to (8) and integrating to obtainthe analytical expression of the deflection πΏ1 (shown inFigure 3(a)) of the cantilever beamwith variable cross-sectionunder load one has
πΏ1 = 12ππ€π2πΈππβ3 β [12 ( 1π½21 + 1) β 1π½1] β 1(π½1 β 1)2 (11)
2.1.2. Shear Deformation. Assume that the distribution ofshear stress on any section is distributed according to the
parabola [31] and the deformation πΏ2 (see Figure 3(b)) causedby the shear force within the width of unit 1 is
πΏ2 = logπ (ππ) β 6 (1 + V) (cosπΌ + π sin πΌ) cot πΌ5πΈπ
β π€π§π sin πΌ + cos πΌ(12)
2.1.3. Inclination Deformation of the Thread Root. Underthe action of the load, the thread surface is subjected to abending moment, and the root of the thread is tilted, asshown in Figure 3(c). Due to the inclination of the thread,axial displacement occurs at the point of action of the threadsurface force, and the axial displacement can be expressed as[31]
πΏ3 = π€π§π sin πΌ + cosπΌ β 12π (1 β V2)ππΈππ2 β [cos πΌ β π
+ π sin πΌ β π β sin πΌ(π2 β π tan πΌ)+ π cos πΌ(π2 β π tan πΌ)]
(13)
2.1.4. Deformation due to Radial Expansion and RadialShrinkage. According to the static analysis, the thread is sub-jected to radial force π€ sin πΌ β π€π cos πΌ (shown in Figure 4),and it is known from the literature [31] that the internal andexternal thread radial deformation (shown in Figure 3(d)) are
πΏ4b = (1 β ]π) ππ2πΈππ tan πΌ β π€π§ (sin πΌ β π cos πΌ)π sin πΌ + cosπΌ (14)
and
πΏ4n = (π·20 + π2ππ·20 β π2π + ]π) ππ2πΈππ β tan πΌ
β π€π§ (sin πΌ β π cos πΌ)π sin πΌ + cos πΌ
(15)
2.1.5. Shear Deformation of the Root. Assuming that theshear stress of the root section is evenly distributed, thedisplacement of the π point in the π₯ direction caused bythe shear deformation (shown in Figure 3(e)) is the sameas the displacement of the thread in the π₯ direction; thisdisplacement can be expressed as [31]
πΏ5 = π€π§π sin πΌ + cos πΌ β 2 (1 β V2) β (cos πΌ + π sin πΌ)ππΈπ
β {ππ logπ (π + π/2π β π/2) + 12 logπ (4π2π2 β 1)}(16)
4 Mathematical Problems in Engineering
For ISO internal threads, the relationship between a, b, c,and pitch π is
π = 0.833ππ = 0.5ππ = 0.289π
(17)
Substituting (17) into (9), (10), (11), (12), (13), (14), and (16)one gets the relation
πΏ1π = 3.468 ππ€ππΈππ2π½3π β [12 ( 1π½2π
+ 1) β 1π½π ]β 1(π½π β 1)2 , π½π = 1.6684
(18)
ππ€π = π€π§π sin πΌ + cos πΌ [0.289π β (cos πΌ + π β sin πΌ)β (0.4165π β 0.289π β tan πΌ) (sin πΌ β π β cos πΌ)]
(19)
πΏ2π = 0.51 β 6 (1 + V) (cos πΌ + π sin πΌ) cot πΌ5πΈπ
β π€π§π sin πΌ + cosπΌ(20)
πΏ3π = π€π§π sin πΌ + cosπΌ β 12π (1 β V2)ππΈππ2 β [0.289π
β (cos πΌ + π β sin πΌ) β (0.4165π β 0.289π β tan πΌ)β (sin πΌ β π β cosπΌ)]
(21)
πΏ4b = (1 β ]π) ππ2πΈππ (sin πΌ β π cos πΌ) tan πΌβ π€π§π sin πΌ + cosπΌ
(22)
πΏ5π = 1.8449π€π§π sin πΌ + cosπΌ β 2 (1 β V2) β (cos πΌ + π sin πΌ)ππΈπ (23)
For ISO internal threads, the relationship between a, b, c,and pitch π is
π = 0.875ππ = 0.5ππ = 0.325π
(24)
Substituting (24) into (9), (10), (11), (12), (13), (15), and (16)type one gets the relation
πΏ1π = 3.784 ππ€ππΈππ2π½2π β [12 ( 1π½2π + 1) β 1π½π ]
β 1(π½π β 1)2 , π½π = 1.751
(25)
ππ€π = π€π§π sin πΌ + cos πΌ [0.325π β (cos πΌ + π β sin πΌ)β (0.4375π β 0.325π β tan πΌ) (sin πΌ β π β cosπΌ)]
(26)
πΏ2π = 0.55962 β 6 (1 + V) (cos πΌ + π sin πΌ) cot πΌ5πΈπ
β π€π§π sin πΌ + cos πΌ(27)
πΏ3π = π€π§π sin πΌ + cos πΌ β 12π (1 β V2)ππΈππ2 β [0.325π
β (cos πΌ + π β sin πΌ)β (0.4375π β 0.325π β tan πΌ) (sin πΌ β π β cosπΌ)]
(28)
πΏ4n = (π·20 + π2ππ·20 β π2π + ]π) ππ2πΈππ β (sin πΌ β π cos πΌ) tan πΌβ π€π§π sin πΌ + cos πΌ
(29)
πΏ5π = 1.7928π€π§π sin πΌ + cos πΌ β 2 (1 β V2) β (cos πΌ + π sin πΌ)ππΈπ (30)
By adding these deformations separately, the total defor-mation (shown in Figure 4) of screw thread and nut threadcan be obtained under the action of force π€π§.
πΏπ = πΏ1π + πΏ2π + πΏ3π + πΏ4π + πΏ5π (31)
πΏπ = πΏ1π + πΏ2π + πΏ3π + πΏ4π + πΏ5π (32)
The unit force per unit width of the axial direction can beexpressed as
πΞ = π€π§π€π§ = 1 (33)
Under the action of unit force of axial unit width, the totaldeformation of external thread and internal thread is
πΏπ1 = πΏ1π + πΏ2π + πΏ3π + πΏ4π + πΏ5ππ€π§ (34)
and
πΏπ1 = πΏ1π + πΏ2π + πΏ3π + πΏ4π + πΏ5ππ€π§ (35)
For threaded connections, at the x-axis of the load F, theaxial deformation of screws and nuts can be expressed as
π§π (π) = πΏπ1 β ππΉππ (36)
π§π (π) = πΏπ1 β ππΉππ (37)
Mathematical Problems in Engineering 5
x
L
NutL
w
Screw
οΏ½read axial force distribution
Fx
Fb
(a)
0.5a
0.5a
w
P
w
y
x
o
Nut
Screw
Partial view
(b)
Figure 1: Thread force.
Here π is the length along the helical direction, and therelation between the axial height π₯ and the length along thehelix direction can be represented by the following formulaaccording to the geometric relation shown in Figure 6.
π = π₯sin π½ (38)
Here, π½ is the lead angle of the thread shown in Figure 6,and then
π§π (π₯) = πΏπ1 β ππΉππ = πΏπ1 β ππΉππ₯ β ππ₯ππ = πΏπ1 β sin π½ β ππΉππ₯ (39)
π§π (π₯) = πΏπ1 β ππΉππ = πΏπ1 β ππΉππ₯ β ππ₯ππ = πΏπ1 β sin π½ β ππΉππ₯ (40)
AssumeππΉππ₯ β π§π (π₯) = 1πΏπ1 β sin π½ = ππ’ππ₯ (π₯) (41)
ππΉππ₯ β π§π (π₯) = 1πΏπ1 β sin π½ = ππ’ππ₯ (π₯) (42)
Here, πππ₯(x) and πππ₯(x) represent the stiffness of the unitaxial length of the nut and the screw, respectively, for the unitforce.
The axial total deformation of the threaded connection atπ₯ is denoted as
π§π₯ (π₯) = π§π (π₯) + π§π (π₯) (43)
The stiffness of the unit axial length of the threadedconnection is expressed as
ππ’π₯ (π₯) = ππΉππ₯ β π§π₯ (π₯)= ππΉππ₯ Γ· (πΏπ1 β sin π½ β ππΉππ₯ + πΏπ1 β sin π½ β ππΉππ₯)= 1(πΏπ1 + πΏπ1) β sin π½
(44)
As shown in Figure 1(a), the threaded connection struc-ture includes a nut body and a screw body. The nut is fixed,
the screw is subjected to pulling force, the total axial forceis πΉπ, and the axial force at the threaded connection screw π₯is F(x). If the position of the bottom end face of the nut isthe origin 0 and, at the π₯ position, the axial force is F(x), thescrew elongation amount ππ and the nut compression ππ canbe obtained from the following:
ππ (π₯) = πΉ (π₯)π΄π (π₯) πΈπ (45)
ππ (π₯) = πΉ (π₯)π΄π (π₯) πΈπ (46)
where π΄π(x) and π΄π(x) are the vertical cross-sectionalareas of screws and nuts at the π₯ position. πΈπ and πΈπ are,respectively, Youngβs modulus of the screw body and Youngβsmodulus of the nut body. Find the displacement gradient forthe expression, which is, respectively, expressed as
ππ§π (π₯)ππ₯ = 1ππ’ππ₯ (π₯) β π2πΉππ₯2 (47)
ππ§π (π₯)ππ₯ = 1ππ’ππ₯ (π₯) β π2πΉππ₯2 (48)
Here, ππ’ππ₯(π₯) = 1/(πΏπ1 β sin π½), and ππ’ππ₯(π₯) = 1/(πΏπ1 β sin π½).
As shown in Figure 1(a), the screw is subjected to thetensile force πΉb, with the bottom of the nut as the coordinateorigin, and the force at the x position is πΉπ₯, and thenthe elongation of the screw at x is β«πΏ
π₯ππ(π₯)ππ₯ = π€π,
and the compressed shortening amount of the nut at x isβ«πΏπ₯ππ(π₯)ππ₯ = π€π. The relationship between π€b, π€n, π§b, andπ§n is β«πΏπ₯ ππ(π₯)ππ₯ + β«πΏ
π₯ππ(π₯)ππ₯ = [π§π(π₯) + π§π(π₯)]π₯=πΏ β [π§π(π₯) +π§π(π₯)]π₯=π₯ (see Figures 7 and 1(a),), and the partial derivative
of this relation can be obtained by the following formula:
ππ (π₯) + ππ (π₯) = ππ§π (π₯)ππ₯ + ππ§π (π₯)ππ₯ (49)
Substituting (45), (46), (47), and (48) into (49) andsimplifying it
(π΄ππΈπ + π΄ππΈπ)π΄ππΈππ΄ππΈπ β ππ’ππ₯ππ’ππ₯(ππ’ππ₯ + ππ’ππ₯)πΉ (π₯) = π2πΉ (π₯)ππ₯2 (50)
6 Mathematical Problems in Engineering
ww Force analysis of thread
Deformation of thread
Total deformation of the thread when
the helix angle is not considered
Total deformation of the thread when considering the helix angle
Calculation of thread stiffness οΏ½read stiffness test FEA of thread stiffness
Comparison of results
Influence of friction coefficient on
stiffness and load distribution
b = 1b + 2b + 3b + 4b + 5b
1b , 2b, 3b, 4b, 5b
1n , 2n, 3n, 4n, 5n
n = 1n + 2n + 3n + 4n + 5n
Kc =L
β«0
kx(x) =L
β«0
f(x)kux(x)dx
zb(x) = b1 Β·F
r= b1 Β·
F
xΒ·x
r= b1 Β· οΌοΌ£οΌ¨ Β·
F
x
zn(x) = n1 Β·F
r= n1 Β·
F
xΒ·x
r= n1 Β· οΌοΌ£οΌ¨ Β·
F
x
Figure 2: Schematic diagram of the article.
Let
π = β (π΄ππΈπ + π΄ππΈπ)π΄ππΈππ΄ππΈπ β ππ’ππ₯ππ’ππ₯(ππ’ππ₯ + ππ’ππ₯) (51)
Then
π2πΉ (π₯) = π2πΉ (π₯)ππ₯2 (52)
From mathematical knowledge, the equation is a differ-ential equation. The general solution of the equation can beexpressed as
πΉ (π₯) = πΆ1 sinh ππ₯ + πΆ2 cosh ππ₯ (53)
As can be seen from Figure 1, the axial force at the firstthread at the connection surface of the nut and the screwis πΉπ, and the axial force at the last thread at the lower endof the thread joint surface of the nut and screw is 0; that is,the boundary condition is F(x=0)=πΉπ and F(x=L)=0. Takingthese boundary conditions into the equation will give πΆ1=-πΉπ(cosh(ππΏ))/sinh(ππΏ) and πΆ2=πΉπ, so we get the expressionof the threaded connection axial load as
πΉ (π₯) = πΉπ (cosh ππ₯ β cosh ππΏsinh ππΏ sinh ππ₯) (54)
Therefore, the axial force distribution density of thethread connection along the π₯ direction can be expressed as
π (π₯) = πΉ (π₯)πΉπ = (cosh ππ₯ β cosh ππΏsinh ππΏ sinh ππ₯) (55)
2.2. Thread Connection Stiffness. The stiffness in the axialdirection π₯ of the bolted connection is equal to the axial forcedistribution of the threaded connectionmultiplied by the unitstiffness; i.e.,
ππ₯ (π₯) = π (π₯) ππ’π₯ (π₯) (56)
The overall stiffness of the bolt connection can beexpressed as
πΎπ = β«πΏ0ππ₯ (π₯) = β«πΏ
0π (π₯) ππ’π₯ (π₯) ππ₯ (57)
Mathematical Problems in Engineering 7
ww
Screw
x
x
Bending deformation
1
(a)
Screw
x
x
ww
Shear deformation
2
(b)
Screw
x
x
ww
3
Inclination deformation of thread root(c)
Screw
ww
x
x
4
Radial direction extended deformation and radial shrinkage deformation
(d)
Screw
x
x
ww
Shear deformation of the root of the thread
5
(e)
Figure 3: Thread deformation caused by various reasons.
Substituting (44) and (55) into (57), the stiffness of thebolt connection is expressed as
πΎπ = β«πΏ0ππ₯ (π₯) = β«πΏ
0π (π₯) ππ’π₯ (π₯) ππ₯
= 1π (πΏπ1 + πΏπ1) β sin π½ β cosh ππΏ β 1sinh ππΏ
(58)
3. FEA Model
A 3D finite element model (shown in Figure 10) was estab-lished, and FEA was performed to analyze the influence ofvarious parameters of the thread on the thread stiffness.Theseparameters include material, thread length, pitch, etc.
The FEA software ANSYS 14.0 was used for analysis.During the analysis, the end face of the nut was fixed (shownin Figure 9), the initial state of themodel is shown in Figure 8,and an axial displacement Ξπ₯ was forced to the end face ofthe screw. Then, the axial force πΉπ₯ of the screw end face wasextracted. The axial stiffness of the threaded connection wascalculated by the FEM.The friction coefficient π of the thread
contact surface is set first. In FEA, the contact algorithmused is Augmented Lagrange. Figures 11(a)β11(c) are the forceconvergence curves for FEA of threaded connections. Figures12(a)β12(c) are the effect of the reciprocal of the mesh sizeon the axial force obtained by FEA. We can see from Figures12(a)β12(c) that as the mesh size decreases, the resulting axialforce gradually decreases, but when the mesh size is small toa certain extent, the resulting axial force will hardly decrease.The axial force at this time is the axial force required bythe author. With known displacements and axial force, thestiffness of the threaded connection can be calculated usingthe formula
πΎπ = πΉπ₯Ξπ₯ (59)
4. Tensile Test of Threaded Connections [1]
In order to verify the effectiveness of this paper method,the experimental data of the experimental device in [1]are used. In [1], the electronic universal testing machine isused to measure the load-defection data of samples, and the
8 Mathematical Problems in Engineering
y
x
o
c
Nut
w
y
x
o
c
Nut
wcos
wsinw
w
wsina
w
wcosa
wsina-wcosa
Comprehensive deformation of nut
n=1n+2n+3n+4n+5n wz=wcos+wsin
(a)
w
w
y o
c
wcos
wsin
w
w
y o
wsina
wcosa
wsina-wcosa
cx x
Screw Screw
Comprehensive deformation of screw
n=1n+2n+3n+4n+5nwz=wcos+wsin
(b)
Figure 4: The comprehensive deformation of the thread.
0.5a
0.5a
y
x
o
h
c-yc
w
w
Figure 5: The force on the thread.
test sample is made of brass. The tension value πΉπ₯π‘ can beread from the test machine. The axial deflection of threadconnection can be represented by the displacement variationπΏπΏ between two lines as shown in Figure 14, which can bemeasured by a video gauge [1]. In [1], in order to obtain the
P
d2
d2
Figure 6: The lead angle of the thread.
most accurate data possible, each size of the thread is in asmall range of deformation during the tensile test, and eachsize of the thread tensile test is performed 10 times, and theaverage value is calculated as the final calculated data. Somesamples in the experiment are shown in Figure 13.
The stiffness calculation formula is
πΎπ = πΉπ₯π‘πΏπΏ . (60)
Mathematical Problems in Engineering 9
L
ScrewL
Nut fixing surface
Nut
Fb
(wn) x
=x
(wb)
x=x
[zοΌ(x
)+z οΌ¨
(x)]
x=x
[zοΌ(x
)+z οΌ¨
(x)]
x=L
Figure 7: Illustration of the elastic deformation of the screwedportion of the threaded connection.
Figure 8: Initial state.
The materials used to make nuts and screws are brass.Youngβsmodulus of brass is 107GPa, and Poissonβs ratio is 0.32[1].
5. Results and Discussion
5.1. Stiffness of Threaded Connections. Croccolo, D. [12],Nassar SA [19], and Zou Q [32] studied the coefficient offriction of the thread. According to the study by Zou Q andNassar SA, in the case of lubricating oil on the thread surface,the friction coefficient of the steel-steel thread connectionthread is 0.08, and the friction coefficient of aluminum-aluminum thread connection thread is 0.1.
In order to verify the correctness of the calculation resultsof the theory presented in this paper, a variety of threadedconnections were used to calculate an experimental test.
In the finite element analysis and theoretical calculationsof this paper, Youngβs modulus of steel is πΈb= πΈn =200Gpa,and Poissonβs ratio of steel is 0.3, and the friction coefficient[12, 19, 32] is set to 0.08.
Fixed support surfaceFx
Figure 9: Threaded connection deformed by axial forces.
The experimental data, FEA data, and Yamamotomethoddata in Tables 1 and 5 are from literature [1]. As can beseen from Tables 1 and 5, the calculated values obtainedin this paper are all higher than the experimental results.Perhaps the error is caused by the presence of a smallamount of impurities on the surface of the thread andpartial deformation of the thread inevitably and there is aslip between the threaded contact surfaces. The theoreticalcalculation results and FEA results in this paper have a smallerror.
In Table 2, the effect of thread length on stiffness ispresented. It can be seen that when the same nominaldiameter M10, the same pitch P=1.5, and the same materialsteel are taken, when the thread engaged length is taken as 14mm, 9mm, and 6mm, respectively, it is found that the longerthe thread engaged length, the greater the stiffness and thesmaller the length of the bond, the smaller the stiffness.
In the FEA, the method of this paper and the Yamamotomethod, Youngβs Modulus of aluminum alloy is E=68.9GPa;Poissonβs ratio of the aluminum alloy is 0.34. The frictioncoefficient of the steel- steel threaded connection [12, 21, 22,32] is set to 0.08, and the friction coefficient of aluminum-aluminum threaded connection [12, 19, 32] is set to 0.1. Table 3shows the effect of different materials on the stiffness ofthreaded connections. The two types of threaded connectionsare made of two different materials, the steel and aluminumalloys. It can be seen from the table that, under the conditionof the same pitch, the same nominal diameter, and the sameengaged length, the stiffness of the steel thread connection islarger than that when the material is aluminum.
In Table 4, it also shows the influence of different pitcheson the stiffness of the thread connection. It can be seenthat with the same engaged length, the same material, andthe same nominal diameter, the pitch π is 1.5, 1.25, and 1,respectively, and we find that the smaller the pitch, the greaterthe stiffness.
When using FEM to analyze the influence of frictionfactors on the stiffness of threaded connections, the thread
10 Mathematical Problems in Engineering
Table1:Stiffnessof
threaded
conn
ectio
nswith
different
engagedleng
ths[1](kN/m
m).
No.
Size
code
ofthreads
Material
Exp.
Theory
FEA
Thisstu
dyYamam
otometho
d1
M36
Γ4Γ32
Brass
3627.6
4201.9
4282.1
3630.9
2M36
Γ4Γ20
2664
.33152.9
3946
.12761.6
3M36
Γ4Γ12
1801.3
2074.2
3129.3
1816.8
Mathematical Problems in Engineering 11
(a) Nut (b) Screw
Figure 10: Finite element meshing of thread-bonded 3D finite element models.
Table 2: Stiffness of threaded connections with different engaged lengths (kN/mm).
No. Size code of threads Material Theory FEAThis study Yamamoto method
1 M10Γ1.5Γ14Steel
2095.26 2076.6 1965.212 M10Γ1.5Γ9 1759.23 2006.2 1770.023 M10Γ1.5Γ6 1354.70 1804.6 1551.27
Table 3: Stiffness of threaded connections with different material (kN/mm).
No. Size code of thread Material Theory FEAThis study Yamamoto method
1 M10Γ1.5Γ9 Steel 1759.23 2006.2 1770.022 M10Γ1.5Γ9 Aluminum alloy 607.51 682.11 597.99
Table 4: Stiffness of threaded connections with different pitch (kN/mm).
No. Size code of threads Material TheoryThis study Yamamoto method
1 M10Γ1.5Γ9Steel
1759.23 2006.22 M10Γ1.25Γ9 2083.17 2177.13 M10Γ1Γ9 2324.30 2375.2
Table 5: Stiffness of threaded connections with different engaged lengths [1] (kN/mm).
No. Size code of threads Material Exp. TheoryThis study Yamamoto method
1 M36Γ3Γ12 Brass 2085.3 2593.2 3525.12 M36Γ2Γ12 Brass 2396.2 3418.3 4008.9
specification isM6Γ1Γ6.1 and the friction coefficients are 0.01,0.05, 0.1, 0.2, 0.25, and 0.3. (as shown in Figures 15β20)
Figures 21 and 22 show the results of stiffness calcu-lations. The thread size is M10Γ1.5Γ9 and M6Γ1Γ6.1, thematerial is steel, Poissonβs ratio of the material is 0.3, YoungβsModulus of the material is 200 GPa, and the thread surfacefriction coefficient is taken as 0.01, 0.05, 0.1, 0.2, 0.25, and
0.3. Calculated using the theories of this paper, FEA andYamamoto, respectively, and from Figures 21 and 22, we cansee that the results of FEA are very similar to the results of thetheoretical calculations of this paper, the variation trend ofstiffness with friction coefficient is the same, and it increaseswith the increase of friction coefficient, and the results ofthe FEA are in good agreement with those of the FEA;
12 Mathematical Problems in Engineering
Force convergence Force Criterion
97.121.44.472.21
0.4480.229
0.05Fo
rce (
N)
Tim
e (s) 1
0
1 2 3 4
1 2 3 4Cumulative Iteration
(a) M10Γ1.5Γ9, π=0.08, Ξπ₯=0.001mm
10820.2
0.7110.4480.1330.025
Forc
e (N
)Ti
me (
s) 10
1 2 3 4
1 2 3 4Cumulative Iteration
Force convergence Force Criterion
(b) M8Γ1.25Γ6.5, π=0.08, Ξπ₯=0.001mm
18.76.7
2.390.8550.3050.109
Forc
e (N
)Ti
me (
s) 10
1 2 3 4
1 2 3 4Cumulative Iteration
Force convergence Force Criterion
(c) M6Γ1.0Γ6.1, π=0.08, Ξπ₯=0.001mm
Figure 11: FEA convergence curve.
however, Yamamoto theory does not consider the influenceof the friction coefficient on stiffness, and this is obviouslyunreasonable.
5.2. Effect of Friction Coefficient on Axial Force Distribution.Take the thread size asM6Γ0.75Γ6.1, the axial load πΉb is takenas 100N, 350N, and 550N, respectively, and take the frictioncoefficients 0, 0.3, 0.6, and 1, respectively, to calculate theaxial force distribution of the thread. As can be seen fromFigure 23, when the friction coefficient is 1, the curve bendingdegree is the greatest, when the friction coefficient is 0, thecurve bending degree is the lightest, the curve bending degree
is greater, indicating that the more uneven the distribution ofaxial force, the smaller the curve bending degree, indicatingthat the more uniform the distribution of axial force. We cansee that the friction coefficient of thread surface has an effecton the distribution of axial force.
6. Conclusion
This study provides a new method of calculating the threadstiffness considering the friction coefficient and analyzes theinfluence of the thread geometry and material parameterson the thread stiffness and also analyzes the influence of
Mathematical Problems in Engineering 13
1750
1800
1850
1900
1950
2000 A
xial
load
F (N
)
0.5 1 1.5 2 2.5 3 3.5 4 4.5 50Reciprocal of mesh size
(a) M10Γ1.5Γ9, π=0.08, Ξπ₯=0.001mm
1300
1350
1400
1450
1500
1550
1600
Axi
al lo
ad F
(N)
1 2 3 4 5 6 7 8 9 100Reciprocal of mesh size
(b) M8Γ1.25Γ6.5, π=0.08, Ξπ₯=0.001mm
2 4 6 8 10 12 14 16 180Reciprocal of mesh size
1090
1100
1110
1120
1130
1140
1150
1160
1170
1180
Axi
al lo
ad F
(N)
(c) M6Γ1.0Γ6.1, u=0.08, Ξπ₯=0.001mm
Figure 12: Influence of the reciprocal of finite element mesh size on axial force.
the friction coefficient on the thread stiffness and axial forcedistribution.
(1) The results of the calculation of the thread stiffnesscalculated by the theoretical calculation method ofthis study are basically consistent with the results ofthe FEA. The results obtained by the test are smallerthan the calculated results.This is due to the influenceof the thread manufacturing on the experimentalresults.
(2) Thread-stiffness is closely related to material proper-ties, pitch, and thread length. We can obtain higherstiffness by increasing Youngβs modulus of the mate-rial, increasing the length of the thread, and reducingthe pitch.
(3) We can also increase the friction coefficient of thethread joint surface to increase the stiffness of thethread connection, but we have found that using thismethod to increase the thread stiffness is limited.
(4) In order to make the axial load distribution of thethread uniform, we can reduce the friction coefficientof the thread surface, but we found that the use of thismethod to improve the distribution of the axial forceof the thread has limited effectiveness.
Nomenclature
π: Contact surface friction coefficientπ€π§: Axial unit width force, NπΏ1: Thread bending deformation, mmπΏ2: Thread shear deformation, mmπΏ3: Thread root inclination deformation, mmπΏ4: Radial direction extended deformation orradial shrinkage deformation, mmπΏ5: Thread root shear deformation, mmπΈ: Bending moment of the unit load beam,Nβmm
14 Mathematical Problems in Engineering
οΏ½read Screwing Length Nut
Screw
Internal thread
External threads
ca
0.125P
0.125H
0.125H
0.167H0.167P
P
P
a
H
c
b
0.25H
b
Figure 13: Part of experimental threaded connection samples and ISO internal thread and ISO external thread.
Collet
Specimen
Initial State End State
L
Figure 14: Tensile test [1].
Mathematical Problems in Engineering 15
0.0010007 Max0.000941740.000882730.000823710.00076470.000705690.000646680.000587670.000528660.000469650.000410640.000351630.000292620.000233610.0001746 Min
Unit:mm
Figure 15: Axial displacement for screws with a friction coefficient of π=0.01. Total force reaction=1086.00 N.
0.0010008Max0.000941570.000882390.00082320.000764020.000704840.000645660.000586480.000527290.000468110.000408930.000349750.000290560.000231380.0001722 Min
Unit:mm
Figure 16: Axial displacement for screws with a friction coefficient of π=0.05. Total force reaction=1092.00 N.
ππ€: Bending moment of the beam under theactual load, NβmmπΌ: Area moment of inertia, mm4πΈπ: Youngβs modulus of the screw material,N/mm2π: Length of the beam, thread pitch lineheight, mmβ: Beam end section height, mmπ΅: Beam section width, mmπ½1: Beam root section height and the beam endsection height ratioπ΅π: Ratio of the height of the screw threadroot section to the section height at themiddiameterπ΅π: Ratio of the height of the nut threadroot section to the section height at themiddiameterπΏ4π: Radial shrinkage deformation of the screwthread, mmπΏ4π: Radial direction extended deformation ofthe nut thread, mmπ: Width of the thread root, mm
D0: Cylinder (nut) outer diameter, mmdp: Effective diameter of the thread, mmVπ: Poissonβs ratio of nut material
P: Pitch, mmππ€π: Bending moment of the screw thread,NβmmπΏ1π: Thread bending deformation of the screwthread, mmπΏ2π: Thread shear deformation of the screwthread, mmπΏ3π: Thread root inclination deformation of thescrew thread, mmπΏ4π: Radial direction extended deformation ofthe screw thread, mmπΏ5π: Thread root shear deformation of the screwthread, mmππ€π: Bending moment of the nut thread, NβmmπΏ1π: Thread bending deformation of the nutthread, mmπΏ2π: Thread shear deformation of the nutthread, mmπΏ3π: Thread root inclination deformation of thenut thread, mmπΏ4π: Radial direction extended deformation ofthe nut thread, mmπΏ5π: Thread root shear deformation of the nutthread, mmπΞ: Unit force per unit width of the axialdirection
16 Mathematical Problems in Engineering
0.0010008Max0.000941470.000882190.00082290.000763620.000704330.000645050.000585760.000526480.000467190.000407910.000348620.000289340.000230050.00017077 Min
Unit:mm
Figure 17: Axial displacement for screws with a friction coefficient of π=0.1. Total force reaction=1098.00 N.
0.0010008Max0.00094140.000882020.000822640.000763250.000703870.000644490.000585110.000525730.000466350.000406970.000347590.000288210.000228820.00016944 Min
Unit:mm
Figure 18: Axial displacement for screws with a friction coefficient of π=0.2. Total force reaction=1108.40 N.
πΏπ1: Total deformation of external (screw)thread, mmπΏπ1: Total deformation of internal (nut) thread,mmπ§π: The load on somewhere on the x-axis is F,where the screw thread axial deformation,mmπ§π: The load on somewhere on the x-axis isF, where the nut thread axial deformation,mmπ: Length along the helical direction, mmπ½: Lead angle of the thread, degreeπππ₯(π₯): Stiffness of the unit axial length of thescrew, N/mmπππ₯(π₯): Stiffness of the unit axial length of the nut,N/mm2π§π₯: Axial total deformation of the threadedconnection, mmππ’π₯(π₯): Stiffness of the unit axial length of thethreaded connection, N/mm2πΉπ: Total axial force (load), Nππ: At the π₯ position, the axial force isπΉ(π₯), thescrew elongation amount
ππ: At the π₯ position, the axialforce is πΉ(π₯), the nutcompression amountπ΄π(π₯): Vertical cross-sectionalareas of screw at the π₯position, mm2π΄π(π₯): Vertical cross-sectionalareas of nut at the π₯position, mm2πΈπ: Youngβs modulus of thescrew body, N/mm2πΈπ: Youngβs modulus of the nutbody, N/mm2ππ₯(π₯): Stiffness in the axialdirection π₯, N/mmπΎπ: Overall stiffness of thethreaded connection,N/mmΞπ₯: Axial displacement, mmπΉπ₯: Total axial force, NπΉπ₯π‘: Axial tension load, NπΎπ‘: Overall stiffness of thethreaded connection,N/mm
Mathematical Problems in Engineering 17
0.0010008Max0.00094370.000881950.000822530.00076310.000703680.000644260.000584840.000525420.0004660.000406570.000347150.000287730.000228310.00016889 Min
Unit:mm
Figure 19: Axial displacement for screws with a friction coefficient of π=0.25. Total force reaction=1110.6 N.
0.0010008Max0.000941350.00088190.000822440.000762990.000703540.000644090.000584640.000525190.000465740.000406280.000346830.000287380.000227930.00016848 Min
Unit:mm
Figure 20: Axial displacement for screws with a friction coefficient of π=0.3. Total force reaction=1114.2 N.
FEMοΏ½eory of this articleSopwith method
Γ 106
1.75
1.8
1.85
1.9
1.95
2
2.05
Stiff
ness
οΌοΌ
(N/m
m)
0.05 0.1 0.15 0.2 0.25 0.30Coefficient of friction
Figure 21: Effect of friction coefficient on stiffness. M10Γ1.5Γ9.
FEMοΏ½eory of this articleSopwith method
Γ 106
1.02
1.04
1.06
1.08
1.1
1.12
1.14
1.16
Stiff
ness
οΌοΌ
(N/m
m)
0.05 0.1 0.15 0.2 0.25 0.30Coefficient of friction
Figure 22: Effect of friction coefficient on stiffness. M6Γ1Γ6.1.
18 Mathematical Problems in Engineering
u=0u=0.3u=0.6u=1
0
10
20
30
40
50
60
70
80
90
100Fo
rce F
(N)
1 2 3 4 5 60Length L (mm)
2.46 2.47 2.482.45Length L (mm)
32
33
34
35
36
Forc
e F (N
)
(a) Fb=100N
0
50
100
150
200
250
300
350
Forc
e F (N
)
1 2 3 4 5 60Length L (mm)
130
135
140
Forc
e F (N
)
2.19 2.2 2.212.18Length L (mm)
u=0u=0.3u=0.6u=1
(b) Fb=350N
u=0u=0.3u=0.6u=1
0
100
200
300
400
500
600
Forc
e F (N
)
1 2 3 4 5 60Length L (mm)
2.71 2.72 2.73 2.742.7Length L (mm)
160
165
170
175
Forc
e F (N
)
(c) Fb=550N
Figure 23: Effect of Friction Coefficient on Axial Force Distribution.
πΏπΏ: Axial deformation of thethread, mmπ»: Thread original trianglehigh, mm.
Data Availability
The data used to support the findings of this study areincluded within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors would like to acknowledge support fromthe National Natural Science Foundation of China [Grantnos. 51675422, 51475366, and 51475146] and Science &
Mathematical Problems in Engineering 19
Technology Planning Project of Shaanxi Province [Grant no.2016JM5074].
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