International Journal on Theoretical and Applied Research in Mechanical Engineering (IJTARME)
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Enhancement of Damping in Layered Cantilever Beams Using FEA
1Biswa Ranjan Samar Ballav Rout, 2R. C. Mohanty
1M. Tech Research Scholar, Dept. of Mechanical Engineering, Centurion University, Bhubaneswar campus, Odisha, India
2 Associate Professor, Dept. of Mechanical Engineering, Centurion University, Bhubaneswar Campus, Bhubaneswar, Odisha, India
Email: 1[email protected],2[email protected] Abstract : A finite element approach based on the Euler-
Bernoulli beam theory has been presented to study the damping mechanism in jointed and layered cantilever structures. The solution considers one-dimensional beam elements with each one consisting of two nodes having two degrees of freedom, i.e. transverse displacement and rotation at each node. It is found from the present investigation that the damping capacity of such structures is influenced by a number of vital parameters such as;
pressure distribution, micro-slip at the interfaces, amplitude, length and thickness of the specimen. Finite element model developed can be utilized effectively in the design of machine tools, automobiles, aerodynamic and space structures, frames and machine members for enhancing their damping capacity.
I. INTRODUCTION
Over the years, the researchers have emphasized their studies on the development of mathematical models for the mechanism of damping along with techniques adopted to improve the damping capacity of layered structures for controlling the adverse effects of vibrations. The characterization and modeling of the dynamic behavior of many built-up structures under vibration conditions is still a subject of current research.
The contribution of damping due to joints to such structures depends largely upon its behavior under dynamic conditions. Gropper [1] and Menq et al. [2]
have shown that joints usually exhibit two types of motions; micro-slip and macro-slip. The micro-slip usually occurs with small relative displacement between the contacting members resulting in energy losses at the
joints causing the damping of the structures. Beards [3]
has performed a series of experiments on the damping of the joints to control the vibration of a structure and established that the damping in joints is much larger compared to the material damping and there exists an optimum clamping force at which the energy dissipation is maximum. Shin et al. [4] have shown that the damping effect is negligible when the joint is very tight.
They further established that the natural frequency decreases and the damping capacity increases with the loosening of the joint. Damisa et al. [5] have studied the effect of non-uniform interface pressure distribution on the mechanism of slip damping for layered beams.
However, their analysis is limited to static load and linear pressure profile. Further, Olunloyo et al. [6] have also studied the damping mechanism with polynomial and hyperbolic forms of interfacial pressure distribution.
Gaul and Lenz [7] have worked in details on the finite element models of different slip mechanisms to study the dynamic response of assembled structures incorporating both the micro- and macro-slip of several bolted/riveted joints. Chen and Deng [8] have proposed a finite element method for understanding and characterizing the non-linear damping behavior of structural joints. Hartwigsen et al. [9] have found out the contact area of bolted joint interfaces using FEM analysis and they further conducted the experiments to verify the same. Sainsbury and Zhang [10] have used finite element analysis through Galerkin element method (GEM) to study the dynamic analysis of damped sandwich beam structures.
II. DIFFERENTIAL EQUATION OF FREE TRANSVERSE VIBRATION OF AN EULER- BERNOULLI BEAM
Fig. 1 Differential analysis of a beam
Most physical phenomena encountered in engineering applications are modeled by differential equations.
Neglecting the effect of axial force acting on the beam and the shear deflection, the equation of beam as shown in Fig. 1 due to transverse vibration is given by;
22dt Adx y x dx
V V
V
2
2
0
V w
x A t
(1)Considering
x V M
, the Eq. (1) is simplified as;2 2
2 2
M w
x A t
(2)From the beam theory, the bending moment
2 2
M EI w x
, and hence the above Eq. (2) is rewritten as;4 2
4 2
EI w w
A x t
4 2
2
4 2
w w
c x t
(3)where ρ,
A
, L, I and E are the usual beam notations and c EI
A
III. FORMULATION USING FINITE ELEMENT METHOD
Fig. 2 Finite element model for the layered and jointed beam
The layered beam is considered to be made up of a number of linear elements of equal size connected at their nodal points. Each element is defined by two nodes and has two degrees of freedom, i.e., transverse displacement and slope at each node. The geometry of an element is shown in Fig. 2 displaying the nodes and
degrees of freedom. Assuming that every layer has the same transverse displacement and deformation, the total set of nodal displacements of an element of the beam is given by;
1 1 2 2
e T
d w w
(4)The transverse deflection of the beam (w) in terms of nodal variables at any section of the element is obtained by considering the Hermitian shape functions as given by;
1 1 2 1 3 2 4 2
( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) w x t S x w t S x t S x w t S x t
(5)where S1, S2, S3, and S4 are the Hermitian shape functions.
Further, the above equation is expressed in the matrix form as;
( , )
ew x t S d
(6)where
S S1 S2 S3 S4
The stiffness matrix of the beam element is found as;
3 2 20
2 2
12 6 12 6
6 4 6 2
12 6 12 6
6 2 6 4
l e T
l l
l l l l
k B EI B dx EI
l l
l
l l l l
(7)where
B d S
22dx
andEI E I
1 1 E I
2 2Similarly, the mass matrix of the element is found as;
2 20
2 2
156 22 54 13
22 4 13 3
54 13 156 22
420
13 3 22 4
l e T
l l
l l l l
m A S S dx Al
l l
l l l l
(8)The individual element mass and stiffness matrices are assembled in the usual way into the global mass matrix [M] and stiffness matrix [K]. Consequently, the free vibration of the beam in the matrix form is given by;
M D K D 0
(9)where D and D are the global acceleration vector and displacement vector respectively.
IV. ENERGY DISSIPATION AND LOGARITHMIC DECREMENT
The nature of the interface pressure profile across the beam layer and its magnitude is to be assessed correctly for determining the damping capacity of a jointed structure. The contact is developed at the interfaces of such structures due to riveting and a separation also occurs beyond a certain distance as shown in Fig. 3.
Minakuchi et al. [11] have found that the interface
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pressure distribution due to this contact is parabolic with a circular influence zone circumscribing the rivet with diameter equal to 5.0 times the diameter of the connecting rivet.
Fig. 3 Free body diagram of a bolt showing the influence zone
The interface pressure under each bolt in a non- dimensional polynomial is given by;
10 8 6 4 2
1 2 3 4 5 6
S B B B B B
p R R R R R
C C C C C C
R R R R R
(10)
where p, σs, R and RB are the interface pressure, surface stress, any radius within the influencing zone and radius of the connecting bolt respectively. The numerical data of Minakuchi et al. [11] are used to evaluate the constants of the polynomial C1 to C6 with a MATLAB
software and are found to be;
1.398793E05,1.737236E070.458813E03, 0.817021E02, 0.877333E01 and 0.488330E00 respectively.
The above pressure distribution given by Eq. (10) is used to find out the total normal force at the interfaces under each connecting rivet as;
12 10 8
3
1 2
6 4 2
5 4
6
1 1 1
6 5 4
2.0625
1 1 1
3 2
M M M
B B B
M M M
B B B
C
C R C R R
R R R
N P
C
C R R R
C
R R R
(11)
where P is the preload in the bolt.
Fig. 4 Mechanism of dynamic slip
The micro-slip at the interfaces of the connecting members occurs under dynamic condition as shown in
Fig. 4 causing energy loss due to frictional force and this relative dynamic slip under a connecting bolt is given by;
1 2
1 2
( , )
( , ) e
r
w x t dS
u x t h h h h d
x dx
(12)
The overall maximum relative dynamic slip of a jointed cantilever beam is found out considering the effect of all the bolts in the jointed structure as;
1 2
rM sum
u h h X
(13)where
1 q
e sum
i
X dS d
dx
and “q” is the number of bolts.The energy loss due to kinematic coefficient of friction (µ) and dynamic slip at the interfaces per cycle of vibration is given by;
f
2
rME Nu
(14)The energy stored per cycle of vibration is given as;
1 2
T
E
n D K D
(15)The logarithmic decrement is the measure of the damping capacity of the structures and is evaluated from the ratio of the energy loss to the input strain energy per cycle of vibration as;
2
f n
E
E
(16)The logarithmic damping decrement is found out combining the Eqs. (14) and (15) with Eq. (16) as;
1
2
2
sumT
N h h X
D K D
(17)V. DISCUSSIONS OF RESULTS
Damping is one of the difficult issues to be dealt with structural dynamics. This analysis assumes the following:
Linear vibration theory is considered.
Both the layers undergo the same transverse deflection.
The local mass of the joint area is not considered as significant in altering the behavior of the beam.
The circular holes for inserting bolts on the test specimens are completely filled by bolts.
The clamped end of the specimen is rigidly fastened to the support.
The logarithmic damping decrements for layered mild steel cantilever beams with diameters 5, 6, 8, and 10 mm connecting bolts have been found out from equation (17) using FEM. The numerical results have been shown in Figs. 5 to 7. The damping of the jointed beams in the present work has been studied for the following variables: amplitude of vibration, thickness ratio, cantilever length and diameter of the bolt. The dependency of the damping on each of these variables is discussed in the following section:
Effect of b e a m length:- There is decrease in the static bending stiffness with an increase in the length of the beam so that the strain energy introduced into the system is decreased. As contact area of the interface increases with the beam length, more energy loss will occur due to overall dynamic slip. The net effect of all these improves the damping with an increase in the cantilever length. The dependence of damping on the beam length is shown in Fig. 5.
Fig. 5 Variation of logarithmic decrement with beam length
Effect of amplitude of vibration:- More input strain
energy to the system is resulted due to increase in the amplitude of vibration. But it is an established fact that the loss in vibration energy depends on the square of amplitude of vibration. It is found from the present investigation that the damping in a jointed beam is enhanced with the increase in the amplitude of vibration which is shown in Fig. 6.
Fig. 6 Variation of logarithmic decrement with the amplitude of excitation
Effect of b o l t diameter:- There will be increase in preload on the b o l t s b y u s i n g b o l t s o f l a r g e r d i a m e t e r , thereby increasing the normal force and the energy loss at the interfaces. Further, the static bending stiffness of the beam increases with the increase in bolt diameter, thereby increasing input strain energy into the system. But the energy dissipation occurs at a higher rate compared to the input energy, thereby producing an increase in damping.
This result is shown in Fig. 7.
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Fig. 7 Variation of logarithmic decrement with the diameter of rivet
VI. CONCLUSION
In the present investigation, the improvement in damping capacity of jointed beams has been presented applying FEM. It is established that the damping capacity of jointed beams can be improved using bolts of larger diameter with larger length and smaller overall thickness of the specimens along with higher amplitude of excitation. This concept can be effectively utilized in real engineering applications where higher damping capacity is desired.
REFERENCES
[1] Groper, M. (1985). “Microslip and macroslip in bolted joints”, Experimental Mechanics, June, pp.
171–174.
[2] Menq, C.H., Bielak, J. and Griffin, J.H. (1996).
"The influence of micro-slip on vibratory response, Part-I: A new micro-slip model”, Journal of Sound and Vibration, Vol. 107, No. 2, pp. 279-293.
[3] Beards, C.F. (1992). “Damping in structural joints”, The Shock and Vibration Digest, Vol. 24, pp. 3-7.
[4] Shin, Y.S., Iverson, J.C. and Kim, K.S. (1991).
“Experimental studies on damping characteristics of bolted joints for plates and shells”, Journal of Pressure Vessel Technology, Vol. 113, No.
August, pp. 402-408.
[5] Damisa, O., Olunloyo, V.O.S., Osheku, C.A. and Oyediran, A.A. (2008). “Dynamic analysis of slip damping in clamped layered beams with non- uniform pressure distribution at the interface”, Journal of Sound and Vibration, Vol. 309, pp.
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[6] Olunloyo, V.O.S., Damisa, O., Osheku, C.A. and Oyediran, A.A. (2007). “Further results on static analysis of slip damping with clamped laminated beams”, European Journal of scientific Research, Vol. 17, No. 4, pp. 491-508.
[7] Gaul, L. and Lenz, J. (1997). “Nonlinear dynamics of structures assembled by bolted joints”, Acta Mechanics, Vol. 125, No. 169-181.
[8] Chen, W. and Deng, X. (2005). “Structural damping caused by micro-slip along frictional interfaces”, International Journal of Mechanical Science, Vol. 47, pp. 1191-1211.
[9] Hartwigsen, C.J., Song, Y., Macfarland, D.M., Bergman, L.A. and Vakakis, L.A. (2004).
“Experimental study of non-linear effects in a typical shear lap joint confifuration”, Journal of Sound and Vibration, Vol. 277, pp. 327-351.
[10] Sainsbury, M.G. and Zhang, Q. J. (1999). “The Galerkin element method applied to the vibration of damped sandwich beams”, Computers &
Structures, Vol. 7, pp. 239-256.
[11] Minakuchi, Y., Koizumi, T. and Shibuya, T.
(1985). “Contact pressure measurement by means of ultrasonic waves using angle probes”, Bulletin of JSME, Vol. 28, No. 243, pp. 1859-1863.