Journal of Vibration Testing and System Dynamics 2(1) (2018) 53-67
Journal of Vibration Testing and System Dynamics Journal homepage: https://lhscientificpublishing.com/Journals/JVTSD-Default.aspx
A Similitude Design Method of Rotating Thin-wall Short Cylindrical Shell Considering Nonlinear Vibration Response Zhong Luo1,2†, Yunpeng Zhu3 , You Wang4 , Fei Wang1,2 , Qingkai Han5 1
School of Mechanical Engineering & Automation, Northeastern University, Shenyang, China Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang, Liaoning, China 3 Department of Automation Control and System Engineering, University of Sheffield, Sheffield S13JD, UK 4 Shenyang Institute of Automation Chinese Academy of Sciences, Shenyang, China 5 School of Mechanical Engineering, Dalian University of Technology, Dalian, China 2
Submission Info Communicated by J.Z. Zhang Received 28 November 2017 Accepted 10 January 2018 Available online 1 April 2018 Keywords Thin-wall short cylindrical shell Non-linear vibration Natural frequencies Dynamic scaling laws Non-linear response
Abstract This study investigates the non-linear dynamic scaling laws for a rotating thin-wall short cylindrical shell. By introducing the geometric non-linear term, corresponding governing equations are employed to establish the non-linear scaling laws. Both the natural frequency and single-point excitation response of the rotating cylindrical shell are investigated. The applicability of the scaling laws of the rotating thin-wall short cylindrical shell is verified numerically. In addition, the scaling laws for linear and non-linear vibrations are compared. Analytical results indicate that the scaled model designed by the non-linear scaling laws are more restrictive than that of using the linear scaling laws. In addition, they predict the characteristics of the prototype with good accuracy. ©2018 L&H Scientific Publishing, LLC. All rights reserved.
1 Introduction A thin-wall short cylindrical shell is a cylindrical structure that has a thin shell and is short in length. These characteristics make the Kirchhoff hypothesis suitable for studying thin-wall short cylindrical shells having an axial half-wave number m = 1. Rotating thin-wall short cylindrical shells are widely applied, for example, in advanced gas turbines, high-powered aircraft jet engines and high-speed centrifugal separators [1, 2]. Vibrations, non-linear vibrations in particular, are of the most significant concern in the study of rotating thin-wall short cylindrical shell structures [3, 4]. As experimental testing is expensive and time consuming, consequently, dynamic scaled down models are often used to predict the non-linear behaviour of the prototype that is built. † Corresponding
author. Email address:
[email protected] ISSN 2475-4811, eISSN 2475-482X/$-see front materials © 2018 L&H Scientific Publishing, LLC. All rights reserved. DOI:10.5890/JVTSD.2018.03.006
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The scaling laws of cylindrical shell structures have been the subject of many investigations. Ungbhakorn [5] derived the scaling laws for symmetric and anti-symmetric cross-ply laminated circular cylindrical shells. To predict linear vibrations, both complete and partial similitude cases were investigated and the model with distortion in material properties was not recommended. Stimitses et al. [6] have studied the scaling laws of a distorted model for the prediction of the free vibration of a laminate shell. In their study of the scaling laws of different material properties, the number of plies and geometric size were derived by using the governing equations for a laminated plate and shell. Bijan [7] studied the load-carrying capacity and energy absorption by using an ‘equivalent’ aluminium tube that has the same flexural stiffness with the prototype. The accuracy of the prediction was verified by experimental data and analytical solutions. Torkamani [8] developed the scaling laws of free vibrations of an orthogonally stiffened cylindrical shell by using the similitude theory. Different examples were solved to validate the scaling laws numerically and experimentally. There are many studies that investigated non-linearity of shell structures. In the study by Oshiro and Alves [9, 10], a new non-linear scaling law for impacted strain-rate sensitive structures was discussed. A robust correction procedure and an indirect similitude method were studied. Wen and Jones [11] studied the scaling laws of metal plates struck by a large mass by using an experimental investigation. A range of experimental parameters were studied in their investigation. Ungbhakorn [12] and Stimitses et al. [13] investigated the buckling of a cross-ply laminated circular cylindrical shell. Both the complete and distorted scaling laws were derived. However, the scaling laws for non-linear vibrations are still being sought. In the present study, the problem associated with the design of a dynamic similarity scaled down model of a rotating thin-wall short cylindrical shell is discussed. The method of deriving governing equations is employed to deduce the corresponding non-linear scaling laws. Furthermore, a numerical analysis is performed to verify the scaling laws.
2 Non-linear governing equations Fig. 1 shows the variables defined for a rotating thin-wall short cylindrical shell. In the oxθ r cylindrical coordinate system, the origin o is one of the centre points of the cylindrical shell’s end face, where the x axis and the central line are overlapped. θ is the circumferential direction angle and r is the length of the radial coordinate. The orthogonal coordinate system is determined by the coordinate system oxθ r, which translates a distance R from the origin o along the r direction to o with R ing the radius of the middle surface. u, v and w represent the displacements of the x-, θ - and z-directions, respectively. ωr is the rotating speed of the thin-wall short cylindrical shell. The Young’s modulus, Poisson’s ratio, and the density of the shell material are denoted by E, μ and ρ , respectively. According to Ref. [14], the following hypotheses need to be satisfied in the thin-wall shell problem: 1) The thickness of the shell is small compared with the other dimensions. 2) The transverse normal stress is small compared with the other normal stress components, thus it can be neglected. 3) Normal to the undeformed middle surface remains straight. It stays normal to the deformed middle surface and suffers no extension. In the non-linear vibration problem of a thin-wall shell, strains and displacements are larger than those in linear vibrations. So the quantities of the second-order magnitude in the strain-displacement relations are considered in the present study.
Zhong Luo et al. / Journal of Vibration Testing and System Dynamics 2(1) (2018) 53–67
v
Ȧr
55
w u
o' r R ș x o
h
l
Fig. 1 Rotating thin-wall short cylindrical shell with clamped-free boundary condition.
2.1
Non-linear governing equations according to Donnell theory
According to the classical shell theory, considering the geometric non-linearity, the strain of a thin-wall short cylindrical shell can be written as:
∂u 1 ∂u 2 + ( ) , ∂x 2 ∂x
(1)
1 ∂v 1 ∂w 2 ( + w) + ( ) , R ∂θ 2 R∂ θ
(2)
∂v 1 ∂u 1 ∂w ∂w + + , ∂x R ∂θ R ∂x ∂θ
(3)
(0)
εx = (0)
εθ = (0)
γxθ =
where the non-linear terms are 12 ( ∂∂ ux )2 , 12 ( R∂∂wθ )2 and R1 ∂∂wx ∂∂ wθ . The curvature and the torsion of the middle surface are:
∂ 2w , ∂ x2
(4)
1 ∂ 2w , R ∂θ2
(5)
χx = − χθ = −
2 ∂ 2w . R ∂ x∂ θ The internal force and the moment of the cylindrical shell can be written as: ⎧ (0) (0) ⎪ Nx = K(εx + μεθ ), ⎪ ⎪ ⎨ (0) (0) Nθ = K(εθ + μεx ), ⎪ ⎪ ⎪ ⎩ Nxθ = K( 1 − μ )γ (0) , xθ 2 ⎧ Mx = D(χx + μ χθ ), ⎪ ⎪ ⎨ Mθ = D(χθ + μ χx ), ⎪ ⎪ ⎩ M = D( 1 − μ )χ , xθ xθ 2
χxθ = −
3
(6)
(7)
(8)
Eh is the bending stiffness and K = 1−Ehμ 2 is the film stiffness. Nx and Nθ are the where D = 12(1− μ2) membrane forces per unit length in the middle surface; Nxθ is the membrane shear force per unit length
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in the middle surface; Mx and Mθ are bending moments per unit length on the middle surface; and Mxθ is the torque per unit length on the middle surface. Considering the geometric non-linear condition, the dynamic governing equations of the rotating thin-wall cylindrical shell are: 1 ∂ 2u 1 ∂ w ∂ Nx 1 ∂ Nxθ ∂ 2u + ) = + Nθ0 ( 2 − ρ h , ∂x R ∂θ R ∂θ2 R ∂x ∂ t2
(9)
1 ∂ Mx Nθ0 ∂ 2 u ∂w ∂ 2v 1 ∂ Nθ ∂ Nxθ 1 ∂ Mxθ + + 2 + ρ hΩ2 v = ρ h 2 , + + − 2ρ hΩ R ∂θ ∂x R ∂x R ∂θ R ∂ x∂ θ ∂t ∂t
(10)
Nθ0 ∂ 2 w ∂ v 1 ∂ 2 Mθ 2 ∂ 2 Mxθ 1 ∂ 2 Mx ∂ 2 w Nθ ∂ 2 w 2Nxθ ∂ 2 w N + + + − + N + + ( − ) θ x ∂ x2 R2 ∂ θ 2 R ∂ x∂ θ R ∂ x2 R2 ∂ θ 2 R2 ∂ θ ∂ x R2 ∂ θ 2 ∂ θ ∂ 2w ∂v = ρ h 2 − 2ρ hΩ − ρ hΩ2 w, (11) ∂t ∂t where Nθ0 represents the initial tangential stress attributed by the centrifugal force, Nθ0 = ρ hΩ2 R2 ; ρ h∂ 2 u/∂ t 2 , ρ h∂ 2 v/∂ t 2 , ρ h∂ 2 w/∂ t 2 are inertia terms; 2ρ hΩ∂ w/∂ t, 2ρ hΩ∂ v/∂ t are the Coriolis forces; ρ hΩ2 v, ρ hΩ2 w are centrifugal forces. Donnell’s hypotheses can be expressed as follows: 1) The curvature of the middle surface θθ = v/R − ∂ w/∂ θ R is simplified as θθ = −∂ w/∂ θ R by ignoring the displacement v. 2) In Eq. (9) and Eq. (10), the internal inertia forces ρ h∂ 2 v/∂ t 2 and ρ h∂ 2 u/∂ t 2 , and the shear force in Eq. (10) are neglected. According to Donnell’s hypotheses, replacing the centrifugal forces, Coriolis forces and the external force as the normal exciting forces, Eq. (9) to Eq. (11) can be simplified as follows [15]: D∇2 ∇2 w + ρ h
∂ 2w = F(t) + Anonlin , ∂ t2
(12)
where ∇2 = ∂∂x2 + R2∂∂ θ 2 is the Laplace operator; Anonlin is the non-linear item; F(t) is the normal exciting force. Note that the point of action is at (x0 , −ωr t), with ωr being the rotating frequency in rad/s as 2
2
F(t) = F0 cos(ω t)δ (x − x0 )δ (θ − ωr t), K ∂ 2w ∂ w 2 μ K ∂ 2w ∂ w 2 K ∂ 2w ∂ w 2 ) + ) ( ) + 2 ( ( 2R4 ∂ θ 2 ∂ θ 2R ∂ θ 2 ∂ x 2 ∂ x2 ∂ x μ K ∂ 2 w ∂ w 2 K(1 − μ ) ∂ w ∂ w ∂ 2 w + 2 2( ) + , 2R ∂ x ∂ θ R2 ∂ x ∂ θ ∂ x∂ θ where δ (x) is the impulse function and ω is the natural frequency of the cylindrical shell.
(13)
Anonlin =
2.2
(14)
Calculation of single mode non-linear vibration response
The axial wave number m of the thin-wall short cylindrical shell is m = 1. Since the Donnell theory is a kind of simplified theory, only when the circumference half-wave number n satisfied n2 ≥ 1 can one expect to obtain accurate results using the theory. The present study employs n ≥ 5 [16] and n = 6. The forward and backward vibration modes (with m = 1 and n = 6) of the rotating thin-wall short cylindrical shell are shown in Fig. 2. Assuming the solution of Eq. (11) is of the following form [17] w(x, θ ,t) = Um (x)[Am,n (t) cos(nθ ) + Bm,n (t) sin(nθ )],
(15)
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(a) Forward vibration mode
57
(b) Backward vibration mode
Fig. 2 Vibration modes (with m=1 and n=6) of the rotating thin-wall short cylindrical shell.
where Um (x) is the vibration mode function of the cylindrical shell and Am,n (t) and Bm,n (t) are the generalized mode values. Substituting Eq. (15) into Eq. (11) to yields ˆ l ˆ 2π ˆ l ˆ 2π ∂ 2w 2 2 (D∇ ∇ w + ρ h 2 )zs (x, θ )Rdxdθ = [F(t) + Anonlin ]zs (x, θ )Rdxdθ , (16) ∂t 0 0 0 0 where zs (x, θ ), a function of x and θ , is the multiplication of the vibration mode functions of the axial and circumference directions. Considering the (m = 1, n = 6) mode vibration, Eq. (15) can be written as: w(x, θ ,t) = U1 (x)[A1,6 (t) cos(6θ ) + B1,6 (t) sin(6θ )],
(17a)
U1 (x) = C1,1 eP1,1 x +C1,2 e−P1,2 x +C1,3 cos(P1,2 x) +C1,4 sin(P1,2 x),
(17b)
where C1,1 , C1,2 , C1,3 , C1,4 , P1,1 , P1,2 are parameters related to the boundary conditions [18]. For the case that is being considered we have U1 (x) cos(6θ ) s = 1, (18) zs (x, θ ) = s = 2. U1 (x) sin(6θ ) Substituting Eq. (18) and Eq. (17) into Eq. (16) to yield two non-linear differential equations 2 A1,6 (t) = F¯ cos(ω t) cos(6ωr t) + HA1,6(t)3 + HA1,6(t)B1,6 (t)2 , A1,6 (t) + ω1,6
(19)
2 B1,6 (t) = −F¯ cos(ω t) cos(6ωr t) + HB1,6(t)3 + HB1,6(t)A1,6 (t)2 , B1,6 (t) + ω1,6
(20)
where D 64 62 2 [ 4 −2 2 ω1,6 = ρh R R
´l
0 U1 (x)U1 (x)dx + ´l 2 0 U1 (x)dx
F¯ =
´l
F0U1 (x0 ) , ´l πρ h 0 U12 (x)dx
(4) 0 U1 (x)U1 (x)dx ], ´l 2 0 U1 (x)dx
´l ´l ´l 162 0 U14 (x)dx 9 0 U1 (x)[U1 (x)]2 dx 45μ 0 U1 (x)[U1 (x)]2 dx K + − H = {− 4 ´ l ´l ´l 2 ρh R R2 0 U12 (x)dx 2R2 0 U12 (x)dx 0 U1 (x)dx ´l ´l 9μ 0 U13 (x)U1 (x)dx 3μ 0 U13 (x)U1 (x)[U1 (x)]2 dx + }. + ´l ´l 2R2 0 U12 (x)dx 8R2 0 U12 (x)dx
(21a) (21b)
(21c)
The Runge-Kutta method is then employed to solve Eq. (19) and Eq. (20) numerically. Substituting the results obtained for A1,6 (t) and B1,6 (t) into Eq. (15), w(x, θ ,t) can be obtained.
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3 Complete scaling laws 3.1 3.1.1
Linear dynamic scaling laws Scaling laws of natural frequency
According to Eq. (9) to Eq. (11) and the similitude theory [19], the followings are true
,
λN 0 λu λN 0 λw λNx λN = xθ = θ 2 = θ λl λR λR λl λR
(22a)
λN 0 λu λNθ λN λM = xθ = 2x = θ = λρ λh λΩ λω λw = λρ λh λΩ2 λv , λR λl λR λl λR
(22b)
λMx λM λM λN λN λw λN λw λN λw λN 0 λw = 2θ = xθ = θ = x 2 = θ 2 = xθ2 = θ 2 2 λR λl λR λl λR λl λR λR λR λN 0 λv = θ2 = λρ λh λω2 λw = λρ λh λΩ2 λw , λR
(22c)
λ λNx = λ λNxθ = λN 0 λu = λN 0 λw = λ λNθ = λMx = λ 3 λΩ λω λw = λ 3 λΩ2 λv = λMθ = λMxθ θ θ = λNx λw = λNθ λw = λNxθ λw = λN 0 λv = λ 3 λω2 λw = λ 3 λω λΩ λv = λ 3 λΩ2 λw
(23)
where, λ j = jp jm represent the scaling law, j = l, R, ρ , and the subscripts p and m represent the prototype and model, respectively. Considering the complete similitude condition and assuming that λl = λR = λh = λ and λρ = λE = λμ = 1, Eq. (22) can be rearranged to look
θ
According to Eq. (1) to Eq. (8) and the similitude theory, it can be shown that ⎧ λu λw ⎪ ⎪ λNx = λu = = λv = , ⎪ ⎪ λ λ ⎪ ⎪ ⎪ ⎪ ⎨ λ = λ = λu = λ = λw , Nθ u v λ λ ⎪ ⎪ λ ⎪ w ⎪ λNxθ = λu = λv = , ⎪ ⎪ ⎪ λ ⎪ ⎩ λMx = λMθ = λMxθ = λ λw .
(24)
Substituting Eq. (24) into Eq. (23) to yield
λu = λv = λw ; λω = λΩ = 3.1.2
1 . λ
(25)
Scaling laws of linear vibration response
According to Eq. (13) we have:
λF0 . λ
(26)
λw λF0 = . λ λ
(27)
λF(t) = Substituting Eq. (26) and Eq. (25) into Eq. (12)
It is evident that Eq. (27) indicates the scaling laws:
λw = λF0 .
(28)
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3.2
59
Non-linear scaling laws
From Section 3.1 above, the scaling law of natural frequency is shown in Eq. (25). According to Eq. (14) and Eq. (13), we have ⎧ λF0 ⎪ ⎪ , ⎨ λF(t) = λ (29) ⎪ λw3 ⎪λ ⎩ Anonlin = 3 . λ Substituting Eq. (29) and Eq. (25) into Eq. (12) to yield
λw λw3 λF0 = 3= . λ λ λ
(30)
It is evident that Eq. (30) indicates the following scaling laws
λw = λF0 = λ .
(31)
Eq. (19) and Eq. (20) along with the similitude theory give 3 2 λF0 λA1,6 λA1,6 λB1,6 = = , λ2 λ4 λ4
(32a)
3 2 λF0 λB1,6 λB1,6 λA1,6 = 2 = 4 = , λ λ λ4
(32b)
λω2 λA1,6 = λω2 λB1,6 while Eq. (30) indicates
λA1,6 = λB1,6 = λF0 = λ .
(33)
Comparing Eq. (25), Eq. (28) and Eq. (31), one can find that the scaling law of natural frequency are the same for both linear and non-linear vibrations. However, in linear vibrations, the amplitude of the vibration response’s scaling law is only related to the scaling law of the exciting force, and they are independent of the size scaling ratio. In non-linear vibrations, the scaling law of vibration response is correlated with the geometric scaling law λ , thus there are more restrictions in the design of non-linear vibrations of cylindrical shells than those of linear vibration.
4 Distorted scaling laws 4.1 4.1.1
Linear distorted scaling laws Scaling laws of natural frequency
In distorted models, the condition λl = λR = λh = λ and λE = λρ = 1 cannot be satisfied. It is noted that deriving the distorted scaling laws from Eq. (22) is very involved. Considering the Clamped-Free (C-F) boundary condition, the solutions of Eq. (9) to Eq. (11) for circumference wave number n and axial wave number m = 1 are as follows [20]: ⎧ ∞ ∞ ⎪ ⎪ θ ,t) = Umn (x, θ )eiωmn t , u(x, ⎪ ∑ ∑ ⎪ ⎪ ⎪ m=0 n=0 ⎪ ⎪ ∞ ∞ ⎨ v(x, θ ,t) = ∑ ∑ Vmn (x, θ )eiωmn t , (m = 1; n = 1, 2, 3...) (34) ⎪ m=0 n=0 ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ ⎪ iω t ⎪ ⎩ w(x, θ ,t) = ∑ ∑ Wmn (x, θ )e mn . m=0 n=0
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Substituting Eq. (34) into Eq. (9), Eq. (10) and Eq. (11), the governing equations of the rotating cylindrical shell can be re-written as L11Umn + L12Vmn + L13Wmn = 0,
(35a)
L21Umn + L22Vmn + L23Wmn = 0,
(35b)
L31Umn + L32Vmn + L33Wmn = 0,
(35c)
where, Li j (i, j = 1, 2, 3) are differential operators defined in the followings L11 = −ρ hω 2 + K
2 1− μ ∂2 ∂2 2 ∂ + K + ρ hΩ ; ∂ x2 2R2 ∂ θ 2 ∂θ2
1+ μ ∂2 ; 2R ∂ x∂ θ μ ∂ ∂ − ρ hΩ2 R ; L13 = K R ∂x ∂x 1+μ ∂2 + ρ hΩ2 R2 ) ; L21 = (K 2R ∂ x∂ θ 1− μ ∂2 1 ∂2 (1 − μ ) ∂ 2 1 ∂2 + K + D + D ; L22 = −ρ hω 2 + ρ hΩ2 + K 2 ∂ x2 R2 ∂ θ 2 2R2 ∂ x2 R4 ∂ θ 2 1 ∂ 1 ∂3 1 ∂3 −D 4 3 −D 2 2 ; L23 = −2ρ hΩω + K 2 R ∂θ R ∂θ R ∂x ∂θ μ ∂ ; L31 = K R ∂x 1 1 ∂3 1 ∂3 ∂ −D 4 3 −D 2 2 ; L32 = −2ρ hΩω + (K 2 + ρ hΩ2 ) R ∂θ R ∂θ R ∂x ∂θ 4 4 1 1 ∂ 2 ∂ ∂4 ∂2 L33 = ρ hω 2 − ρ hΩ2 + K 2 + D 4 + D 4 4 + D 2 2 2 − ρ hΩ2 2 . R ∂x R ∂θ R ∂x ∂θ ∂θ L12 = K
Of these differential operators, only L11 , L22 , L33 and L23 , L32 are related to the frequency, ω . Considering L11 , L22 , L33 , one has
λE λh λE λh = 2 = λρ λh λΩ2 , λl2 λR
(36a)
λL22 = λρ λh λω2 =
λE λh λE λh λE λh3 λE λh3 = 2 = 2 2= = λρ λh λΩ2 , λl2 λR λR λl λR4
(36b)
λL33 = λρ λh λω2 =
λE λh λE λh3 λE λh3 λE λh3 = = = 2 2 = λρ λh λΩ2 . λR2 λl4 λR4 λR λl
(36c)
λL11 = λρ λh λω2 =
As a result, the equations below can be obtained based on Eq. (17) λE 1 λω = λΩ = , λρ λl
λω = λΩ =
λω = λΩ =
(37a)
λE 1 , λρ λR
(37b)
λE λh , λρ λl2
(37c)
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λE λh , λρ λl λR λE λh λω = λΩ = . λρ λR2
λω = λΩ =
(37d)
(37e)
Although L23 and L32 contain frequency information, however, they follow the same scaling laws in Eq. (37). The scaling law can be obtained by taking a geometrical mean between Eq. (37a) and Eq. (37d) to yield Eq. (38) (The detail of the particular approach is found in Ref. [19, 21]) 12 λE λh / λω = λΩ = , (38) λρ λl λ 1/2 R
Equation (38) is a first-order distorted scaling law of the cylindrical shell with the C-F boundary condition. In most situations, the condition λl = λR = λ is satisfied. This leads to 12 λE λh / λω = λΩ = . (39) λρ λ 3/2 4.1.2
Scaling laws of linear vibration response
Considering the Clamped-Free (C-F) boundary condition, the solutions of Eq. (1) for circumference wave number n and a axial wave number m=1 are [22]: ⎧ ∞ ∞ ⎪ ⎪ θ ,t) = u(x, ⎪ ∑ ∑ Umn(x, θ )Tmn (t), ⎪ ⎪ ⎪ m=1 n=1 ⎪ ⎪ ∞ ∞ ⎨ v(x, θ ,t) = ∑ ∑ Vmn (x, θ )Tmn (t), (40) ⎪ m=1 n=1 ⎪ ⎪ ⎪ ∞ ∞ ⎪ ⎪ ⎪ ⎪ θ ,t) = w(x, ⎩ ∑ ∑ Wmn (x, θ )Tmn (t). m=1 n=1
Substituting Eq. (40) into Eq. (35) and adding an excitation, one has Tmn (t)LU − ρ hT¨mn (t)U = Q,
(41)
where ⎤ L11 L12 L13 L = ⎣ L21 L22 L23 ⎦ , L31 L32 L33 T
U = Umn Vmn Wmn , T
Q = −qX (x, θ ,t) −qθ (x, θ ,t) −qZ (x, θ ,t) . ⎡
Consider the free vibration equation LU= − ρ hω 2 U,
(42)
and substitute Eq. (42) into Eq. (41) to yield
ρ h T¨mn (t) + Tmn (t)ω 2 U = Q.
(43)
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Considering the orthogonality of the vibrational mode [21]: ˆ lˆ 0
where δsk =
2π
(UmnsUmnk +VmnsVmnk +WmnsWmnk )Rdθ dx = δsk Mmn ,
(44)
0
´ l ´ 2π 1s=k 2 +V 2 + , Mmn is the generalized mass of the (m, n) order that Mmn = 0 0 ρ h(Umn mn 0 s = k
2 )Rdθ dx. Wmn Multiply Umn , Vmn and Wmn on both side of Eq. (44) to generate
Gmn (t) ψ , T¨mn (t) − ω T˙mn (t) − ω 2 Tmn (t) = 2π Mmn
(45)
´ l ´ 2π where Gmn (t) = 0 0 (Umn qx +Vmn qθ +Wmn qz )Rdθ dx is the generalized force of the (m, n) order. According to Eq. (10), Tmn (t) can be formed by the Duhamel integration as follows 1 Tmn (t) = ω¯ Mmn
ˆ
t
Gmn (τ ) sin ω¯ (t − τ )d τ .
(46)
0
Considering the impulse force f (t) that applies at (x0 , θ0 ) along the radial direction, the impact can be written as [19]: ⎧ ⎪ ⎨ qx (x, θ ,t) = 0, qθ (x, θ ,t) = 0, (47) ⎪ ⎩ qz (x, θ ,t) = − f (t)δ (θ − θ0 )δ (x − x0 ), f (t) = F0 sin(ω0 t + φ ),
(48)
As the initial velocity and initial acceleration of the coated cylindrical shell under forced vibration are both equal to 0, according to Eq. (5), the amplitude of the force response is: ∞
∞
1 w(x, θ ,t) = ∑ ∑ [− ω¯ Mmn m=1 n=1 ∞
=
∑
∞
∑ {−
m=1 n=1
ˆ 0
t
f (τ )Δ−1 S
ˆ lˆ 0
2π
Wmn δ (θ − θ0 )δ (x − x0 )Rdθ dx·sin ω¯ (t − τ )d τ ]Wmn (x, θ )
0
B∗ F0 Δ−1 S (1 − cos ω¯ 2 t)}Wmn (x, θ ), ∗ 2 2 2 2 A ρ hω (q − ωo /ω )
(49)
´ l ´ 2π 2 ´ ´ 2 +W 2 dθ dx and B∗ = l 2π W δ (θ − θ )δ (x − x )dθ dx, with Δ being the +Vmn where A∗ = 0 0 Umn mn 0 0 S mn 0 0 impact area. According to Eq. (14), Wmn (x, θ )B∗ A∗ is dimensionless. Thus the impulse response of the cantilever coated thin-wall cylindrical shell is only correlated to the impact, natural frequency and mass per unit area. These parameter are represented by Ξ: Ξ=
F0 Δ−1 S sin(ω0 t + φ ). 2 ρ hω 2 1 − ω02 ω 2
(50)
From Eq. (50), the scaling law of the response amplitude can be obtained
λw = λΞ =
λF0 λ −2 λF λ = 0 2. 2 λρ λh λω λE λh
(51)
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4.2
63
Non-linear distorted scaling laws
According to Eq. (19) and Eq. (20) Eq. (21), the followings can be derived
λA3 λA1,6 λB21,6 λF0 = 1,6 = , λρ λh λ λ4 λ4
(52a)
λB31,6 λB1,6 λA21,6 λF0 = = 4 = . λρ λh λ λ λ4
(52b)
λω2 λA1,6 = λω2 λB1,6 Considering also Eq. (17) we have
λw = λB1,6 = λA1,6 = λω λ = 2
λE 1/2 1/2 λ λ , λρ h
λF0 = λρ λh λ λω2 λA1,6 = λρ λh λ 3 λω3 =
52 λE3 λh / . λρ λ 3/2
(53a)
(53b)
Eq. (51) and Eq. (53) indicate that in linear vibrations, the amplitude of the vibration response’s scaling law is related to the scaling law of the exciting force, and they are independent of the size scaling ratio. In non-linear vibrations, the scaling law of vibration response is only related to the geometric scaling laws and materials. However, Eq. (53a) is true only if Eq. (53b) is satisfied, making it more restrictive in designing non-linear scaled models than linear ones.
5 Examples 5.1
Complete similitude case
The parameters used for the prototype rotating thin-wall short cylindrical shell are provided in Table 1. The linear vibration frequencies of the prototype are: Table 1 Parameters of the prototype. Parameters
Value
Density ρ (kg·m−1 )
7850
Yong’s modulus
E 1 (N·m2 )
2.06×1011
Poisson’s ratio μ
0.3
Exciting force F0 (N)
10
Rotating speed ωr
(rad·s−1 )
10
Radius R(mm)
150
Length l(mm)
335
Thickness h(mm)
1
Point of action x0 (mm)
330
ω f ,1,6 = 388.02Hz, ωb,1,6 = 407.13Hz,
where subscript f and b represent the forward and backward frequency, respectively.
(54)
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Table 2 Parameters of the model. Parameters Density
ρ (kg·m−1 )
Value 7850
Yong’s modulus E 1 (N·m2 )
2.06×1011
Poisson’s ratio μ
0.3
Exciting force F0 (N)
5
Rotating speed ωr
(rad·s−1 )
20
Radius R(mm)
75
Length l(mm)
167.5
Thickness h(mm)
0.5
Point of action x0 (mm)
165
Fig. 3 Responses of the prototype rotating cylindrical shell.
The responses of the linear and non-linear vibrations of the prototype rotating cylindrical shell are shown in Fig. 3. Assuming λl = 2, according to Eq. (25) and Eq. (31), the parameters of the scaled model are shown in Table. 2 The predicted natural frequency and the non-linear response are shown in Fig. 4. The figure shows a high level of accuracy for the predicted result, thus indicating that Eq. (25) and Eq. (28) are applicable to be employed as the scaling laws of rotating thin-wall cylindrical shell under non-linear vibration. 5.2
Distorted similitude case
The parameters of the distorted rotating cylindrical shell model with λ = 2 are shown in Table 3. The values of hm seen in Table 3 are highlighted in Table 4. In Table 3, ωrm and F0m can be calculated by using Eq. (39) and Eq. (53b). The predicted natural frequency and non-linear response are shown in Fig. 5. The figure indicates that the non-linear distorted scaling laws are applicable to predicting non-linear vibration responses.
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Fig. 4 Predictive results of the model rotating cylindrical shell. Table 3 Parameters of the model. Parameters
Value
Density ρ (kg·m−1 )
2770
Yong’s modulus
E 1 (N·m2 )
7×1010
Poisson’s ratio μ
0.3
Exciting force F0 (N)
F0m
Rotating speed ωr
(rad·s−1 )
ωrm
Radius R(mm)
750
Length l(mm)
167.5
Thickness h(mm)
hm
Point of action x0 (mm)
165
Table 4 Thicknesses of the distorted models. Models
Thickness of the model shell /mm
Models
Thickness of the model shell /mm
M1
0.45
M3
0.55
M2
0.5
M4
0.6
6 Conclusions This study presented the non-linear dynamic scaling laws for the prediction of the dynamic characteristics of a rotating thin-wall short cylindrical shell. The complete and distorted scaling laws were derived from the corresponding governing equations. Prediction of natural frequency and non-linear vibration response were both numerically verified. The scaling laws of linear vibrations were also derived and compared with the non-linear scaling laws. The results showed that the non-linear scaling laws imposes restrictions on model design. It can be summarized as follows i. Corresponding governing equations were employed to derive the scaling law. The complete scaling law for frequency, λω = 1 λ , gave accurate predicted results for both forward and backward frequencies.
Zhong Luo et al. / Journal of Vibration Testing and System Dynamics 2(1) (2018) 53–67
Amplitude w /mm
66
Fig. 5 Predicted results of the distorted model rotating cylindrical shell.
ii. A geometric non-linear term was introduced in the governing equations. Using the non-linear governing equations, the scaling law of the single-point excitation response was shown to be λw = λF0 = λ . iii. The dynamic scaling laws for the linear vibration response were also discussed. In linear vibrations, the scaling law of excitation is independent of the geometric scaling law, λw = λF0 . By contrast, the non-linear scaling law of excitation was limited by the geometric scaling law, λ . iv. The distorted scaling laws for the frequency and response of the shell were also derived. Compared with the linear scaling laws, the non-linear scaling law for the vibration response was independent of the excitation force’s scaling law λF0 , although λF0 was a parameter in the distorted model. Acknowledgements This work was supported by the National Science Foundation of China under the grant number 11572082; the Fundamental Research Funds for the Central Universities of China under the grant numbers N160312001 and N150304004; and the Excellent Talents Support Program in Institutions of Higher Learning in Liaoning Province of China under the grant number LJQ2015038. References [1] Qin, Z.Y., Han, Q.K., and Chu, F.L. (2013), Analytical model of bolted disk-drum joints and its application to dynamic analysis of jointed rotor, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 228, 646-663. [2] Luo, Z., Sun, N., Wang, Y., and et al. (2013), Study of vibration characteristics of the short thin cylindrical shells and its experiment, Journal of Vibroengineering, 15(3), 1270-1283. [3] Yao, M.H., Chen, Y.P., and Zhang, W. (2012), Nonlinear vibrations of blade with varying rotating speed, Nonlinear Dynamics, 68, 487-504. [4] Yao, M.H. and Zhang, W. (2014), Using the extended Melnikov method to study multi-pulse chaotic motions of a rectangular thin plate, International Journal of Dynamics and Control, 2, 365-385.
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