mme.modares.ac.ir
115
[email protected]: 29 1394 : 25 1394 : 19 1394
.
.
.
.
.
.
.
.
.
.
Free vibration of micro rotating Euler-Bernoulli beams based on the strain
gradient theory
Hadi Arvin Boroujeni
Department of Mechanical Engineering, Shahrekord University, Shahrekord, Iran P.O.B. 115 Shahrekord, Iran, [email protected]
A
RTICLE
I
NFORMATION
A
BSTRACT
Original Research Paper Received 20 November 2015 Accepted 15 January 2016 Available Online 08 February 2016
In this paper, free vibration of rotating microbeams based on the strain gradient theory and Euler-Bernoulli beam assumptions is investigated. The Hamilton's Principle is applied on the attained strain and kinetic energy relations to obtain the equations of motion for the rotating microbeam. Then, by employing the adimensional parameters, the nondimensional form of the equations of motion is derived. After that, by applying the Galerkin approach on the dynamic equations of motion, the flapping and axial natural frequencies are calculated. Subsequently, the current results are validated by the existed papers results. After validation of the present results, the effects of the thickness to the material length scale parameter ratio, rotation speed and Poisson's coefficient on the flapping and axial frequencies are studied and the strain gradient theory results are compared with the modified couple stress and classical theories. The results show that the type of theory that is appointed has essential effects on the predicted natural frequencies. The effect of rotation speed on the possibility of the occurrence of internal resonances is also examined. In addition, for the first time, the effect of different mentioned theories on the axial natural frequencies are inspected. The presented results illustrate that, by considering the strain gradient theory, varying the Poisson's coefficient changes the axial frequencies, while the modified couple stress and classical theories are ineffectual in predicting any variations on the axial frequencies and the mentioned theories predict the same results for axial frequencies.
Keywords:
Micro Rotating Beams Euler-Bernoulli Beams Strain Gradient Theory Poisson's Coefficient
1
-
.
]
.
1
[
121
.
]
.
2
[
.
]
.
3
[
.
]
.
4
[
]
.
5
[
.
.
]
6
[
.
.
]
7
[
)
Pull-in
(
)
(
.
.
]
8
[
.
.
]
.
9
[
.
]
10
.
[
.
]
11
[
.
.
]
.
12
[
.
-]
.
13
[
.
]
.
14
[
.
.
]
15
[
.
.
]
16
[
.
]
17 18
[
.
MIT
]
19
]
[
20
[
]
21
[
.
.
]
22
.
[
]
23
[
.
-]
24
[
.
-]
25
[
]
26
[
]
.
27
[
3
.
.
-]
28
[
]
29
[
.
]
.
30
[
.
]
.
31
[
.
.
.
.
.
2
-
2
-1
-
1
)
.
( )
)
(
( )
(
.
-
)
1
(
]
29
:[
)
1
(
= ( , )
( , ),
= ( , )
Fig. 1 A schematic of a rotating microbeam [29]
1
]
29
[
)
2
(
]
6
[:
)
2
(
=
+
+
( ) ( )+
, , = 1,2,3
))
3
(
]
6
[:
)
3
-=
1
2
,+
,)
3
-=
,)
3
-( )
=
,+
,+
, ,+
2
+
+ 2
+
,+ 2
,)
3
-=
1
2
,+
,)
4
(
)
4
(
=
1
2 ×
)
2
(
))
5
(
:
)
5
-=
+
)
5
-=
)
5
-( )
=
( ))
5
-=
)
1
(
)
3
)
6
(
=
·
( )
.
-)
6
(
)
7
(
]
29
:[
123
)
7
(
=
·
+
1
2
)
1
(
)
4
(
)
8
(
=
)
7
)
(
3
-) (
3
(
)
8
(
)
3
(
)
5
(
.
)
2
(
)
9
(
=
+
+
+
+
+
=
=
2 +
+
= 2
+
= 2
+
= + 2
.
)
10
(
]
29
:[
)
10
(
=
+
+
( + + )
.
)
10
(
-)
11
(
=
+
+
+
+
( + + )
(
11
)
=
=
)
9
) (
11
(
]
32
[
)
12
(
(
+
)
= 0
)
13
(
+
=
)
14
(
+
+
( + + ) =
= = = 0
-]
29
[
.
-]
6
[
= , = , = , = ,
= · ,
=
,
=
,
=
, =
, r =
(
15
)
=
.
= ( ) + ( , )
=
( , )
.
"
^
"
( , )
( )
)
16
) (
17
(
)
16
(
+
=
)
17
(
+
+
=
"
"
2
-2
-
.
)
18
(
]
32
:[
)
18
(
=
( ) ( ) , =
( ) ( )
.
)
18
)
(
16
) (
17
(
)
19
(
· + · = 0
=
.
( , ) =
,
( , ) =
+
, = 1, … , , = 1, … ,
-
20
)
( , ) =
,
( , ) =
+
+
, = 1 + , … ,2 , , = 1 + , … ,2
-
20
)
3
-
3
-1
-
.
.
]
30
[
.
]
6
[
.
= = 0
]
28
[
]
30
[
GPa
200
GPa
69
.
rpm
1000
2
0.05
0.005
0.1
.
1
.
.
]
30
[
]
30
[
]
30
.[
.
]
6
[
.
2
.
= = =
= 2
= 20
3
m20
m
50
.
]
6
[
]
.
28
[
.
GPa1.4
= 0.3
= 1220
= 17.6
m1
]
30
[
( )Table 1 Current first three flapping and axial mode frequencies in
comparison with those of [30] ( )
]
30
[
114.51 114.46 280.41 280.34 462.06 461.93 3997.68 4003.17 11996.7 11998.53 19994.98 19996.082
]
6
[
Table 2 Material and geometric properties of micro cantilever beam
made of epoxy [6]
l
(µm) ( ) E (GPa) 17.6 0.38 1000 1.443
]
6
[
Table 3 Current first three flapping mode frequencies in comparison
with those of [6] ( )
)
m(
]
6
[
20 546.76 535.13 2.13 3425.15 3353.7 2.09 9581.86 9419.8 1.69 50 103.77 102.99 0.76 649.88 638.89 1.69 1816.74 1803.4 0.73= = 0
=
= 1
= 2
= 20
= 4
r = 0
4
.
.
.
]
28
[
3
5
3
-2
-
.
2
= 1220
= 2
= 20
=
0.1086
2
3
4
.
= = 0
= = = 0
.
.
.
125
4
]
28
[
Table 4 Current first two adimensional flapping mode frequencies in
comparison with those of [28]
]
28
[
9.417 9.377 0.42 52.815 52.257 1.06=
5
6
7
8
9
.
1
.
Fig. 2 Effect of the thickness to the material length scale ratio on the
first flapping frequency of the rotating microbeam
2
Fig. 3 Effect of the thickness to the material length scale ratio on the
second flapping frequency of the rotating microbeam
3 Fig. 4 Effect of the thickness to the material length scale ratio on the
third flapping frequency of the rotating microbeam
4
Fig. 5 Effect of the thickness to the material length scale ratio on the
ratio of the first flapping frequency predicted by strain gradient theory and modified couple stress theory to that calculated based on the classical theory of the rotating microbeam
5
Fig. 6 Effect of the thickness to the material length scale ratio on the
ratio of the second flapping frequency predicted by strain gradient theory and modified couple stress theory to that calculated based on the classical theory of the rotating microbeam
6
Fig. 7 Effect of the thickness to the material length scale ratio on the
ratio of the third flapping frequency predicted by strain gradient theory and modified couple stress theory to that calculated based on the classical theory of the rotating microbeam
7
.
.
)
]
30
.(
[
.
Fig. 8 Effect of non-dimensional rotation speed of the microbeam on
the first flapping frequency
8
10
.
3
1
1
1
.
.
.
= 0.01, 0.5, 1
11
.
.
.
)
= 0
= 0.5
.(
.
.
= 0.01, 0.5, 1
12
.
Fig. 9 Effect of non-dimensional rotation speed of the microbeam on
the first axial frequency
9
127
Fig. 10 Possibility of 1:1 (between the second flapping and the first
axial modes) and 3:1 (between the first flapping and the first axial or second flapping modes) internal resonances for the rotating microbeam predicted by, respectively, the strain gradient, modified couple and classical theories
10
1:1
)
1:3
(
)
.
.
Fig. 11 The Poisson's effect on the first flapping frequency of the
rotating microbeam
11
Fig. 12 The Poisson's effect on the ratio of the first axial frequency
predicted by strain gradient theory and modified couple stress theory to that calculated based on the classical theory of the rotating microbeam
12
4
-
.
.
.
.
1
.
2
-3
.
4
-
3
1
1
1
.
.
5
.
.
.
5
-
[1] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science, Vol. 46, No. 5, pp. 427–437, 2008.
[2] L. L. Ke, Y. S. Wang, Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory, Composite Structures, Vol. 93, No. 2, pp. 342–350, 2011.
[3] A. R. Alashti, A. H. Abolghasemi, A size-dependent Bernoulli-Euler beam formulation based on a new model of couple stress theory, International
Journal of Engineering TRANSACTIONS C, Vol. 27, No. 6, pp. 951–960,
2014.
[4] A. M. Dehrouyeh-Semnani, M. Nikkhah-Bahrami, A discussion on incorporating the Poisson effect in microbeam models based on modified couple stress theory, International Journal of Engineering Science, Vol. 86, No. 1, pp. 20–25, 2015.
[5] A. Ghanbari, A. Babaei, The new boundary condition effect on the free vibration analysis of micro-beams based on the modified couple stress theory,
International Research Journal of Applied and Basic Sciences, Vol. 9, No. 3,
pp. 274-279, 2015.
[6] S. Kong, S. Zhou, Z. Nie, K. Wang, Static and dynamic analysis of micro beams based on strain gradient elasticity theory, International Journal of
Engineering Science, Vol. 47, No. 4, pp. 487-498, 2009.
[7] Y. Tadi Beni, I. Karimipour, Static pull-in instability analysis of beam type NEMS under molecular force using strain gradient theory, Modares
Mechanical Engineering, Vol. 12, No. 3, pp. 37-49, 2012 (in Persian ). [8] R. Ansari, R. Gholami, S. Sahmani, Free vibration analysis of size-dependent functionally graded microbeams based on the strain gradient Timoshenko beam theory, Composite Structures, Vol. 94, No. 1, pp. 221-228, 2011. [9] M. Asghari, M. H. Kahrobaiyan, M. Nikfar, M. T. Ahmadian, A
size-dependent nonlinear Timoshenko microbeam model based on the strain gradient theory, Acta Mechanica, Vol. 223, No. 6, pp. 1233-1249, 2012. [10] M. H. Ghayesh, M. Amabili, H. Farokhi, Nonlinear forced vibrations of a
microbeam based on the strain gradient elasticity theory, International
Journal of Engineering Science, Vol. 63, No. 1, pp. 52-60, 2013.
[11] R. Ansari, R. Gholami, S. Sahmani, Size-dependent vibration of functionally graded curved microbeams based on the modified strain gradient elasticity theory, Archive of Applied Mechanics, Vol. 83, No. 10, pp. 1439-1449, 2013. [12] H. Mohammadi, M. Mahzoon, Investigating thermal effects in nonlinear buckling analysis of micro beams using modified strain gradient theory, IJST,
Transactions of Mechanical Engineering, Vol. 38, No. M2, pp. 303-320,
2014.
[13] R. Ansari, R. Gholami, S. Sahmani, Free vibration of size-dependent functionally graded microbeams based on the strain gradient reddy beam theory, International Journal for Computational Methods in Engineering
Science and Mechanics, Vol. 15, No. 5, pp. 401-412, 2014.
[14] D. Karli , P. Kozi , R. Pavlovi , K. Mazaheri, Flexural vibration and
buckling analysis of single-walled carbon nanotubes using different gradient elasticity theories based on Reddy and Huu-Tai formulations, Journal of
Theoretical and Applied Mechanics, Vol. 53, No. 1, pp. 217-233, 2015.
[15] I. Karimipour, A.R. Karimipour,Y. Tadi Beni, Determination of size-dependent non-linear pull-in instability of clamped nano-Beam based on the modified strain gradient theory using HPM, Modares Mechanical
Engineering, Vol. 15, No. 2, pp. 101-112, 2015 (in Persian ). [16] A. Mehra, X. Zhang, A. A. Ayón, I. A. Waitz, M. A. Schmidt, C. M.
Spadaccini, A Six-Wafer Combustion System for a Silicon Micro Gas Turbine Engine, Journal of Microelectromechanical Systems, Vol. 9, No. 4, pp. 517-527, 2000.
[17] W. Y. Sim, G. Y. Kim, S. S. Yang, Fabrication of micro power source (MPS) using a micro direct methanol fuel cell (DMFC) for the medical application,
Proceedings of the 14th IEEE International Conference on Micro Electro Mechanical Systems, Interlaken, Switzerland, January 21–25, 2001.
[18] J. Peirs, D. Reynaerts, F. Verplaetsen, A microturbine for electric power generation, Sensors and Actuators A: Physical, Vol. 113, No. 1, pp. 86–93, 2004.
[19] A. H. Epstein, S. A. Jacobson, J. M. Protz, L.G. Fréchette, Shirtbuttonsized gas turbines: the engineering challenges of micro high speed rotating machinery, Proceedings of the 8th International Symposium on Transport
Phenomena and Dynamics of Rotating Machinery, Honolulu, Hawaii, March
26–30, 2000.
[20] S. Kang, S.-J.J. Lee, F.B. Prinz, Size does matter, the pros and cons of miniaturization, ASEA Brown Boveri Review, Vol. 2, No. 2, pp. 54–62, 2001. [21] E. Matsuo, H. Yoshiki, T. Nagashima, C. Kato, Development of Ultra Gas
Turbines, Technical Digest Power MEMS 2002, International Workshop on
Power MEMS, Tsukuba, Japan, November 12–13 2002.
[22] S. C. Pradhan, T. Murmu, Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever, Physica E:
Lowdimensional Systems and Nanostructures, Vol. 42, No. 7, pp. 1944–1949,
2010.
[23] S. Narendar, Differential quadrature based nonlocal flapwise bending vibration analysis of rotating nanotube with consideration of transverse shear deformation and rotary inertia, Applied Mathematics and Computation, Vol. 219, No. 3, pp. 1232–1243, 2012.
[24] J. Aranda-Ruiz, J. Loya, J. Fernández-Sáez, Bending vibrations of rotating nonuniform nanocantilevers using the Eringen nonlocal elasticity theory,
Composite Structures, Vol. 94, No. 9, pp. 2990–3001, 2012.
[25] M. A. Torkaman-Asadi, M. Rahmanian, R.D. Firouz-Abadi, Free vibrations and stability of high-speed rotating carbon nanotubes partially resting on Winkler foundations, Composite Structures, Vol. 126, No. 1, pp. 52-61, 2015.
[26] S. Dutta, V. Giurgiutiu, Health monitoring and quality assurance for rotary micromachines and active sensors, 8th International Symposium on
Transport Phenomena and Dynamics of Rotating Machinery, Honolulu,
Hawaii, March 26-30 2000.
[27] M. Hashemi, M. Asghari, Investigation of the small-scale effects on the three-dimensional flexural vibration characteristics of a basic model for micro-engines, Acta Mechanica, Vol. 226, No. 9, pp. 3085–3096, 2015. [28] A. M. Dehrouyeh-Semnani, The influence of size effect on flapwise
vibration of rotating microbeams, International Journal of Engineering
Science, Vol. 94, No. 1, pp. 150-163, 2015.
[29] H. Arvin, F. Bakhtiar--Nejad, Nonlinear modal analysis of a rotating beam,
International Journal of Nonlinear Mechanics, Vol. 46, No. 6, pp. 877-897,
2011.
[30] W. Lacarbonara, H. Arvin, F. Bakhtiari-Nejad, A geometrically exact approach to the overall dynamics of elastic rotating blades - part 1: linear modal properties, Nonlinear Dynamics, Vol. 70, No. 1, pp. 659-675, 2012. [31] A. Najafi, M. R. Ghazavi, A. A. Jafari, Nonlinear analysis of resonance
between the blade and rotor by the bifurcation equation and numerical method, Modares Mechanical Engineering, Vol. 14, No. 12, pp. 43-51, 2014
(in Persian ).
[32] L. Meirovitch, Principles and Techniques of Vibrations, First Edittion, pp. 368-370, 400-405, 544-547, New Jersey: Prentice-Hall, 1997.
