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Flapwise Vibration of Rotating Functionally Graded Beam
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ScienceDirect Materials Today: Proceedings 4 (2017) 3736–3744
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Flapwise Vibration of Rotating Functionally Graded Beam P. Ravi Kumara *, K. Mohana Raob, N. Mohan Raoc a
Assistant Professor, Mech. Engg. Dept., PVP Siddhartha Institute of Technology, Vijayawada,INDIA b Professor, Mech. Engg. Dept., VR Siddhartha Engineering College., Vijayawada,INDIA c Professor, Mech. Engg. Dept., University College of Engg., JNTUK., Vizianagaram, INDIA
Abstract
Functionally Graded materials have been widely used in engineering applications due to the advantage of smooth and continuous variation in material properties, better fatigue life, less stress concentration, lower thermal stresses and attenuation of stress waves. By intelligently designing material constitutions and gradient distributions of FGMs, structures made of them can have excellent dynamic and thermodynamic characteristics. In the present study Flapwise vibration of rotating functionally graded beam is studied. The effective material properties are determined using Mori Tanaka Method. The natural frequencies are obtained using differential transform method (DTM). The effect of rotating speed, gradient index and hub radius on natural frequency are discussed. ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:FGM, FG beam, Free vibration, Mori Tanaka Method, Differential Transform Method
1. Introduction Functionally graded materials (FGM) are made by combining two or more materials and posses positiondependent microstructure and chemical composition resulting in smooth gradients in the properties like mechanical strength and thermal conductivity. In majority applications, material properties change through the thickness of the material. Due to smooth variation of properties, functionally graded materials possess a number advantages like
* Corresponding author. Tel.: +919885519328 E-mail address:[email protected] 2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
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reduction in stresses through the thickness, uniform residual stress distribution leads less stress concentration, high Large number of studies carried out by researchers on functionally graded beams. Aydogdu and Taskin [1] fracture toughness and high temperature resistance. To design rotating components like turbine blades, compressor blades, aircraft propeller blades and helicopter rotor blades properly their natural frequencies estimated accurately. carried out free vibration analysis of functionally graded beam with simply supported edges. Governing equations are found by Hamilton principle and natural frequencies are obtained by Navier type solution method. Kapuria et. al [2] validated a third order zigzag theory based model through experiments for static and free vibration response. Sina et. al. [3] carried out free vibration analysis of functionally graded beams using a new beam theory different from traditional first order shear deformation theory. Governing equations are derived using Hamilton principle and the natural frequencies are obtained by an exact method. Li Yang and Shi [4] extended state space based differential quadrature method to study free vibration of functionally graded piezoelectric beam with different boundary conditions. The influence of gradient index on fundamental frequency is discussed. Piovan and Sampaio [5] developed a rotating non linear beam model made of functionally graded material. Simsek [6] carried out fundamental frequency analysis of functionally graded beams having different boundary conditions. The governing frequency equation is obtained using Lagrange’s equation the effect of slenderness ratio, material variations and different beam theories on the fundamental frequencies are examined. Alshorbagy [7] performed free vibration analysis of a functionally graded beam by finite element method. The system of equations of motion is derived by using the principle of virtual work and the effects of different material distribution, slenderness ratios, and boundary conditions on the frequency are presented. Giunta et. al [8] investigated free vibration of functionally graded beams using various beam theories and presented the effect of different material distributions on natural frequencies. Anandrao et. al [9] carried out free vibration analysis of functionally graded beams with classical boundary conditions. Principle of virtual work is used to obtain finite element equations effect of transverse shear, length to thickness ratio and volume fraction exponent on natural frequency of functionally graded material demonstrated. Ramesh and Mohan Rao[10] derived equations of motion for a FG beam from a modelling method by employing the hybrid deformation variable, determined natural frequencies using Raleigh Ritz method and examined effect of various parameters through numerical studies. Sarmila Sahoo [11] applied Finite Element Method to solve free vibration problem of laminated composite shell with cutouts and analyzed to suggest optimum size and position of the cutout. Ramu and Mohanty [12] carried out free vibration analysis on functionally graded plate using finite element method. The governing equation of free vibration of rotating beams is a fourth order differential equation with variable constants. The DTM is a technique that uses Taylor series for the solution of differential equations in the form of a polynomial. Zhou [13] was the first one used the DTM in engineering applications to solve electrical circuit problem. Presently DTM has spread to solve problem include partial differential equations and system of differential equations. Ozdemir and Kaya [14] calculated natural frequencies for non prismatic beam whose cross section vary according to linear variation of area and cubical variation of moment of inertia. Metin O. Kaya [15] used DTM to obtain the solution for flap wise bending vibration analysis of a rotating tapered Timoshenko beam. Keivan Torabi et al.[16] applied differential transform method for longitudinal vibration analysis with non uniform cross section. To the author’s knowledge studies on natural frequencies for FG beam have been sparsely reported and very few have concentrated on DTM to find natural frequencies. Hence modeling of FG beam with DTM is considered in the present work.
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Nomenclature Ke Km Kc Vc Vm Ge Gm Gc z b h n E(z) μ(z) ρc ρm ρ(z) x r Ω A I ω ζ δ γ
equivalent bulk modulus bulk modulus of metal bulk modulus of ceramic volume fraction of ceramic volume fraction of metal equivalent modulus of rigidity metal rigidity modulus ceramic rigidity modulus lateral coordinate width of the beam total thickness of the beam gradient index equivalent young’s modulus equivalent poisons ratio density of ceramic density of metal equivalent density longitudinal coordinate hub radius Constant rotational speed cross sectional area moment of inertia circular natural frequency beam length parameter hub radius parameter angular speed parameter
2. Statement of the problem In the present work Free vibration Analysis of Functionally Graded Rotating beam is carried out. The beam consists of metallic core with ceramic surfaces as shown in Fig. 1 the material properties vary symmetrically along the thickness direction from core towards the surface. The beam constructed with Functional graded material which is ceramic rich at the top and bottom surfaces (at z = -h/2 and h/2) with a protective metallic core at mid surface (at z = 0).
Fig. 1: Rotating Functionally Graded Beam
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Steel was considered as metallic constituent and alumina was used as the ceramic constituent with mechanical properties as shown in Table 1. The beam has the following dimensions Length = 1000mm, breadth = 20mm, height = 20mm. Table 1: Properties of Metal and Ceramic Materials Properties
Steel (Metal)
Alumina (Ceramic)
Young’s Modulus (GPa)
214
390
Density (kg/m3)
7800
3200
Poisson’s ratio
0.3
0.34
Modulus of Rigidity (GPa)
82.2
137
According to Mori Tanaka method the effective bulk modulus Ke and shear Modulus Ge of the functionally graded beam are taken from the literature [17] 𝐾𝑒 −𝐾𝑚 𝐾𝑐 −𝐾𝑚
𝐺𝑒 −𝐺𝑚 𝐺𝑐 −𝐺𝑚
=
=
𝑉𝑐
(1)
1+𝑉𝑚 (𝐾𝑐 −𝐾𝑚 )/(𝐾𝑚 −4𝐺𝑚 /3) 𝑉𝑐
(2)
1+𝑉𝑚 (𝐺𝑐 −𝐺𝑚 )/[𝐺𝑚 +𝐺𝑚 (9𝐾𝑚 +8𝐺𝑚)/6(𝐾𝑚 +2𝐺𝑚 )] 2𝑧 𝑛
𝑉𝐶 (𝑧) = � � , 𝑉𝑚 = 1 − 𝑉𝐶 (3) ℎ
Where n is called gradient index. Now the effective young’s modulus E and poisons ratio μ are expressed as 𝐸(𝑧) = μ(𝑧) =
9𝐾𝑒 𝐺𝑒
(4)
3𝐾𝑒 −2𝐺𝑒
(5)
3𝐾𝑒 +𝐺𝑒
6𝐾𝑒 +2𝐺𝑒
The effective mass density ρ from rule of mixture as 𝜌(𝑧) = 𝜌𝑐 𝑉𝑐 + 𝜌𝑚 𝑉𝑚 (6)
Infig. 2, a cantilever beam of length L, which is fixed at point ‘O’ to a rigid hub with radius r, is shown. The beam is assumed to be rotating at a constant angular velocityΩ. The X -axis coincides with the neutral axis of the beam in the undeflected position, the Z-axis is parallel to the axis of rotation and the Y-axis lies in the plane of rotation. Let ‘v’ is the deflection and pv is the applied force per unit length in the flap wise direction (Z direction) of the rotating beam.
Fig. 2: Configuration of Rotating Beam
The centrifugal tension force at a distance x L
T = ∫x mΩ2 (r + x)dx (7)
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The governing differential equation for the flap-wise bending motion for Euler Bernoulli beam is given by [14] 𝜌𝐴
𝜕2 𝑣 𝜕𝑡 2
+
𝜕2
𝜕𝑥 2
�𝐸𝐼
𝜕2 𝑣
𝜕𝑥 2
�−
𝜕
𝜕𝑥
�𝑇
𝜕𝑣
𝜕𝑥
� = 𝑝𝑣 (8)
The flap-wise displacement v(x,t) can be assumed as the product of the function of v(x) which depends only on the spatial coordinate x and a time dependent harmonic function as v(x, t) = v(x)eiωt . Now equation (8) becomes −𝜔2 𝜌𝐴𝑣(𝑥) +
𝜕2
𝜕𝑥 2
�𝐸𝐼
𝜕2 𝑣(𝑥)
�−
𝜕𝑥 2
𝜕
𝜕𝑥
�𝑇
𝜕𝑣(𝑥) 𝜕𝑥
� = 𝑝𝑣 (9)
The rotating beam which is fixed at hub free at other end will be considered as a cantilever beam. The boundary conditions for cantilever beam 𝑣= 𝜕2 𝑣
𝜕𝑣
𝜕𝑥
=
𝜕𝑥 2
= 0 𝑎𝑡𝑥 = 0 (10)
𝜕3 𝑣
𝜕𝑥 3
= 0 𝑎𝑡𝑥 = 𝐿(11)
To simplify the equations the dimensionless parameters are used [14] as follows 𝑥
𝜁=
𝐿
𝜌𝐴Ω2 L4
𝑟
, 𝛿 = , 𝛾2 =
𝐸𝐼
𝐿
, 𝜆2 =
𝜔2 Ω2
(12)
Using the dimensionless parameters in the centrifugal tension equation (7), differential equation of motion(9) and boundary conditions (10) & (11) can be expressed as 𝑇(𝜁) = 𝜌𝐴Ω2 L2 (2δ + 1 − 2δ𝜁 − 𝜁 2 )(13) 1
𝛾2
�
𝑑 4 𝑣(𝜁)
𝑣= 𝑑2𝑣 𝑑𝜁 2
𝑑𝜁 4 𝑑𝑣 𝑑𝜁
=
� − 𝜆2 𝑣(𝜁) −
𝑑
𝑑𝜁
1
��𝛿 + − 𝛿𝜁 − 2
= 0 𝑎𝑡𝜁 = 0 (15)
𝑑3𝑣 𝑑𝜁 3
𝜁 2 𝑑𝑣(𝜁) 2
�
𝑑𝜁
� = 0 (14)
= 0 𝑎𝑡𝜁 = 1(16)
3. Solution with Differential Transform Method The Differential Transform Method (DTM) useful techniques to get the solution of differential equation based on Taylor series expansion. The basic theorem used for transformation procedure and boundary conditions are shown in Tabl2 and Table 3 respectively Table 2: Basic theorems of Differential Transformation Original Function
Transformed Functions
𝑓(𝑥) = 𝑔(𝑥) ± ℎ(𝑥)
F[k] = G[k] ± H[k]
𝑓(𝑥) = 𝑔(𝑥)ℎ(𝑥)
𝐹[𝑘] = �
𝑓(𝑥) = 𝜆𝑔(𝑥) 𝑓(𝑥) =
𝑚
𝑑 𝑔(𝑥) 𝑑𝑥 𝑚
𝑓(𝑥) = 𝑋 𝑚
𝐹[𝑘] = 𝜆𝐺[𝑘] 𝑘
𝐺[𝑘 − 𝑙] ± 𝐻[𝑙]
𝑙=0
(𝑘 + 𝑚)! 𝐹[𝑘] = 𝐺[𝑘 + 𝑚] 𝑘! 0 𝑖𝑓𝑘 ≠ 𝑚 𝐹[𝑘] = 𝛿(𝑘 − 𝑛) = � 1 𝑖𝑓𝑘 = 𝑚
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Table 3: Differential Transformation theorems for boundary conditions x=0
x=1
Original BC
Transformed BC
Original BC
𝑓(0) = 0
𝐹[0] = 0
𝑓(1) = 0
𝑑𝑓 (0) = 0 𝑑𝑥 2 𝑑 𝑓 (0) = 0 𝑑𝑥 2 3 𝑑 𝑓 (0) = 0 𝑑𝑥 3
Transformed BC �
𝑑𝑓 (1) = 0 𝑑𝑥 2 𝑑 𝑓 (1) = 0 𝑑𝑥 2 3 𝑑 𝑓 (1) = 0 𝑑𝑥 3
𝐹[1] = 0 𝐹[2] = 0 𝐹[3] = 0
�
Applying the Differential Transformation to the boundary conditions
�
∞
𝑘=0
�
∞
𝑘=0
∞
𝑘=0 ∞
𝐹[𝑘] = 0
𝑘𝐹[𝑘] = 0
𝑘=0
𝑘 (𝑘 − 1)𝐹[𝑘] = 0
𝑘(𝑘 − 1)(𝑘 − 2)𝐹[𝑘] = 0
at ζ = 0 ⟹ V[0] = 0, V[1] = 0 (17)
∑∞ k=2 k (k − 1) V[k] = 0 (18)
∑∞ K=3 k (k − 1)(k − 2) V[k] = 0 (19)
LetV[2] = c1 and V[3] = c2 ,where c1 and c2 are the constants. The differential Transformation is applied to the equation (14) 1 (k + 1)(k + 2)(k + 3)(k + 4) V[k + 4] − γ2 �δ + � (k + 1)(k + 2) V[k + 2] 2 + γ2 δ(k + 1)2 V [k + 1] + �−λ2 ω2 + γ2 �
k2 +k 2
�� V[k] = 0 (20)
Here V[k] is the differential transform ofv(ζ). Using equation (20), V[k] values for k = 4, 5, 6, 7….. can be evaluated in terms of ω, c1 and c2 . Substituting expressions of V[k] terms into boundary conditions, i.e. equation (18) and (19), the following equation is obtained (m) (m) Aj1 (ω)c1 + Aj2 (ω)c2 = 0, where j = 1,2 (21)
th m Where Am j1 (ω), A j2 (ω) are polynomials of ω corresponding to m term. Equation (20) is written in matrix form, we get
Am 11 (ω) m (ω) A21
Am 0 12 (ω) c1 m (ω)� �c � = � �(22) A22 2 0
Am 11 (ω) Am 21 (ω)
Am 12 (ω) � = 0 (23) Am 22 (ω)
�
The Eigen value equation is obtained from equation (22) as follows �
(m)
(m)
Solving equation (23), we get ω = ωj where j = 1, 2. Here ωj m and the value of m is obtained by the following equation (m)
�ωj
(m−1)
− ωj
(m−1)
� ≤ ε(24)
is the jth estimated Eigen value corresponding to
is the jth estimated Eigen value corresponding to m-1 and ε is the tolerance parameter. If equation Where ωj (m) (24) is satisfied, then we have jth eigne value ωj is obtained.
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Table 4: Comparison of Natural Frequencies (Hz) Frequency
Present Method
First
96.90
Ref [5]
Ref [5]
(Analytical)
(Experimental)
96.9
97.0
Second
607.30
607.6
610.0
Third
1700.46
1699.0
1693.0
For further analysis the beam parameters considered are Length = 1000mm, breadth = 20mm, height = 20mm. The properties are obtained using Mori Tanaka Method and solution is obtained using Differential transform Method. The first three fundamental frequencies are obtained and presented in Table 5. It represents that frequency increases with increase in speed and hub radius due to increased stiffness of beam. The frequency is decreased with increase in gradient index or metal composition due to increase of softness. Table 5: First three natural frequencies of a rotating functionally graded beam First Frequency (Hz) δ
0
0.5
2
n
Second Frequency (Hz)
Third Frequency (Hz)
Ω=2
Ω = 25
Ω = 50
Ω=2
Ω = 25
Ω = 50
Ω=2
Ω = 25
Ω = 50
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
0
34.784
44.119
64.244
217.618
226.674
252.052
609.229
618.257
644.670
4
19.122
33.068
56.587
119.159
134.991
174.177
333.451
349.645
394.124
8
18.155
32.500
56.189
113.061
129.640
170.043
316.366
333.387
379.721
0
34.829
49.332
77.767
217.658
232.549
272.511
609.270
624.513
668.159
4
19.204
39.684
71.341
119.231
144.600
202.246
333.526
360.520
430.665
8
18.241
39.203
71.009
113.138
139.611
198.641
316.445
344.763
417.409
0
34.964
62.361
108.453
217.777
249.294
325.752
609.392
642.849
733.307
4
19.448
54.840
103.442
119.449
169.981
268.160
333.750
391.006
522.522
8
18.497
54.472
103.148
113.367
165.721
265.337
316.682
376.463
511.013
The first three natural frequencies are obtained with respect to speed for various gradient index values. and represented in fig. 3. It is observed that natural frequency increases with increase in rotating speed. This is due to increase centrifugal force on beam which causes increment in stiffness of the beam.
P Ravi Kumar et.al./ Materials Today: Proceedings 4 (2017) 3736–3744
800
First
Natural Frequency (Hz)
700
Second
3743
Third
600 500 400 300 200 100 0 0
20
40
60
80
100
Rotating Speed (Rps) Fig. 3: Variation of Natural Frequency with respect to Rotating Speed for various gradient indexes
The variation of first, second, third natural frequencies with respect to Gradient index at various speeds are shown in fig.4, fig.5 and fig.6 respectively. It shows that natural frequenciesdecrease with increase in gradient index and become asymptotic. It represents the increment of metal composition which leads to increased softness due to which there is a drop in natural frequencies. The effect of gradient index is more in higher modes. First Natural Frequency (Hz)
120 100 Rps
100
80 Rps
80
60 Rps
60
40 Rps
40
20 Rps
20
0 Rps
0 0
2
4 Gradient Index (n)
6
8
Second Natural Frequency (Hz)
Fig. 4: Variation of First Natural Frequency with respect to Gradient Index 400 350
100 Rps
300
80 Rps
250 200
60 Rps
150
40 Rps
100
20 Rps
50
0 Rps
0 0
2
4 Gradient Index (n)
6
8
Fig. 5: Variation of Second Natural Frequency with respect to Gradient Index
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Third Natural Frequency (Hz)
800 700 600
100 Rps
500
80 Rps
400
60 Rps
300
40 Rps
200
20 Rps
100
0 Rps
0 0
2
4
6
8
Gradient Index (n) Fig. 6: Variation of Third Natural Frequency with respect to Gradient Index
5. Conclusions In the present work free vibration analysis of rotating functionally graded beam is carried out. Material properties are evaluated using Mori Tanaka Method and natural frequencies were obtained using Differential transform method.The effect of rotating speed, gradient index and hub radius ratio on natural frequencies was studied. It was observed that natural frequency is affected by rotating speed and hub radius ratio due to increase in centrifugal force on beam which causes increase in stiffness of the beam. Due to increase in gradient index beam material the morphology changes from ceramic to steel which causes the decrease in stiffness of the beam, hence natural frequency is decreased. From the study it is concluded that gradient index plays an important role in order to obtain an economical and reliable design as well as control the vibration of rotating beam structures. References [1]Metin Aydogdu and Vedat Taskin, Materials and Design, 28 (2007)1651–1656. [2]S. Kapuria, M. Bhattacharyya and A.N. Kumar, Composite Structures, 82 (2008) 390–402. [3] S.A. Sina, H.M. Navazi and H. Haddadpour, Materials and Design 30 (2009) 741–747. [4] Li Yang and Shi Zhifei, Composite Structures 87 (2009) 257–264. [5] M. T. Piovan and R. Sampaio, Journal of Sound and Vibration 327 (2009)134–143. [6] Mesut Simsek, Nuclear Engineering and Design, 240 (2010) 697–705. [7] Amal E. Alshorbagy, M.A. Eltaher and F.F. Mahmoud, Applied Mathematical Modelling, 35 (2011) 412–425. [8] G. Giunta, D. Crisafulli, S. Belouettar and E. Carrera, Composite Structures, 94 (2011) 68–74. [9] K. Sanjay Anandrao, R.K. Gupta, P. Ramachandran and G. Venkateswara Rao, Defence Science Journal, 62/ 3 (2012) 139-146 [10] M. N. V. Ramesh and N. Mohan Rao, International Journal of Acoustics and Vibration, 19/1 (2014) 31-41. [11] Sarmila Sahoo, Advanced Materials Manufacturing & Characterization, 4/2 (2014)113-124. [12] Ramu I and Mohanty S.C., Procedia Materials Science 6 ( 2014 ) 460 – 467. [13] Zhou J. K., Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuan, China, (1986). [14] Ozdemir O. and Kaya M. O., Journal of Sound and Vibration, 289 (2005) 413-420. [15] Metin O. Kaya, 78/3 (2006), 194–203. [16] Keivan Torabi, Hassan Afshari and Ehsan Zafari,, Journal of Materials Science and Engineering B3 /1 (2013)63-69. [17]Liao-Liang Ke, Yue-Sheng Wang, composite strucutres, 93 (2011) 342-3.
ScienceDirect Materials Today: Proceedings 4 (2017) 3736–3744
www.materialstoday.com/proceedings
5th International Conference of Materials Processing and Characterization (ICMPC 2016)
Flapwise Vibration of Rotating Functionally Graded Beam P. Ravi Kumara *, K. Mohana Raob, N. Mohan Raoc a
Assistant Professor, Mech. Engg. Dept., PVP Siddhartha Institute of Technology, Vijayawada,INDIA b Professor, Mech. Engg. Dept., VR Siddhartha Engineering College., Vijayawada,INDIA c Professor, Mech. Engg. Dept., University College of Engg., JNTUK., Vizianagaram, INDIA
Abstract
Functionally Graded materials have been widely used in engineering applications due to the advantage of smooth and continuous variation in material properties, better fatigue life, less stress concentration, lower thermal stresses and attenuation of stress waves. By intelligently designing material constitutions and gradient distributions of FGMs, structures made of them can have excellent dynamic and thermodynamic characteristics. In the present study Flapwise vibration of rotating functionally graded beam is studied. The effective material properties are determined using Mori Tanaka Method. The natural frequencies are obtained using differential transform method (DTM). The effect of rotating speed, gradient index and hub radius on natural frequency are discussed. ©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016). Keywords:FGM, FG beam, Free vibration, Mori Tanaka Method, Differential Transform Method
1. Introduction Functionally graded materials (FGM) are made by combining two or more materials and posses positiondependent microstructure and chemical composition resulting in smooth gradients in the properties like mechanical strength and thermal conductivity. In majority applications, material properties change through the thickness of the material. Due to smooth variation of properties, functionally graded materials possess a number advantages like
* Corresponding author. Tel.: +919885519328 E-mail address:[email protected] 2214-7853©2017 Elsevier Ltd. All rights reserved. Selection and peer-review under responsibility of Conference Committee Members of 5th International Conference of Materials Processing and Characterization (ICMPC 2016).
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reduction in stresses through the thickness, uniform residual stress distribution leads less stress concentration, high Large number of studies carried out by researchers on functionally graded beams. Aydogdu and Taskin [1] fracture toughness and high temperature resistance. To design rotating components like turbine blades, compressor blades, aircraft propeller blades and helicopter rotor blades properly their natural frequencies estimated accurately. carried out free vibration analysis of functionally graded beam with simply supported edges. Governing equations are found by Hamilton principle and natural frequencies are obtained by Navier type solution method. Kapuria et. al [2] validated a third order zigzag theory based model through experiments for static and free vibration response. Sina et. al. [3] carried out free vibration analysis of functionally graded beams using a new beam theory different from traditional first order shear deformation theory. Governing equations are derived using Hamilton principle and the natural frequencies are obtained by an exact method. Li Yang and Shi [4] extended state space based differential quadrature method to study free vibration of functionally graded piezoelectric beam with different boundary conditions. The influence of gradient index on fundamental frequency is discussed. Piovan and Sampaio [5] developed a rotating non linear beam model made of functionally graded material. Simsek [6] carried out fundamental frequency analysis of functionally graded beams having different boundary conditions. The governing frequency equation is obtained using Lagrange’s equation the effect of slenderness ratio, material variations and different beam theories on the fundamental frequencies are examined. Alshorbagy [7] performed free vibration analysis of a functionally graded beam by finite element method. The system of equations of motion is derived by using the principle of virtual work and the effects of different material distribution, slenderness ratios, and boundary conditions on the frequency are presented. Giunta et. al [8] investigated free vibration of functionally graded beams using various beam theories and presented the effect of different material distributions on natural frequencies. Anandrao et. al [9] carried out free vibration analysis of functionally graded beams with classical boundary conditions. Principle of virtual work is used to obtain finite element equations effect of transverse shear, length to thickness ratio and volume fraction exponent on natural frequency of functionally graded material demonstrated. Ramesh and Mohan Rao[10] derived equations of motion for a FG beam from a modelling method by employing the hybrid deformation variable, determined natural frequencies using Raleigh Ritz method and examined effect of various parameters through numerical studies. Sarmila Sahoo [11] applied Finite Element Method to solve free vibration problem of laminated composite shell with cutouts and analyzed to suggest optimum size and position of the cutout. Ramu and Mohanty [12] carried out free vibration analysis on functionally graded plate using finite element method. The governing equation of free vibration of rotating beams is a fourth order differential equation with variable constants. The DTM is a technique that uses Taylor series for the solution of differential equations in the form of a polynomial. Zhou [13] was the first one used the DTM in engineering applications to solve electrical circuit problem. Presently DTM has spread to solve problem include partial differential equations and system of differential equations. Ozdemir and Kaya [14] calculated natural frequencies for non prismatic beam whose cross section vary according to linear variation of area and cubical variation of moment of inertia. Metin O. Kaya [15] used DTM to obtain the solution for flap wise bending vibration analysis of a rotating tapered Timoshenko beam. Keivan Torabi et al.[16] applied differential transform method for longitudinal vibration analysis with non uniform cross section. To the author’s knowledge studies on natural frequencies for FG beam have been sparsely reported and very few have concentrated on DTM to find natural frequencies. Hence modeling of FG beam with DTM is considered in the present work.
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Nomenclature Ke Km Kc Vc Vm Ge Gm Gc z b h n E(z) μ(z) ρc ρm ρ(z) x r Ω A I ω ζ δ γ
equivalent bulk modulus bulk modulus of metal bulk modulus of ceramic volume fraction of ceramic volume fraction of metal equivalent modulus of rigidity metal rigidity modulus ceramic rigidity modulus lateral coordinate width of the beam total thickness of the beam gradient index equivalent young’s modulus equivalent poisons ratio density of ceramic density of metal equivalent density longitudinal coordinate hub radius Constant rotational speed cross sectional area moment of inertia circular natural frequency beam length parameter hub radius parameter angular speed parameter
2. Statement of the problem In the present work Free vibration Analysis of Functionally Graded Rotating beam is carried out. The beam consists of metallic core with ceramic surfaces as shown in Fig. 1 the material properties vary symmetrically along the thickness direction from core towards the surface. The beam constructed with Functional graded material which is ceramic rich at the top and bottom surfaces (at z = -h/2 and h/2) with a protective metallic core at mid surface (at z = 0).
Fig. 1: Rotating Functionally Graded Beam
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Steel was considered as metallic constituent and alumina was used as the ceramic constituent with mechanical properties as shown in Table 1. The beam has the following dimensions Length = 1000mm, breadth = 20mm, height = 20mm. Table 1: Properties of Metal and Ceramic Materials Properties
Steel (Metal)
Alumina (Ceramic)
Young’s Modulus (GPa)
214
390
Density (kg/m3)
7800
3200
Poisson’s ratio
0.3
0.34
Modulus of Rigidity (GPa)
82.2
137
According to Mori Tanaka method the effective bulk modulus Ke and shear Modulus Ge of the functionally graded beam are taken from the literature [17] 𝐾𝑒 −𝐾𝑚 𝐾𝑐 −𝐾𝑚
𝐺𝑒 −𝐺𝑚 𝐺𝑐 −𝐺𝑚
=
=
𝑉𝑐
(1)
1+𝑉𝑚 (𝐾𝑐 −𝐾𝑚 )/(𝐾𝑚 −4𝐺𝑚 /3) 𝑉𝑐
(2)
1+𝑉𝑚 (𝐺𝑐 −𝐺𝑚 )/[𝐺𝑚 +𝐺𝑚 (9𝐾𝑚 +8𝐺𝑚)/6(𝐾𝑚 +2𝐺𝑚 )] 2𝑧 𝑛
𝑉𝐶 (𝑧) = � � , 𝑉𝑚 = 1 − 𝑉𝐶 (3) ℎ
Where n is called gradient index. Now the effective young’s modulus E and poisons ratio μ are expressed as 𝐸(𝑧) = μ(𝑧) =
9𝐾𝑒 𝐺𝑒
(4)
3𝐾𝑒 −2𝐺𝑒
(5)
3𝐾𝑒 +𝐺𝑒
6𝐾𝑒 +2𝐺𝑒
The effective mass density ρ from rule of mixture as 𝜌(𝑧) = 𝜌𝑐 𝑉𝑐 + 𝜌𝑚 𝑉𝑚 (6)
Infig. 2, a cantilever beam of length L, which is fixed at point ‘O’ to a rigid hub with radius r, is shown. The beam is assumed to be rotating at a constant angular velocityΩ. The X -axis coincides with the neutral axis of the beam in the undeflected position, the Z-axis is parallel to the axis of rotation and the Y-axis lies in the plane of rotation. Let ‘v’ is the deflection and pv is the applied force per unit length in the flap wise direction (Z direction) of the rotating beam.
Fig. 2: Configuration of Rotating Beam
The centrifugal tension force at a distance x L
T = ∫x mΩ2 (r + x)dx (7)
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The governing differential equation for the flap-wise bending motion for Euler Bernoulli beam is given by [14] 𝜌𝐴
𝜕2 𝑣 𝜕𝑡 2
+
𝜕2
𝜕𝑥 2
�𝐸𝐼
𝜕2 𝑣
𝜕𝑥 2
�−
𝜕
𝜕𝑥
�𝑇
𝜕𝑣
𝜕𝑥
� = 𝑝𝑣 (8)
The flap-wise displacement v(x,t) can be assumed as the product of the function of v(x) which depends only on the spatial coordinate x and a time dependent harmonic function as v(x, t) = v(x)eiωt . Now equation (8) becomes −𝜔2 𝜌𝐴𝑣(𝑥) +
𝜕2
𝜕𝑥 2
�𝐸𝐼
𝜕2 𝑣(𝑥)
�−
𝜕𝑥 2
𝜕
𝜕𝑥
�𝑇
𝜕𝑣(𝑥) 𝜕𝑥
� = 𝑝𝑣 (9)
The rotating beam which is fixed at hub free at other end will be considered as a cantilever beam. The boundary conditions for cantilever beam 𝑣= 𝜕2 𝑣
𝜕𝑣
𝜕𝑥
=
𝜕𝑥 2
= 0 𝑎𝑡𝑥 = 0 (10)
𝜕3 𝑣
𝜕𝑥 3
= 0 𝑎𝑡𝑥 = 𝐿(11)
To simplify the equations the dimensionless parameters are used [14] as follows 𝑥
𝜁=
𝐿
𝜌𝐴Ω2 L4
𝑟
, 𝛿 = , 𝛾2 =
𝐸𝐼
𝐿
, 𝜆2 =
𝜔2 Ω2
(12)
Using the dimensionless parameters in the centrifugal tension equation (7), differential equation of motion(9) and boundary conditions (10) & (11) can be expressed as 𝑇(𝜁) = 𝜌𝐴Ω2 L2 (2δ + 1 − 2δ𝜁 − 𝜁 2 )(13) 1
𝛾2
�
𝑑 4 𝑣(𝜁)
𝑣= 𝑑2𝑣 𝑑𝜁 2
𝑑𝜁 4 𝑑𝑣 𝑑𝜁
=
� − 𝜆2 𝑣(𝜁) −
𝑑
𝑑𝜁
1
��𝛿 + − 𝛿𝜁 − 2
= 0 𝑎𝑡𝜁 = 0 (15)
𝑑3𝑣 𝑑𝜁 3
𝜁 2 𝑑𝑣(𝜁) 2
�
𝑑𝜁
� = 0 (14)
= 0 𝑎𝑡𝜁 = 1(16)
3. Solution with Differential Transform Method The Differential Transform Method (DTM) useful techniques to get the solution of differential equation based on Taylor series expansion. The basic theorem used for transformation procedure and boundary conditions are shown in Tabl2 and Table 3 respectively Table 2: Basic theorems of Differential Transformation Original Function
Transformed Functions
𝑓(𝑥) = 𝑔(𝑥) ± ℎ(𝑥)
F[k] = G[k] ± H[k]
𝑓(𝑥) = 𝑔(𝑥)ℎ(𝑥)
𝐹[𝑘] = �
𝑓(𝑥) = 𝜆𝑔(𝑥) 𝑓(𝑥) =
𝑚
𝑑 𝑔(𝑥) 𝑑𝑥 𝑚
𝑓(𝑥) = 𝑋 𝑚
𝐹[𝑘] = 𝜆𝐺[𝑘] 𝑘
𝐺[𝑘 − 𝑙] ± 𝐻[𝑙]
𝑙=0
(𝑘 + 𝑚)! 𝐹[𝑘] = 𝐺[𝑘 + 𝑚] 𝑘! 0 𝑖𝑓𝑘 ≠ 𝑚 𝐹[𝑘] = 𝛿(𝑘 − 𝑛) = � 1 𝑖𝑓𝑘 = 𝑚
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Table 3: Differential Transformation theorems for boundary conditions x=0
x=1
Original BC
Transformed BC
Original BC
𝑓(0) = 0
𝐹[0] = 0
𝑓(1) = 0
𝑑𝑓 (0) = 0 𝑑𝑥 2 𝑑 𝑓 (0) = 0 𝑑𝑥 2 3 𝑑 𝑓 (0) = 0 𝑑𝑥 3
Transformed BC �
𝑑𝑓 (1) = 0 𝑑𝑥 2 𝑑 𝑓 (1) = 0 𝑑𝑥 2 3 𝑑 𝑓 (1) = 0 𝑑𝑥 3
𝐹[1] = 0 𝐹[2] = 0 𝐹[3] = 0
�
Applying the Differential Transformation to the boundary conditions
�
∞
𝑘=0
�
∞
𝑘=0
∞
𝑘=0 ∞
𝐹[𝑘] = 0
𝑘𝐹[𝑘] = 0
𝑘=0
𝑘 (𝑘 − 1)𝐹[𝑘] = 0
𝑘(𝑘 − 1)(𝑘 − 2)𝐹[𝑘] = 0
at ζ = 0 ⟹ V[0] = 0, V[1] = 0 (17)
∑∞ k=2 k (k − 1) V[k] = 0 (18)
∑∞ K=3 k (k − 1)(k − 2) V[k] = 0 (19)
LetV[2] = c1 and V[3] = c2 ,where c1 and c2 are the constants. The differential Transformation is applied to the equation (14) 1 (k + 1)(k + 2)(k + 3)(k + 4) V[k + 4] − γ2 �δ + � (k + 1)(k + 2) V[k + 2] 2 + γ2 δ(k + 1)2 V [k + 1] + �−λ2 ω2 + γ2 �
k2 +k 2
�� V[k] = 0 (20)
Here V[k] is the differential transform ofv(ζ). Using equation (20), V[k] values for k = 4, 5, 6, 7….. can be evaluated in terms of ω, c1 and c2 . Substituting expressions of V[k] terms into boundary conditions, i.e. equation (18) and (19), the following equation is obtained (m) (m) Aj1 (ω)c1 + Aj2 (ω)c2 = 0, where j = 1,2 (21)
th m Where Am j1 (ω), A j2 (ω) are polynomials of ω corresponding to m term. Equation (20) is written in matrix form, we get
Am 11 (ω) m (ω) A21
Am 0 12 (ω) c1 m (ω)� �c � = � �(22) A22 2 0
Am 11 (ω) Am 21 (ω)
Am 12 (ω) � = 0 (23) Am 22 (ω)
�
The Eigen value equation is obtained from equation (22) as follows �
(m)
(m)
Solving equation (23), we get ω = ωj where j = 1, 2. Here ωj m and the value of m is obtained by the following equation (m)
�ωj
(m−1)
− ωj
(m−1)
� ≤ ε(24)
is the jth estimated Eigen value corresponding to
is the jth estimated Eigen value corresponding to m-1 and ε is the tolerance parameter. If equation Where ωj (m) (24) is satisfied, then we have jth eigne value ωj is obtained.
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Table 4: Comparison of Natural Frequencies (Hz) Frequency
Present Method
First
96.90
Ref [5]
Ref [5]
(Analytical)
(Experimental)
96.9
97.0
Second
607.30
607.6
610.0
Third
1700.46
1699.0
1693.0
For further analysis the beam parameters considered are Length = 1000mm, breadth = 20mm, height = 20mm. The properties are obtained using Mori Tanaka Method and solution is obtained using Differential transform Method. The first three fundamental frequencies are obtained and presented in Table 5. It represents that frequency increases with increase in speed and hub radius due to increased stiffness of beam. The frequency is decreased with increase in gradient index or metal composition due to increase of softness. Table 5: First three natural frequencies of a rotating functionally graded beam First Frequency (Hz) δ
0
0.5
2
n
Second Frequency (Hz)
Third Frequency (Hz)
Ω=2
Ω = 25
Ω = 50
Ω=2
Ω = 25
Ω = 50
Ω=2
Ω = 25
Ω = 50
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
(rps)
0
34.784
44.119
64.244
217.618
226.674
252.052
609.229
618.257
644.670
4
19.122
33.068
56.587
119.159
134.991
174.177
333.451
349.645
394.124
8
18.155
32.500
56.189
113.061
129.640
170.043
316.366
333.387
379.721
0
34.829
49.332
77.767
217.658
232.549
272.511
609.270
624.513
668.159
4
19.204
39.684
71.341
119.231
144.600
202.246
333.526
360.520
430.665
8
18.241
39.203
71.009
113.138
139.611
198.641
316.445
344.763
417.409
0
34.964
62.361
108.453
217.777
249.294
325.752
609.392
642.849
733.307
4
19.448
54.840
103.442
119.449
169.981
268.160
333.750
391.006
522.522
8
18.497
54.472
103.148
113.367
165.721
265.337
316.682
376.463
511.013
The first three natural frequencies are obtained with respect to speed for various gradient index values. and represented in fig. 3. It is observed that natural frequency increases with increase in rotating speed. This is due to increase centrifugal force on beam which causes increment in stiffness of the beam.
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800
First
Natural Frequency (Hz)
700
Second
3743
Third
600 500 400 300 200 100 0 0
20
40
60
80
100
Rotating Speed (Rps) Fig. 3: Variation of Natural Frequency with respect to Rotating Speed for various gradient indexes
The variation of first, second, third natural frequencies with respect to Gradient index at various speeds are shown in fig.4, fig.5 and fig.6 respectively. It shows that natural frequenciesdecrease with increase in gradient index and become asymptotic. It represents the increment of metal composition which leads to increased softness due to which there is a drop in natural frequencies. The effect of gradient index is more in higher modes. First Natural Frequency (Hz)
120 100 Rps
100
80 Rps
80
60 Rps
60
40 Rps
40
20 Rps
20
0 Rps
0 0
2
4 Gradient Index (n)
6
8
Second Natural Frequency (Hz)
Fig. 4: Variation of First Natural Frequency with respect to Gradient Index 400 350
100 Rps
300
80 Rps
250 200
60 Rps
150
40 Rps
100
20 Rps
50
0 Rps
0 0
2
4 Gradient Index (n)
6
8
Fig. 5: Variation of Second Natural Frequency with respect to Gradient Index
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Third Natural Frequency (Hz)
800 700 600
100 Rps
500
80 Rps
400
60 Rps
300
40 Rps
200
20 Rps
100
0 Rps
0 0
2
4
6
8
Gradient Index (n) Fig. 6: Variation of Third Natural Frequency with respect to Gradient Index
5. Conclusions In the present work free vibration analysis of rotating functionally graded beam is carried out. Material properties are evaluated using Mori Tanaka Method and natural frequencies were obtained using Differential transform method.The effect of rotating speed, gradient index and hub radius ratio on natural frequencies was studied. It was observed that natural frequency is affected by rotating speed and hub radius ratio due to increase in centrifugal force on beam which causes increase in stiffness of the beam. Due to increase in gradient index beam material the morphology changes from ceramic to steel which causes the decrease in stiffness of the beam, hence natural frequency is decreased. From the study it is concluded that gradient index plays an important role in order to obtain an economical and reliable design as well as control the vibration of rotating beam structures. References [1]Metin Aydogdu and Vedat Taskin, Materials and Design, 28 (2007)1651–1656. [2]S. Kapuria, M. Bhattacharyya and A.N. Kumar, Composite Structures, 82 (2008) 390–402. [3] S.A. Sina, H.M. Navazi and H. Haddadpour, Materials and Design 30 (2009) 741–747. [4] Li Yang and Shi Zhifei, Composite Structures 87 (2009) 257–264. [5] M. T. Piovan and R. Sampaio, Journal of Sound and Vibration 327 (2009)134–143. [6] Mesut Simsek, Nuclear Engineering and Design, 240 (2010) 697–705. [7] Amal E. Alshorbagy, M.A. Eltaher and F.F. Mahmoud, Applied Mathematical Modelling, 35 (2011) 412–425. [8] G. Giunta, D. Crisafulli, S. Belouettar and E. Carrera, Composite Structures, 94 (2011) 68–74. [9] K. Sanjay Anandrao, R.K. Gupta, P. Ramachandran and G. Venkateswara Rao, Defence Science Journal, 62/ 3 (2012) 139-146 [10] M. N. V. Ramesh and N. Mohan Rao, International Journal of Acoustics and Vibration, 19/1 (2014) 31-41. [11] Sarmila Sahoo, Advanced Materials Manufacturing & Characterization, 4/2 (2014)113-124. [12] Ramu I and Mohanty S.C., Procedia Materials Science 6 ( 2014 ) 460 – 467. [13] Zhou J. K., Differential Transformation and Its Application for Electrical Circuits, Huazhong University Press, Wuan, China, (1986). [14] Ozdemir O. and Kaya M. O., Journal of Sound and Vibration, 289 (2005) 413-420. [15] Metin O. Kaya, 78/3 (2006), 194–203. [16] Keivan Torabi, Hassan Afshari and Ehsan Zafari,, Journal of Materials Science and Engineering B3 /1 (2013)63-69. [17]Liao-Liang Ke, Yue-Sheng Wang, composite strucutres, 93 (2011) 342-3.