Dynamic analysis of grouped turbo blades using radial basis function

Abstract

Eigenvalue problems of grouped turbo blades were numerically formulated using a radial basis function. This study investigated different boundary conditions, which accompany the radial basis function, for transforming partial differential equations of grouped turbo blades into a discrete eigenvalue problem. The numerical results show the validity and efficiency of the radial basis function for solving this problem. The crack depth considerably affects the dynamic behavior of grouped turbo blades.

Keywords

Radial basis function (RBF)blade edge-cracked beam vibration analysis

Introduction

With the increasing availability of various numerical methods, such as the finite difference, finite element, and boundary element methods, static and dynamic solutions for numerous complex structures are now achievable. However, exploring alternative efficient techniques remains necessary. For designing turbo-machinery, high efficiency is required; therefore, the complex shape of a blade in turbomachinery is unavoidable. Because of the complex shape of a turbo blade, a pretwisted taper beam has been used to approximate it. Rao,¹ Hodges et al.,² Abrate,³ Dawson,⁴ and Dawson and Carneige⁵ have calculated the natural frequencies of single-tapered and pretwisted turbo blades by using the Rayleigh–Ritz method. Gupta and Rao⁶ applied the finite element method to locate natural frequencies of vibrations in double-tapered and twisted beams. Swaminathan and Rao⁷ solved the vibration problem of a rotating, pretwisted, and tapered blade. Subrahmanyam et al.^8,9 have determined the natural frequencies and the mode shapes of a uniform pretwisted cantilevered blade by using the Reissner method. Chen and Keer¹⁰ analyzed the transverse vibrations of a rotating pretwisted Timoshenko beam subjected to axial loading. Storiti and Aboelnaga¹¹ studied the transverse deflections of a straight-tapered symmetric beam attached to a rotating hub and used it as a model for examining the bending vibration of blades in turbo machinery. Wagner¹² calculated the forced vibration response of subsystems with various natural frequencies and damping; these subsystems were attached to a foundation with finite stiffness or mass. Griffin et al.^13–15 has determined the dynamic responses of frictionally damped turbine blades. Wagner and Griffin^16,17 have analyzed the harmonic responses of grouped-blade and flexible-disk systems. For simplicity, tapered pretwisted beams have been used to approximate blades.^18,19 To improve the dynamic behavior of a rotating shroud blade disk, the blades are shrouded and grouped frequently. The dynamic characteristics of grouped blades are crucial in turbo machinery designs. To analyze the blade-group effect on the dynamic behavior of a turbo disk, it was assumed that blades are grouped periodically. The integrity and computational efficiency of the radial basis functions (RBFs) in this problem were demonstrated through a series of case studies.

Formulation of the eigenvalue problem

The periodic shrouded-blade structure, as shown in Figures 1 and 2, consists of a rigid hub and a cyclic assembly of $N_{G}$ -grouped blades.¹⁸ In each blade group, the blade is coupled by a spring k with an adjacent blade through a shroud. The massless spring $k$ is assumed to model the function of this shroud. The dynamic characteristics of edge-cracked beams are crucial in numerous designs. The flexibility $G$ caused by a crack of depth $\bar{a}$ can be determined using the Broek approximation^19–21 as follows:

\frac{(1 - μ^{2}) K_{I}^{2}}{E} = \frac{{(P_{b})}^{2}}{2 b} \frac{dG}{d \bar{a}}

(1)

where $K_{I}$ is the stress intensity factor under mode I loading, μ is the Poisson’s ratio, $P_{b}$ denotes the bending moment at the crack, and $G$ is the flexibility of the blade. The magnitude of the stress intensity factor can be determined using the Tada formula^20,21 as follows:

K_{I} = \frac{6 P_{b}}{{bt}_{0}^{2}} \sqrt{π r^{*} t_{0}} F_{I} (r^{*})

(2)

where

F_{I} (r^{*}) = \sqrt{\frac{2}{π r^{*}} \tan (\frac{π r^{*}}{2})} \frac{0.923 + 0.199 {(1 - \sin (\frac{π r^{*}}{2}))}^{4}}{\cos (\frac{π r^{*}}{2})}

(3)

and

r^{*} = \frac{\bar{a}}{t_{0}}

(4)

Figure 1.

Geometry of the grouped turbo blade disk system.

Figure 2.

Geometry of a cracked, pretwisted, tapered blade.

Substituting the stress intensity factor $K_{I}$ into equation (1) yields

G = \frac{6 (1 - μ^{2}) t_{0} \int_{0}^{r^{*}} π r^{*} F_{I}^{2} (r^{*}) {dr}^{*}}{{EI}_{0}}

(5)

Because

k_{T} = \frac{1}{G}

(6)

the bending stiffness $k_{T}$ of the cracked section of a blade can be expressed as

k_{T} = \frac{{EI}_{0}}{6 (1 - μ^{2}) t_{0} \int_{0}^{r^{*}} π r^{*} F_{I}^{2} (r^{*}) {dr}^{*}}

(7)

The equations of motion of the $s th$ blade can be derived as follows¹⁸:

\begin{array}{l} E \frac{\partial^{2} I_{y y}}{\partial r^{2}} \frac{\partial^{2} u_{s}}{\partial r^{2}} + 2 E \frac{\partial I_{y y}}{\partial r} \frac{\partial^{3} u_{s}}{\partial r^{3}} + E I_{y y} \frac{\partial^{4} u_{s}}{\partial r^{4}} + E \frac{\partial^{2} I_{x y}}{\partial r^{2}} \frac{\partial^{2} v_{s}}{\partial r^{2}} + \\ 2 E \frac{\partial I_{x y}}{\partial r} \frac{\partial^{3} v_{s}}{\partial r^{3}} + E I_{x y} \frac{\partial^{4} v_{s}}{\partial r^{4}} - ρ Ω^{2} \frac{\partial}{\partial r} [\int_{r}^{L} A (r + r_{0}) d r] \frac{\partial u_{s}}{\partial r} - \\ ρ Ω^{2} [\int_{r}^{L} A (r + r_{0}) d r] \frac{\partial^{2} u_{s}}{\partial r^{2}} + ρ A \frac{\partial^{2} u_{s}}{\partial t^{2}} = 0 \\ for s = 1, 2, \dots, N_{G} \end{array}

(8)

\begin{array}{l} E \frac{\partial^{2} I_{x x}}{\partial r^{2}} \frac{\partial^{2} v_{s}}{\partial r^{2}} + 2 E \frac{\partial I_{x x}}{\partial r} \frac{\partial^{3} v_{s}}{\partial r^{3}} + E I_{x x} \frac{\partial^{4} v_{s}}{\partial r^{4}} + E \frac{\partial^{2} I_{x y}}{\partial r^{2}} \frac{\partial^{2} u_{s}}{\partial r^{2}} + \\ 2 E \frac{\partial I_{x y}}{\partial r} \frac{\partial^{3} u_{s}}{\partial r^{3}} + E I_{x y} \frac{\partial^{4} u_{s}}{\partial r^{4}} - ρ Ω^{2} \frac{\partial}{\partial r} [\int_{r}^{L} A (r + r_{0}) d r] \frac{\partial v_{s}}{\partial r} \\ - ρ Ω^{2} A v_{s} - ρ Ω^{2} [\int_{r}^{L} A (r + r_{0}) d r] \frac{\partial^{2} v_{s}}{\partial r^{2}} + ρ A \frac{\partial^{2} v_{s}}{\partial t^{2}} = 0 \\ f o r s = 1, 2, \dots, N_{G} \end{array}

(9)

We assume that the cross section of the blade symmetrically encompasses two principal axes, and only flexural bending can occur. Consider the cross-sectional area of the tapered pretwisted blade at position $r$ to be

A (r) = b_{0} t_{0} (1 - α \frac{r}{L}) (1 - β \frac{r}{L})

(10)

where ρ is the density of the blade, and $b_{0}$ and $t_{0}$ denote, respectively, the breadth and thickness at the root of the blade, as shown in Figures 1 and 2, and

α = \frac{b_{0} - b_{1}}{b_{0}}

(11)

β = \frac{t_{0} - t_{1}}{t_{0}}

(12)

where $b_{1}$ and $t_{1}$ are, respectively, the breadth and thickness at the blade tip. In this equation, $r_{0}$ is the radius of the hub; Ω is the rotational speed; and $I_{xx}$ , $I_{yy}$ , and $I_{xy}$ are the moments of the area. Consider the tapered blade to be pretwisted with a uniform twist angle θ; the moments of the area at position $r$ can then be derived as follows¹⁸:

I_{xx} = I_{XX} \cos^{2} (\frac{r}{L} θ) + I_{YY} \sin^{2} (\frac{r}{L} θ)

(13)

I_{yy} = I_{XX} \sin^{2} (\frac{r}{L} θ) + I_{YY} \cos^{2} (\frac{r}{L} θ)

(14)

I_{xy} = (I_{YY} - I_{XX}) \sin (\frac{r}{L} θ) \cos (\frac{r}{L} θ)

(15)

where $L$ is the length of the tapered blade

I_{XX} = \frac{b_{0} t_{0}^{3}}{12} (1 - α \frac{r}{L}) {(1 - β \frac{r}{L})}^{3}

(16)

I_{YY} = \frac{b_{0}^{3} t_{0}}{12} {(1 - α \frac{r}{L})}^{3} (1 - β \frac{r}{L})

(17)

The corresponding boundary conditions are

u_{s} (0, t) = 0 for s = 1, 2, \dots, N_{G}

(18)

v_{s} (0, t) = 0 for s = 1, 2, \dots, N_{G}

(19)

\frac{\partial u_{s} (0, t)}{\partial r} = 0 for s = 1, 2, \dots, N_{G}

(20)

A crack can be represented as a spring of zero length and zero mass.¹⁸ The corresponding boundary conditions are

EI \frac{\partial^{2} v_{1} (0, t)}{\partial r^{2}} = k_{T} \frac{\partial v_{1} (0, t)}{\partial r}

(21)

\frac{\partial v_{s} (0, t)}{\partial r} = 0 for s = 2, 3, \dots, N_{G}

(22)

\frac{\partial^{2} u_{s} (L, t)}{\partial r^{2}} = 0 for s = 1, 2, \dots, N_{G}

(23)

\frac{\partial^{2} v_{s} (L, t)}{\partial r^{2}} = 0 for s = 1, 2, \dots, N_{G}

(24)

\frac{\partial^{3} u_{s} (L, t)}{\partial r^{3}} = 0 for s = 1, 2, \dots, N_{G}

(25)

and

\begin{array}{l} E I_{x x} \frac{\partial^{3} v_{s} (L, t)}{\partial r^{3}} + k v_{s - 1} (L, t) - 2 k v_{s} (L, t) + k v_{s + 1} (L, t) = 0 \\ for s = 1, 2, \dots, N_{G} \end{array}

(26)

Consider the displacements to be of the following form:

u_{s} (r, t) = U_{s} (r) \exp (i ω t) for s = 1, 2, \dots, N_{G}

(27)

v_{s} (r, t) = V_{s} (r) \exp (i ω t) for s = 1, 2, \dots, N_{G}

(28)

Equations (8) and (9) can be simplified as follows:

\begin{array}{l} E \frac{d^{2} I_{y y}}{d r^{2}} \frac{d^{2} U_{s}}{d r^{2}} + 2 E \frac{d I_{y y}}{d r} \frac{d^{3} U_{s}}{d r^{3}} + E I_{y y} \frac{d^{4} U_{s}}{d r^{4}} \\ + E \frac{d^{2} I_{x y}}{d r^{2}} \frac{d^{2} V_{s}}{d r^{2}} + 2 E \frac{d I_{x y}}{d r} \frac{d^{3} V_{s}}{d r^{3}} + E I_{x y} \frac{d^{4} V_{s}}{d r^{4}} \\ - ρ Ω^{2} \frac{d}{d r} [\int_{r}^{L} A (r + r_{0}) d r] \frac{d U_{s}}{d r} - ρ Ω^{2} [\int_{r}^{L} A (r + r_{0}) d r] \frac{d^{2} U_{s}}{d r^{2}} = ω^{2} ρ A U_{s} \\ for s = 1, 2, \dots, N_{G} \end{array}

(29)

\begin{array}{l} E \frac{d^{2} I_{x x}}{d r^{2}} \frac{d^{2} V_{s}}{d r^{2}} + 2 E \frac{d I_{x x}}{d r} \frac{d^{3} V_{s}}{d r^{3}} + E I_{x x} \frac{d^{4} V_{s}}{d r^{4}} + E \frac{d^{2} I_{x y}}{d r^{2}} \frac{d^{2} U_{s}}{d r^{2}} + \\ 2 E \frac{d I_{x y}}{d r} \frac{d^{3} U_{s}}{d r^{3}} + E I_{x y} \frac{d^{4} U_{s}}{d r^{4}} - ρ Ω^{2} A V_{s} - \\ ρ Ω^{2} \frac{d}{d r} [\int_{r}^{L} A (r + r_{0}) d r] \frac{d V_{s}}{d r} - ρ Ω^{2} [\int_{r}^{L} A (r + r_{0}) d r] \frac{d^{2} V_{s}}{d r^{2}} = ω ρ A V_{s} \\ for s = 1, 2, \dots, N_{G} \end{array}

(30)

For simplicity, equations (29) and (30) can be rewritten in a dimensionless form as

\begin{matrix} \frac{d^{2} {\bar{I}}_{yy} (\bar{r})}{d {\bar{r}}^{2}} \frac{d^{2} {\bar{U}}_{s}}{d {\bar{r}}^{2}} + 2 \frac{d {\bar{I}}_{yy} (\bar{r})}{d \bar{r}} \frac{d^{3} {\bar{U}}_{s}}{d {\bar{r}}^{3}} + {\bar{I}}_{yy} (\bar{r}) \frac{d^{4} {\bar{U}}_{s}}{d {\bar{r}}^{4}} \\ + \frac{d^{2} {\bar{I}}_{xy} (\bar{r})}{d {\bar{r}}^{2}} \frac{d^{2} {\bar{V}}_{s}}{d {\bar{r}}^{2}} + 2 \frac{d {\bar{I}}_{xy} (\bar{r})}{d \bar{r}} \frac{d^{3} {\bar{V}}_{s}}{d {\bar{r}}^{3}} + {\bar{I}}_{xy} (\bar{r}) \frac{d^{4} {\bar{V}}_{s}}{d {\bar{r}}^{4}} \\ - {\bar{Ω}}^{2} \frac{d}{d \bar{r}} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d {\bar{U}}_{s}}{d \bar{r}} \\ - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} {\bar{U}}_{s}}{d {\bar{r}}^{2}} = {\bar{ω}}^{2} {\bar{U}}_{s} \\ for s = 1, 2, \dots, N_{G} \end{matrix}

(31)

\begin{matrix} \frac{d^{2} {\bar{I}}_{xx} (\bar{r})}{d {\bar{r}}^{2}} \frac{d^{2} {\bar{V}}_{s}}{d {\bar{r}}^{2}} + 2 \frac{d {\bar{I}}_{xx} (\bar{r})}{d \bar{r}} \frac{d^{3} {\bar{V}}_{s}}{d {\bar{r}}^{3}} + {\bar{I}}_{xx} (\bar{r}) \frac{d^{4} {\bar{V}}_{s}}{d {\bar{r}}^{4}} \\ + \frac{d^{2} {\bar{I}}_{xy} (\bar{r})}{d {\bar{r}}^{2}} \frac{d {\bar{U}}_{s}}{d \bar{r}} \\ + 2 \frac{d {\bar{I}}_{xy} (\bar{r})}{d \bar{r}} \frac{d^{3} {\bar{U}}_{s}}{d {\bar{r}}^{3}} + {\bar{I}}_{xy} (\bar{r}) \frac{d^{4} {\bar{U}}_{s}}{d {\bar{r}}^{4}} - {\bar{Ω}}^{2} \bar{A} (\bar{r}) {\bar{V}}_{s} \\ - {\bar{Ω}}^{2} \frac{d}{d \bar{r}} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d {\bar{V}}_{s}}{d \bar{r}} \\ - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} {\bar{V}}_{s}}{d {\bar{r}}^{2}} = {\bar{ω}}^{2} {\bar{V}}_{s} \\ for s = 1, 2, \dots, N_{G} \end{matrix}

(32)

with

\bar{r} = \frac{r}{L}

(33)

{\bar{r}}_{0} = \frac{r_{0}}{L}

(34)

{\bar{I}}_{xx} = \frac{I_{xx}}{I_{o}}

(35)

I_{o} = \frac{b_{0} t_{0}^{3}}{12}

(36)

{\bar{I}}_{yy} = \frac{I_{yy}}{I_{o}}

(37)

{\bar{I}}_{xy} = \frac{I_{xy}}{I_{o}}

(38)

\bar{A} = \frac{A}{A_{0}}

(39)

A_{0} = b_{0} t_{0}

(40)

{\bar{U}}_{s} = \frac{U_{s}}{L} for s = 1, 2, \dots, N_{G}

(41)

{\bar{V}}_{s} = \frac{V_{s}}{L} for s = 1, 2, \dots, N_{G}

(42)

\bar{ω} = ω \sqrt{\frac{ρ A_{0} L^{4}}{{EI}_{o}}}

(43)

\bar{Ω} = Ω \sqrt{\frac{ρ A_{0} L^{4}}{{EI}_{o}}}

(44)

The corresponding nondimensional boundary conditions are

{\bar{U}}_{s} (0) = 0 for s = 1, 2, \dots, N_{G}

(45)

{\bar{V}}_{s} (0) = 0 for s = 1, 2, \dots, N_{G}

(46)

\frac{d {\bar{U}}_{s} (0)}{d \bar{r}} = 0 for s = 1, 2, \dots, N_{G}

(47)

\frac{\partial^{2} {\bar{V}}_{1} (0, t)}{\partial {\bar{r}}^{2}} = {\bar{k}}_{t} \frac{\partial {\bar{V}}_{1} (0, t)}{\partial \bar{r}}

(48)

where ${\bar{k}}_{t} = k_{t} L / EI$

\frac{d {\bar{V}}_{s} (0)}{d \bar{r}} = 0 for s = 1, 2, \dots, N_{G}

(49)

\frac{d^{2} {\bar{U}}_{s} (1)}{d {\bar{r}}^{2}} = 0 for s = 1, 2, \dots, N_{G}

(50)

\frac{d^{2} {\bar{V}}_{s} (1)}{d {\bar{r}}^{2}} = 0 for s = 1, 2, \dots, N_{G}

(51)

\frac{d^{3} {\bar{U}}_{s} (1)}{d {\bar{r}}^{3}} = 0 for s = 1, 2, \dots, N_{G}

(52)

and

\begin{array}{l} \frac{d^{3} {\bar{V}}_{s} (1)}{d {\bar{r}}^{3}} + \bar{k} {\bar{V}}_{s - 1} (1) - 2 \bar{k} {\bar{V}}_{s} (1) + \bar{k} {\bar{V}}_{s + 1} (1) = 0 \\ for s = 1, 2, \dots, N_{G} \end{array}

(53)

The dimensionless shroud stiffness is $\bar{k} = {kL}^{3} / EI$ .

Radial basis function

Mesh-free approximation methods are a new area of research in the mathematics and engineering and have gained much attention in recent years. An RBF is a real-valued function and its value depends on its distance from its point of origin. Kansa^22,23 studied a given function or the partial derivatives of a function with respect to a coordinate direction, which is expressed as a linear weighted sum of all functional values at all mesh points along the direction that was initiated on the basis of the concept of the RBF. The node distribution in their algorithm was completely unstructured. Elfelsoufi²⁴ investigated the buckling, fluttering, and vibration of beams by using RBFs. Wang and Liu²⁵ proposed a point interpolation meshless method that was based on RBFs and incorporated the Galerkin weak form for solving partial differential equations. Hon et al.²⁶ employed RBFs for function fitting and solving partial differential equations using global nodes and collocation procedures. Liu et al.²⁷ constructed shape functions with the delta function property based on radial and polynomial basis functions. In this study, shape functions are constructed using RBFs. A RBF can be expressed as follows^28,29:

B_{i} (\bar{r}) = \sqrt{{(\bar{r} - {\bar{r}}_{i})}^{2} + c^{2}}

(54)

where $c$ is a constant. The RBF is typically used to develop the functional approximations^28,29:

{\bar{U}}_{s} (\bar{r}) = \sum_{i = 1}^{N} a_{Usi} B_{i} (\bar{r})

(55)

{\bar{V}}_{s} (\bar{r}) = \sum_{i = 1}^{N} a_{Vsi} B_{i} (\bar{r})

(56)

where $a_{Usi}$ and $a_{Vsi}$ are the coefficients that must be determined. The blade deflections ${\bar{U}}_{s} (\bar{r})$ and ${\bar{V}}_{s} (\bar{r})$ denote a sum of $N$ RBFs, each associated with a different center ${\bar{r}}_{i}$ . The domain contains $N$ collocation points. Although these nonlinear equations of the grouped turbo blades do not have an analytical solution, numerical approaches can be adopted to solve it. These nonlinear partial differential equations are solved numerically using the RBF approach, which does not require a mesh. By using the RBF approach, equations (55) and (56) can be substituted into equations (31)–(53). The equation of motion of the grouped turbo blades can be rearranged into a formula based on the RBF approach

\begin{matrix} [\begin{matrix} \frac{d^{2} {\bar{I}}_{yy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} {\bar{I}}_{yy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \dots \frac{d^{2} {\bar{I}}_{yy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} \\ + [\begin{matrix} 2 \frac{d {\bar{I}}_{yy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} 2 \frac{d {\bar{I}}_{yy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \dots 2 \frac{d {\bar{I}}_{yy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \end{matrix}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} \\ + [\begin{matrix} {\bar{I}}_{yy} ({\bar{r}}_{i}) \frac{d^{4} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} {\bar{I}}_{yy} ({\bar{r}}_{i}) \frac{d^{4} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \dots {\bar{I}}_{yy} ({\bar{r}}_{i}) \frac{d^{4} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \end{matrix}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} \\ + [\begin{matrix} \frac{d^{2} {\bar{I}}_{xy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} {\bar{I}}_{xy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \dots \frac{d^{2} {\bar{I}}_{xy} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix}] {[\begin{matrix} a_{Vs 1} a_{Vs 2} \dots a_{VsN} \end{matrix}]}^{T} \\ + [\begin{matrix} 2 \frac{d {\bar{I}}_{xy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} 2 \frac{d {\bar{I}}_{xy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \dots 2 \frac{d {\bar{I}}_{xy} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \end{matrix}] {[\begin{matrix} a_{Vs 1} a_{Vs 2} \dots a_{VsN} \end{matrix}]}^{T} \\ + [\begin{matrix} {\bar{I}}_{xy} ({\bar{r}}_{i}) \frac{d^{4} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} {\bar{I}}_{xy} ({\bar{r}}_{i}) \frac{d^{4} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \dots {\bar{I}}_{xy} ({\bar{r}}_{i}) \frac{d^{4} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \end{matrix}] {[\begin{matrix} a_{Vs 1} a_{Vs 2} \dots a_{VsN} \end{matrix}]}^{T} \\ + [\begin{matrix} - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{{dB}_{1} ({\bar{r}}_{i})}{d \bar{r}} - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{{dB}_{2} ({\bar{r}}_{i})}{d \bar{r}} \end{matrix} \dots - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{{dB}_{N} ({\bar{r}}_{i})}{d \bar{r}}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} \\ + [\begin{matrix} - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix} - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \dots - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} \\ = [\begin{matrix} {\bar{ω}}^{2} B_{1} ({\bar{r}}_{i}) {\bar{ω}}^{2} B_{2} ({\bar{r}}_{i}) \dots {\bar{ω}}^{2} B_{N} ({\bar{r}}_{i}) \end{matrix}] {[\begin{matrix} a_{Us 1} a_{Us 2} \dots a_{UsN} \end{matrix}]}^{T} for s = 1, 2, \dots, N_{G} and i = 1, 2, \dots, N \end{matrix}

(57)

\begin{array}{l} [\begin{matrix} \frac{d^{2} {\bar{I}}_{x x} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & \frac{d^{2} {\bar{I}}_{x x} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & \dots & \frac{d^{2} {\bar{I}}_{x x} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ + [\begin{matrix} 2 \frac{d {\bar{I}}_{x x} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} & 2 \frac{d {\bar{I}}_{x x} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} & \dots & 2 \frac{d {\bar{I}}_{x x} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ + [\begin{matrix} {\bar{I}}_{x x} ({\bar{r}}_{i}) \frac{d^{4} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} & {\bar{I}}_{x x} ({\bar{r}}_{i}) \frac{d^{4} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} & \dots & {\bar{I}}_{x x} ({\bar{r}}_{i}) \frac{d^{4} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ + [\begin{matrix} \frac{d^{2} {\bar{I}}_{x y} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & \frac{d^{2} {\bar{I}}_{x y} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & \dots & \frac{d^{2} {\bar{I}}_{x y} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix}] {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} \\ + [\begin{matrix} 2 \frac{d {\bar{I}}_{x y} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} & 2 \frac{d {\bar{I}}_{x y} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} & \dots & 2 \frac{d {\bar{I}}_{x y} ({\bar{r}}_{i})}{d \bar{r}} \frac{d^{3} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{3}} \end{matrix}] {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} \\ + [\begin{matrix} {\bar{I}}_{x y} ({\bar{r}}_{i}) \frac{d^{4} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} & {\bar{I}}_{x y} ({\bar{r}}_{i}) \frac{d^{4} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} & \dots & {\bar{I}}_{x y} ({\bar{r}}_{i}) \frac{d^{4} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{4}} \end{matrix}] {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} \\ + [\begin{matrix} - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{d B_{1} ({\bar{r}}_{i})}{d \bar{r}} & - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{d B_{2} ({\bar{r}}_{i})}{d \bar{r}} & \dots & - {\bar{Ω}}^{2} \frac{d [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}]}{d \bar{r}} \frac{d B_{N} ({\bar{r}}_{i})}{d \bar{r}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ + [\begin{matrix} - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{1} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{2} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} & \dots & - {\bar{Ω}}^{2} [\int_{\bar{r}}^{1} \bar{A} (\bar{r}) (\bar{r} + {\bar{r}}_{0}) d \bar{r}] \frac{d^{2} B_{N} ({\bar{r}}_{i})}{d {\bar{r}}^{2}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ = [\begin{matrix} {\bar{ω}}^{2} B_{1} ({\bar{r}}_{i}) & {\bar{ω}}^{2} B_{2} ({\bar{r}}_{i}) & \dots & {\bar{ω}}^{2} B_{N} ({\bar{r}}_{i}) \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} for s = 1, 2, \dots, N_{G} and i = 1, 2, \dots, N \end{array}

(58)

Based on the RBF approach, the boundary conditions of a clamped-free blade with an edge crack can be rearranged into matrix forms as follows:

\begin{array}{l} [\begin{matrix} B_{1} ({\bar{r}}_{1}) & B_{2} ({\bar{r}}_{1}) & \dots & B_{N} ({\bar{r}}_{1}) \end{matrix}] \\ {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(59)

\begin{array}{l} [\begin{matrix} B_{1} ({\bar{r}}_{1}) & B_{2} ({\bar{r}}_{1}) & \dots & B_{N} ({\bar{r}}_{1}) \end{matrix}] \\ {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(60)

\begin{array}{l} [\begin{matrix} \frac{\partial B_{1} ({\bar{r}}_{1})}{\partial \bar{r}} & \frac{\partial B_{2} ({\bar{r}}_{1})}{\partial \bar{r}} & \dots & \frac{\partial B_{N} ({\bar{r}}_{1})}{\partial \bar{r}} \end{matrix}] \\ {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(61)

\begin{array}{l} [\begin{matrix} \frac{\partial^{2} B_{1} ({\bar{r}}_{1})}{\partial {\bar{r}}^{2}} & \frac{\partial^{2} B_{2} ({\bar{r}}_{1})}{\partial {\bar{r}}^{2}} & \dots & \frac{\partial^{2} B_{N} ({\bar{r}}_{1})}{\partial {\bar{r}}^{2}} \end{matrix}] \\ {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ = [\begin{matrix} {\bar{k}}_{t} \frac{\partial B_{1} ({\bar{r}}_{1})}{\partial \bar{r}} & {\bar{k}}_{t} \frac{\partial B_{2} ({\bar{r}}_{1})}{\partial \bar{r}} & \dots & {\bar{k}}_{t} \frac{\partial B_{N} ({\bar{r}}_{1})}{\partial \bar{r}} \end{matrix}] {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} \\ for s = 1 \end{array}

(62)

\begin{array}{l} [\begin{matrix} \frac{\partial B_{1} ({\bar{r}}_{1})}{\partial \bar{r}} & \frac{\partial B_{2} ({\bar{r}}_{1})}{\partial \bar{r}} & \dots & \frac{\partial B_{N} ({\bar{r}}_{1})}{\partial \bar{r}} \end{matrix}] \\ {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} = [0] \\ for s = 2, 3, \dots, N_{G} \end{array}

(63)

\begin{array}{l} [\begin{matrix} \frac{\partial^{2} B_{1} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} & \frac{\partial^{2} B_{2} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} & \dots & \frac{\partial^{2} B_{N} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} \end{matrix}] \\ {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(64)

\begin{array}{l} [\begin{matrix} \frac{\partial^{2} B_{1} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} & \frac{\partial^{2} B_{2} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} & \dots & \frac{\partial^{2} B_{N} ({\bar{r}}_{N})}{\partial {\bar{r}}^{2}} \end{matrix}] \\ {[\begin{matrix} a_{V s 1} & a_{V s 2} & \dots & a_{V s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(65)

\begin{array}{l} [\begin{matrix} \frac{\partial^{3} B_{1} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} & \frac{\partial^{3} B_{2} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} & \dots & \frac{\partial^{3} B_{N} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} \end{matrix}] \\ {[\begin{matrix} a_{U s 1} & a_{U s 2} & \dots & a_{U s N} \end{matrix}]}^{T} = [0] \\ for s = 1, 2, \dots, N_{G} \end{array}

(66)

and

\begin{matrix} [\begin{matrix} \frac{\partial^{3} B_{1} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} \frac{\partial^{3} B_{2} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} \dots \frac{\partial^{3} B_{N} ({\bar{r}}_{N})}{\partial {\bar{r}}^{3}} \end{matrix}] \\ {[\begin{matrix} a_{Vs 1} a_{Vs 2} \dots a_{VsN} \end{matrix}]}^{T} \\ + [\begin{matrix} \bar{k} B_{1} ({\bar{r}}_{N}) \bar{k} B_{2} ({\bar{r}}_{N}) \dots \bar{k} B_{N} ({\bar{r}}_{N}) \end{matrix}] \\ {[\begin{matrix} a_{V (s - 1) 1} a_{V (s - 1) 2} \dots a_{V (s - 1) N} \end{matrix}]}^{T} \\ - [\begin{matrix} 2 \bar{k} B_{1} ({\bar{r}}_{N}) 2 \bar{k} B_{2} ({\bar{r}}_{N}) \dots 2 \bar{k} B_{N} ({\bar{r}}_{N}) \end{matrix}] \\ {[\begin{matrix} a_{Vs 1} a_{Vs 2} \dots a_{VsN} \end{matrix}]}^{T} \\ + [\begin{matrix} \bar{k} B_{1} ({\bar{r}}_{N}) \bar{k} B_{2} ({\bar{r}}_{N}) \dots \bar{k} B_{N} ({\bar{r}}_{N}) \end{matrix}] \\ {[\begin{matrix} a_{V (s + 1) 1} a_{V (s + 1) 2} \dots a_{V (s + 1) N} \end{matrix}]}^{T} \\ = [0] for s = 1, 2, \dots, N_{G} \end{matrix}

(67)

Results and discussion

A model of a bladed disk with a rigid hub attached to 60 blades was assembled. The shroud ring segments on the blade tips coupled the blades. A group consisted of six blades. Figure 3 shows the calculated natural frequencies of the grouped turbo blade disk with different values of ${\bar{k}}_{t}$ . The nondimensional parameters of the rotating disk system are ${\bar{b}}_{0} = 0.1$ , ${\bar{t}}_{0} = 0.02$ , and ${\bar{r}}_{0} = 0.2$ .¹⁸ We used constant c = 1. A higher nondimensional natural frequency was calculated for the grouped-blade system when high ${\bar{k}}_{t}$ . The crack depth considerably affects the dynamic behavior of grouped turbo blades. They also suggested that nondimensional natural frequencies calculated using the RBF approach are extremely close to the solutions obtained using the differential quadrature method. Figure 4 shows nondimensional natural frequencies of a rotating turbo disk with different pretwist angles. Higher second and fourth nondimensional natural frequencies were calculated for the grouped-blade system when the total pretwisted angle was smaller. Figure 5 shows calculated nondimensional natural frequencies of the grouped-turbo disk with different values of shroud stiffness. A higher nondimensional natural frequency was calculated for the grouped-blade system when the shroud stiffness was higher. The results show that a strong shroud may considerably increase the nondimensional natural frequencies of a grouped turbo blade system. Figure 6 shows calculated nondimensional natural frequencies of the grouped turbo disk for various rotating speeds. A higher nondimensional natural frequency was calculated for the grouped-blade system when the rotating speed was higher. Figure 7 shows the calculated nondimensional natural frequencies of the grouped turbo disk for various values of α. A higher nondimensional natural frequency was calculated for the grouped-blade system when α was higher. Figure 8 shows the calculated nondimensional natural frequencies of the grouped turbo disk for various values of β, and a higher first nondimensional natural frequency and lower fourth nondimensional natural frequency were calculated for the grouped-blade system when β was higher.

Figure 3.

Calculated nondimensional natural frequencies of a grouped turbo blade disk with different values of ${\bar{k}}_{t}$ .

Figure 4.

Calculated nondimensional natural frequencies of a rotating turbo disk with different pretwist angles.

Figure 5.

Calculated nondimensional natural frequencies of a grouped turbo disk with different values of shroud stiffness $\bar{k}$ .

Figure 6.

Calculated nondimensional natural frequencies of the grouped turbo disk for various rotating speeds.

Figure 7.

Calculated nondimensional natural frequencies of the grouped turbo disk for various values of α.

Figure 8.

Calculated nondimensional natural frequencies of the grouped turbo disk for various values of β.

Concluding remarks

The developed approach is convenient for solving problems associated with fourth-order differential equations. In addition to the problem examined here, numerous other applications can be modeled using the RBF. Furthermore, this method can be used to solve a wide variety of boundary value problems, and the simplicity of the formulation renders it a suitable candidate for modeling more complex applications. The crack depth considerably affects the dynamic behavior of grouped turbo blades.

Footnotes

Academic Editor: Hamid Reza Shaker

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

The authors thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract Nos. NSC 100-2221-E-346-011.

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