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Modeling and nonlinear dynamic analysis of bolt joints considering fractal surfaces

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Abstract

The surface morphology has an important influence on the dynamic response and bolt loosening of bolted joints. In this paper, based on the fractal contact model, an original nonlinear dynamic model of the bolted joint structure including the piecewise restoring force and dry friction is proposed to predict the normal and tangential coupling vibration and investigate the dynamic characteristics. The relative sliding behavior between bolts and connected parts is considered for avoiding the bolt loosening failure, which is rarely mentioned in previous studies. Furthermore, normal and tangential fractal contact stiffness models are experimentally validated by comparing the modal test and finite element simulation with virtual material layer. The joint interface is equalized as the equivalent layer in the FE model, and its material parameters are deduced by the measured surface profiles. Numerical simulation is employed to investigate the effects of harmonic excitation, bolt preload and fractal parameters on the dynamic characteristics. The abundant nonlinear phenomena, such as quasi-period motion, chaotic motion and jump discontinuous, are captured. The predicted nonlinear vibration response is helpful for preventing the resonance and instability of system.

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Data availability

The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

a e :

Real contact area at an asperity in elastic deformations

\(a^{\prime}\) :

Truncated microcontact area

\(a^{\prime}_{{{\text{cn}}}}\) :

Critical truncated microcontact area

\(a^{\prime}_{{\text{l}}}\) :

Largest truncated microcontact area

A a :

Nominal contact area

\(A^{\prime}_{{\text{r}}}\) :

Total truncated contact area of the initial loading procedure

C n(C τ):

Normal (tangential) damping factors of the system

d 2 :

Basic pitch diameter of the thread

D :

Fractal dimension of the equivalent surface

D e :

Effective diameter of the bolt head

E 1, E 2, \(E^{\prime}\), E vm :

Elastic moduli of the two connected parts, elastic modulus of the equivalent surface and elastic modulus of the equivalent layer

f :

Excitation frequency

f pre, F pre :

Bolt pre-tightening force and preload of the bolts

F :

Amplitude of external excitation

g :

Acceleration of gravity

G :

Fractal roughness parameter of the equivalent surface

G 1, G 2, \(G^{\prime}\), G vm :

Shear moduli of the two connected parts, shear modulus of the equivalent surface and shear modulus of the equivalent layer

h :

Thickness of the equivalent layer

h nbolts :

Criterion of the preloaded bolts

H :

Brinell hardness of softer material

k, k max :

\(k = {{\left| {P_{{{\tau s}}} } \right|} \mathord{\left/ {\vphantom {{\left| {P_{{{\tau s}}} } \right|} {\left( {\mu_{{\text{s}}} P_{{{\text{ne}}}} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\mu_{{\text{s}}} P_{{{\text{ne}}}} } \right)}}\) And maximum of k per excitation force cycle

k nbolt :

Normal stiffness of a bolt

k τ s :

Tangential stiffness at an asperity

K n, K npre :

Overall normal stiffness and value of Kn when the system is static equilibrium and preloaded

K nbases(K τ bases):

Total normal (tangential) stiffness of bases

K nbolts(K τ bolts):

Total normal (tangential) stiffness of bolts

K ns(K τ s):

Normal (tangential) contact stiffness of the interface

K τ spre :

Value of Kτs when the system is static equilibrium and preloaded

l 1, l 2 :

Thickness of bases

L :

Sample length

m :

Number of bolts

M :

Equivalent mass of the system

\(n\left( {a^{\prime}} \right)\) :

Size distribution of the truncated microcontact area

p :

Pitch of the thread

p ne :

Elastic contact force at an asperity

P 0 :

Contact force when the bolts are pre-tightened

P ne, P ne0, P nepre :

Total elastic contact force of the interface, initial value of Pne and value of Pne when the system is static equilibrium and preloaded

P nbolts(P τ bolts):

Normal (horizontal) force on the bases produced by the bolts

P τ s :

Total tangential contact force of the interface

r :

Radius of the real contact area at an asperity

S 1(τ), S 2(τ), S(τ):

Structure functions of contact surfaces and structure functions of the equivalent surface

T :

Torsional moment

x :

Distance in the horizontal direction

z, z pre :

Separation between the rigid parts and value of z when the system is static equilibrium and preloaded

z s, z s0, z spre :

Separation between the mean planes of the joint interface, initial value of zs and value of zs when the system is static equilibrium and preloaded

Z(x):

Surface profile height

α :

Angle between the harmonic force and the horizontal direction

γ :

Scaling parameter

Δx :

Relative horizontal displacement between the rigid parts

Δx c :

Critical horizontal displacement of the bolt at a certain time

Δx s :

Variation of relative horizontal displacement between the two contact bodies

Δx sc :

Critical relative horizontal displacement between the two contact bodies

Δx sl :

Equivalent cumulative sliding distance between the bolts and the rigid parts

Δx ssl :

Equivalent cumulative sliding distance between the surfaces

Δz :

Relative vertical displacements between the rigid parts

Δz bpre :

Elongation of the bolts when the system is static equilibrium and preloaded

Δz s, Δz spre :

Reduction in separation between the mean planes of the joint interface and value of Δzs when the system is static equilibrium and preloaded

ζ n :

Normal damping ratio of the system

η τ s :

Tangential contact damping dissipation factor

μ b :

Sliding friction coefficient between the bolts and the bases

μ h :

Coefficient of the under-head friction

μ s :

Sliding friction coefficient of the contact surfaces

μ t :

Coefficient of the thread friction

ν 1, ν 2, ν, ν vm :

Poisson ratios of the two connected parts, Poison ratio of softer material and Poison ratio of the equivalent layer

σ :

Root-mean-square height of the equivalent rough surface

φ :

φ = P0/T

φ 1 n :

Random phase

ψ :

Domain extension factor

ω :

Angular frequency of excitation force

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Acknowledgements

We would like to express our appreciation to Chinese National Natural Science Foundation (U1708254) and National Key R & D Program of China (2019YFB2004400) for supporting this research.

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, China

    Zhenyuan Li, Changyou Li, Mengtao Xu, Weibing Dai & Zhendong Liu

  2. College of Mechanical and Automotive Engineering, Zhaoqing University, Zhaoqing, 526061, China

    Yimin Zhang

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Li, Z., Zhang, Y., Li, C. et al. Modeling and nonlinear dynamic analysis of bolt joints considering fractal surfaces. Nonlinear Dyn 108, 1071–1099 (2022). https://doi.org/10.1007/s11071-022-07255-3

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