Research Article Analytical Model of Bolted Joint ...

Research ArticleAnalytical Model of Bolted Joint Structure and Its NonlinearDynamic Characteristics in Transient Excitation

Xin Liao,1 Jianrun Zhang,1 and Xiyan Xu2

1School of Mechanical Engineering, Southeast University, Nanjing 211189, China2School of Science, University Autonoma of Madrid, 28049 Madrid, Spain

Correspondence should be addressed to Jianrun Zhang; [email protected]

Received 25 July 2016; Revised 19 September 2016; Accepted 4 October 2016

Academic Editor: Emiliano Mucchi

Copyright © 2016 Xin Liao et al.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The dynamic response of crucial components often depends upon the dynamic behavior of bolted connections. As is usually thecase, the accuratemodeling of structureswithmanymechanical joints remains a challengework.Thenonlinear behavior included inassembled structures strongly depends on the interface properties. In this paper, an analytical model of the simple bolted joint beamin tangential direction is first proposed for transient excitation, based on phenomenological model. The fourth-order Runge-Kuttamethod is employed to calculate the transient response, where the dynamic response of the nonlinear stiffness on system is alsoinvestigated.The simulation results show that natural frequency has a certain dependence on cubic stiffness term and cubic stiffnessismore suitable formodeling of nonlinear systemof awider frequency range.Thereby, a series Iwanmodel containing cubic stiffnessterm is established to describe nonlinear behaviors of bolted joint beams in shear vibration.The amplitude-frequency curves showthat the influence of interaction between nonlinear stiffness and damping mechanism on dynamic response characteristics is moreobvious. Finally, a new type of nonlinear model is applied into finite element analysis. The results of proposed transient excitationexperiment are discussed qualitatively, indicating that nonlinear effects observed agree with the numerical simulation results.

1. Introduction

The mechanical behavior of the bolted joint structure undervibration has long been characterized and studied in thecontext of engineering machinery, including modeling andexperiments methods. Dynamic analysis and evaluation havebeen studied the most in order to achieve expected effects ofvibration or minimize the undesirable behaviors [1–4]. Thenonlinear characteristics of bolted joint interface are of greatinterest because of their influence on structural vibration,which can even be detrimental to overall dynamics. Mostof researchers have investigated some complicated charac-teristics by building different dynamic systems. And in thecontribution, measured and simulated frequency responsefunctions (FRFs) are commonly considered to identify non-linear joint characteristics or analyze the dynamics of anassembly for a specific input-output behavior.

Bolted joint provides localized high stress during fasten-ing, which brings more considerations for modeling works

and design. For specific structural designs and practicalworking conditions, researchers require simulating and pre-dicting the dynamic behaviors of such a bolted joint structure,which is also a crucial safety assessment. The finite elementmethod is widely used in all kinds of research for boltedjoint properties. Pratt and Pardoen [5] proposed nonlinearfinite elementmodels to predict the load-elongation behaviorof conical-head bolted lap joints, and simulation resultswere compared with test data. Kim et al. [6] developedfour kinds of finite element models to analyze the preten-sion effect and contact behavior of bolted joint structure.Shuguo et al. [7] studied the dynamic effects of bolted andspline joints in the aeroengine rotor using finite elementmethod and evaluated the influence of typical parameterson critical speed and vibrationmodes. Additionally, interfacemodeling approaches are also studied by many researchers;for example, Miller and Quinn [8] used a series-seriesIwan model together with an elastic chain to carry out thesimulation of frictional damping for bolted joint structure,

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 8387497, 11 pageshttp://dx.doi.org/10.1155/2016/8387497

2 Shock and Vibration

which significantly reduced the computational requirements.Grosse and Mitchell [9] demonstrated that external loadingcould soften or stiffen the joint for various test configurations.Jalali et al. [10] established a detailed parameter model ofbolted joint interface and identified the parameters of modelby force-state mapping method. Although there have beenalready many accepted modeling approaches, more detaileddescriptions for the structure with bolted joint in modelingare still inadequate due to nonlinear joint properties and thelack of theoretical analytical relationship [11].

The bolted joint interface of interest between two bodiesintroduces both linear and nonlinear dynamic at the sametime, which is a common challenge in experimental precisionand design. Modal testing and analysis have always beenwidely used in studies [12–14]. Lawlor et al. [15] presented anexperimental study on the effects of variable bolt-hole clear-ance in double-lap, multibolt joints. For bolt self-looseningphenomenon, impulse tests of bolted joint are also measuredby some researchers. Although there have been already someexperimental researches of bolted joint structure for variousloading conditions, few reports have been found on thetransient excitation in tangential direction, which plays animportant role in shear vibration.

Until recently researchers have focused on nonlineardamping and stiffness characteristics in bolted joint inter-face and also nonlinear dynamic behavior of bolted jointstructure. Through some reviews about theory and experi-ment, Ouyang et al. [16] studied damping behavior evolvingfrom linear viscous type to nonlinear frictional type witha bolted joint test fixture. The experiments conducted byHartwigsen et al. [17] also revealed that damping behaviorhad large influence on the dynamic response of bolted jointstructure. Furthermore, some nonlinear damping models areestablished and applied in numerical simulation. Similarly,nonlinear stiffnessmodels have also been developed and usedgradually [18]. The expression of stiffness term and its corre-lation coefficient in system are particularly important due tosimulation calculation totally depending on equations withthe parameters.Thus, the types of stiffness applied within thecoupling element affect dynamic response characteristics ofsystem directly and determine if nonlinear behavior can bedescribed accurately. Ahmadian and Jalali [19] simulated astiffness-softening effect, which was symptomatic of a cubicstiffness term in the model, in a joint structure. Experimentalstudies [16] also showed that odds number of superharmonicsin the frequency spectra seem to suggest that a cubic stiffnessterm would be useful in simulating dynamic behavior ofbolted joint. Budnitzki et al. [20] discussed the effect of cubicstiffness on fatigue characterization resonator performanceand used a parametric optimization strategy to minimize thecubic stiffness.

Even though various nonlinearmodels for the bolted jointstructure have been applied in the dynamic researches, ana-lytical modeling approach in transient excitation still needsto be further developed due to complexity of interface. Inaddition, the transient excitation of shear direction can causedeflection of vibration for bolted joint structure, resultingin changes of whole dynamic performance, or eventuallyleading to bolt relaxation and fatigue fracture caused by

nonuniform force. Up to now, there have been few publishedstudies that integrated nonlinear stiffness with dampingin modeling of bolted joint interface to capture system’sdynamic characteristics under mutual interactions of the twoin transient excitation. Therefore, the purpose of the presentwork is to develop a nonlinear dynamic model of bolted jointstructure and investigate the influence of nonlinear factorsof bolted joint interface on system dynamic characteristicsin transient excitation. To this end, an analytical modelfor simple bolted joint structure is first proposed based onphenomenological model, and the dynamic responses ofsystem under different nonlinear stiffness parameters areanalyzed. Then, considering the tangential stiffness and fric-tion damping of interface related to bolt preload, a series Iwanmodel and cubic spring stiffness are combined to describedifferent dynamic behaviors, stick-slip, and macroslips. Theamplitude-frequency curves for different excitation levels andpreload levels in shear vibration are analyzed, respectively.Meanwhile, the new series Iwan model is added to the finiteelement analysis. Finally, the correlative transient excitationexperiments are carried out to validate good agreement withthe simulation results.

2. Nonlinear Modeling of Bolted Joint Beam

2.1. Dynamics Model. The proper theoretical modeling fora system assemblies is a key part of evaluating dynamiccharacteristics. The aim of modeling approach diversity is tohelp understand the behavior of the system better, especiallythe types on nonlinearities in play. For some experimentalresults [17, 21], a linear model cannot match data acquiredaccurately. The tangential stiffness is often neglected orassumed to be rigid in modeling, such as the content inliterature [22]. Therefore, it can be noticed that how to buildthe analytical model accurately in tangential direction isvery important for the dynamics analysis under transientexcitation, but the related content has not been described indetail for bolted joint structures in earlier investigations. Inthis part, a kind of common bolted joint structure in practicalengineering is first introduced, as shown in Figure 1.The fourplates, which are made of low-carbon steel, are employed tobe bolted together by the way of a double shear lap joint. Allthe rectangular steel plates have a length of 250mmandwidthof 100mm and a thickness of 5mm. The shear lap regionlocates in its center.The study considersM12 boltswithmetricmechanical property class 8.8.

In order to analyze the dynamics characteristics of thetest case in Figure 1 in transient excitation, the double shearlap joint structure is simplified to be a discrete model with2 degrees of freedom (DOF) based on phenomenologicalmodel, as seen in Figure 2, where 𝑚1 and 𝑚2, respectively,represent the mass of the upper and lower plates. Thetransient excitation is loaded in tangential direction.

In this system, the equation of motion can be written as

𝑚1��1 + 𝑐1��1 + 𝑘1𝑥1 + 𝑐2 (��1 − ��2) + 𝑘2 (𝑥1 − 𝑥2)

+ 𝑘3 (𝑥1 − 𝑥2)3= 0,

Shock and Vibration 3

(a)

100

70

500150

(b)

Figure 1: The double shear lap joint structure. Green: top plate; red: bottom plate; yellow and blue: middle plate.

Fexc

(a)

Fexc

k1

m1

c1

c2

x1 x2k3

k2

m2

(b)

Figure 2: Two-degree-of-freedom (DOF) nonlinear model.

𝑚2��2 + 𝑐2 (��2 − ��1) + 𝑘2 (𝑥2 − 𝑥1)

+ 𝑘3 (𝑥2 − 𝑥1)3= 𝐹exc,

(1)where 𝑥1 and 𝑥2 are the displacement values, dots representderivatives with respect to time, 𝑐1 and 𝑐2 are the viscousdamping coefficients, 𝑘1 and 𝑘2 are linear spring stiffnessvalues, 𝑘3 is cubic spring stiffness, and 𝐹exc is the excitationforce. Here, a cubic stiffness nonlinearity in tangential direc-tion is employed to allow the system modal frequencies toincrease with impact levels, which is just reflected in theresults of modal tests. The cubic stiffness term has a verysmall effect for small deflections when linear behavior occurs,because maybe linear and nonlinear factors exist at the sametime in the system. In addition, for the transient excitation,when impact amplitude becomes much larger suddenly, thedeflection amount of cubic spring will bring about morenonlinearity, especially in the case of low preload. Actually,choosing the cubic stiffness term contributes to supplement-ing an extra increase of stiffness. The transient excitationis easier to excite the cubic stiffness due to instantaneousincreased preload caused by instant impact. Furthermore,from the point of view of the working principle of construc-tionmachinery, when establishing the dynamicsmodel underspecial working conditions, such as large impact level, cubicstiffness term just makes up for the lack of total stiffness. Forexample, the excavator cab is susceptible to impact with largeramplitude in working; thus analytical model of vibrationisolation system of cab contains cubic stiffness term, and inaddition cubic stiffness provides more elastic support [23].Similarly, in the study of dynamics modeling of wind turbineblade, the calculation result of differential equation includingcubic stiffness term appears more reliable [24]. Hence, basedon the reasons above, the cubic spring stiffness is chosen

Table 1: Simulation parameters of the numerical system.

Parameter Value𝑚1 = 𝑚2 (kg) 0.5𝑐1 = 𝑐2 (Ns/m) 0.02𝑘1 = 𝑘2 (N/m) 1𝑘3 (N/m

3) 0.05𝐹exc (N) 40, 100, 150

in modeling. To study the influence of bolt preloads andexcitation levels on the amplitude response depending ontime, the fourth-order Runge-Kutta method is employed tosolve the coupled differential motion equation (1) to obtainthe frequency responses of the two-DOF nonlinear systemunder the transient excitation. The calculated time step is2 × 10−4 seconds. The simulation parameters of the systemare given in Table 1.

2.2. Transient Excitation. The expression of transient excita-tion normally has a small difference in shape when impulseis much faster than the response speed. Here, the impactlevels are adjusted by the hammer as excitation inputs andthe duration of the impact is approximately one millisecond.Then, taking 40N and 100N, for example, of low and highexcitation amplitudes, respectively, their impact excitationcurves for high and low level hits are shown in Figure 3.

3. Numerical Results and Analysis

When the bolted joint beam is subjected to transient excita-tion, the magnitude of excitation amplitude maybe changesthe dynamic response properties of the whole under theaction of complicated factors.Therefore, investigating various


15

35

55

75

95

115

2.59% 2.73% 2.88% 2.92% 3.08% 3.22% 3.37% 3.52%

Forc

e am

plitu

de (N

)

Time (s)High level hitLow level hit

−5

Figure 3: Impact force for simulation.

0 20 40 60 80

0

0.2

0.4

Time (s)

−0.2

−0.4

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

m1

m2

Figure 4: Mass acceleration time histories.

excitation amplitude levels and bolt preloads contribute todescribing nonlinear dynamic behavior. Figure 4 shows thecalculated acceleration of mass in the transient response. Inthis system response, it can be found that the amplitude ofacceleration of the first mass decreases more than that of thesecond mass during the period of first twenty seconds. Andthen the decline rate of their amplitude acceleration becomesconsistent basically.

It can be seen that the amplitudes of acceleration for fre-quency response curves change a lot under different impactlevels in Figure 5, and the corresponding frequencies vari-ation basically keeps unchanged. However, Figure 6 showsthat the nonlinear effects are more visible in the accelerationamplitude curves normalized by the impact level. In Figure 7,the bolt torque is gradually increased by means of a constantincrement. Similarly, it can be found that there are somechanges in the amplitudes of acceleration and frequenciescorresponding to the peaks during an impact.This simulatingresult illustrates that dynamic response characteristics havedependence on bolt preload and impact amplitude level, andthe cubic spring is very sensitive to the preload changeswithin a certain range. When the bolts are forced with

0 0.2 0.4 0.6

0.02

0.04

0.06

0.08

0.1

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

Fexc = 40NFexc = 100NFexc = 150N

Figure 5:The comparison of frequency response curves for differentexcitation amplitudes.

Frequency (Hz)

102

101

100

10−1

Am

plitu

de (m

/s2/N

)

0.60.40.20


Figure 6:The normalized acceleration amplitude curves for variousexcitation amplitudes at torque = 8N⋅m.

increasing torque, damping and stiffness parameters of theinterface are changed and especially the cubic stiffness term isexcited by normal compression that increases joint stiffness.

Based on the datum analysis above, the bolt preloadcontributes to the changes of cubic stiffness term, which alsohas a certain impact on the dynamic properties of system.Similarly, to study the effect of the cubic stiffness term indetail, somenewnumerical analysiswith different parametersare carried out. From Figure 8, it can be observed thatthe amplitude-frequency curves with various cubic stiffnessparameters are different when the values of 𝑘1 and 𝑘2 arekept unchanged. The frequency domains corresponding tothe peaks seem to become wider, even though the peak


0 0.2 0.4 0.6 0.8 10

0.01

0.02

0.03

0.04

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

Torque = 6N·mTorque = 7N·mTorque = 8N·m

Figure 7:The comparison of frequency response curves for differenttorques when the force amplitude 𝐹exc = 100N.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

k3 = 0.01N/m3

k3 = 0.1N/m3k3 = 1.5N/m3

k3 = 2.5N/m3

Figure 8:The comparison of frequency response curves for varyingcubic stiffness parameters.

frequencies change little. Its nonlinear phenomenon is notvery obvious.

However, when we change 𝑘1 and 𝑘2 in process of variedcubic stiffness, two peak values of amplitude-frequencycurves and natural frequencies occur changing obviously;meanwhile the frequency domains for peak values are gettingwider, which can be seen in Figure 9. But the broadeningextent decreases as the values of 𝑘1 and 𝑘2 increase. It alsoindicates that natural frequency has smaller dependence oncubic stiffness term. However, by the effect of cubic springstiffness, the amount of frequency variation corresponding tothe second peaks is more than that corresponding to the firstpeaks. The results reveal that, comparing with low frequencyregion, cubic stiffness term has more influence on dynamicresponse characteristics or becomes more sensitive to non-linear system at some extent, in relative higher frequencyregion. Therefore, the cubic stiffness nonlinearity can beemployed accurately in modeling, especially when nonlinearsystem is excited in a larger frequency range. Moreover, inpractical engineering application, the impact forces across the

interface often change, especially when instantaneous strongimpact converts low frequency vibration into high frequencyvibration; it will be easier to cause slappingmechanism.Thus,the interaction between the cubic stiffness term and dampingmechanism and their influence on nonlinear dynamic char-acteristics will be discussed next.

4. Series Iwan-Type Model

Mechanical joints have an important influence on structuraldynamic response, in which nonlinear stiffness and dampingare primarily concerned. Also, damping is the importantsource of system energy dissipation. Therefore, in order todescribe dynamic behavior of bolted joint structure betterand accurately, a series-series Iwan model can be appliedinto the modeling of the whole system for stick-slip andmacroslip phase. Figure 10 shows the Iwan series-seriesmodel comprised of a series of Jenkins elements. EachJenkins unit consists of a linear spring and a Coulomb slider.However, combined with the cubic spring, a new Iwanmodelis proposed, as shown in Figure 11. A single Jenkins elementis in parallel with a cubic spring. The friction contact ischaracterized with the Coulomb friction coefficient 𝜇. Thedifferent kinematic states of the Iwan element, stick-slip, ormacroslip are determined by a critical slipping force 𝐹𝑓,which indicates the onset of macroslip.

In this model, two different states are, respectively, con-sidered. In the stick-slip phase, the stiffness is determined by𝐾1 and 𝐾2. In the macroslip phase, the stiffness of originalIwan model’s system will become zero when the systemreaches its ultimate force 𝑓𝑦, and𝐾2 will disappear. However,the remaining cubic spring stiffness just makes up the rigidityof the system at the onset of macroslip. Also, there is anequivalent slipping friction force instead, and the 𝐾1 springelements are in series with each other, which is representedas in the discrete system

1

∑𝐾1=1

𝐾1,1+1

𝐾1,2+ ⋅ ⋅ ⋅ +

1

𝐾1,𝑛, (2)

where we assume that all of stiffness values𝐾1,𝑛 are equal.Theextended nonlinear model is shown in Figure 12. The motionequation can be rewritten as

𝑚1��1 + 𝑐��1 + 𝑘𝑥1 + 𝐹𝐼 = 0,

𝑚2��2 + 𝐹𝐼 = 𝐹exc,(3)

where𝐹𝐼 is the nonlinear inner force calculated in series Iwanmodel due to deformation, 𝑘 is linear spring stiffness, and 𝑐 isviscous damping coefficient.

On the basis of the above discretemodels, new amplitude-frequency curves including friction damping are obtained.Figure 13 shows the frequency response results of variableinteractions. It can be seen that the peak frequencies andamplitudes did change slightly with the variation of ultimateforce and cubic stiffness parameter, which indicates thatthe nonlinear effects can be visible by modeling in stick-slip phase and closely related to the preload. In addition,these curves seem not to be very smooth and there are


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

k1 = k2 = 1.0N/m, k3 = 0.1N/m3

k1 = k2 = 1.0N/m, k3 = 1.5N/m3

k1 = k2 = 1.5N/m, k3 = 0.1N/m3

k1 = k2 = 1.5N/m, k3 = 1.5N/m3

k1 = k2 = 3.0N/m, k3 = 0.1N/m3

k1 = k2 = 3.0N/m, k3 = 1.5N/m3

(a)

0.05 0.1 0.15 0.2 0.25

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

k1 = k2 = 1.0N/m, k3 = 0.1N/m3

k1 = k2 = 1.0N/m, k3 = 1.5N/m3

k1 = k2 = 1.5N/m, k3 = 0.1N/m3

k1 = k2 = 1.5N/m, k3 = 1.5N/m3

k1 = k2 = 3.0N/m, k3 = 0.1N/m3

k1 = k2 = 3.0N/m, k3 = 1.5N/m3

(b)

0.4 0.45 0.5 0.55 0.60

0.005

0.01

0.015

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

k1 = k2 = 1.0N/m, k3 = 0.1N/m3

k1 = k2 = 1.0N/m, k3 = 1.5N/m3

k1 = k2 = 1.5N/m, k3 = 0.1N/m3

k1 = k2 = 1.5N/m, k3 = 1.5N/m3

k1 = k2 = 3.0N/m, k3 = 0.1N/m3

k1 = k2 = 3.0N/m, k3 = 1.5N/m3

(c)

Figure 9: The comparison of frequency response curves for varying cubic stiffness when 𝑘1 and 𝑘2 are changed. (a) Effect of cubic stiffnessterm. (b) Local enlarged view from left circle. (c) Local enlarged view from right circle.

some small peaks appearing compared with the previ-ous graphs. It illustrates that the addition of series-seriesIwan model excites the smaller amplitudes on frequencyregion in nonlinear dynamic system. From another point ofview, when the nonlinear damping and nonlinear stiffnesselement are simultaneously included in the series Iwanmodel, the nonlinear stiffness can represent more accurate

dynamic characteristics by simulation modeling in the caseof the transient excitation once the jointed structure startsmacroslip. Furthermore, for the bolted joint structure withlarger damping under transient excitation, the simulationmodel with series Iwan element has an important influenceon judgment and evaluation working of whole machinedynamic response.


Figure 10: Iwan’s one-dimensional series-series model.

F

K1,1

K2,1

K1,2K1,n

K2,n

Figure 11: A new series Iwan model.

Series-seriesIwan model

k

c

X1 X2

m2m1 Fexc

Figure 12: Two-degree-of-freedom nonlinear model with seriesIwan element.

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

Frequency (Hz)

Am

plitu

de o

f acc

eler

atio

n (m

/s2)

10.90.80.70.60.50.40.30.20.10

fy = 40N, K1,n = 0.1N/m3

fy = 50N, K1,n = 0.2N/m3

fy = 60N, K1,n = 0.3N/m3

Figure 13: The comparison of frequency response curves fordifferent parameters when the force amplitude 𝐹exc = 100N.

Figure 14 is the schematic of the transfer function betweenthe acceleration of mass 𝑚1 and the input excitation. Withthe addition of the nonlinear damping model to the dynamicsystem, it can be observed that the natural frequency andamplitude change with bolt preloads.

The resulting FRFs for each torque show that thenonlinear effects are obvious. When the magnitude of torqueis tuned to increase, the peak values change a little larger

Frequency (Hz)

FRF

(m/s

2/N

)

10.90.60.30

102

101

100

10−1


Figure 14: Comparison of frequency response functions for variousbolt torque levels when the force amplitude 𝐹exc = 100N.

Frequency (Hz)10.90.60.30

FRF

(m/s

2/N

)102

101

100

10−1


Figure 15: Comparison of frequency response functions for variousexcitation levels at torque = 8N⋅m.

than before and the width of peaks also becomes more widerunder the interaction of nonlinear coupling element, whichindicates that the model takes on stiffening trend gradually.Figure 15 shows FRFs for different excitation levels underthe same tightening force. These plots reveal that the naturalfrequency first does not change much significantly with theincreasing excitation amplitudes, but the increments of peakvalues become a little larger when the excitation level reachesa greater value. It indicates that the dependence of naturalfrequency on the impact level is small even though the systemnonlinearity behavior is stronger in low preload level. Also, itcan be found that the peaks for impact level 100N are higherthan those for impact level 150N. This seems to result froma certain mode excited with increase in damping and theassumed extent of hit is certainly not large for bolted jointstructure in tangential direction. Besides, in this above case,the calculation is only gone on under relative lower tighteningforce of bolts.


(a) (b)

Figure 16: Finite element model: (a) global model; (b) local mesh.

5. Modeling of the Double Shear Lap JointBeam with Bolts

In order to investigate the influence of bolted joint interfaceon the dynamics of the connected structure, a detailed finiteelement model for composite beam structure with boltedjoints is developed in ABAQUS, as shown in Figure 16(a),which totally consists of 67306 elements and 79493 nodes.The thickness of each plate is represented by double rows ofmesh elements. In addition, the finite element model for asimilar beam structure consisting of narrower plates jointedwith 4 screws is also built. The tangential stiffness of thejoint is closely related to the normal load, which plays animportant role in shear vibration of transient excitation.Thusthe tangential and normal stiffness are both considered in ourfinite element analysis. First, since the contact nonlinearityis primary consideration in the prestressed model of boltedjoint, the contact relationship between components is builtthrough using surface-to-surface contact and the master-slave algorithm. The main contact pairs are defined asthe annular regions around bolt holes, whose outer radiusincreases by about 85% compared with radius of bolt hole, asshown in Figure 16(b).This is because stress distribution areain linear system basically concentrates in the range, and alsodifferentmesh densities are assigned, respectively, on annularregions and other regions on the plates, which is helpful toobtain reliable simulation results.

According to the relationship between selected torque 𝑇and bolt tension 𝐹, the equivalent normal load applied to thebolt and nut can be calculated by the following equation [25]:

𝑇 = 𝐾𝐹𝐷, (4)

where 𝐾 is the dimensionless nut factor, which is approxi-mately equal to 0.2 for all cases, and𝐷 is the nominal diameterof the bolt.

In this model, there are the calculations of multiple loadsteps, including the bolt preload as the first analysis step andthe subsequent transient load step imposed on compositebeam structure. By application of Lagrange’s approach, thegeneral form of equation for the composite beam system isgiven by

M�� + C�� + K𝜑 + FI = Fexc, (5)

where 𝜑 is vector of generalized displacements, M is massmatrix, C is damping matrix, K is stiffness matrix, FI isnonlinear inner force matrix, and Fexc is vector of transientexcitation force. The dynamic explicit reduced integrationis used to mesh the whole model designated element typeC3D8R. The normal behavior in contact is considered tobe hard contact. The properties of the elastic elements andseries Iwan elements are as follows: elastic modulus 𝐸 =2.1 × 105MPa, Poisson ratio 𝜆 = 0.3, and density 𝜌 =7.85 × 103 kg/m3. Based on the contact algorithm, nonlinearstiffness matrix is inserted into the computational modeland the initial forcing vector is gradually updated whenevery Iwan element undergoes a transition from stick toslip in transient excitation. The obtained simulation resultsare, respectively, compared with experimental data in thefollowing section.

6. Transient Excitation Experiments

To evaluate the effect of various preloads and impact lev-els on the nonlinear dynamic behavior in bolted joints, aseries of experiments are performed to understand systemresponse characteristics under transient excitation. Then,this measured dynamic response data can contribute to thecomparison with a theoretical model, thereby predictingthe accuracy or extracting the relevant data to facilitate thesimulation calculation and parameter identification, and soforth. For this purpose, the experimental system is set up,as shown in Figure 17. There are two triaxial sensors whichare, respectively, arranged near the front and back ends. Thesensors chosen are PCB Model 356B08 with a 10mV/m/s2sensitivity and the hammer is PCB Model 086C03 witha 2.25mV/N sensitivity. The data acquisition and analysissystem used in this study is𝑚 + 𝑝 international SO Analyzerwith 8 channels (VXI Mainframe rack unit, Agilent E8491B).In order to ensure the reliability of the experiment, threeimpacts have been carried out for each excitation forceand then averaged [26]. The four bolts are tightened to apredefined torquewith a torquewrench, and the cross sectionof right end of the double shear lap beam is impacted withthe hammer. During this period of testing, bolt preload levelsare adjusted in turn to be 4N⋅m, 8N⋅m, and 12N⋅m, andcoherence function is kept well in experiments.


Sensor Impact location

Figure 17: Double shear lap beam experimental setup.

245 250 255 260 265 2700

20

40

60

80

100

120

Frequency (Hz)

Am

plitu

de (m

/s2)/

N

Fexc = 40NFexc = 100N

Fexc = 150NFexc = 200N

Figure 18: Frequency response functions of the secondmode underdifferent impact levels.

Figure 18 is a plot of the transfer function betweenthe accelerometer on the left side of the beam structureand the input force, as shown in Figure 17. The majormodal frequency is about 258Hz and almost identical inthe four cases. With increasing of the impact level, thevariation of natural frequency is very small and the changetendency of amplitude of acceleration is generally similarto the results in Figure 5. It can be conjectured that thecubic stiffness in tangential direction has small influenceon dynamic behavior of system, in the process of impact.Moreover, when the impact level is higher, we can see that theamplitude value suddenly becomes more smaller than thosein other cases due to the effect of the nonlinear damping.In Figures 19 and 20, it is evident that as the applied

impact level is increased, the response characteristics in thesecond mode have obvious changes, especially the naturalfrequency whose trend increases with the preload levels.Similarly, the correlative nonlinear stiffness parameter anddamping coefficient will change at the same time when thebolt torque is increased slowly. In particular, the influence ofcubic stiffness on dynamic response characteristics mainly isreflected in a certain range in the process of increasing torque.If the torque is too small, the coupling damping is small which

240 260 2800

50

100

150

200

250

Frequency (Hz)

Am

plitu

de (m

/s2)/

N


Figure 19: Frequency response functions of the second mode withimpact levels 𝐹exc = 100N.

240 260 2800

20

40

60

80

100

120

140

Frequency (Hz)

Am

plitu

de (m

/s2)/

N


Figure 20: Frequency response functions of the second mode withimpact levels 𝐹exc = 200N.

can cause extra rotation of local system at some extent, andthe effect of cubic stiffness term is almost insignificant innonlinear system. If the bolt is screwed too tightly, the cubicstiffness term becomes insensitive to the preload. Also, insome specific dynamic cases, for example, when the doubleshear lap joint structure is subjected to large impact force or

10 Shock and VibrationA

mpl

itude

(m/s

2)/

N

Experimental dataSimulation data

1000

100

10

1

0.1

0.01

1E − 3800700600500400300200

Frequency (Hz)

(a)

Am

plitu

de (m

/s2)/

N

1000

100

10

1

0.1

0.01

1E − 3

Experimental dataSimulation data

800700600500400300200

Frequency (Hz)

(b)

Figure 21: Comparison of frequency response functions obtained by experiential data and finite element analysis results at torque = 8N⋅m.(a) The double shear lap beam jointed with 4 screws. (b) The similar beam structure consisting of narrower plates jointed with 2 screws.

high frequency excitation, the clapping damping mechanismwill play a major role rather than the cubic stiffness term.Moreover, these two figures even illustrate that experimentphenomena agree with the numerical results in Section 3,except for the orders of magnitude. In addition, the responseamplitudes are also examined for various preload levels inthe double shear lap beam. The magnitude of the peaks doesnot always keep decreasing with increasing of preload levels,however, which can be seen in Figure 19. It shows that theacceleration amplitude has a slight increase when torqueincreases from 8N⋅m to 12N⋅m in the experiments, whichappears as if certain natural frequency can be excited betterby impulse force in some cases. Figure 20 indicates that dueto the increase of impact level the peaks for low preloadlevel appreciably become smaller compared with those inFigure 19, and structural damping is also increased, resultingfrom more relative motion between plates. Finally, for theabove two kinds of double shear lap joint beam structures,the comparisons of finite element analysis results with thefull experimental data containing all modes of the systemare, respectively, shown in Figure 21. It can be seen in theplot that the curves match well for several major modes ofinterest, where the accuracy of the simulation results is alsoverified and the resulting FRFs accurately reflect the dynamicbehaviors.

7. Conclusion

In this work, the dynamic response characteristics of a doubleshear lap joint structure in transient excitation are inves-tigated. Based on phenomenological model, an analyticalmodel of simple bolted joint beam in tangential direction isdeveloped under transient excitation. The system’s nonlineardynamic characteristics are studied and the effect of nonlin-ear stiffness parameters on frequency response of bolted jointstructure is also investigated.

In the analysis of transient excitation, the numericalsimulation results show that the cubic spring is very sensitive

to the preload changes only within a certain range in theprocess of increasing toque. When only changing the cubicstiffness parameter, the variety of natural frequency is notvery obvious in nonlinear system, which indicates thatnatural frequency has a smaller dependence on cubic stiffnessterm. However, for a wider frequency range, especially inthe higher frequency region, the cubic stiffness term ismore suitable formodeling of nonlinear system.Additionally,nonlinear influence in transient excitation is also observedby a new series Iwan model containing cubic stiffness term,which is also applied in the finite element calculation. Theresult indicates that overall nonlinear performance of doubleshear lap joint beam for the low preload case is reflectedmoreevidently than before.

A series of transient excitation experiments are conductedon the bolted joint beam, and the nonlinear behavior forvariable preloads and impact force levels is investigated.The results show that the changes of amplitude and naturalfrequency in the frequency domain plot agree with thenumerical simulation results through qualitative discussionand analysis.

The numerical model for bolted joint structure proposedin this paper can be adopted to develop complicated modelwithmultiple degrees of freedom, and a detailed investigationon dynamic response characteristics of nonlinear systemcan provide more help for critical structural design andmechanical application in future works.

Competing Interests

The authors declare no competing interests regarding thepublication of this paper.

Acknowledgments

The authors acknowledge the Technology Grand Special Pro-ject (Approval nos. 2013ZX04012032 and 2012ZX04002032),


Research and Development of the Key Technology for theOptimization and Reliability Design of Large High-EndPaving Machine (Approval no. BE2014133), and 2014 Scienceand Technology Support Plan of Jiangsu China (Approval no.BE2014004-3).

References

[1] C. Antonios, D. J. Inman, and A. Smaili, “Experimental andtheoretical behavior of self-healing bolted joints,” Journal ofIntelligent Material Systems and Structures, vol. 17, no. 6, pp.499–509, 2006.

[2] A. Bouzid and A. Chaaban, “An accurate method of evaluatingrelaxation in bolted flanged connections,” ASME Journal ofPressure Vessel Technology, vol. 119, no. 1, pp. 10–17, 1997.

[3] L. Gaul and R. Nitsche, “Friction control for vibration suppres-sion,” Mechanical Systems and Signal Processing, vol. 14, no. 2,pp. 139–150, 2000.

[4] S. Bograd, P. Reuss, A. Schmidt, L. Gaul, and M. Mayer,“Modeling the dynamics of mechanical joints,” MechanicalSystems and Signal Processing, vol. 25, no. 8, pp. 2801–2826, 2011.

[5] J. D. Pratt and G. Pardoen, “Numerical modeling of bolted lapjoint behavior,” Journal of Aerospace Engineering, vol. 15, no. 1,pp. 20–31, 2002.

[6] J. Kim, J.-C. Yoon, and B.-S. Kang, “Finite element analysis andmodeling of structure with bolted joints,”AppliedMathematicalModelling, vol. 31, no. 5, pp. 895–911, 2007.

[7] L. Shuguo,M.Yanhong, Z.Dayi, andH. Jie, “Studies ondynamiccharacteristics of the joint in the aero-engine rotor system,”Mechanical Systems and Signal Processing, vol. 29, pp. 120–136,2012.

[8] J. D. Miller and D. D. Quinn, “A two-sided interface model fordissipation in structural systems with frictional joints,” Journalof Sound and Vibration, vol. 321, no. 1-2, pp. 201–219, 2009.

[9] I. R. Grosse and L.D.Mitchell, “Nonlinear axial stiffness charac-teristics of axisymmetric bolted joints,” Journal of Mechanisms,Transmissions, and Automation in Design, vol. 112, no. 3, pp.442–449, 1990.

[10] H. Jalali, H. Ahmadian, and J. E. Mottershead, “Identification ofnonlinear bolted lap-joint parameters by force-state mapping,”International Journal of Solids and Structures, vol. 44, no. 25-26,pp. 8087–8105, 2007.

[11] T. F. Lehnhoff and B. A. Bunyard, “Effects of bolt threads on thestiffness of bolted joints,” Journal of Pressure Vessel Technology,vol. 123, no. 2, pp. 161–165, 2001.

[12] P. Mohanty and D. J. Rixen, “Operational modal analysis inthe presence of harmonic excitation,” Journal of Sound andVibration, vol. 270, no. 1-2, pp. 93–109, 2004.

[13] E. Mucchi, “Experimental evaluation of modal damping inautomotive components with different constraint conditions,”Meccanica, vol. 47, no. 4, pp. 1035–1041, 2012.

[14] R. J. Allemang and D. L. Brown, “A complete review of thecomplex mode indicator function (CMIF) with applications,”in Proceedings of the International Conference on Noise andVibration Engineering (ISMA ’06), vol. 1–8, pp. 3209–3246,Katholieke Universiteit, Leuven, Belgium, September 2006.

[15] V. P. Lawlor,M. A.McCarthy, andW. F. Stanley, “An experimen-tal study of bolt-hole clearance effects in double-lap, multi-boltcomposite joints,” Composite Structures, vol. 71, no. 2, pp. 176–190, 2005.

[16] H. Ouyang,M. J. Oldfield, and J. E.Mottershead, “Experimentaland theoretical studies of a bolted joint excited by a torsionaldynamic load,” International Journal ofMechanical Sciences, vol.48, no. 12, pp. 1447–1455, 2006.

[17] C. J. Hartwigsen, D. M. McFarland, Y. Song, L. Bergman, andA. F. Vakakis, “Experimental study of nonlinear effects in atypical shear lap joint configuration,” inProceedings of theASMEDesign Engineering Technical Conferences and Computers andInformation in Engineering Conference, vol. 37033, pp. 1109–1116,September 2003.

[18] S. A. Nassar and A. Abboud, “An improved stiffness model forbolted joints,” Journal of Mechanical Design, Transactions of theASME, vol. 131, no. 12, pp. 1210011–12100111, 2009.

[19] H. Ahmadian and H. Jalali, “Identification of bolted lap jointsparameters in assembled structures,” Mechanical Systems andSignal Processing, vol. 21, no. 2, pp. 1041–1050, 2007.

[20] M. Budnitzki, M. C. Scates, R. O. Ritchie, E. A. Stach, C. L.Muhlstein, and O. N. Pierron, “The effects of cubic stiffnesson fatigue characterization resonator performance,” Sensors andActuators A: Physical, vol. 157, no. 2, pp. 228–234, 2010.

[21] H.Nouira, E. Foltete, B. Ait Brik, L. Hirsinger, and S. Ballandras,“Experimental characterization and modeling of microslidingon a small cantilever quartz beam,” Journal of Sound andVibration, vol. 317, no. 1-2, pp. 30–49, 2008.

[22] Z. Y. Qin, Q. K. Han, and F. L. Chu, “Analytical model ofbolted disk-drum joints and its application to dynamic analysisof jointed rotor,” Proceedings of the Institution of MechanicalEngineers, Part C: Journal of Mechanical Engineering Science,vol. 228, no. 4, pp. 646–663, 2014.

[23] D.Qiu, S. Seguy et, andM. Paredes, “Design of cubic stiffness forthe absorber of nonlinear energy sink (NES),” inCFA/VISHNO,pp. 2295–2300, Le Mans, France, April 2016.

[24] V. Ramakrishnan and B. F. Feeny, “In-plane nonlinear dynamicsof wind turbine blades,” in Proceedings of the ASME Interna-tional Design Engineering Technical Conferences & Computersand Information in Engineering Conference (IDETC/CIE ’11), pp.761–769, Washington, DC, USA, August 2011.

[25] N. Motosh, “Development of design charts for bolts preloadedup to the plastic range,” Journal of Engineering for Industry, vol.98, no. 3, pp. 849–851, 1976.

[26] J. Bendat, Engineering Applications of Correlation and SpectralAnalysis, John Wiley & Sons, New York, NY, USA, 1993.

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation http://www.hindawi.com

Journal ofEngineeringVolume 2014

Submit your manuscripts athttp://www.hindawi.com

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014

SensorsJournal of


Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks


Research Article Analytical Model of Bolted Joint ...

Documents

Transcript of Research Article Analytical Model of Bolted Joint ...