A simplified nonlinear dynamic model for the analysis of pipe structures with bolted flange joints

Journal of Sound and Vibration 331 (2012) 325–344

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A simplified nonlinear dynamic model for the analysis of pipe structures with bolted flange joints Yu Luan, Zhen-Qun Guan n, Geng-Dong Cheng, Song Liu State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, Dalian 116024, China

a r t i c l e i n f o

abstract

Article history: Received 31 May 2011 Received in revised form 30 August 2011 Accepted 1 September 2011 Handling Editor: H. Ouyang Available online 28 September 2011

Bolted flange joints are widely used in engineering structures; however, the dynamic behavior of this connection is complex in nature. In this paper, a simplified nonlinear dynamic model with bi-linear springs is proposed and validated for pipe structures with bolted flange joints. First, static mechanical properties of the bolted flange joint are investigated. The analytical solution reveals that the axial stiffness of the bolted flange joint is different in tension and compression. Then, nonlinear springs with different stiffness in tension and compression are employed to represent the bolted flange joint. A special type of dynamic behavior, coupling vibration in the transverse and longitudinal directions, is observed in analytical derivation. Finally, relevant physical experiments and numerical simulations are performed. The physical experiments confirm the existence of the coupling vibration behavior. The relationship of longitudinal and transverse vibration frequencies is discussed. The numerical solutions reveal that the simplified nonlinear dynamic model better fits the physical response than conventional reduced linear beam model. & 2011 Elsevier Ltd. All rights reserved.

1. Introduction Since its simple configuration and reliable operation, the bolted flange joint is the primary connection in engineering structures such as launch vehicles and main shaft of nuclear coolant pumps. However, due to discontinuity of the structure and mechanical contact and friction of the connecting interface, the connection becomes the main source of nonlinearity and uncertainty for the assembled structure [1], especially under dynamic loads, such as cross-wind forces for launch vehicles and unbalanced hydrodynamic forces/unbalanced centrifugal forces for pump shafts. Static responses of bolted flange connections are generally solved via contact algorithm based on detailed finite element (FE) models, with relevant analysis strategy well established [2,3]. On the other hand, limited by multiple nonlinearities, nonlinear dynamic algorithm and computational cost, the paradigm of dynamic analysis for structures with bolted flange is difficult to establish [4]. Existing works mainly concentrate on approximate dynamic modeling [5–8] and model updating for reduced-order dynamic models [9,10]. Amongst proposed models, linear beam model is most employed in operational modal analysis (OMA) for launch vehicles [11,12]. In this model, the joint is modeled as a node fixing the connected components, and the connection stiffness is homogenized in the equivalent bending stiffness of adjacent beam elements [13].

n

Corresponding author. Tel./fax: þ86 411 84709730. E-mail address: [email protected] (Z.-Q. Guan).

0022-460X/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2011.09.002

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Merits of the linear beam model are its simplicity, adaptability for modal analysis and low computational cost. Nevertheless, this model also has some significant drawbacks. The first drawback is it allows no axial deformation under transverse loads owing to the basic assumptions of beam theory [14]. This makes it difficult to obtain internal longitudinal forces at the joint, which serve as key parameters in the design for bolts and flanges. The second drawback is its constant joint stiffness, which actually varies with the contact states of connecting interface. Thirdly, the homogenization of joint stiffness, in return, affects the accuracy of connected components. In this paper, to characterize the behavior of bolted flange joints, static responses of the assembled structure are firstly investigated with FE analysis. Then, the different tensile and compressive axial behavior of the joint is modeled by bi-linear springs, which transform the contact nonlinearity into the material nonlinearity. Accordingly, a mass–spring system is developed, and a special type of dynamic response, coupling vibration of transverse and longitudinal direction, is obtained in analytical derivation. Besides, the relation between longitudinal and transverse responses is also discussed. Finally, physical experiments are performed to validate the theoretical prediction and the proposed model. 2. Static characteristic of bolted flange joint The structure discussed in this paper, two cylindrical shells assembled by bolted flange (Fig. 1), is a typical application of the bolted flange joint. To establish an accurate description for the assembled structure, static responses subjected to transverse and axial loads are investigated at first. 2.1. Deformation under transverse loads Deformation of the assembled structure subjected to bending moment and transverse force (Fig. 2(b)) is solved by FE analysis. With the increase of the loads, tensile side of the connection starts to separate, and, consequently, a clearance arises between the contact interfaces. Furthermore, expansion of the clearance raises the length of the axis, shown in Fig. 2(c). In other words, due to the existence of mechanical contact at the joint, transverse loads produce not only transverse deformation but also axial deformation [15]. Axial deformation caused by transverse loads is an important feature for bolted flange joints. Nevertheless, limited by its constant axial length under transverse loads, the linear beam model fails to simulate this distinct behavior (Fig. 3). 2.2. Different tensile and compressive axial stiffness One remarkable feature of the bolted flange assembled structure is its different tensile and compressive axial stiffness. To study this property, the uni-axial responses of a pair of rectangular flanges fixed with one bolt (Fig. 4(a)) are studied. First, the case under axial tensile load is considered. By analogy with the deformed configuration of the flange (Fig. 4(b)), a beam–spring model (Fig. 5(b)) is developed based on the following assumptions: 1. the bolt and flange are linear elastic with small deformation; 2. since the stiffness in thickness direction is extremely large compared with the bending stiffness, the variation for the thickness of the flange is ignored, and according to this assumption, the following deductions are derived (Fig. 5(a)): 2.1. the deflection of flange can be described by a bending beam; 2.2. the contact state is considered as point to point contact, and, hence, the contact pressure is taken as a concentrated force;

Fig. 1. Typical bolted flange assembled structure.

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Fig. 2. Deformation of the bolted flange (four bolts) under transverse loads: (a) detailed FE model; (b) deformation (in gray) with origin configuration (full line); (c) deformation (in gray) for the joint with origin configuration (full line); and (d) contour plot of the axial displacement for the joint.

Fig. 3. Linear beam model for the bolted flange joint: (a) deformation of the beam and (b) deformation with beam section displayed.

Fig. 4. The case under axial tensile force: (a) isometric view of bolted flange and (b) cross-section of bolted flange with its deformed configuration (dash lines).

2.3. neither rotation nor transverse displacement exists at the start point of clearance (C in Figs. 4 and 5), which is idealized as a fixed end; 2.4. the bending moment on the cross-section of the flange at the start of the clearance is zero; 3. the role of the bolt, FB, is equal with a spring with equivalent stiffness. According to the assumptions and deductions above, the equilibrium equation for the transverse forces on the flange and that for moment at the start of the clearance are obtained (1) F þ RC FB ¼ 0

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Fig. 5. Beam–spring model for bolted flange: (a) force boundaries for the flange and (b) beam–spring model for bolted flange joint.

FB ðlC lB ÞFlC ¼ 0

(2)

where F stands for the external force, RC stands for the contact force, and FB stands for the bolt force. Deformation compatibility condition for the beam–spring model can be written as

dB ¼ oB

(3)

where oB is the deflection of the beam at the spring, dB is the axial deformation of the spring. The expression of oB and dB are given in Eqs. (4) and (5), respectively

oB ¼

FðlC lB Þ2 ½3lC ðlC lB Þ FB ðlC lB Þ3  6EFL IFL 3EFL IFL

(4)

ðFB FB0 ÞtFL EB AB

(5)

dB ¼

where EFL is Young’s modulus for the flange, EB is Young’s modulus for the bolt, AB is the unthreaded sectional area of the bolt, IFL is the moment of inertia of the flange section about the middle surface, FB0 is the pretension of the bolt. By solving Eqs. (1)–(3) simultaneously, all the unknown variables, lC, RC and FB, are determined. Therefore, deflection of the flange at the open end, d0, can be derived by superposition of the deflections caused by the axial force and that by the bolt force

d0 ¼

Fl3C FB ðlC lB Þ2 ½3lC ðlC lB Þ  6EFL IFL 3EFL IFL

(6)

For the case under axial compression (Fig. 6(a)), since the compressed region is within the thickness of connected vertical shells (Fig. 6(b)), the axial deformation can be simply expressed as

d0 ¼

FtF EFL tC

where tF and tC stand for the thickness of the flange and that of the connected vertical shell, respectively. Combining Eqs. (6) and (7), the axial behavior of the bolted flange is described by 8 3 2 < FlC  FB ðlC lB Þ ½3lC ðlC lB Þ , F 4 0 3E I 6EFL IFL d0 ¼ FtFL FL : F , F r0 EFL tC

(7)

(8)

Eq. (8) reveals that the axial stiffness for bolted flange joint is bi-linear with a single change at the equilibrium position. The analytical results meets the FE results with an error lower than 2%, as is shown in Fig. 7. Furthermore, this also

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Fig. 6. The case under axial compressive force: (a) isometric view of bolted flange and (b) cross-section of bolted flange.

Fig. 7. Uni-axial force–deflection curve: (a) geometry of the tested joint; (b) FE model of the tested joint (flange: E¼ 70 GPa, v¼0.3; bolt: E ¼210 GPa, v¼ 0.28); and (c) axial force–deflection curve.

explains the axial displacement under transverse loads, since the stiffness on the tensile side of the joint is smaller than that on the compressive side. 3. Simplified nonlinear dynamic model for bolted flange structure 3.1. Nonlinear spring with different tensile and compressive stiffness To represent the different tensile and compressive axial stiffness for bolted flange assembled structure, unidirectional bi-linear spring is introduced. Constitutive property for the spring is defined by the axial force–deflection curve of the joint (Fig. 8), which can be derived ether from FE results or from static experiments.

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Fig. 8. Force–deflection curve for the bi-linear spring (k þ stands for the tensile stiffness and k  stands for the compressive stiffness).

Fig. 9. Measurement of deflection: (a) measurement for tensile deformation and (b) measurement for compressive deformation.

Specifically, axial relative displacement of the joint is measured with an appropriate distance from the top and bottom surface of the flanges, which indicates that the springs stand for whole connecting zone but not the region within the thickness of the flanges. This is aimed to eliminate the local deformations on the flange (Fig. 2(d)). A detailed illustration for stiffness acquisition can be found in Fig. 9, in which the tensile stiffness k þ and compressive stiffness k  of the springs can be determined by 8 T < k þ ¼ L FL T 0 (9) C : k ¼ L FL 0 C

3.2. Simplified nonlinear model By replacing the node for the joint with rigid beams and bi-linear springs, a simplified nonlinear dynamic model is established based on linear beam model (Fig. 10(a)). Specifically, beam elements are used to model the cylinder shells. As the warping deformation of the cylinder shells under lateral loads near the joint is relative small, the plane section of beam theory is assumed to be satisfied. Accordingly, rigid beams are employed to construct rigid planes as the boundaries of the joint. Additionally, to transmit shear forces, the lateral dofs between the center nodes of the rigid plane are coupled using constrain equations. Finally, bi-linear springs are applied to each pair of the outer nodes of the rigid planes to constrain the axial relative motion between the rigid planes. Equipped with springs with different tensile and compressive stiffness, the proposed model can accurately describe the axial property of the joint. Furthermore, for the case under transverse loads, since stiffness of the tensile side is different from that of the compressive side, ‘‘clearance’’ appears (Fig. 10(b)), and the axial length rises with increase of the loads. In other words, the bi-linear springs also bring flexibility in the axial direction when the model is subjected to transverse loads. 3.3. Mass–spring system for simplified nonlinear dynamic model To investigate the impact behavior of the joint, a schematic model composed of a rigid mass block and two bi-linear springs is constructed (Fig. 11(a)). The properties of the joint are characterized by two bi-linear springs, the tensile and compressive stiffness of which are k þ and k  , respectively. Bottom nodes of the springs are fully constrained, while their top nodes are fixed on the block. The length of the spring equals L0 in Fig. 9. By assuming the springs stay vertical, the mass block representing the connected component can be described by two generalized coordinates: longitudinal displacement u and rotation about origin y (Fig. 11(b)). 3.3.1. Governing equation Damped free vibration of the system can be expressed as My€ þ Ky ¼ 0 where y¼(u,y)T is the displacement vector; M is the mass matrix; and K is the stiffness matrix.

(10)

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Fig. 10. Deformation of the simplified nonlinear model for bolted flange joint: (a) simplified nonlinear model; (b) deformation of the simplified nonlinear model; and (c) deformed configuration of the joint.

Fig. 11. Mass–spring system: (a) mass–spring system and (b) deformation of the system.

The kinetic energy of the mass block is T¼

1 1 2 mu_ 2 þ J y_ 2 2

(11)

where m is the mass of the block and J is the inertia of moment of the block about z-axis. Elastic potential energy of the springs is U¼

1 n 2 2 ðk d þ kn2 d2 Þ 2 1 1 n

(12)

n

where d1 and d2 are the axial deformations of the springs, k1 and k2 are stiffnesses of the springs. The superscript ‘‘n’’ n indicates that the value of ki depends on state of the spring. Refer to Fig. 11(b), the relation between the springs’ deformation and the displacement of the block is ! " # 1 2b  u  d1 (13) ¼ d2 1 2b y Substitute Eq. (13) into Eq. (12), the elastic potential energy is rearranged with u and y   1 b2 2 ðkn1 þ kn2 Þu2 þbðkn1 kn2 Þuy þ ðkn1 þ kn2 Þy U¼ 2 4

(14)

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Finally, by using Lagrange’s equations, the governing equation of damping free vibration is derived 3 " #  2 n   b n n k1 þ kn2 m 0 u u€ 2 ðk1 k2 Þ 4 5 þ ¼0 2 b 0 J y y€ ðkn kn Þ b ðkn þ kn Þ 2

n

1

2

4

1

(15)

2

n

3.3.2. k1 and k2 The determination of spring stiffness can be defined as ( k , kni ¼ kþ ,

di o0 di Z0

(16)

where i¼1 or 2. n n To acquire an explicit expression of k1 and k2, sign function is employed 8 > < 1, di o0 sgnðdi Þ ¼

0, > : 1,

di ¼ 0 di 40

(17)

The combination of Eqs. (16) and (17) yields kni ¼

k þ k þ k k þ  sgnðdi Þ 2 2

(18) n

n

Then, substitution of Eq. (13) into Eq. (18) leads to the explicit expressions for k1 and k2 with u and y 8 < kn ¼ k þ k þ  k k þ sgnðu þ b yÞ 1 2 2 2 : kn2 ¼

k þ k þ 2

þ  k k sgnðu 2b yÞ 2

(19)

3.3.3. Zone division According to Eq. (19), stiffness of the system essentially depends on the positive–negative value of u þ ðb=2Þy and uðb=2Þy, the critical state of which is ( u þ 2b y ¼ 0 (20) u 2b y ¼ 0 Taking Eq. (20) as the dividing lines, motion of the mass block is divided into four independent areas (Fig. 12), each of which contains a unique combination of the springs’ states. Accordingly, the stiffness matrix in each zone is determined, and, by solving the characteristic equation of motion, natural frequencies and mode shapes are obtained (Tables 1 and 2). For Zones I and III, rotational and longitudinal vibrations are independent of each other. However, for Zones II and IV, due to the emergence of coupling elements in stiffness matrix, independent vibration of longitudinal and rotational direction no longer exists but replaced by coupling vibrations of the two directions. Besides, noticed by definition of the term k2, the frequencies in Zone II equal those in Zone IV, while the mode shapes in Zone IV are symmetric with those in Zone II. Thus, six natural frequencies have been recognized, while eight mode shapes are obtained.

Fig. 12. Division of the motion.

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Table 1 Stiffness in each zone. Zone

I

II

III

IV

n

Boundary of zones

n

k1, k2

[d1, d2]

[u, y]

d1 40 d2 40

u 40 2 2  uo y o u b b

k1 ¼ k þ n k2 ¼ k þ

d1 40 d2 o 0

 2b y o u o 2b y y40

k1 ¼ k þ n k2 ¼ k 

d1 o 0 d2 o 0

uo0 2 2 uo y o  u b b

k1 ¼ k  n k2 ¼ k 

d1 o 0 d2 40

b b 2 youo 2 y

k1 ¼ k  n k2 ¼ k þ

yo0

n

n

n

n

Stiffness matrix, K

"

2k þ 0

2 4 "

2 4

#

0 b2 2

kþ

3

b 2 ðk k þ Þ 5 b2 4 ðk þ k þ Þ

k þ k þ b 2 ðk k þ Þ

2k 0

b2 2

0 k

k þ k þ b 2 ðk þ k Þ

#

3

b 2 ðk þ k Þ 5 b2 4 ðk þ k þ Þ

Table 2 Natural frequencies and mode shapes. Zone

Circular frequencies (o21 , o22 )

I

2k þ m

II

ðmk3 þ Jk1 Þ

2

, b 2Jk þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðmk3 þ Jk1 Þ þ

ðmk3 Jk1 Þ2 þ 4mJk22 2mJ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Mode shape (u1, u2)

Additional terms

   1 0 , 0 1 ! ! 1 0 , r1 r2

k2 ¼ 2b ðk k þ Þ

k1 ¼ k þ þ k

r1 ¼

b2 4 ðk þ þ k Þ k1 mo21 k2 ¼ k J k2 o2 3

r2 ¼

k1 mo22 k2

k3 ¼

ðmk3 Jk1 Þ2 þ 4mJk22 2mJ

1

III

2k m

IV

ðmk3 þ Jk1 Þ

2

, b 2Jk , pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðmk3 þ Jk1 Þ þ

ðmk3 Jk1 Þ2 þ 4mJk22 2mJ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmk3 Jk1 Þ2 þ 4mJk22

    1 0 , 0 1 ! ! 0 1 , r2 r1

2mJ

¼

k2 k3 Jo22

k1 ¼ k þ þ k k2 ¼ 2b ðk þ k Þ k3 ¼ r1 ¼ r2 ¼

b2 4 ðk þ þ k Þ k1 mo21 k2 ¼ k J k2 o21 3 k1 mo22 k2

¼

k2 k3 Jo22

3.3.4. General solution Assume the solution to Eq. (10) is in the form y ¼ A sinðot þ jÞ where A and j denote the amplitude and phase angle, respectively. Initial conditions for the system are defined as ( _ uð0Þ ¼ u0 , uð0Þ ¼ u_ 0 yð0Þ ¼ y0 , y_ ð0Þ ¼ y_ 0

(21)

(22)

By applying the initial values to Eq. (21), motion of the mass block in all zones is determined. Specifically, for Zones I and III, there are 8 u_ 0 sinðo1 tÞ < uðtÞ ¼ u0 cosðo1 tÞ þ o 1 (23) _ y : yðtÞ ¼ y0 cosðo2 tÞ þ 0 sinðo2 tÞ o2 and, for Zones II and IV, there are ( uðtÞ ¼ A01 cosðo1 tÞ þ A02 sinðo1 tÞ þ A001 cosðo2 tÞ þA002 sinðo2 tÞ

yðtÞ ¼ r1 ½A01 cosðo1 tÞ þ A02 sinðo1 tÞ þ r2 ½A001 cosðo2 tÞ þ A002 sinðo2 tÞ

(24)

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where A01 ¼

y0 r2 u0 r1 r2

,

A02 ¼

y_ 0 r2 u_ 0

o1 ðr1 r2 Þ

,

A001 ¼

u0 r2 y0 , r1 r2

A002 ¼

u_ 0 r2 y_ 0

o2 ðr1 r2 Þ

:

3.3.5. Impact response Consider a longitudinal impact along y axis is applied to the mass block. In this case, the springs will be either stretched or compressed, which leads the motion into Zone I (d1 40, d2 40) or Zone III (d1 o0, d2 o0), respectively. Idealized as constant force acting during an extremely short period, the impact is transformed into initial velocity based on the conservation of momentum ( u0 ¼ 0, u_ 0 a0 (25) y0 ¼ 0, y_ 0 ¼ 0 For transverse impact, the initial condition can be expressed as ( u0 ¼ 0, u_ 0 ¼ 0 y ¼ 0, y_ a0 0

(26)

0

In this case, one spring is stretched while the other is compressed. Accordingly, motion of the system begins in Zone II (d1 40, d2 o0) under clockwise impact or Zone IV (d1 o0, d2 40) under anticlockwise impact. The impact responses for the schematic model are solved with the strategy of initial value problems. Numerical solutions are programmed in C language, and the moments, at which bounds of the zones are crossed, is obtained using recursion method. Then, applying the given initial values, dynamic responses under axial and transverse impacts are calculated, which is shown in Fig. 13. 3.4. Discussion For axial impacts, the displacement responses are confined in Zones I and III (Fig. 13(e)). Since none coupling element exists in the stiffness matrix, just longitudinal vibration is excited, with y(t)  0 (Fig. 13(a)). In other words, once uni-axial dynamic loads are applied, the block will vibrate along the u axis and periodically cross the y-axis (Fig. 13(c) and (e)). In this case, the system contains two frequencies, which are axial frequencies of Zones I and III (Fig. 13(g)). For the responses under rotational impact, both axial and rotational vibrations are excited, as is shown in Fig. 13(b). The vibration of the mass block covers all zones in u–y plane (Fig. 13(d)). Specifically, the motion begins in Zone IV, and, as time goes, the other three zones are all entered (Fig. 13(f)) with more frequencies excited (Fig. 13(h)). Furthermore, another important feature under transverse impacts is recognized. With regard to Fig. 13(b), it can be expressly found that, during one rotational vibration cycle, the system experiences two vibration cycles in the axial direction. As a result, the axial frequency should double the transverse frequency in the frequency spectrogram. Based on the discussion above, the following conclusions are drawn: 1. When the joint is subjected to uni-axial impacts, only axial vibration is excited with two frequencies: the axial frequencies of Zone I and that of Zone III. 2. When transverse impacts are applied, both rotational and axial vibrations will be excited, and the rotational and axial vibrations are no longer independent of each other but replaced by coupling vibrations. Moreover, the frequency in axial direction doubles that in transverse direction. 4. Results for a typical bolted flange assembled structure To validate the simplified nonlinear dynamic model, a dynamic experiment is carried out. The geometry and dimensions of the specimen are shown in Fig. 14. The cylinder shells are built with carbon structural steel (E¼200 GPa, n ¼0.27, r ¼7800 kg m  3), and the flanges are fixed by four low carbon steel (E¼210 GPa, n ¼0.27, r ¼7850 kg m  3) M10 bolts (diameter: 10 mm). 4.1. Dynamic experiment 4.1.1. Experiment setup The experiment setup is shown in Figs. 15 and 16. The accelerators and input module are manufactured by DK, Denmark. The type number of the input module is 3038, through which the acceleration responses within 1 s are recorded with 6000 Hz sampling frequency. The triangular impulse is applied to the experiment structure by impact hammer, and the recorded input signal is shown in Fig. 17. Additionally, to evaluate the influence of the bolt torques, 10 and 20 N m are applied to the bolts, respectively.

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Fig. 13. Impact responses of the mass–spring system: (a) displacement response under axial impact; (b) displacement response under transverse impact; (c) displacement response in u–y plane under axial impact; (d) displacement response in u–y plane under transverse impact; (e) state of the motion under axial impact; (f) state of the motion under transverse impact; (g) vibration frequencies under axial impact; and (h) vibration frequencies under transverse impact.

4.1.2. Experiment results The acceleration responses under transverse impulse are shown in Fig. 18, and the response frequencies at the top plane are extracted using fast Fourier transformation (FFT), which are listed in Table 3.

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Fig. 14. Geometry and dimensions for experiment: (a) front view; (b) sectional view with dimensions; and (c) vertical view with dimensions.

Fig. 15. Experiment setup.

According to the test results, the initial transverse impulse excites both transverse and longitudinal vibrations with the same order of magnitude but with different frequencies, which meets the theoretical prediction. Specifically, the axial frequency is the doubles transverse frequency. This might be induced by collision behavior between the junction interfaces: the interfaces collide twice during one transverse vibration cycle, which, in return, explains the coupling mechanism between rotational and axial vibration. This is further employed in the validation for the reduced models in the following discussion. Besides, since the results with different bolt torques are similar with slight changes in frequency, the results with 10 N m bolt torque are used as the experiment data in the following discussion. Additionally, since the longitudinal accelerators are fixed on the edge of the top plane (Fig. 15), the base transverse frequency, 405 Hz, also emerges in the longitudinal FFT spectrogram under transverse impact.

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Fig. 16. Photo of the dynamic experiment.

Fig. 17. Recorded impulse.

Fig. 18. Acceleration response under transverse impact of experiment with 10 N m pre-torsion in the bolts.

Table 3 Acceleration responses under transversal impacts. Bolt pre-torsion (N m) 2

10

20

Transversal acceleration

Scale (m s ) Frequency (Hz)

 300 to 400 405

 300 to 300 402

Longitudinal acceleration

Scale (m s  2) Frequency (Hz)

 300 to 400 405, 809

 250 to 200 403, 804

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4.2. Reduced-order dynamic models The parameters for reduced-order dynamic models, such as the stiffness for the bi-linear springs, are important, and these are conventionally derived from the results of static physical experiments. As the contact algorithm and FE method have been well established, FE method is employed instead. A detailed FE model of the experiment structure is constructed in advance (Figs. 19(a) and 21(a)). Then, the static analysis under transverse loads and axial forces are carried out, respectively, results of which are shown in Fig. 22. 4.2.1. Linear beam model The linear beam model is constructed based on beam theory, in which mechanical behavior of the joint is modeled with equivalent bending stiffness of beam elements. Specifically, the equivalent bending stiffness is determined by static responses under transverse loads (Table 4), and the cylinder shells are modeled by beam elements with designed crosssections and equivalent properties (Fig. 19(b) and (c)). Mass properties of the linear beam model are listed in Table 5, and the modal results are listed in Table 6. 4.2.2. Simplified nonlinear dynamic model In the simplified nonlinear dynamic model, characteristics of the joint are modeled by nonlinear springs with different tensile and compressive stiffness. Specifically, the cylinder shells are modeled by beam elements with designed sections

Fig. 19. Linear beam modeling: (a) detailed FE model using solid elements; (b) linear beam model; and (c) linear beam model with beam section displayed.

Table 4 Parameters for linear beam model. Symbol

Meaning

Unit

Value

E

Young’s module (equivalent module) Poisson ratio Density

GPa

46.8 0.27 7800

n r

kg m  3

Table 5 Mass properties of the models. Parameter

Unit

Design value

Linear beam model

Simplified nonlinear dynamic model

Mass Mass center relative to origin Inertia of moment about origin

kg m kg m2

26.25 0.0.0 0.84

26.27 0.0.0 0.82

26.26 0.0.0 0.82

Table 6 Modal analysis for linear beam model. Order

Frequency (Hz)

Mode shape

1 2 3 4 5

379 737 937 1336 1353

Transverse vibration: bended once along the axis Torsional vibration along the axis Transverse vibration: bended twice along the axis Torsional vibration along the axis Longitudinal vibration

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and real material properties, while the joint is modeled by two rigid planes and four springs with different tensile and compressive stiffness (Figs. 20 and 21). Detailed demonstration for joint modeling is shown in Fig. 22. The rigid planes are constructed with rigid beams to fix four bi-linear springs between them. The springs are placed at circumferential position of the bolts with 60 mm radial distance from center of the rigid plane, which equals the outer radius of cylinder shell. The length of the springs is defined as 20 mm to eliminate the local deflection on the flange, which is 10 mm thicker than the flanges. Then, constrain equations are employed, which only allows relative axial displacement between the centers of rigid planes. Finally, relevant mass and inertia of moment are added on the central node of rigid planes, and the mass properties of the model are also listed in Table 5. The tensile and compressive stiffness of the springs are firstly derived from the static responses under axial loads (Fig. 22(a)) in the approach mentioned in Section 3.1, and, then, verified with the static responses under transverse loads (Fig. 22(b)–(d)). Material properties and the derived parameters are listed in Table 7. Moreover, according to Fig. 22, the simplified nonlinear model meet the FE results in all directions; whereas, the linear beam model fails to fit the longitudinal displacement under longitudinal (Fig. 22(a)) and transverse loads (Fig. 22(c)). In other words, equipped with bi-linear springs, the nonlinear model better describes the static behavior of the assembled structure. 4.2.3. Dynamic analysis and numerical results With the recorded impacts applied to the top node, dynamic responses of the reduced models are calculated using implicit time integral method (Newmark algorithm). Responding to the bi-linear module in the simplified nonlinear model, an extremely small time step (Dt ¼10  5 s) is used to minimize the energy dissipation when the boundaries of zones are crossed [16]. The numerical results are outputted every 10  4 s, the sampling frequency of which is 10,000 Hz. The numerical results are shown in Figs. 23 and 24. Response frequencies are derived using FFT, which are listed in Tables 8 and 9. 4.3. Result comparison Comparison can be drawn between the reduced-order dynamic models and the dynamic experiment under transverse impacts (Table 10). For the linear beam model (Fig. 23), transverse impact only excites the transverse vibration, while axial impact only excites the axial vibration, and no coupling behavior between the transverse and axial response is detected. Besides, the transverse frequency of linear beam model is 376 Hz, which equals the transverse base frequency derived from modal analysis (Table 7), and its error relative to the experiment result (405 Hz) is 6.9%.

Fig. 20. Simplified nonlinear dynamic modeling: (a) detailed FE model using solid elements; (b) simplified nonlinear dynamic model; and (c) simplified nonlinear dynamic model with beam section displayed.

Fig. 21. Joint modeling: (a) detailed FE model with solid elements; (b) simplified nonlinear dynamic model; and (c) simplified nonlinear dynamic model with beam section displayed.

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Fig. 22. Static responses of the models: (a) axial displacement under uni-axial force; (b) transverse displacement of the top plane under bending moment; (c) axial displacement of the top plane under bending moment; and (d) rotation of the top plane under bending moment.

Table 7 Parameters for simplified nonlinear dynamic model. Part Joint modeling

Cylinder shells

Symbol

Meaning

Unit

Value

kþ k R L

Tensile stiffness of the spring Compressive stiffness of the spring Radius for the rigid plane Length of the springs

N mm N mm  1 mm mm

4.08  106 6.65  104 60 20

E

Young’s module Poisson ratio Density

GPa

200 0.27 7800

n r

1

kg m

3

For the simplified nonlinear dynamic model (Fig. 24), axial impact only excites the axial vibration, while transverse impact excites both of the transverse vibration and longitudinal vibration with the same amplitude. This is another significant improvement compared with the linear beam model. Furthermore, under transverse impact, the axial frequency (755 Hz) doubles the transverse frequency (377 Hz), which is consistent with the experiment phenomenon. In order to reduce the error of reduced-order dynamic models, dynamic model modification is performed according to the results with 10 N m bolt torques: the equivalent stiffness E of the linear beam model is increased from 46.8 to 56.5 GPa; the tensile stiffness k þ of the simplified nonlinear dynamic model is increased from 6.65  104 N mm  1 to 3.13  105 N mm  1. Then, results of the updated models subjected to transverse impact are shown in Figs. 25 and 26, and relevant frequency spectrograms are shown in Figs. 27 and 28. Through dynamic modification, the accuracy of the simplified nonlinear dynamic model is significantly improved in both the transverse and the longitudinal directions; nevertheless, limited by the assumptions of beam theory, the linear beam model fails to offer accurate responses in the axial directions under transverse impact.

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Fig. 23. Acceleration response of linear beam model: (a) acceleration response under transverse impact and (b) acceleration response under axial impact.

Fig. 24. Acceleration response of simplified nonlinear dynamic model: (a) acceleration response under transverse impact and (b) acceleration response under axial impact.

Table 8 Response of linear beam model. Impact direction

Transversal

Longitudinal

Transversal acceleration

Scale (m s  2) Frequency (Hz)

 600 to 600 376

0 –

Longitudinal acceleration

Scale (m s  2) Frequency (Hz)

0 –

 300 to 50 1278

Transversal

Longitudinal

Table 9 Response of simplified nonlinear dynamic model. Impact direction 2

Transversal acceleration

Scale (m s ) Frequency (Hz)

 400 to 400 377

0 –

Longitudinal acceleration

Scale (m s  2) Frequency (Hz)

 600 to 400 751

 250 to 20 1401

5. Conclusion Upon the investigation of the static behavior of bolted flange connections, a simplified nonlinear dynamic model is proposed, in which the mechanical properties of the joint are modeled by bi-linear springs. The different tensile and

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Table 10 Response under transversal impact. Experiment (pre-torsion¼ 10 N m)

Linear beam model

Simplified nonlinear dynamic model

Transversal acceleration

Scale (m s  2) Frequency (Hz)

 300 to 400 405

 600 to 600 376

 600 to 400 377

Longitudinal acceleration

Scale (m s  2) Frequency (Hz)

 300 to 400 405, 809

0 –

 400 to 400 751

Fig. 25. Transverse acceleration response under transverse impact.

Fig. 26. Longitudinal acceleration response under transverse impact.

compressive modules of bi-linear springs not only present accurate axial stiffness, but also bring flexibility in axial direction under transverse loads, which is an significant improvement compared with the linear beam model. To simply illustrate the dynamic behavior of bolted flanges, a 2-dof mass–spring system is developed. Then, coupling vibration of longitudinal vibration and transverse vibration is recognized, which is caused by the emergence of coupling elements in stiffness matrix. Then, study on the impact behaviors of the mass–spring system reveals that transverse impact can excite the coupling longitudinal vibrations, while longitudinal impact only excites longitudinal vibrations. Furthermore, the relation between longitudinal and transverse frequencies under transverse impact is predicted: the longitudinal frequency doubles the transverse one.

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Fig. 27. Transverse frequencies under transverse impact.

Fig. 28. Longitudinal frequencies under transverse impact.

The impact behaviors of a typical bolted flange assembled structure are tested, which confirms the existence of coupling response. Besides, the double frequency of transverse vibration is expressly detected in the longitudinal direction, which verifies the theoretical prediction. Finally, numerical solutions indicate the simplified nonlinear dynamic model can fit the test results in both longitudinal and transverse directions, while the linear beam model fails to offer accurate longitudinal response under transverse impact.

Acknowledgment This work is financially supported by the Chinese 973 program, ‘‘Long term run features and safety evaluation of the reactor coolant pump’’ (Grant no. 2009CB724302, Project no. GZ0809). Prof. Xing-Lin Guo, Assoc. Prof. Ming-Fa Ren and Assoc. Prof. Xiao-Peng Zhang are acknowledged for their guidance and assistance to the dynamic experiment. References [1] L. Gaul, J. Lenz, Nonlinear dynamics of structures assembled by bolted joints, Acta Mechanica 125 (1997) 169–181. [2] T. Tsuta, S. Yamaji, Finite element analysis of contact problem, Theory and Practice in Finite Element Structural Analysis, University of Tokyo Press, Japan, 1973. [3] H. Zerres, Y. Gue´rout, Present calculation methods dedicated to bolted flanged connections, International Journal of Pressure Vessels and Piping 81 (2004) 211–216. [4] R.A. Ibrahim, C.L. Pettit, Uncertainties and dynamic problems of bolted joints and other fasteners, Journal of Sound and Vibration 279 (2005) 857–936. [5] M.J. Oldfield, H. Ouyang, J.E. Mottershead, Simplified models of bolted joints under harmonic loading, Computers and Structures 84 (2005) 25–33.

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[6] W.H. Semke, G.D. Bibel, S. Jerath, S.B. Gurav, A.L. Webster, Efficient dynamic structural response modelling of bolted flange piping systems, International Journal of Pressure Vessels and Piping 83 (2006) 767–776. [7] H. Ouyang, M.J. Oldfield, J.E. Mottershead, Experimental and theoretical studies of a bolted joint excited by a torsional dynamic load, International Journal of Mechanical Sciences 48 (2006) 1447–1455. [8] V. Aurelian, L. Dimitri, G. Jean, Bolted joints for very large bearings—numerical model development, Finite Elements in Analysis and Design 42 (2006) 298–313. [9] J.E. Mottershead, M.I. Friswell, Model updating in structure dynamics: a survey, Journal of Sound and Vibration 167 (2) (1993) 347–375. [10] J.-B. Qiu, J.-M. Wang, Z.-Y. Tan, Some new technologies for structural dynamic analysis of launch Vehicle, Chinese Journal of Missiles and Space Vehicles 252 (2001) 16–21 (in Chinese). [11] J. Chen, T. Rose, M. Trubert, B. Wada, F. Shaker, Modal test/analysis correlation for the Centaur G prime launch vehicle, Journal of Spacecraft and Rockets 24 (1987) 423–429. [12] Y. Morino, Vibration test of scale H-I launch vehicle, 27th AIAA SDM, 1987. [13] Y. Song, C.J. Hartwigsen, D.M. McFarland, A.F. Vakakis, L.A. Bergman, Simulation of dynamics of beam structures with bolted joints using adjusted Iwan beam elements, Journal of Sound and Vibration 273 (2004) 249–276. [14] S.M. Han, H. Benaroya, T. Wei, Dynamics of transversely vibrating beams using four engineering theories, Journal of Sound and Vibration 225 (5) (1999) 935–988. [15] Y. Luan, Z.-Q. Guan, G.-D. Cheng, The study on nonlinear dynamic behaviors of the structures with bolted flange joint, IOP Conference Series: Materials Science and Engineering 10 (2010) 012172. [16] H.-T. Yang, B. Wang, An analysis of longitudinal vibration of bimodular rod via smoothing function approach, Journal of Sound and Vibration 317 (2008) 419–431.