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Parametric instability of a rotating truncated conical shell subjected to periodic axial loads
Accepted Manuscript Title: Parametric instability of a rotating truncated conical shellsubjected to periodic axial loads Author: Han Qinkai Chu Fulei PII: DOI: Reference:
S0093-6413(13)00119-5 http://dx.doi.org/doi:10.1016/j.mechrescom.2013.08.005 MRC 2770
To appear in: Received date: Revised date: Accepted date:
17-6-2013 31-7-2013 15-8-2013
Please cite this article as: Han Qinkai, Chu Fulei, Parametric instability of a rotating truncated conical shellsubjected to periodic axial loads, Mechanics Research Communications (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.08.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
The “Research Highlights” for the manuscript are given as follows:
Parametric instability of periodic axial loaded conical shells is
Current rotating conical shell model is corrected.
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studied.
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Primary instability regions for various natural modes are obtained.
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Effects of various system parameters on the instability regions are
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ed
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examined.
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*Manuscript
Noname manuscript No. (will be inserted by the editor)
Han Qinkai · Chu Fulei
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cr
ip t
Parametric instability of a rotating truncated conical shell subjected to periodic axial loads
Received: date / Accepted: date
an
conical shells received relatively less attention. Kornecki [6] first studied the dynamic stability of truncated conical shells under pulsating pressure. The influence of deformations prior to instability on the dynamic stability of conical shells was studied by Tani [7; 8] for the system under periodic axial loading and periodic pressure, respectively. Later, Tani [9] also investigated the dynamic stability characteristics of conical shells under pulsating torsion excitations. Massalas, Dalamangas and Tzivanidis [10] employed Galerkin’s method to reduce the base equations but used Bolotin’s method to obtain the principal instability regions. In their investigation, the formulation considered the clamped conical shells with variable modulus of elasticity. Using the Marguerre type dynamic equations, Ye [11] analyzed the non-linear vibration and dynamic instability of thin shallow conical shells subjected to periodic transverse and in-plane loads. Ng, Hua, Lam and et.al. [12] utilized the Generalized Differential Quadrature (GDQ) method to examine the effects of boundary conditions on the parametric instability of truncated conical shells under periodic edge loading. Studies have also been carried out on orthotropic truncated conical shells. Ganapathia, Patela, Sambandamb and et.al. [13] conducted dynamic stability analysis of laminated conical shells under periodic in-plane load. The influences of various parameters such as cone orthotropicity, cone angle, ply-angle and elastic edge restraint on dynamic stability were brought out. Utilizing the perturbation method, thermally induced dynamic instability of laminated composite conical shells was investigated by Wu and Chiu [14]. Recently, Sofiyev [15; 16; 17] carried out a series of work on the nonlinear stability and buckling behaviors of functionally graded truncated conical shells. The articles mentioned above concentrated mainly on the non-rotating conical shell. To the knowledge of the authors, no publication is available in the open literature that reports the effect of rotation on the dynamic stability of rotating conical shells. For the rotating cylindrical shell under periodic axial loads, Ng, Lam and Reddy [18] and Liew, Hu and Ng [19] have reported that the coriolis Page 2 of 11
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Abstract Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love’s thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the Generalized Differential Quadrature (GDQ) method. Hill’s method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.
Ac ce p
Keywords rotating conical shell · parametric instability · periodic axial loads · GDQ method 1 Introduction
Shell structures have many applications in civil, mechanical and aerospace engineering. Such elements subjected to in-plane periodic forces may undergo unstable transverse vibrations, leading to parametric instability, due to certain combinations of the values of load parameters and natural frequency of transverse vibration. The parametric instability of shell structures has been extensively studied over the last four decades. Early research progress could be found in [1; 2; 3; 4]. Recently, in depth reviews on dynamic stability behavior of plates and shells with various geometries, boundary conditions and load types were published by Sahu and Datta [5]. As the literature shown, many studies were focused on cylindrical shells, while the parametric instability of Han Qinkai Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China Tel.: +861062788308 Fax: +861062788308 E-mail: [email protected]
cr us
an
Fig. 1 Geometry and co-ordinate of a rotating thin truncated circular conical shell under periodic loading in the meridional direction.
constant load as
Na (t) = ηa Ncr (1 + ǫ cos ωa t)
(1)
M
and centrifugal forces induced by rotation have significant impacts on the instability regions. Lam [20] pointed out that the amount of analysis required to extend the study from rotating cylindrical shell to rotating conical shell is quite considerable. Also, the conical shell is a more general type of shell structures. Thus, it is necessary and meaningful to investigate the effect of rotation on the parametric instability of rotating conical shells under periodic axial loads. The GDQ method was first presented by Shu and Richards [21] to directly solve the governing equations of engineering problems. Presently, the GDQ method has been widely used for vibration analysis of shell structures [22; 23; 24]. In this paper, the method will be utilized for establishing the vibration model of rotating conical shell. As long as the rotation is considered, the rotating conical shell under periodic axial loads becomes a parametrically excited gyroscopic system. The Bolotin’s method [1] could not be used for instability analysis for such system, as the assumption of Floquet multipliers in Bolotin’s method cannot be satisfied for the gyroscopic system [25; 26]. Thus, the Hill’s method will be used for parametric instability analysis. The content of this paper is organized as follows: first, the equations of motion of a rotating conical shell subjected to periodic axial loads are derived based upon the Love’s thin shell theory and GDQ method. Then, the Hill’s method is introduced for parametric instability analysis. Based upon these, the instability regions for various natural modes are computed and discussed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are also examined in detail. Finally, some conclusions are given.
ip t
2
where ηa and ǫ, respectively, denote the relative amplitudes of constant and perturbed loads. Ncr represents the buckling load of intermediate length cylindrical shells 2 given by Timoshenko and Gere [27], i.e. Ncr = √Eh 2
Ac ce p
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R
2 Formulations for the rotating conical shell subjected to periodic axial load The truncated conical shell is isotropic and has Young’s modulus E, mass density ρ and Poisson’s ratio ν. Fig. 1 shows the geometry and co-ordinate system for a truncated circular conical shell rotating about its symmetrical and horizonal axis at an angular velocity Ω. In the figure, α is the cone angle, L the length, h the thickness, and a and b are the radii at the two ends. The reference surface of the conical shell is taken to be at its middle surface where an orthogonal co-ordinate system (x, θ, z) is fixed, and r = r(x) is a radius at any co-ordinate point (x, θ, z). The deformations of the rotating conical shell in the meridional x, circumferential θ and normal z directions are defined by u, v, w, respectively. The rotating conical shell is subjected to periodic loading in the meridional direction as shown in Fig. 1. The time-dependent axial load Na (t) is assumed to be a small and sinusoidal perturbation superimposed upon a
3(1−ν )
in which R is the radius of the cylindrical shell. In present analysis, we set R = a. The governing equations of motion of the rotating conical shell have been given in Ref. [20]. However, the expressions of inertial force and initial hoop tension are found inadequate and some terms are missing. In the following, the actual expressions of both inertial and initial hoop tension are derived, and a corrected equations of motion for the rotating conical shell are then presented.
2.1 Inertial forces, initial hoop tension and translational force induced by axial loads A unit vector (i, j, k) is defined for the co-ordinate (x, θ, z). The position vector of any point on the shell from the rotating axis is expressed as r = ui + vj + wk
(2)
And its velocity r˙ = ui ˙ + vj ˙ + wk ˙ +u
dj dk di +v +w dt dt dt
(3)
= ρ˙ + Ω × r where ρ˙ = ui ˙ + vj ˙ + wk, ˙ Ω represents the rotating speed vector. The derivative of ρ˙ with respect to time t is expressed as d ¨ + Ω × ρ˙ ρ˙ = ρ (4) dt Page 3 of 11
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Ω = −Ω cos αi + Ω sin αk
(6)
Substituting Eq. (6) into Eq. (5), the acceleration vector could be derived as ¨r = u¨ − 2vΩ ˙ sin α − (u sin α + w cos α)Ω 2 sin α i + v¨ + 2(uΩ ˙ sin α + wΩ ˙ cos α) − vΩ 2 j (7) 2 + w ¨ − 2vΩ ˙ cos α − (u sin α + w cos α)Ω cos α k
in which LT11 = ρhΩ 2
∂ ∂2 , LT12 = 0, LT13 = −ρhrΩ 2 cos α (12) 2 ∂θ ∂x
∂ ∂2 + ρhΩ 2 sin α , ∂x∂θ ∂θ ∂ T 2 T L22 = ρhrΩ sin α , L23 = 0 ∂x ∂2 ∂ LT31 = 0, LT32 = −ρhΩ 2 cos α , LT33 = ρhΩ 2 2 ∂θ ∂θ LT21 = ρhrΩ 2 cos α
where r = a + x sin α. and LI differential operators for inertial as follows I I L11 L12 LI = LI21 LI22 LI31 LI32
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The inertial force of micro unit of the rotating shell are expressed as FI = −ρh¨r (8)
ip t
˙ = 0. From Fig. For constant rotating speed, one has Ω 1, it is easy to find
cr
(5)
us
d ˙ × r + Ω × r˙ ρ˙ + Ω dt ˙ × r + Ω × ρ˙ + Ω × r ¨ + Ω × ρ˙ + Ω =ρ ˙ × r + 2Ω × ρ˙ + Ω × Ω × r ¨+Ω =ρ
¨r =
where u = [u v w]T is the displacement vector. The L represents the matrix of differential operators for the non-rotational shell, and could be found in Ref. [20]. The LT is induced by the initial hoop tension, and defined by T T T L11 L12 L13 LT = LT21 LT22 LT23 (11) LT31 LT32 LT33
an
¨=u where ρ ¨i + v¨j + wk. ¨ Thus, the acceleration vector is derived accordingly as
where
Ac ce p
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d
where the FI includes the mass and vibration inertial forces and gyroscopic force induced by the rotation. In Ref. [20], the third terms of inertial force in the i (meridional x) and j (circumferential θ) directions are not taken into account, and thus the obtained inertial force is inaccurate. It is well known that the rotation would not only bring centrifugal and gyroscopic forces, but also the initial hoop tension. The resulted internal force is expressed as follows ∂w ∂2u − r cos α i 2 ∂θ ∂x ∂u ∂v ∂2u + sin α + r sin α j + ρhΩ 2 ∂x∂θ ∂θ ∂x ∂v ∂2w − cos α k + ρhΩ 2 ∂θ2 ∂θ
FT =ρhΩ 2
(9)
In Ref. [20], the terms of direction k are expressed as 2 ρhΩ 2 ∂∂θw2 − r cos α ∂u ∂x . For the rotating cylindrical shell (α = 0), only using the expression of Eq. (9) could degenerate into the classical results [18; 19]. The internal force induced by the axial load is given by FN = ∂ ∂w ∂x Na (t) ∂x k. 2.2 Corrected formulations Considering the internal forces of the non-rotating conical shell besides the above-derived forces, and applying the force balancing theorem, one could gain the equations of motion for the rotating truncated conical circular shell as L + LT + LI + LN u = 0 (10)
LI11 = ρhΩ 2 sin2 α − ρh
(13)
(14)
denotes the matrix of force, and is expressed LI13 (15) LI23 LI33
∂2 I ∂ , L12 = 2ρhΩ sin α , 2 ∂t ∂t (16)
LI13 = ρhΩ 2 sin α cos α ∂ I ∂2 , L22 = ρhΩ 2 − ρh 2 , ∂t ∂t ∂ I L23 = −2ρhΩ cos α ∂t ∂ LI31 = ρhΩ 2 sin α cos α, LI32 = 2ρhΩ cos α , ∂t ∂2 LI33 = ρhΩ 2 cos2 α − ρh 2 ∂t
LI21 = −2ρhΩ sin α
(17)
(18)
and LN denotes the matrix of differential operators induced by axial load, and is expressed as follows N N N L11 L12 L13 N N LN = LN (19) 21 L22 L23 N N LN L L 31 32 33 where N N N N N N N LN 11 = L12 = L13 = L21 = L22 = L23 = L31 = L32 = 0 (20) 2 ∂ (21) LN 33 = Na (t) ∂x2 Compared with the model of Lam and Hua [20], the LT31 , LT32 , LI11 , LI13 , LI22 of present model have different expressions. Detailed comparisons are given in Tab. 1. Four Page 4 of 11
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Table 1 Comparisons between the Lam&Hua’s model [20] and present model.
i = 1, 2, · · · , N . Obviously, the closer both the edges of the shell, the closer is the distribution of the discrete mesh points. It is noted that the distribution includes the discrete mesh points at two edges of the shell, namely x1 = 0 and xN = L. The discrete expressions for the derivatives of V (x), W (x) could be obtained similarly with that of U (x) in Eq. (24). Imposing these expressions of GDQ method on the set of ordinary differential governing equation (Eq. (23)), a set of linear algebraic equations is obtained as L∗ u∗ x=x = R3×15 U∗15×3 x=x = 0 (25)
3 Equations of motion for the rotating conical shell by GDQ method
i
us
boundary conditions, including: clamped small edge and clamped large edge (Cs-Cl), simply supported small edge and clamped large edge (Ss-Cl), clamped small edge and simply supported large edge (Cs-Sl) and simply supported small edge and simply supported large edge (SsSl) are considered in the analysis.
cr
ip t
Present model 0 ∂ −ρhΩ 2 cos α ∂θ 2 2 ∂2 ρhΩ sin α − ρh ∂t 2 ρhΩ 2 sin α cos α ∂2 ρhΩ 2 − ρh ∂t 2
i−1 1−cos( N −1 )π L 2
i
where R3×15 is the 3 × 15 variable coefficient matrix and varies with the co-ordinate of discrete mesh point at x = xi . U∗15×3 is a column vector of mode shape on discrete mesh points and is given by U∗15×3 x=x =
an
LT31 LT32 LI11 LI13 LI22
Lam&Hua’s model [20] ∂ −ρhΩ 2 r cos α ∂x 0 ∂2 −ρh ∂t 2 0 ∂2 −ρh ∂t 2
points in the meridional x direction is chosen as xi =
i
te
W (x) cos(nθ + ̟t)]T
(22)
(xi ), U (2) (xi ), U (3) (xi ), U (4) (xi ),
V (xi ), V (1) (xi ), V (2) (xi ), V (3) (xi ), V (4) (xi ),
(26)
W (xi ), W (1) (xi ), W (2) (xi ), W (3) (xi ), W (4) (xi )]T
PN PN m m where U (m) (xi ) = j=1 Cij U (xj ), V (m) (xi ) = j=1 Cij V (xj ), P N (m) m W (xi ) = j=1 Cij W (xj ) with m = 1, 2, 3, 4. In general, there are eight boundary conditions at both ends of the rotating conical shell. Considering the given boundary conditions, and imposing Eq. (25) on all the discrete mesh points and then rearranging the resulting equation according to the natural frequency ̟, one obtains the polynomial eigenvalue problem as
d
u = [u(x, θ, t), v(x, θ, t), w(x, θ, t)] = [U (x) cos(nθ + ̟t), V (x) sin(nθ + ̟t),
[U (xi ), U
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Before the application of GDQ method, the displacement field of the conical shell for free vibration analysis could be taken as
(1)
Ac ce p
where ̟ is the natural circular frequency of the rotating conical shell, and n represents the circumferential wave number of the vibration shell. Substituting the displacement field Eq. (22) into Eq. (10), a set of ordinary differential governing equations with variable coefficients in the meridional x direction in only spatial domain is derived as L ∗ u∗ = 0 (23) where u∗ = [U (x), V (x), W (x)]T is an unknown spatial function vector of mode shape, L∗ is a 3 × 3 differential operator matrix of u∗ , and can be derived easily from the expressions of the previous section. The concept of the GDQ method is that the derivative of a sufficiently smooth function with respect to a co-ordinate direction at a discrete point can be expressed approximately by a weighted linear sum of the functional values at all the discrete points. Taking the function U (x) as an example, the basic concept of the GDQ method is written as follows N X ∂ m U (x) m = Cij U (xj ) i = 1, 2, · · · , N ∂xm x=xi
(24)
j=1
where N is the number of total discrete mesh points in m the x direction. Cij is the weighting coefficient related to the mth order derivative and can be obtained easily from Ref. [12]. A cosine distribution of discrete mesh
2 ̟ H2 + ̟H1 + (H0 + η0 Ncr Ha ) d = 0
(27)
where H0 , H1 , H2 , Ha are the J × J coefficient matrices (J = 3N − 8). It is noted that the Ha is induced by the axial load solely. d is the Jth order column vector of mode shape on the discrete mesh points and written as d =[U (x2 ), U (x3 ), · · · , U (xN −2 ), U (xN −1 ), V (x2 ), V (x3 ), · · · , V (xN −2 ), V (xN −1 ),
(28)
W (x3 ), W (x4 ), · · · , W (xN −3 ), W (xN −2 )]T
By solving the polynomial eigenvalue problem Eq. (27), the backward and forward frequencies of the rotating conical shell with certain rotating speed could be obtained. For the derivation of equations of motion, the displacement field of the conical shell could be taken as u =[U (x)(cos nθq(t) − sin nθp(t)), V (x)(sin nθq(t) + cos nθp(t)), W (x)(cos nθq(t) − sin nθp(t))]
(29) T
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M¨f + Gf˙ + (K0 + ǫKa cos ωa t)f = 0
(30)
.. . · · · 0 · · · 0 · · · I · · · 0 · · · .. . ···
(37)
us
cr
Substituting Eqs. (33-36) into Eq. (30) and applying the harmonic balance theorem, the following algebraic equations are gained 2 2 λ ωa IM + 2iλωa2 IM J1 − ωa2 IM J2 + λωa IG + iωa IG J1 1 + ¯IK + ǫ˜IK J3 An = 0 2 (38) where IM = diag[· · · M · · · ], IG = diag[· · · G · · · ], ¯IK = diag[· · · K0 · · · ] and ˜IK = diag[· · · Ka · · · ]. In order for Eq. (30) to admit a non-trivial solution of form Eq. (32), the determinant of the coefficients’s matrix of Eq. (38) must vanish (ωa 6= 0)
d
M
Obviously, considering the periodical axial load, the governing equations of the conical thin shell would have time-periodic coefficients of the Mathieu-Hill type. Such system is also called the parametrically excited system, and the parametric instability behaviors of the system is of the primary concern.
.. . 0 I 0 I .. .
an
where f = [q1 (t), q2 (t), · · · , qJ (t), p1 (t), p2 (t), · · · , pJ (t)]T is the 2J×1 column vector of the generalized co-ordinates, and the mass, gyroscopic and stiffness matrices are written as follows H2 0 0 −H1 M= , G= , 0 H2 H1 0 H0 + η0 Ncr Ha 0 K0 = , (31) 0 H0 + η0 Ncr Ha η N H 0 Ka = 0 cr a 0 η0 Ncr Ha
.. .. . . 0I I0 0I 00 . . · · · .. ..
· · · · · · · · · J3 = · · · · · ·
ip t
Similar with the derivation of polynomial eigenvalue prob- in which lem, the equations of motion for the rotating conical shell subjected to periodic axial loads could be obtained as
4 Parametric instability analysis
f = eλωa t
Ac ce p
te
The stability of the solutions of Eq. (30) will be studied via the Hill’s Method. This method is based on the Floquet theory, and the main concept is that a solution of Eq. (30) can be written as a product of an exponential part and a 2π periodic part. Representing the periodic part by its complex Fourier series expansion, this solution can be written as ∞ X
k=−∞
fk eikωa t ≈ eλωa t
Nk X
fk eikωa t
(32)
k=−Nk
1 IG det λ2 IM + λ 2iIM J1 + ωa 1 1 ˜ 1 + − IM J2 + i IG J1 + 2 ¯IK + ǫIK J3 = 0 2 ωa ωa 2ωa (39)
This equation can be used to calculate the λ values corresponding to stability for given frequency ωa and relative amplitude ǫ of periodic axial loads. If the system is stable, the real part of all eigenvalues λ is negative and the exponential part diminishes as the time passes. In another hand, if at least one of the eigenvalues has a positive part, the system is unstable. In order to get approximate numerical eigenvalues for the stability analysis, only a few number of Nk is needed to meet the precision requirement.
√ where where i = −1, λ represents the Floquet (or characteristic) exponent and fk are the complex Fourier coefficients vectors. Introducing Φ = [· · · Ie−iωa t I Ieiωa t · · · ] 5 Computations and discussions and An = [· · · f−1 f0 f1 · · · ]T (in which I is the iden5.1 Model validation and comparisons tify matrix), one obtains f = eλωa t ΦAn
(33)
f˙ = λωa eλωa t ΦAn + iωa eλωa t ΦJ1 An
(34)
¨f = λ2 ω 2 eλωa t ΦAn +2iλω 2 eλωa t ΦJ1 An −ω 2 eλωa t ΦJ2 An a a a (35) where J1 = diag[· · · kI · · · ] and J2 = diag[· · · k 2 I · · · ]. Through trigonometric calculations, the cos ωa tΦ could be rewritten as cos ωa tΦ =
1 1 −iωa t e + eiωa t Φ = ΦJ3 2 2
(36)
Before the discussion, the present model and the results by GDQ method should be verified. Let ωs denote frequencies of the non-rotational shell, and backward and forward frequencies of certain mode (m, n) are represented by ωb and ωf , where n = 1, 2, · · · represents the number of circumferential waves and m = 1, 2, · · · is the number of axial half-waves in the corresponding standing wave pattern. are defined p p The non-dimensional values as: Ω ∗ = Ω ρa2 (1 − ν 2 )/E, ωa∗ = ωa ρa2 (1 − ν 2 )/E, p p ωs∗ = ωs pρa2 (1 − ν 2 )/E, ωb∗ = ωb ρa2 (1 − ν 2 )/E and ωf∗ = ωf ρa2 (1 − ν 2 )/E. Tab. 2 gives comparisons of Page 6 of 11
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Table 2 Comparisons of frequency parameter ωs∗ for nonrotating isotropic conical shells with the Cs-Cl and Ss-Sl boundary conditions (m = 1, ν = 0.3, α = 30o , h/b = 0.01).
2 3 4 5 6 7
0.3094 0.2078 0.1494 0.1267 0.1306 0.1498
0.3015 0.1938 0.1321 0.1156 0.1314 0.1596
0.3103 0.2102 0.1533 0.1323 0.1375 0.1582
0.3100 0.2094 0.1524 0.1315 0.1366 0.1569
f b
0
0.02
0.04
(d)
0.22
Backward
0.16
0
Table 3 Comparisons of backward and forward frequencies ωb∗ , ωf∗ for rotating isotropic cylindrical shells with the Ss-Sl boundary condition (ν = 0.3, α = 0o , h = a/500, L = 5a). Sun [29] 0.1866 0.1376 0.0887 0.0382 0.0399 0.0673 0.1026
Present 0.1875 0.1385 0.0868 0.0386 0.0402 0.0674 0.1027
The present results are obtained for NGP=16.
0.04
Forward
0
Ω*
0.02
0.04
Ω*
Fig. 2 Variation of backward and forward frequencies with rotating speed Ω ∗ for the conical shell (m = n = 1, ν = 0.3, α = 30o , h/a = 0.01, L/a = 6) with various boundary conditions: (a) Cs-Cl; (b) Ss-Sl; (c) Cs-Sl and (d) Ss-Cl. Solid and dash lines represent the frequency results of present model and Lam&Hua’s model, respectively. Table 4 Frequency comparisons between the Lam&Hua’s model and present model with various boundary conditions (m = n = 1, ν = 0.3, α = 30o , h/a = 0.01, L/a = 6). Ω∗ 0.05
te
d
frequency parameter ωs∗ for non-rotating isotropic conical shells with the Cs-Cl and Ss-Sl boundary conditions. The results obtained by numerical integration method in Ref. [28] are utilized as benchmark values. Moreover, when the rotation is considered, both the backward and forward frequencies with various rotating speeds for the rotating isotropic cylindrical shell (α = 0) are compared with the results of Sun, Chu and Cao [29], as shown in Tab. 3. It is notedp that the benchmark values are all dimensionless with ρa2 (1 − ν 2 )/E in order for comparisons with the present study. Tab. 2 shows good agreement and convergence of the GDQ method with increasing discrete mesh points for the non-rotating isotropic conical shell. Similar good agreement could also be found in Tab. 3, indicating that the present model is correct and the frequency results obtained by GDQ method are believable. In the following, number of total gird points is set to be 16 for precision requirements. Fig.2 gives the variation of backward and forward frequencies with rotating speed for the conical shell with four boundary conditions. Both the frequency results of present model and Lam&Hua’s model are given in the figure. It is shown that for Cs-Cl (or Ss-Cl) the values of backward and forward frequencies of present model are lower than that of Lam&Hua’s model [20]. With the increasing of rotating speed, the difference becomes more and more distinct. However, if the shell is under Ss-Sl (or Cs-Sl) boundary condition, only the forward frequencies have greater differences for high rotating speed, while the backward frequencies of the two model almost maintain the same. Thus, one can say that for the conical shell sys-
Ac ce p
0.16
an
(1, 3)
0 0.05 0.1 0 0.01 0.03 0.05
∗ ωf Present 0.1875 0.2336 0.2765 0.0386 0.0523 0.1036 0.1631
0.18
M
(1, 1)
∗ ωb Sun [29] 0.1866 0.2326 0.2787 0.0382 0.0519 0.1035 0.1630
0.02
Backward
0.2
us
0.14
0.04
*
(c)
0.18
0.02
Ω
0.2
Forward
Ω∗
0
*
Ω
NGP represents the number of total gird points.
Modes (m, n)
Forward 0.14
0.16
ip t
0.1883 0.1169 0.0798 0.0663 0.0679 0.0760
cr
0.1889 0.1185 0.0814 0.0673 0.0684 0.0765
0.16
f
0.1987 0.1339 0.0953 0.0770 0.0757 0.0839
Forward
b
0.1881 0.1165 0.0791 0.0655 0.0667 0.0738
0.18
ω* ω*
1 2 3 4 5 6
0.2
ω* ω*
f
NGP=16
Backward
0.18
Backward
b
Present NGP=10
ω* ω*
NGP=6
b
Ss-Sl L/b sin α = 0.5
Irie [28]
(b)
f
n
0.22
ω* ω*
Cases Cs-Cl L/b sin α = 0.75
(a)
Boundary condition Cs-Cl Ss-Sl Cs-Sl Ss-Cl
0.075
Cs-Cl Ss-Sl Cs-Sl Ss-Cl
Lam&Hua’s model 0.1548(f) 0.2234(b) 0.1408(f) 0.1834(b) 0.1420(f) 0.1938(b) 0.1536(f) 0.2222(b) 0.1415(f) 0.2325(b) 0.1174(f) 0.1903(b) 0.1252(f) 0.1979(b) 0.1404(f) 0.2326(b)
Present model 0.1505 0.2211 0.1341 0.1847 0.1353 0.1950 0.1489 0.2193 0.1317 0.2322 0.1093 0.1923 0.1103 0.1994 0.1302 0.2318
Difference (%) 2.8% 1.0% 4.9% -0.7% 4.9% -0.6% 3.1% 1.3% 7.4% 0.1% 7.4% -1.0% 13.5% -0.8% 7.8% 0.3%
The ”b” represents the backward frequency, and ”f” denotes the forward frequency.
tem with Cs-Cl (or Ss-Cl) boundary condition the values of both backward and forward frequencies are overestimated by using the Lam&Hua’s model, especially for the higher speed range. When the Ss-Sl (or Cs-Sl) boundary condition is considered, only the forward frequencies would be overestimated. In order to quantitatively illustrate the frequency differences between the present model and Lam&Hua’s model, Tab. 4 gives the detailed frequency comparisons for the rotating conical shell with two rotating speeds (Ω ∗ = 0.2, 0.3) and various boundary conditions. It is shown that for Ω ∗ = 0.2 the maximal difference is lower than 5%. However, when the rotating speed is Ω ∗ = 0.3, the maximal difference is 13.5%, which is unacceptable in engineering applications. Thus, for higher rotating speeds, our model should be used to avoid distinct errors. Page 7 of 11
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−3
−4
x 10
Nk=1
Max(ℜ(λ))
Nk=2
1.5 1 0.5 0 0.1364
1
0.1366
0.1368
0.137
0.1372
0.1374
0.1368
0.137
0.1372
0.1374
−3
1.5
x 10
0.25
0.2502
0.2504
0.2506
0.2508
0.251
ω*a
0 0.1364
(b)
7
N =1
6
Nk=2
k
1.5
5 4 3
x 10
(c) 1 0.5 0 0.1364
2
0.1366
0.1368
0.137
0.1372
0.138
0.1382
0.1374
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0.1376
0.1378
0.138
0.1382
0.1384
0.1376
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0.138
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0.1384
* a
ω
1 0.2055
0.206
0.2065
0.207
0.2075
0.208
0.2085
ω*
a
Fig. 3 Variation of Max(ℜ(λ)) with excitation frequency for the non-rotating conical shell (m = n = 1, ν = 0.3, α = 20o , h/b = 0.008, L/a = 10) with (a) Cs-Cl boundary condition and (b) Ss-Cl boundary condition. Dash lines represent the instability region obtained by Ng et.al. [12].
Fig. 4 Variation of primary instability region with rotating speed Ω ∗ for the conical shell (m = 1, n = 2, ν = 0.3, α = 20o , h/b = 0.008, L/a = 10) with Cs-Cl boundary condition and tensile axial load η0 = 0.2: (a) Ω ∗ = 0; (b) Ω ∗ = 0.002; (c) Ω ∗ = 0.005. From inside to outside are: ǫ = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, respectively.
M
ωa∗
an
0 0.205
0.1366 −3
Max(ℜ(λ))
Max(ℜ(λ))
8
0.5
0.1378
cr
0.2498
−4
x 10
1
us
0.2496
Max(ℜ(λ))
(b)
0 0.2494
0.1376
ip t
(a)
x 10
(a) Max(ℜ(λ))
2
2
Ac ce p
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d
ditions and tensile axial load with η0 = 0.2 and ǫ = From Section 4, one can see that the choose of Nk 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 are considered, and presented in should be analyzed for getting relatively accurate eigen- Fig. 4, respectively. It is shown from Fig. 4 that with the values for the stability analysis. Considering the non- increasing of ǫ the instability region is enlarged symmetrotating conical shell and Nk = 1, 2, variations of max- rically. The excitation frequency corresponding to the imal real parts of Floquet exponents (Max(ℜ(λ))) with symmetrical axis is just the sum of backward and forexcitation frequency ωa∗ are given in Fig. 3. The results ward frequencies, i.e. ω ∗ = ω ∗ + ω ∗ . Obviously, for the a b f obtained by Ng et.al. [12] are also given in the figures. Ac- non-rotating conical shell, one has the ωa∗ = 2ωs∗ . As cording to the parametric stability theory, for Max(ℜ(λ)) > long as the rotation is considered, the instability region 0 the system is unstable and parametric resonance oc- moves towards the higher frequency range, especially for curs. Thus, the frequency ranges with Max(ℜ(λ)) > 0 the greater rotating speed Ω ∗ = 0.005. However, for cerin the figures are instability regions, and are also called tain ǫ, the range of instability region seems to be little the primary instability regions because their ranges are affected by the rotating speed. Similar phenomena could around the twice of natural frequency. It is shown from also be found for the other natural modes with Ss-Sl, Fig. 3 that the results of Nk = 1 and Nk = 2 are almost Cs-Sl and Ss-Cl boundary conditions. Due to the space the same. Thus, for the primary instability regions, the limitations, the results are not given here. In the followNk = 1 could meet the precision requirements. ing, only the Cs-Cl boundary condition is considered and Compared with the results of Ng. et.al.[12], the re- various system parameters are discussed for their effects sults of present study are in good agreement generally upon the instability regions. and the instability regions slightly move towards the lower frequency range. From Fig. 3 (a) and (b), one can see that the movement is about 0.0002 for Cs-Cl boundary condition and 0.0005 for Ss-Cl boundary condition. 5.3 Effects of system parameters upon instability As the rotation is not considered (Ω ∗ = 0), so our model regions is the same with that of Ng. et.al.[12]. The slight movements of instability regions might be caused by numerical 5.3.1 Constant axial load calculation errors. Variations of the primary instability regions for various modes with constant axial loads upon are plotted in Fig. 5.2 Parametric Instability under various rotating speeds 5. Three values of constant axial loads are considered (η0 = 0.1, 0.2, 0.3) and the relative amplitude is fixed Variation of primary instability region with rotating speed as ǫ = 0.5. The rotating speed maintains the same, i.e. Ω ∗ = 0, 0.002, 0.005 are computed. Four boundary con- Ω ∗ = 0.005. With the increasing of η0 , the instability rePage 8 of 11
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−4 −3
6 η =0.1 0
η =0.2
2
0
η =0.3 0
1
0 0.2475
0.248
0.2485
0.249
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(a) 4
6
0.14
0.16
η =0.1 0
4
η =0.2 0
η =0.3
2
0
0.0845
0.085
0.0855
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0.087
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0.088
0.0885
0.2
x 10
(b) 4
Max(ℜ(λ))
η =0.1 0
η =0.2
0.005
0
η =0.3 0
0.065
0.066
0.067
0.068
0.069
0.26
α=60
0.07
0.075
(c) 0.005
0.08
0.085
0.09
α=20 α=40 α=60
0 0.056
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α=40
0 0.065
0.089
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α=20
2
0.01 (c)
0 0.064
0.18
cr
(b)
Max(ℜ(λ))
Max(ℜ(λ))
α=60
0.12 −3
0.01 Max(ℜ(λ))
α=40
0
0.25
x 10
0 0.084
α=20
2
−3
6
x 10
ip t
(a)
us
3
x 10
Max(ℜ(λ))
Max(ℜ(λ))
4
* a
ω
0.058
0.06
0.062
0.064
0.066
0.068
0.07
0.072
* a
an
ω
Fig. 6 Effects of cone angle upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with CsCl boundary condition are: ν = 0.3, h/b = 0.008, L/a = 10 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
gion moves towards the higher frequency range. This is because increasing the constant axial load will increase the values of natural frequencies, and then makes the starting points of instability regions shift to the higher speed range. For the given ǫ, increasing the constant axial load also increase the amplitude of periodic axial load and the parametric stiffness excitation is enhanced. Thus, the instability ranges are also widened significantly, as show in the figure. With the number of circumferential waves increasing (from mode (1, 1) to (1, 4)), similar movement and enhancement of instability region could still be found.
5.3.3 Thickness-to-radius ratio
M
Fig. 5 Effects of constant axial loads upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , h/b = 0.008, L/a = 10 and Ω ∗ = 0.005, ǫ = 0.5.
Ac ce p
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Effects of thickness-to-radius ratio upon the instability regions for various modes are investigated and presented in Fig. 7. Three values of thickness-to-radius ratio are analyzed, i.e. h/a = 0.005, 0.008, 0.012. It is shown that increasing the value of h/a will not only cause the instability region shift to higher frequency range, but also enlarge the unstable range significantly. For greater number of circumferential waves, such as mode (1, 4), the movement and enlargement of instability with h/a does not reduced. 5.3.4 Length-to-radius ratio
5.3.2 Cone angle
Here, the cone angle is considered for its effects upon the primary instability regions. Cone angles with three values (α = 20o , 40o , 60o ) are taken into account, and the results are shown in Fig. 6, respectively. For mode (1, 1), increasing the value of α makes the instability region move towards the lower frequency range, and greatly increases the instable range. For the modes (1, 3) and (1, 4), it first makes the instability region move towards the higher and then lower frequency range, as shown in Fig. 6(b) and (c). Moreover, it also increase the instability ranges. However, movement and increasing of the instability region for modes (1, 3) and (1, 4) with cone angle are much weaker than the case of mode (1, 1). It is indicated that the influence of cone angle upon the instability region becomes more and more weaker with the the number of circumferential waves increasing. This phenomena was also found by Ng. et.al. [12].
In this section, the influence of length-to-radius ratio upon the instability region is discussed, and the results for L/a = 10, 15, 20 are shown in Fig. 8. One can find that with the increasing of L/a the instability regions tends to appear in lower frequency range. For the instability mode with lower number of circumferential waves, such as mode (1, 1), besides the movement of instability region, the instability width seems to be little affected. However, for the instability mode with greater number of circumferential waves, increasing the value of L/a will also reduce the unstable width, as shown in Fig. 8(c).
6 Conclusions The equations of motion for the rotating conical shell are presented, and the parametric instability for the system under periodic axial load is analyzed based upon Page 9 of 11
9
stability region along the frequency axis, while it has little effects upon the instability width. Increasing the constant axial load would not only enhance the instability width significantly, but also leads to the movement of instability region towards to the higher (for tensile loading) frequency range. In addition, effects of various geometrical parameters upon the instability regions are examined in detail. The variation of cone angle, thickness-to-radius ratio or length-to-radius ratio all leads to the moving of instability regions along the frequency axis. Besides, the instability width will be increased for cone angle and thicknessto-radius ratio (or decreased for length-to-radius ratio). With the number of circumferential waves increasing, the influence of cone angle upon the instability region becomes weaker, while the influence of thickness-to-radius ratio almost has no change.
−4
x 10
(a)
h/a=0.005 h/a=0.008
2
h/a=0.012
0.249
0.2495
0.25
ip t
0 0.2485 −3
x 10
(b)
h/a=0.005
4
h/a=0.008 h/a=0.012
2 0 0.084
0.085
0.086
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cr
Max(ℜ(λ))
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(c)
h/a=0.005 h/a=0.008
0.005
us
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0 0.064
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0.066
0.067
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0.074
ω*
a
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x 10
L/a=10
0.14
0.16
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6
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Max(ℜ(λ))
x 10
0.5
0.24
d
L/a=20
0 0.12
Max(ℜ(λ))
References
L/a=15
2
1.5
The research work described in the paper was supported by the National Science Foundation of China under Grant No. 51075224 and 11102095.
(a)
te
Max(ℜ(λ))
4
Acknowledgments
M
Fig. 7 Effects of thickness-to-radius ratio upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , L/a = 10 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
an
Max(ℜ(λ))
4
x 10 (c)
4
0.09
0.1
0.11
0.12
0.13
0.14
0.15
L/a=10 L/a=15
2 0 0.04
0.08
L/a=20
0.045
0.05
0.055
0.06
0.065
0.07
0.075
* a
ω
Fig. 8 Effects of length-to-radius ratio upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 2) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , h/b = 0.008 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
the GDQ method and Hill’s method. Compared with the Lam&Hua’s model [20], both backward and forward frequencies might be overestimated for the Cs-Cl or Ss-Cl boundary condition. Only the forward frequencies would be overestimated for the Ss-Sl or Cs-Sl boundary condition. Based upon these, the instability regions for various natural modes are computed and discussed. Increasing the rotating speed would lead to the movements of in-
1. V. V. Bolotin, The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964. 2. R. M. Ewan-Iwanowski, On the Parametric Response of Structures. Applied Mechanics Reviews 18(9) (1965) 699702. 3. R. A. Ibrahim, Parametric Vibration, Part-III, Current Problems (1). Shock and Vibration Digest 10(3) (1978) 41-57. 4. G. J. Simitses, Instability of Dynamically Loaded Structures. Applied Mechanics Reviews 40(10) (1987) 14031408. 5. S.K. Sahu, P.K. Datta, Research advances in the dynamic stability behavior of plates and shells: 1987-2005, Part I: Conservative systems. Trans. ASME, Applied Mechanics Reviews 60 (2007) 65-75. 6. A. Kornecki, Dynamic stability of truncated conical shells under pulsating pressure, Israel Journal of Technology 4 (1966) 110-120. 7. J. Tani, Influence of deformations prior to instability on the dynamic instability of conical shells under periodic axial load. Trans. ASME, Journal of Applied Mechanics 43 (1976) 87-91. 8. J. Tani, Influence of deformation before instability on the dynamic instability of conical shells under periodic pressure. Journal of Sound and Vibration 45 (1976) 253258. 9. J. Tani, Dynamic stability of truncated conical shells under pulsating torsion. Trans. ASME, Journal of Applied Mechanics 48 (1981) 391-398. 10. C. Massalas, A. Dalamangas and G. Tzivanidis, Dynamic instability of truncated conical shells, with variable modulus of elasticity, under periodic compressive forces. Journal of Sound and Vibration 79(4) (1981) 519-528. 11. Z.M. Ye, The non-linear vibration and dynamic instability of thin shallow shells. Journal of Sound and Vibration 202(3) (1997) 303-311.
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12. T.Y. Ng, L. Hua, K.Y. Lam, C.T. Loy, Parametric instability of conical shells by the generalized differential quadrature method. International Journal for Numerical Methods in Engineering 44 (1999) 819-837. 13. M. Ganapathia, B.P. Patela, C.T. Sambandamb, M. Touratierc, Dynamic Instability Analysis of Circular Conical Shells. Composite Structures 46(1) (1999) 59-64. 14. C.P. Wu, S.J. Chiu, Thermally induced dynamic instability of laminated composite conical shells. International Journal of Solids and Structures 39(11) (2002) 3001-3021. 15. A.H. Sofiyev, The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading. International Journal of Solids and Structures 41(13) (2004) 3411-3424. 16. A.H. Sofiyev, The buckling of functionally graded truncated conical shells under dynamic axial loading. Journal of Sound and Vibration 305(4-5) (2007) 808-826. 17. A.H. Sofiyev, The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure. Composite Structures 89(3) (2009) 356-366. 18. T.Y. Ng, K.Y. Lam, J.N. Reddy, Parametric resonance of a rotating cylindrical shell subjected to perodic axial loads, Journal of Sound and Vibration 214(3) (1998) 513529. 19. K.M. Liew, Y.G. Hu, T.Y. Ng, Dynamic stability of rotating cylindrical shells subjected to periodic axial loads. International Journal of Solids and Structures 43 (2006) 7553-7570. 20. K.Y. Lam, L. Hua, Vibration analysis of a rotating truncated circular conical shell. International Journal of Solids and Structures 34(17) (1997) 2183-2197. 21. C. Shu and B.E. Richards, Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids 15 (1992) 791-798. 22. C.T. Loy, K.Y. Lam and C. Shu, Analysis of cylindrical shells using generalized quadrature. Shock and Vibration 4 (1997) 193-198. 23. L. Hua, K.Y. Lam, The generalized differential quadrature method for frequency analysis of a rotating conical shell with initial pressure. International Journal for Numerical Methods in Engineering 48 (2000) 1703-1722. 24. K.Y. Lam, L. Hua, Generalized differential quadrature for frequency of rotating multilayered conical shell. Journal of Engineering Mechanics 126 (2000) 1156-1162. 25. Y.C. Pei, Stability boundaries of a spinning rotor with parametrically excited gyroscopic system. European Journal of Mechanics A/Solids 28 (2009) 891-896. 26. Y.C. Pei, Q.C. Tan, Parametric instability of flexible disk rotating at periodically varying angular speed. Meccanica 44 (2009) 711-720. 27. S.P. Timoshenko and J.M. Gere, Theory of Elastic Stability, McGraw-Hill, New York, 1961. 28. T. Irie, G. Yamada, K. Tanaka, Natural frequencies of truncated conical shells. Journal of Sound and Vibration 92 (1984) 447-453. 29. S. Sun, S. Chu and D.Q. Cao, Vibration characteristics of thin rotating cylindrical shells with various boundary conditions. Journal of Sound and Vibration 331 (2012) 4170-4186.
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S0093-6413(13)00119-5 http://dx.doi.org/doi:10.1016/j.mechrescom.2013.08.005 MRC 2770
To appear in: Received date: Revised date: Accepted date:
17-6-2013 31-7-2013 15-8-2013
Please cite this article as: Han Qinkai, Chu Fulei, Parametric instability of a rotating truncated conical shellsubjected to periodic axial loads, Mechanics Research Communications (2013), http://dx.doi.org/10.1016/j.mechrescom.2013.08.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Highlights (for review)
The “Research Highlights” for the manuscript are given as follows:
Parametric instability of periodic axial loaded conical shells is
Current rotating conical shell model is corrected.
ip t
studied.
cr
Primary instability regions for various natural modes are obtained.
us
Effects of various system parameters on the instability regions are
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examined.
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*Manuscript
Noname manuscript No. (will be inserted by the editor)
Han Qinkai · Chu Fulei
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Parametric instability of a rotating truncated conical shell subjected to periodic axial loads
Received: date / Accepted: date
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conical shells received relatively less attention. Kornecki [6] first studied the dynamic stability of truncated conical shells under pulsating pressure. The influence of deformations prior to instability on the dynamic stability of conical shells was studied by Tani [7; 8] for the system under periodic axial loading and periodic pressure, respectively. Later, Tani [9] also investigated the dynamic stability characteristics of conical shells under pulsating torsion excitations. Massalas, Dalamangas and Tzivanidis [10] employed Galerkin’s method to reduce the base equations but used Bolotin’s method to obtain the principal instability regions. In their investigation, the formulation considered the clamped conical shells with variable modulus of elasticity. Using the Marguerre type dynamic equations, Ye [11] analyzed the non-linear vibration and dynamic instability of thin shallow conical shells subjected to periodic transverse and in-plane loads. Ng, Hua, Lam and et.al. [12] utilized the Generalized Differential Quadrature (GDQ) method to examine the effects of boundary conditions on the parametric instability of truncated conical shells under periodic edge loading. Studies have also been carried out on orthotropic truncated conical shells. Ganapathia, Patela, Sambandamb and et.al. [13] conducted dynamic stability analysis of laminated conical shells under periodic in-plane load. The influences of various parameters such as cone orthotropicity, cone angle, ply-angle and elastic edge restraint on dynamic stability were brought out. Utilizing the perturbation method, thermally induced dynamic instability of laminated composite conical shells was investigated by Wu and Chiu [14]. Recently, Sofiyev [15; 16; 17] carried out a series of work on the nonlinear stability and buckling behaviors of functionally graded truncated conical shells. The articles mentioned above concentrated mainly on the non-rotating conical shell. To the knowledge of the authors, no publication is available in the open literature that reports the effect of rotation on the dynamic stability of rotating conical shells. For the rotating cylindrical shell under periodic axial loads, Ng, Lam and Reddy [18] and Liew, Hu and Ng [19] have reported that the coriolis Page 2 of 11
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Abstract Parametric instability of a rotating truncated conical shell subjected to periodic axial loads is studied in the paper. Through deriving accurate expressions of inertial force and initial hoop tension, a rotating conical shell model is presented based upon the Love’s thin shell theory. Considering the periodic axial loads, equations of motion of the system with periodic stiffness coefficients are obtained utilizing the Generalized Differential Quadrature (GDQ) method. Hill’s method is introduced for parametric instability analysis. Primary instability regions for various natural modes are computed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are examined.
Ac ce p
Keywords rotating conical shell · parametric instability · periodic axial loads · GDQ method 1 Introduction
Shell structures have many applications in civil, mechanical and aerospace engineering. Such elements subjected to in-plane periodic forces may undergo unstable transverse vibrations, leading to parametric instability, due to certain combinations of the values of load parameters and natural frequency of transverse vibration. The parametric instability of shell structures has been extensively studied over the last four decades. Early research progress could be found in [1; 2; 3; 4]. Recently, in depth reviews on dynamic stability behavior of plates and shells with various geometries, boundary conditions and load types were published by Sahu and Datta [5]. As the literature shown, many studies were focused on cylindrical shells, while the parametric instability of Han Qinkai Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China Tel.: +861062788308 Fax: +861062788308 E-mail: [email protected]
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Fig. 1 Geometry and co-ordinate of a rotating thin truncated circular conical shell under periodic loading in the meridional direction.
constant load as
Na (t) = ηa Ncr (1 + ǫ cos ωa t)
(1)
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and centrifugal forces induced by rotation have significant impacts on the instability regions. Lam [20] pointed out that the amount of analysis required to extend the study from rotating cylindrical shell to rotating conical shell is quite considerable. Also, the conical shell is a more general type of shell structures. Thus, it is necessary and meaningful to investigate the effect of rotation on the parametric instability of rotating conical shells under periodic axial loads. The GDQ method was first presented by Shu and Richards [21] to directly solve the governing equations of engineering problems. Presently, the GDQ method has been widely used for vibration analysis of shell structures [22; 23; 24]. In this paper, the method will be utilized for establishing the vibration model of rotating conical shell. As long as the rotation is considered, the rotating conical shell under periodic axial loads becomes a parametrically excited gyroscopic system. The Bolotin’s method [1] could not be used for instability analysis for such system, as the assumption of Floquet multipliers in Bolotin’s method cannot be satisfied for the gyroscopic system [25; 26]. Thus, the Hill’s method will be used for parametric instability analysis. The content of this paper is organized as follows: first, the equations of motion of a rotating conical shell subjected to periodic axial loads are derived based upon the Love’s thin shell theory and GDQ method. Then, the Hill’s method is introduced for parametric instability analysis. Based upon these, the instability regions for various natural modes are computed and discussed. Effects of rotational speed, constant axial load, cone angle and other geometrical parameters on the location and width of various instability regions are also examined in detail. Finally, some conclusions are given.
ip t
2
where ηa and ǫ, respectively, denote the relative amplitudes of constant and perturbed loads. Ncr represents the buckling load of intermediate length cylindrical shells 2 given by Timoshenko and Gere [27], i.e. Ncr = √Eh 2
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2 Formulations for the rotating conical shell subjected to periodic axial load The truncated conical shell is isotropic and has Young’s modulus E, mass density ρ and Poisson’s ratio ν. Fig. 1 shows the geometry and co-ordinate system for a truncated circular conical shell rotating about its symmetrical and horizonal axis at an angular velocity Ω. In the figure, α is the cone angle, L the length, h the thickness, and a and b are the radii at the two ends. The reference surface of the conical shell is taken to be at its middle surface where an orthogonal co-ordinate system (x, θ, z) is fixed, and r = r(x) is a radius at any co-ordinate point (x, θ, z). The deformations of the rotating conical shell in the meridional x, circumferential θ and normal z directions are defined by u, v, w, respectively. The rotating conical shell is subjected to periodic loading in the meridional direction as shown in Fig. 1. The time-dependent axial load Na (t) is assumed to be a small and sinusoidal perturbation superimposed upon a
3(1−ν )
in which R is the radius of the cylindrical shell. In present analysis, we set R = a. The governing equations of motion of the rotating conical shell have been given in Ref. [20]. However, the expressions of inertial force and initial hoop tension are found inadequate and some terms are missing. In the following, the actual expressions of both inertial and initial hoop tension are derived, and a corrected equations of motion for the rotating conical shell are then presented.
2.1 Inertial forces, initial hoop tension and translational force induced by axial loads A unit vector (i, j, k) is defined for the co-ordinate (x, θ, z). The position vector of any point on the shell from the rotating axis is expressed as r = ui + vj + wk
(2)
And its velocity r˙ = ui ˙ + vj ˙ + wk ˙ +u
dj dk di +v +w dt dt dt
(3)
= ρ˙ + Ω × r where ρ˙ = ui ˙ + vj ˙ + wk, ˙ Ω represents the rotating speed vector. The derivative of ρ˙ with respect to time t is expressed as d ¨ + Ω × ρ˙ ρ˙ = ρ (4) dt Page 3 of 11
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Ω = −Ω cos αi + Ω sin αk
(6)
Substituting Eq. (6) into Eq. (5), the acceleration vector could be derived as ¨r = u¨ − 2vΩ ˙ sin α − (u sin α + w cos α)Ω 2 sin α i + v¨ + 2(uΩ ˙ sin α + wΩ ˙ cos α) − vΩ 2 j (7) 2 + w ¨ − 2vΩ ˙ cos α − (u sin α + w cos α)Ω cos α k
in which LT11 = ρhΩ 2
∂ ∂2 , LT12 = 0, LT13 = −ρhrΩ 2 cos α (12) 2 ∂θ ∂x
∂ ∂2 + ρhΩ 2 sin α , ∂x∂θ ∂θ ∂ T 2 T L22 = ρhrΩ sin α , L23 = 0 ∂x ∂2 ∂ LT31 = 0, LT32 = −ρhΩ 2 cos α , LT33 = ρhΩ 2 2 ∂θ ∂θ LT21 = ρhrΩ 2 cos α
where r = a + x sin α. and LI differential operators for inertial as follows I I L11 L12 LI = LI21 LI22 LI31 LI32
M
The inertial force of micro unit of the rotating shell are expressed as FI = −ρh¨r (8)
ip t
˙ = 0. From Fig. For constant rotating speed, one has Ω 1, it is easy to find
cr
(5)
us
d ˙ × r + Ω × r˙ ρ˙ + Ω dt ˙ × r + Ω × ρ˙ + Ω × r ¨ + Ω × ρ˙ + Ω =ρ ˙ × r + 2Ω × ρ˙ + Ω × Ω × r ¨+Ω =ρ
¨r =
where u = [u v w]T is the displacement vector. The L represents the matrix of differential operators for the non-rotational shell, and could be found in Ref. [20]. The LT is induced by the initial hoop tension, and defined by T T T L11 L12 L13 LT = LT21 LT22 LT23 (11) LT31 LT32 LT33
an
¨=u where ρ ¨i + v¨j + wk. ¨ Thus, the acceleration vector is derived accordingly as
where
Ac ce p
te
d
where the FI includes the mass and vibration inertial forces and gyroscopic force induced by the rotation. In Ref. [20], the third terms of inertial force in the i (meridional x) and j (circumferential θ) directions are not taken into account, and thus the obtained inertial force is inaccurate. It is well known that the rotation would not only bring centrifugal and gyroscopic forces, but also the initial hoop tension. The resulted internal force is expressed as follows ∂w ∂2u − r cos α i 2 ∂θ ∂x ∂u ∂v ∂2u + sin α + r sin α j + ρhΩ 2 ∂x∂θ ∂θ ∂x ∂v ∂2w − cos α k + ρhΩ 2 ∂θ2 ∂θ
FT =ρhΩ 2
(9)
In Ref. [20], the terms of direction k are expressed as 2 ρhΩ 2 ∂∂θw2 − r cos α ∂u ∂x . For the rotating cylindrical shell (α = 0), only using the expression of Eq. (9) could degenerate into the classical results [18; 19]. The internal force induced by the axial load is given by FN = ∂ ∂w ∂x Na (t) ∂x k. 2.2 Corrected formulations Considering the internal forces of the non-rotating conical shell besides the above-derived forces, and applying the force balancing theorem, one could gain the equations of motion for the rotating truncated conical circular shell as L + LT + LI + LN u = 0 (10)
LI11 = ρhΩ 2 sin2 α − ρh
(13)
(14)
denotes the matrix of force, and is expressed LI13 (15) LI23 LI33
∂2 I ∂ , L12 = 2ρhΩ sin α , 2 ∂t ∂t (16)
LI13 = ρhΩ 2 sin α cos α ∂ I ∂2 , L22 = ρhΩ 2 − ρh 2 , ∂t ∂t ∂ I L23 = −2ρhΩ cos α ∂t ∂ LI31 = ρhΩ 2 sin α cos α, LI32 = 2ρhΩ cos α , ∂t ∂2 LI33 = ρhΩ 2 cos2 α − ρh 2 ∂t
LI21 = −2ρhΩ sin α
(17)
(18)
and LN denotes the matrix of differential operators induced by axial load, and is expressed as follows N N N L11 L12 L13 N N LN = LN (19) 21 L22 L23 N N LN L L 31 32 33 where N N N N N N N LN 11 = L12 = L13 = L21 = L22 = L23 = L31 = L32 = 0 (20) 2 ∂ (21) LN 33 = Na (t) ∂x2 Compared with the model of Lam and Hua [20], the LT31 , LT32 , LI11 , LI13 , LI22 of present model have different expressions. Detailed comparisons are given in Tab. 1. Four Page 4 of 11
4
Table 1 Comparisons between the Lam&Hua’s model [20] and present model.
i = 1, 2, · · · , N . Obviously, the closer both the edges of the shell, the closer is the distribution of the discrete mesh points. It is noted that the distribution includes the discrete mesh points at two edges of the shell, namely x1 = 0 and xN = L. The discrete expressions for the derivatives of V (x), W (x) could be obtained similarly with that of U (x) in Eq. (24). Imposing these expressions of GDQ method on the set of ordinary differential governing equation (Eq. (23)), a set of linear algebraic equations is obtained as L∗ u∗ x=x = R3×15 U∗15×3 x=x = 0 (25)
3 Equations of motion for the rotating conical shell by GDQ method
i
us
boundary conditions, including: clamped small edge and clamped large edge (Cs-Cl), simply supported small edge and clamped large edge (Ss-Cl), clamped small edge and simply supported large edge (Cs-Sl) and simply supported small edge and simply supported large edge (SsSl) are considered in the analysis.
cr
ip t
Present model 0 ∂ −ρhΩ 2 cos α ∂θ 2 2 ∂2 ρhΩ sin α − ρh ∂t 2 ρhΩ 2 sin α cos α ∂2 ρhΩ 2 − ρh ∂t 2
i−1 1−cos( N −1 )π L 2
i
where R3×15 is the 3 × 15 variable coefficient matrix and varies with the co-ordinate of discrete mesh point at x = xi . U∗15×3 is a column vector of mode shape on discrete mesh points and is given by U∗15×3 x=x =
an
LT31 LT32 LI11 LI13 LI22
Lam&Hua’s model [20] ∂ −ρhΩ 2 r cos α ∂x 0 ∂2 −ρh ∂t 2 0 ∂2 −ρh ∂t 2
points in the meridional x direction is chosen as xi =
i
te
W (x) cos(nθ + ̟t)]T
(22)
(xi ), U (2) (xi ), U (3) (xi ), U (4) (xi ),
V (xi ), V (1) (xi ), V (2) (xi ), V (3) (xi ), V (4) (xi ),
(26)
W (xi ), W (1) (xi ), W (2) (xi ), W (3) (xi ), W (4) (xi )]T
PN PN m m where U (m) (xi ) = j=1 Cij U (xj ), V (m) (xi ) = j=1 Cij V (xj ), P N (m) m W (xi ) = j=1 Cij W (xj ) with m = 1, 2, 3, 4. In general, there are eight boundary conditions at both ends of the rotating conical shell. Considering the given boundary conditions, and imposing Eq. (25) on all the discrete mesh points and then rearranging the resulting equation according to the natural frequency ̟, one obtains the polynomial eigenvalue problem as
d
u = [u(x, θ, t), v(x, θ, t), w(x, θ, t)] = [U (x) cos(nθ + ̟t), V (x) sin(nθ + ̟t),
[U (xi ), U
M
Before the application of GDQ method, the displacement field of the conical shell for free vibration analysis could be taken as
(1)
Ac ce p
where ̟ is the natural circular frequency of the rotating conical shell, and n represents the circumferential wave number of the vibration shell. Substituting the displacement field Eq. (22) into Eq. (10), a set of ordinary differential governing equations with variable coefficients in the meridional x direction in only spatial domain is derived as L ∗ u∗ = 0 (23) where u∗ = [U (x), V (x), W (x)]T is an unknown spatial function vector of mode shape, L∗ is a 3 × 3 differential operator matrix of u∗ , and can be derived easily from the expressions of the previous section. The concept of the GDQ method is that the derivative of a sufficiently smooth function with respect to a co-ordinate direction at a discrete point can be expressed approximately by a weighted linear sum of the functional values at all the discrete points. Taking the function U (x) as an example, the basic concept of the GDQ method is written as follows N X ∂ m U (x) m = Cij U (xj ) i = 1, 2, · · · , N ∂xm x=xi
(24)
j=1
where N is the number of total discrete mesh points in m the x direction. Cij is the weighting coefficient related to the mth order derivative and can be obtained easily from Ref. [12]. A cosine distribution of discrete mesh
2 ̟ H2 + ̟H1 + (H0 + η0 Ncr Ha ) d = 0
(27)
where H0 , H1 , H2 , Ha are the J × J coefficient matrices (J = 3N − 8). It is noted that the Ha is induced by the axial load solely. d is the Jth order column vector of mode shape on the discrete mesh points and written as d =[U (x2 ), U (x3 ), · · · , U (xN −2 ), U (xN −1 ), V (x2 ), V (x3 ), · · · , V (xN −2 ), V (xN −1 ),
(28)
W (x3 ), W (x4 ), · · · , W (xN −3 ), W (xN −2 )]T
By solving the polynomial eigenvalue problem Eq. (27), the backward and forward frequencies of the rotating conical shell with certain rotating speed could be obtained. For the derivation of equations of motion, the displacement field of the conical shell could be taken as u =[U (x)(cos nθq(t) − sin nθp(t)), V (x)(sin nθq(t) + cos nθp(t)), W (x)(cos nθq(t) − sin nθp(t))]
(29) T
Page 5 of 11
5
M¨f + Gf˙ + (K0 + ǫKa cos ωa t)f = 0
(30)
.. . · · · 0 · · · 0 · · · I · · · 0 · · · .. . ···
(37)
us
cr
Substituting Eqs. (33-36) into Eq. (30) and applying the harmonic balance theorem, the following algebraic equations are gained 2 2 λ ωa IM + 2iλωa2 IM J1 − ωa2 IM J2 + λωa IG + iωa IG J1 1 + ¯IK + ǫ˜IK J3 An = 0 2 (38) where IM = diag[· · · M · · · ], IG = diag[· · · G · · · ], ¯IK = diag[· · · K0 · · · ] and ˜IK = diag[· · · Ka · · · ]. In order for Eq. (30) to admit a non-trivial solution of form Eq. (32), the determinant of the coefficients’s matrix of Eq. (38) must vanish (ωa 6= 0)
d
M
Obviously, considering the periodical axial load, the governing equations of the conical thin shell would have time-periodic coefficients of the Mathieu-Hill type. Such system is also called the parametrically excited system, and the parametric instability behaviors of the system is of the primary concern.
.. . 0 I 0 I .. .
an
where f = [q1 (t), q2 (t), · · · , qJ (t), p1 (t), p2 (t), · · · , pJ (t)]T is the 2J×1 column vector of the generalized co-ordinates, and the mass, gyroscopic and stiffness matrices are written as follows H2 0 0 −H1 M= , G= , 0 H2 H1 0 H0 + η0 Ncr Ha 0 K0 = , (31) 0 H0 + η0 Ncr Ha η N H 0 Ka = 0 cr a 0 η0 Ncr Ha
.. .. . . 0I I0 0I 00 . . · · · .. ..
· · · · · · · · · J3 = · · · · · ·
ip t
Similar with the derivation of polynomial eigenvalue prob- in which lem, the equations of motion for the rotating conical shell subjected to periodic axial loads could be obtained as
4 Parametric instability analysis
f = eλωa t
Ac ce p
te
The stability of the solutions of Eq. (30) will be studied via the Hill’s Method. This method is based on the Floquet theory, and the main concept is that a solution of Eq. (30) can be written as a product of an exponential part and a 2π periodic part. Representing the periodic part by its complex Fourier series expansion, this solution can be written as ∞ X
k=−∞
fk eikωa t ≈ eλωa t
Nk X
fk eikωa t
(32)
k=−Nk
1 IG det λ2 IM + λ 2iIM J1 + ωa 1 1 ˜ 1 + − IM J2 + i IG J1 + 2 ¯IK + ǫIK J3 = 0 2 ωa ωa 2ωa (39)
This equation can be used to calculate the λ values corresponding to stability for given frequency ωa and relative amplitude ǫ of periodic axial loads. If the system is stable, the real part of all eigenvalues λ is negative and the exponential part diminishes as the time passes. In another hand, if at least one of the eigenvalues has a positive part, the system is unstable. In order to get approximate numerical eigenvalues for the stability analysis, only a few number of Nk is needed to meet the precision requirement.
√ where where i = −1, λ represents the Floquet (or characteristic) exponent and fk are the complex Fourier coefficients vectors. Introducing Φ = [· · · Ie−iωa t I Ieiωa t · · · ] 5 Computations and discussions and An = [· · · f−1 f0 f1 · · · ]T (in which I is the iden5.1 Model validation and comparisons tify matrix), one obtains f = eλωa t ΦAn
(33)
f˙ = λωa eλωa t ΦAn + iωa eλωa t ΦJ1 An
(34)
¨f = λ2 ω 2 eλωa t ΦAn +2iλω 2 eλωa t ΦJ1 An −ω 2 eλωa t ΦJ2 An a a a (35) where J1 = diag[· · · kI · · · ] and J2 = diag[· · · k 2 I · · · ]. Through trigonometric calculations, the cos ωa tΦ could be rewritten as cos ωa tΦ =
1 1 −iωa t e + eiωa t Φ = ΦJ3 2 2
(36)
Before the discussion, the present model and the results by GDQ method should be verified. Let ωs denote frequencies of the non-rotational shell, and backward and forward frequencies of certain mode (m, n) are represented by ωb and ωf , where n = 1, 2, · · · represents the number of circumferential waves and m = 1, 2, · · · is the number of axial half-waves in the corresponding standing wave pattern. are defined p p The non-dimensional values as: Ω ∗ = Ω ρa2 (1 − ν 2 )/E, ωa∗ = ωa ρa2 (1 − ν 2 )/E, p p ωs∗ = ωs pρa2 (1 − ν 2 )/E, ωb∗ = ωb ρa2 (1 − ν 2 )/E and ωf∗ = ωf ρa2 (1 − ν 2 )/E. Tab. 2 gives comparisons of Page 6 of 11
6
Table 2 Comparisons of frequency parameter ωs∗ for nonrotating isotropic conical shells with the Cs-Cl and Ss-Sl boundary conditions (m = 1, ν = 0.3, α = 30o , h/b = 0.01).
2 3 4 5 6 7
0.3094 0.2078 0.1494 0.1267 0.1306 0.1498
0.3015 0.1938 0.1321 0.1156 0.1314 0.1596
0.3103 0.2102 0.1533 0.1323 0.1375 0.1582
0.3100 0.2094 0.1524 0.1315 0.1366 0.1569
f b
0
0.02
0.04
(d)
0.22
Backward
0.16
0
Table 3 Comparisons of backward and forward frequencies ωb∗ , ωf∗ for rotating isotropic cylindrical shells with the Ss-Sl boundary condition (ν = 0.3, α = 0o , h = a/500, L = 5a). Sun [29] 0.1866 0.1376 0.0887 0.0382 0.0399 0.0673 0.1026
Present 0.1875 0.1385 0.0868 0.0386 0.0402 0.0674 0.1027
The present results are obtained for NGP=16.
0.04
Forward
0
Ω*
0.02
0.04
Ω*
Fig. 2 Variation of backward and forward frequencies with rotating speed Ω ∗ for the conical shell (m = n = 1, ν = 0.3, α = 30o , h/a = 0.01, L/a = 6) with various boundary conditions: (a) Cs-Cl; (b) Ss-Sl; (c) Cs-Sl and (d) Ss-Cl. Solid and dash lines represent the frequency results of present model and Lam&Hua’s model, respectively. Table 4 Frequency comparisons between the Lam&Hua’s model and present model with various boundary conditions (m = n = 1, ν = 0.3, α = 30o , h/a = 0.01, L/a = 6). Ω∗ 0.05
te
d
frequency parameter ωs∗ for non-rotating isotropic conical shells with the Cs-Cl and Ss-Sl boundary conditions. The results obtained by numerical integration method in Ref. [28] are utilized as benchmark values. Moreover, when the rotation is considered, both the backward and forward frequencies with various rotating speeds for the rotating isotropic cylindrical shell (α = 0) are compared with the results of Sun, Chu and Cao [29], as shown in Tab. 3. It is notedp that the benchmark values are all dimensionless with ρa2 (1 − ν 2 )/E in order for comparisons with the present study. Tab. 2 shows good agreement and convergence of the GDQ method with increasing discrete mesh points for the non-rotating isotropic conical shell. Similar good agreement could also be found in Tab. 3, indicating that the present model is correct and the frequency results obtained by GDQ method are believable. In the following, number of total gird points is set to be 16 for precision requirements. Fig.2 gives the variation of backward and forward frequencies with rotating speed for the conical shell with four boundary conditions. Both the frequency results of present model and Lam&Hua’s model are given in the figure. It is shown that for Cs-Cl (or Ss-Cl) the values of backward and forward frequencies of present model are lower than that of Lam&Hua’s model [20]. With the increasing of rotating speed, the difference becomes more and more distinct. However, if the shell is under Ss-Sl (or Cs-Sl) boundary condition, only the forward frequencies have greater differences for high rotating speed, while the backward frequencies of the two model almost maintain the same. Thus, one can say that for the conical shell sys-
Ac ce p
0.16
an
(1, 3)
0 0.05 0.1 0 0.01 0.03 0.05
∗ ωf Present 0.1875 0.2336 0.2765 0.0386 0.0523 0.1036 0.1631
0.18
M
(1, 1)
∗ ωb Sun [29] 0.1866 0.2326 0.2787 0.0382 0.0519 0.1035 0.1630
0.02
Backward
0.2
us
0.14
0.04
*
(c)
0.18
0.02
Ω
0.2
Forward
Ω∗
0
*
Ω
NGP represents the number of total gird points.
Modes (m, n)
Forward 0.14
0.16
ip t
0.1883 0.1169 0.0798 0.0663 0.0679 0.0760
cr
0.1889 0.1185 0.0814 0.0673 0.0684 0.0765
0.16
f
0.1987 0.1339 0.0953 0.0770 0.0757 0.0839
Forward
b
0.1881 0.1165 0.0791 0.0655 0.0667 0.0738
0.18
ω* ω*
1 2 3 4 5 6
0.2
ω* ω*
f
NGP=16
Backward
0.18
Backward
b
Present NGP=10
ω* ω*
NGP=6
b
Ss-Sl L/b sin α = 0.5
Irie [28]
(b)
f
n
0.22
ω* ω*
Cases Cs-Cl L/b sin α = 0.75
(a)
Boundary condition Cs-Cl Ss-Sl Cs-Sl Ss-Cl
0.075
Cs-Cl Ss-Sl Cs-Sl Ss-Cl
Lam&Hua’s model 0.1548(f) 0.2234(b) 0.1408(f) 0.1834(b) 0.1420(f) 0.1938(b) 0.1536(f) 0.2222(b) 0.1415(f) 0.2325(b) 0.1174(f) 0.1903(b) 0.1252(f) 0.1979(b) 0.1404(f) 0.2326(b)
Present model 0.1505 0.2211 0.1341 0.1847 0.1353 0.1950 0.1489 0.2193 0.1317 0.2322 0.1093 0.1923 0.1103 0.1994 0.1302 0.2318
Difference (%) 2.8% 1.0% 4.9% -0.7% 4.9% -0.6% 3.1% 1.3% 7.4% 0.1% 7.4% -1.0% 13.5% -0.8% 7.8% 0.3%
The ”b” represents the backward frequency, and ”f” denotes the forward frequency.
tem with Cs-Cl (or Ss-Cl) boundary condition the values of both backward and forward frequencies are overestimated by using the Lam&Hua’s model, especially for the higher speed range. When the Ss-Sl (or Cs-Sl) boundary condition is considered, only the forward frequencies would be overestimated. In order to quantitatively illustrate the frequency differences between the present model and Lam&Hua’s model, Tab. 4 gives the detailed frequency comparisons for the rotating conical shell with two rotating speeds (Ω ∗ = 0.2, 0.3) and various boundary conditions. It is shown that for Ω ∗ = 0.2 the maximal difference is lower than 5%. However, when the rotating speed is Ω ∗ = 0.3, the maximal difference is 13.5%, which is unacceptable in engineering applications. Thus, for higher rotating speeds, our model should be used to avoid distinct errors. Page 7 of 11
7
−3
−4
x 10
Nk=1
Max(ℜ(λ))
Nk=2
1.5 1 0.5 0 0.1364
1
0.1366
0.1368
0.137
0.1372
0.1374
0.1368
0.137
0.1372
0.1374
−3
1.5
x 10
0.25
0.2502
0.2504
0.2506
0.2508
0.251
ω*a
0 0.1364
(b)
7
N =1
6
Nk=2
k
1.5
5 4 3
x 10
(c) 1 0.5 0 0.1364
2
0.1366
0.1368
0.137
0.1372
0.138
0.1382
0.1374
0.1384
0.1376
0.1378
0.138
0.1382
0.1384
0.1376
0.1378
0.138
0.1382
0.1384
* a
ω
1 0.2055
0.206
0.2065
0.207
0.2075
0.208
0.2085
ω*
a
Fig. 3 Variation of Max(ℜ(λ)) with excitation frequency for the non-rotating conical shell (m = n = 1, ν = 0.3, α = 20o , h/b = 0.008, L/a = 10) with (a) Cs-Cl boundary condition and (b) Ss-Cl boundary condition. Dash lines represent the instability region obtained by Ng et.al. [12].
Fig. 4 Variation of primary instability region with rotating speed Ω ∗ for the conical shell (m = 1, n = 2, ν = 0.3, α = 20o , h/b = 0.008, L/a = 10) with Cs-Cl boundary condition and tensile axial load η0 = 0.2: (a) Ω ∗ = 0; (b) Ω ∗ = 0.002; (c) Ω ∗ = 0.005. From inside to outside are: ǫ = 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, respectively.
M
ωa∗
an
0 0.205
0.1366 −3
Max(ℜ(λ))
Max(ℜ(λ))
8
0.5
0.1378
cr
0.2498
−4
x 10
1
us
0.2496
Max(ℜ(λ))
(b)
0 0.2494
0.1376
ip t
(a)
x 10
(a) Max(ℜ(λ))
2
2
Ac ce p
te
d
ditions and tensile axial load with η0 = 0.2 and ǫ = From Section 4, one can see that the choose of Nk 0.05, 0.1, 0.2, 0.3, 0.4, 0.5 are considered, and presented in should be analyzed for getting relatively accurate eigen- Fig. 4, respectively. It is shown from Fig. 4 that with the values for the stability analysis. Considering the non- increasing of ǫ the instability region is enlarged symmetrotating conical shell and Nk = 1, 2, variations of max- rically. The excitation frequency corresponding to the imal real parts of Floquet exponents (Max(ℜ(λ))) with symmetrical axis is just the sum of backward and forexcitation frequency ωa∗ are given in Fig. 3. The results ward frequencies, i.e. ω ∗ = ω ∗ + ω ∗ . Obviously, for the a b f obtained by Ng et.al. [12] are also given in the figures. Ac- non-rotating conical shell, one has the ωa∗ = 2ωs∗ . As cording to the parametric stability theory, for Max(ℜ(λ)) > long as the rotation is considered, the instability region 0 the system is unstable and parametric resonance oc- moves towards the higher frequency range, especially for curs. Thus, the frequency ranges with Max(ℜ(λ)) > 0 the greater rotating speed Ω ∗ = 0.005. However, for cerin the figures are instability regions, and are also called tain ǫ, the range of instability region seems to be little the primary instability regions because their ranges are affected by the rotating speed. Similar phenomena could around the twice of natural frequency. It is shown from also be found for the other natural modes with Ss-Sl, Fig. 3 that the results of Nk = 1 and Nk = 2 are almost Cs-Sl and Ss-Cl boundary conditions. Due to the space the same. Thus, for the primary instability regions, the limitations, the results are not given here. In the followNk = 1 could meet the precision requirements. ing, only the Cs-Cl boundary condition is considered and Compared with the results of Ng. et.al.[12], the re- various system parameters are discussed for their effects sults of present study are in good agreement generally upon the instability regions. and the instability regions slightly move towards the lower frequency range. From Fig. 3 (a) and (b), one can see that the movement is about 0.0002 for Cs-Cl boundary condition and 0.0005 for Ss-Cl boundary condition. 5.3 Effects of system parameters upon instability As the rotation is not considered (Ω ∗ = 0), so our model regions is the same with that of Ng. et.al.[12]. The slight movements of instability regions might be caused by numerical 5.3.1 Constant axial load calculation errors. Variations of the primary instability regions for various modes with constant axial loads upon are plotted in Fig. 5.2 Parametric Instability under various rotating speeds 5. Three values of constant axial loads are considered (η0 = 0.1, 0.2, 0.3) and the relative amplitude is fixed Variation of primary instability region with rotating speed as ǫ = 0.5. The rotating speed maintains the same, i.e. Ω ∗ = 0, 0.002, 0.005 are computed. Four boundary con- Ω ∗ = 0.005. With the increasing of η0 , the instability rePage 8 of 11
8
−4 −3
6 η =0.1 0
η =0.2
2
0
η =0.3 0
1
0 0.2475
0.248
0.2485
0.249
0.2495
(a) 4
6
0.14
0.16
η =0.1 0
4
η =0.2 0
η =0.3
2
0
0.0845
0.085
0.0855
0.086
0.0865
0.087
0.0875
0.088
0.0885
0.2
x 10
(b) 4
Max(ℜ(λ))
η =0.1 0
η =0.2
0.005
0
η =0.3 0
0.065
0.066
0.067
0.068
0.069
0.26
α=60
0.07
0.075
(c) 0.005
0.08
0.085
0.09
α=20 α=40 α=60
0 0.056
0.07
0.24
α=40
0 0.065
0.089
0.22
α=20
2
0.01 (c)
0 0.064
0.18
cr
(b)
Max(ℜ(λ))
Max(ℜ(λ))
α=60
0.12 −3
0.01 Max(ℜ(λ))
α=40
0
0.25
x 10
0 0.084
α=20
2
−3
6
x 10
ip t
(a)
us
3
x 10
Max(ℜ(λ))
Max(ℜ(λ))
4
* a
ω
0.058
0.06
0.062
0.064
0.066
0.068
0.07
0.072
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an
ω
Fig. 6 Effects of cone angle upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with CsCl boundary condition are: ν = 0.3, h/b = 0.008, L/a = 10 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
gion moves towards the higher frequency range. This is because increasing the constant axial load will increase the values of natural frequencies, and then makes the starting points of instability regions shift to the higher speed range. For the given ǫ, increasing the constant axial load also increase the amplitude of periodic axial load and the parametric stiffness excitation is enhanced. Thus, the instability ranges are also widened significantly, as show in the figure. With the number of circumferential waves increasing (from mode (1, 1) to (1, 4)), similar movement and enhancement of instability region could still be found.
5.3.3 Thickness-to-radius ratio
M
Fig. 5 Effects of constant axial loads upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , h/b = 0.008, L/a = 10 and Ω ∗ = 0.005, ǫ = 0.5.
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Effects of thickness-to-radius ratio upon the instability regions for various modes are investigated and presented in Fig. 7. Three values of thickness-to-radius ratio are analyzed, i.e. h/a = 0.005, 0.008, 0.012. It is shown that increasing the value of h/a will not only cause the instability region shift to higher frequency range, but also enlarge the unstable range significantly. For greater number of circumferential waves, such as mode (1, 4), the movement and enlargement of instability with h/a does not reduced. 5.3.4 Length-to-radius ratio
5.3.2 Cone angle
Here, the cone angle is considered for its effects upon the primary instability regions. Cone angles with three values (α = 20o , 40o , 60o ) are taken into account, and the results are shown in Fig. 6, respectively. For mode (1, 1), increasing the value of α makes the instability region move towards the lower frequency range, and greatly increases the instable range. For the modes (1, 3) and (1, 4), it first makes the instability region move towards the higher and then lower frequency range, as shown in Fig. 6(b) and (c). Moreover, it also increase the instability ranges. However, movement and increasing of the instability region for modes (1, 3) and (1, 4) with cone angle are much weaker than the case of mode (1, 1). It is indicated that the influence of cone angle upon the instability region becomes more and more weaker with the the number of circumferential waves increasing. This phenomena was also found by Ng. et.al. [12].
In this section, the influence of length-to-radius ratio upon the instability region is discussed, and the results for L/a = 10, 15, 20 are shown in Fig. 8. One can find that with the increasing of L/a the instability regions tends to appear in lower frequency range. For the instability mode with lower number of circumferential waves, such as mode (1, 1), besides the movement of instability region, the instability width seems to be little affected. However, for the instability mode with greater number of circumferential waves, increasing the value of L/a will also reduce the unstable width, as shown in Fig. 8(c).
6 Conclusions The equations of motion for the rotating conical shell are presented, and the parametric instability for the system under periodic axial load is analyzed based upon Page 9 of 11
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stability region along the frequency axis, while it has little effects upon the instability width. Increasing the constant axial load would not only enhance the instability width significantly, but also leads to the movement of instability region towards to the higher (for tensile loading) frequency range. In addition, effects of various geometrical parameters upon the instability regions are examined in detail. The variation of cone angle, thickness-to-radius ratio or length-to-radius ratio all leads to the moving of instability regions along the frequency axis. Besides, the instability width will be increased for cone angle and thicknessto-radius ratio (or decreased for length-to-radius ratio). With the number of circumferential waves increasing, the influence of cone angle upon the instability region becomes weaker, while the influence of thickness-to-radius ratio almost has no change.
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References
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The research work described in the paper was supported by the National Science Foundation of China under Grant No. 51075224 and 11102095.
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Acknowledgments
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Fig. 7 Effects of thickness-to-radius ratio upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 3) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , L/a = 10 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
an
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x 10 (c)
4
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L/a=20
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0.06
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Fig. 8 Effects of length-to-radius ratio upon the instability regions for various modes: (a) mode (1, 1); (b) mode (1, 2) and (c) mode (1, 4). Other parameters of the rotating conical shell with Cs-Cl boundary condition are: ν = 0.3, α = 20o , h/b = 0.008 and Ω ∗ = 0.005, η0 = 0.2, ǫ = 0.5.
the GDQ method and Hill’s method. Compared with the Lam&Hua’s model [20], both backward and forward frequencies might be overestimated for the Cs-Cl or Ss-Cl boundary condition. Only the forward frequencies would be overestimated for the Ss-Sl or Cs-Sl boundary condition. Based upon these, the instability regions for various natural modes are computed and discussed. Increasing the rotating speed would lead to the movements of in-
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