CC BY
20Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 1, pp. 3-18. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220101
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 74H45 74A55 70K25 70K40
Analysis of the Vibrational Behavior of a Bolted Beam
in the Presence of Friction
F. Chekirou, K. Brahimi, H. Bournine, K. Hamouda, M. Haddad, T. Benkajouh, A. Le Bot
This paper presents the effect of friction-induced vibration between two beams in relative motion according to Timoshenko's beam theory. This system is composed of two cantilever beams screwed together, allowing friction force to occur in the contact interface. The nonlinear behavior can be divided into two phases: stick and slip. The differential equations of motion in the two phases are developed, with the precision of the transition condition between each phase. A number of experiments are carried out in order to validate the theoretical model, the main contribution of which is to test these specimens in modes greater than one. The experiments demonstrate the influence of changing the clamping force on the stiffness of the structure and thus on its frequency and damping ratio. The comparison between theory and experiments reveals a good agreement. In addition, the tests show an increase in the modal damping ratio when the frequencies are increased. This leads to a considerable increase in energy dissipation by the structure, making it a good choice as a friction damper.
Keywords: bolted beam, Timoshenko beam, damping, stick-slip phenomenon
Received July 10, 2021 Accepted November 29, 2021
The work is financed by the Algerian Ministry of Higher Education and Scientific Research.
Fatine Chekirou
fchekirou@usthb.dz,chekirou.amel@gmail.com
Khaled Brahimi
khaled.brahimi@usthb.edu.dz Hadjila Bournine hadjila15@hotmail.com Khaled Hamouda hamoudakhaled_2000@ya.hoo.fr
Université des Sciences et de la Technologie Houari Boumediene BP 32 Bab Ezzouar, 16111, Algiers, Algeria
Moussa Haddad haddad.emp@gmail .com Tarek Benkajouh bktarek@gmail.com
Laboratoire de Mécanique des Structures (LMS), EMP BP 17 Bordj El Bahri, 16111, Algiers, Algeria
Alain Le Bot
alain.le-bot@ec-lyon.fr
Laboratoire de Tribologie et Dynamique des Systèmes (LTDS), Ecole Centrale de Lyon 36, Avenue Guy de Collongue, 69130, Ecully, Lyon, France
1. Introduction
Civil engineering structures are subjected to strong vibration such as wind and earthquakes that can lead to their failure. Mechanical systems are also subjected to important vibration amplitudes and this affects their service life and the surface quality of the manufactured parts. Over the past decades, a number of sophisticated mechanisms, known as friction dampers, have been developed in order to reduce the force applied to the structure, the vibration amplitude and the ground acceleration [1]. These devices dissipate or absorb energy by means of the frictional forces that are developed within them. The addition of a friction damper is classified into four groups according to their energy requirement: active, passive, semiactive and hybrid dampers [2]. Bolted connections are the typical joints used as a source of damping in several types of mechanical systems and civil structures due to their efficient installation, low cost and acceptable reliability. The functioning of these structures depends on [3, 4]: the joint geometry, the nature of the load and the value of the bolt clamping force, the influence of materials surface treatment and cut hole of the plates [5].
A bolted beam has been used as a vibration damper [6] and can be placed in a wide range of structures by using one or more bolts to press the beams on top of each other. This structure has been studied by Popp et al. [7], who have proved the optimum clamping force for maximum energy dissipation. Bournine et al. [8] have focused on the effect of friction on the dynamic behavior of the bolted beam in its first bending mode. Also, they analyzed the energy dissipated by this structure in order to optimize the bolt tension and then dissipate the maximum vibration energy. Sedighi et al. [9] have considered the dynamic behavior of two-layer beams with a friction interface and including nonlinear stick-slip phenomena. These studies have chosen the model of Euler Bernoulli as the main beam theory. However, this model has certain limitations for thick beams and even slender beams vibrating in high frequency order [10, 11]. The effect of rotatory inertia and shear deformation for two-layer beams with a small clearance have been taken into account by V. A. Krysko et al. [12] for the upper beam, whereas the lower beam is governed by the Euler Bernoulli model. Furthermore, O. A. Saltykova et al. [13] have concentrated on the contact interaction between two Timoshenko beams which are subjected to a transverse force acting on one of the beams. They investigated the transition from harmonic motion to chaotic motion by using dynamic analysis and qualitative methods. Some researchers have proposed an improvement in the clamping force, by proffering a semiactive method for the double beam [14]; others have designed a new test bench in order to measure the damping induced by partial slip and friction in a planar joint [15] or have introduced new friction dampers [16, 17] to limit structure damage under seismic conditions.
With regard to experiments on a bolted beam, no one has excited this structure above its first mode according to the literature search we consulted. Whereas the authors of [6] applied snap back excitation on this structure with a fixed weight on it free end for exclusively one clamping force, the authors of [8] went further in experiments when they studied the dynamic behavior of this structure in the first bending mode only.
The main objective of this paper is to study the dynamic behavior of a bolted beam in the presence of friction taking into account the shear effect and the rotatory inertia. This makes the proposed approach more accurate than the previous approximations for high-frequency excitation, thick beams, and wavelength approaches to the thickness of the beam. Therefore, the analytical model is established in stick and slip cases as well as the transition condition from stick to slip motion. In addition, the prediction of the vibration envelope will be done as a function of the bending slope. We have also set up experiments to validate the proposed model
by testing the specimens in the four first modes for different clamping forces. What could also be relevant is that determining the dynamic behavior of the bolted beam in the higher modes provides effective protection of structures against failure.
2. Differential equation of motion
Consider two prismatic beams assembled with four bolts. The boundary conditions of beams are clamp-free. Assuming that the section of the beams is constant, the two beams are compatible, i. e., they have the same transverse displacement and bending slope. Suppose that each beam here is a Timoshenko beam. The relative beams displacement u in movement can be expressed as
u = u2 — ui + Tha(x,t), (2.1)
where ui is the longitudinal displacement of the neutral fiber of the beam i, i = 1, 2, Th is the distance between the neutral axes of the beams, a(x, t) is the bending slope, x is the spatial variable according to the length of the beam, t is time.
The deformation due to the normal force is given by
P-
i EiS^
where Pi is the axial force, Ei is Young's modulus, Si is the cross section. Deriving (2.1) with respect to x and using (2.2), we get
du P2 Pi „ da
(2.2)
dx E2S2 E1S1 ' hdx
The element dx of the bolted beam Fig. 1, where Mi is the bending moment, Vi is the shear force, P is the axial force acting on the bolted beam,
P = Pl + P2, (2.4)
f (x) is the linear friction force, q(x) is the linear lateral force due to bolt tension, r(x) is the interface linear reaction force, hi is the distance between the neutral fibers of beam i, $i is the rotational inertia:
d 2a
<t>i = (2-5)
where pi is the density of material, Ii is the second moment of area of the cross section with respect to the bending axis.
The differential equation of motion is determined using the free body diagram of an infinitesimal element (Figs. 1 and 2).
The equilibrium equation gives Separating the two beams:
dP
/(,) = (2.6)
dVi „ d2w dx
where w(x, t) is the transverse displacement.
Neutral fiber 1
Structure's neutral fiber
Neutral fiber 2
M + dM
M1 + dM1
P1+dP1 A -A
P+dP
M2 + dM2
P2+dP2
V + dV
dx
Fig. 1. Free body diagrams of the dx element of the system, bolted beam
V\ can be written according to Timoshenko's beam theory as
Vi = K'íGiSi - a(x, t)
(2.9)
where Ki is the shear correction factor which depends on the shape of the cross section, Gi is the shear modulus.
Combining (2.8) and (2.9), we obtain
cfiw da\ _ cßw
(2.10)
The bending moment can be expressed in terms of bending deflection, bending slope and friction force as
( dw \ d2 a dM-
KiGiSi ( — " ) " PJi-QjJ ~ WW = "XT- i2-11)
dx
On the other hand, the moment Mi is given by
dx
M. = —EI —
dx
(2.12)
M1
q{x)
M9
+
m
Mt + dM1
r(x)
q(x)
M2+ +dM2
dx
Fig. 2. Free body diagrams of the dx element of the system, separated elements
The total bending moment M is calculated with respect to the neutral axis of the structure (Fig. 1) as follows:
M = M! + M2 + P(Th - hi) - P^h. (2.13)
Deriving (2.13) and using (2.12) and (2.6) and (2.7), the friction force can be expressed in terms of bending slope and total bending moment:
The Coulomb friction model is used here (Fig. 3):
f(x) = Ru + nq(x) sign ( — j , (2.15)
where R is the contact stiffness, it tends to infinity in the case of stick and to zero in the case of f 0,
slip: R = < n is the friction coefficient.
I
m' u(x)
Fig. 3. Coulomb friction model
Depending on the movement phase, the friction force will take different values, i.e., it will be equal to Ru in stick phase and to ¡j,q(x) sign in slip phase. Using (2.3), (2.6), (2.7) and (2.15), we get
d2f ( f(x) f(x) d2a dx2 \E2S2 ElSl hdx2
(2.16)
Using (2.10) and combining (2.11), (2.14) and (2.16) with the supposition of two identical beams, the equation of motion becomes Stick phase:
'd 2w da\ d2 w
— — I = 2po-
ill-
(2.17)
2kGS
dx2
dw
2k,GS ( t)J ~2pl m2
dx J d2a
dt2 '
T2ES + AEI\ d2a
2
dx2
Slip phase:
dx2 dx J dt2
dw \ d2 a d2 a
(2.18)
Galerkin's method is used in (2.17) and (2.18) separately to transform the system of partial differential equations into a finite number of ordinary differential equations, each of which has only one degree of freedom [18, 19]. The spatial modes are determined as shown in [20].
The transfer from stick to slip motion occurs once the friction force exceeds the static friction force:
f (x) > W(x). (2.19)
From (2.13), (2.14) and (2.15) the friction force can be expressed as a function of bending slope as follows:
f(r)__ (2 20) f[X)- (E2S2 + ElSl)dx2- (2"20)
Therefore, the transfer from stick to slip will appear when (2.21) is checked, for each clamping force:
ThES d2 a
**> * ~%rw (2-21)
2.1. Energy balance model
Bending slope and transverse displacement can be expressed as
a(x, t) = A(x)Xt (2.22)
w(x, t) = W(x)Xt. (2.23)
The energy lost between two successive instants ti and ti+l by friction effect can be written as follows:
i
Eloss = -J uf (x) dx, (2.24)
0
where l is the length of the beam. Replacing u from(2.1), we obtain
i ti+i
Eloss Th J J dx dt. (2.25)
0t
Finally, the energy lost by the friction force between ti and ti+l is
Eloss = -Thf (x)IA(Xti+i - Xh), (2.26)
where
i
IA = J A(x) dx. (2.27)
0
Xt is the displacement time function and it is estimated in each half period since the kinetic energy will be zero in this case, i.e., Xt represents the envelope points in free vibration. On the other hand, the potential energy for the structure is
e>=/ Hs2+kgs(-q(-t' j>+^ <2-28) 0
The potential energy can be given by
Ep = X2 EE, (2.29)
where
7
2 (—)
dx J "" \dx J
0
The conservation of energy between the successive instants ti and ti+l gives
f(x)ThIA
X, =-jy ' h--xt . 2.31
*i+1 EE ** y '
EE = j («GS (_A(x) + d-§)2 + E, (f )2] d,. (2.30)
3. Experimental setup
Fig. 4. Experimental setup
Fig. 5. The bolted beam with the shaker
The experimental setup consists of a vibration exciter Bruel and Kaejer type 4809; Bruel and Kaejer accelerometer type 4507 B 001, serial number: 31057 with sensitivity: 0.9853 mV/(m • s2) in order to measure acceleration; Bruel and Kaejer power amplifier type 2706. Acquisition card: Bruel and Kjaer LAN I/F 3560C 4CH 25 kHz Sound Vibration Acoustic FFT Analyzer, Software used for data acquisition: Pulse Lab Shop 21. The experiments have been carried out at the same ambient temperature. Its average value is 20°C.
Shaker
Excitation Power
amplifier
Pulse software
/
Cantilever bolted beam
Accelerometer
Acqusition card B&K
Response
Fig. 6. Schematic of the experimental setup
Fig. 7. Block diagram of the process
The tested beams are galvanized beams with the dimensions: 0.368 m x 0.02 m x 0.008 m. Young modulus: E = 235 GPa. Poisson coefficient: v = 0.3, friction coefficient: i = 0.4, density: p = 7800 kg/m3. These beams are identical with some manufacturing uncertainties.
Five different configuration cases are tested: single beam, bolted beam with zero clamping force (the glue is used between the screw and the nut to keep the two beams together during the excitation), bolted beam with maximum clamping force (an open-ended wrench is used so as to completely lock the assembly), bolted beam with clamping force of 5 N/bolt, bolted beam with clamping force of 11 N/bolt. Control of the clamping force by springs of a known stiffness.
Excitation characteristics:
1. Excitation type: sinusoidal excitation.
2. Excitation frequency interval: 3-200 Hertz.
3. Duration in excitation: 40 seconds.
4. Sampling frequency: 4096 Hertz.
5. Recording frequency: 1600 Hertz.
4. Results and discussion
4.1. Forced vibration analysis
For the cases of zero clamping force, clamping forces of 5 N/bolt and 11 N/bolt are used. The band [3200 Hz] allows one to get three natural frequencies, while for the case of the maximum clamping force we need to sweep the interval [3400 Hz] to get the first four natural frequencies.
fl o ■ 1—1 400
0) G O '—- 200
M 0
pi o 400
-s SH (1) CN Cfi 200
CI) ti
CJ
<1 0
0 20 40 60 80 100 120 140 160 180 200 Frequency (Hz)
(b)
d o ■ i-H 400
® > O) H o -—- 200
0
0 20 40 60 80 100 120 140 160 180 200 Frequency (Hz)
_(c)_
0 20 40 60 80 100 120 140 160 180 200 Frequency (Hz)
Fig. 8. Acceleration as a function of frequencies: (a) bolted beam with zero clamping force, (b) bolted beam with clamping force 5 N/bolt, (c) bolted beam with clamping force 11 N/bolt
Frequency (Hz)
Fig. 9. Bolted beam with the maximum clamping force
Figures 8 and 9 show how changes in the clamping force affect the stiffness of the bolted beam and therefore its natural frequencies and modal damping ratio. The complete slip motion will take place in the case of zero clamping force since the bolted beam has twice the stiffness and twice the mass of a single beam. Once the bolts have started to be tightened, both stick and slip motion will occur, whereas in case of maximum frictional force the structure will vibrate in stick mode since the structure has quadrature stiffness and the double mass of a single beam.
Two unexpected peaks are shown in the bolted beam with zero clamping force in the vicinity of (120 Hertz) and the maximum clamping force (250 Hertz) because of the excitation of the torsional modes with the shaker.
Table 1. Average natural frequencies F.i for bending with standard deviation ai obtained from the tests
Test cases F1 (Hz) a! (Hz) F2 (Hz) <t2 (Hz) F3 (Hz) a3 (Hz) F4 (Hz) a4 (Hz)
Single beam 5.0000 0.0000 31.5000 0.7071 91.5000 0.7071 180.5000 3.5355
Beam with zero clamping force 6.0000 0.0000 33.6667 0.5774 89.6667 0.5774 181.0000 4.5826
Beam with clamping force 5 N/bolt 5.6667 0.5774 34.3333 0.5774 93.0000 1.0000 190.6667 3.0551
Beam with clamping force 11 N/bolt 8.0000 0.0000 35.5000 1.3229 95.3333 3.0551 190.8333 4.9329
Beam with max clamping force 11.0000 0.0000 60.0000 1.0000 159.3333 0.5774 338.6667 2.8868
Each frequency (Table 1) is obtained from three successive tests. It is clear that the natural frequencies of the bolted beam with zero clamping force are close to those of the single beam. On the other hand, the natural frequencies of the structure with the maximum clamping force are close to those of the double thickness. The standard deviation is relatively small for the first three natural frequencies. But this rate increases from the fourth natural frequency.
4.2. Free vibration analysis
To analyze the bolted beam vibration in the presence of friction, we consider the case of free vibration.
Fig. 10. Free vibration response including the stick-slip phase and the stick phase
The bolted beam vibrates in the stick-slip regime until it disappears, and then the stick regime continues until the movement stops.
The logarithmic decrement method is used to calculate the modal damping ratio in order to study the amplitude as a function of time, the transition between the stick and slip phases and to simplify data processing.
Table 2. Average damping ratio £i with standard deviation ai for each test case
Test case % % ¿2, % aii % ¿3, % <73,% % <T4, %
Single beam 1.7200 0.0017 2.1950 0.0127 2.2640 0.0020 6.0000 0.0212
Beam with zero clamping force 3.9400 0.0023 3.1200 0.0036 6.4300 0.0159 6.2000 0.0194
Beam with clamping force 5 N/bolt 14.3600 0.0233 3.2400 0.0018 5.6700 0.0258 3.9400 0.0042
Beam with clamping force 11 N/bolt 2.7600 0.0047 5.1000 0.0050 10.2800 0.0075 8.1900 0.0426
Beam with max clamping force 1.2900 5.5076e-04 3.7200 0.0071 22.3800 0.0233 3.6567 0.8868
Each damping ratio is obtained from three successive tests (Table 2). We notice an increase in the modal damping ratio for the bolted beam compared to that of the single beam, except the case beam with the maximum clamping force for its first natural frequency.
Each clamping force shows an increase in the modal damping ratio from the first to the third natural frequency.
0.05 0.1 Time (s)
0.15
0.05 0.1 Time (s)
0.15
Single beam Max clamping force #Zero clamping force ■Load 5 N/bolt Load 11 N/bolt
0.005 0.01 Time (s)
0.015
Fig. 11. Effect of the clamping force on the damping ratio for each case of the test: (a) the first natural frequency, (b) the second natural frequency, (c) the third natural frequency
Figure 11 shows that, when the bolts are tightened, the maximum displacement decreases due to the dissipation of energy under the action of friction.
4.3. Comparison between theory and experiments
The proposed model gives the natural frequencies for the case: beam with zero clamping force and beam with the maximum clamping force.
Table 3. Comparison between experimental and theoretical frequencies
Frequency order. Beam with zero clamping force Beam with maximum clamping force
Theoretical results (Hz) Experimental results (Hz) Relative error, % Theoretical results (Hz) Experimental results (Hz) Relative error, %
1 5.2379 6.0000 14.5497 10.4757 11.0000 5.0049
2 32.8248 33.6667 2.5648 65.64589 60.0000 8.6003
3 91.9072 89.6667 2.4377 183.7892 159.3333 13.3064
4 180.0922 181.0000 0.5015 360.0931 338.6667 5.9502
Table 3 represents a comparison between theoretical frequencies according to Timoshenko's theory and experimental frequencies. All frequencies have a relative error less than 10 %, except two cases, which demonstrates a good agreement between theory and experiments.
0.2 0.3 0.4 Time (s)
ö
CP
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Time (s)
Experimental results ......Theoretical results
Time (s) Time (s)
Fig. 12. The dynamics of the bolted beam depending on the applied clamping force for the second natural frequency: (a) zero clamping force, (b) 5 N/bolt clamping force, (c) 11 N/bolt clamping force, (d) maximum clamping force
Figure 12 depicts a good agreement between theory and experiments. On the other hand, in the measured displacement curve a certain exponential decay can be observed, this is due to other phenomena that are not taken into account such as: bolt and sensor weight, friction between bolts and beam, etc.
5. Conclusion
This work is focused on the characterization of friction-induced vibrations between two simple bolted beams taking into account the effect of rotatory inertia and shear deformation. These then affect the free vibration amplitude and the transition from stick to slip motion.
The experimental analyses are presented in order to discuss the role of the clamping force in the friction-induced vibrations. The bolted beam vibrates in slip motion for zero clamping force, in stick motion for the maximum clamping force and in stick-slip motion in the intermediate stage.
The vibration amplitude has changed according to different parameters such as the clamping force and the excitation frequency.
The comparison between experiments and theory is made at low frequency order and shows a good agreement. There is a significant increase in the modal damping ratio from the first to the third mode, which generates considerable energy dissipation. Besides, its low cost makes the bolted beam a good choice as a friction damper.
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