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On Antiresonance in the Forced Response of Mistuned Bladed Disks

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On Antiresonance in the Forced Response of Mistuned Bladed Disks Shock and Vibration 10 (2003) 135–146 135 IOS Press On antiresonance in the forced response of mistuned bladed disks Keith Jones and Charles Cross Propulsion Directorate, Air Force Research Laboratory, 1950 Fifth Street, Wright Patterson AFB, OH 45433-7251, USA Received 14 January 2002 Revised 2 September 2002 Abstract. Mistuning in bladed disks usually increases the forced response of the maximum responding blade leading to shortened component life in turbine engines. This paper investigates mistuning using a transfer function approach where the frequency response functions (FRFs) are described by natural frequencies and antiresonant frequencies. Using this approach, antiresonant frequencies are shown to be a critical factor in determining the maximum blade response. Two insights are gained by formulating antiresonant frequencies as the eigenvalues of reduced system matrices: 1) Mistuning a particular blade has no effect on that blade’s antiresonant frequencies. 2) Engine orders N and N/2, where N is the number of blades on the disk, tend to produce the highest maximum local response. Numerical examples are given using a spring-mass-oscillator model of a bladed disk. Pole-zero loci of mistuned bladed disks show that increased maximum blade response is often due to the damping of antiresonant frequencies. An important conclusion is that antiresonant frequencies can be arranged such that a mistuned bladed disk has a lower maximum blade response than a tuned bladed disk. Nomenclature x Displacement degree of freedom A Mistuning amplification factor y Measured output B Input shape matrix z Transfer function zeros (antiresonant C Output shape matrix frequencies) E Engine order zu Undamped zeros F Vector of applied forces u Harmonic input excitation th th FRF Frequency response function (20 log |h|) φin The i element in the n mode shape th G Transfer function gain φn The n mode shape h Blade frequency response ω Excitation frequency I Identity matrix ωb Nominal blade natural frequency i √Blade number θ Interblade phase angle j −1 ζ Damping factor K Global stiffness matrix kb Blade stiffness kc Coupling stiffness M Global mass matrix 1. Background m Nominal blade mass mi Individual blade mass 1.1. Mistuning N Number of blades p System poles (natural frequencies) As bladed disks rotate in turbine engines, the air- pu Undamped poles foils experience alternating forces resulting from dis- R Coupling ratio turbances in the flow field. This excitation is often ISSN 1070-9622/03/$8.00 2003 – IOS Press. All rights reserved 136 K. Jones and C. Cross / On antiresonance in the forced response of mistuned bladed disks called engine order excitation where the engine order that for discrete spring-mass systems antiresonant fre- refers to the number of equally spaced disturbances due quencies are the resonant frequencies of some substruc- to struts, vanes, or stators either upstream or down- ture. These substructures and their constraints are de- stream of the bladed disk. The resulting blade vibra- fined by La Civita [8] in continuous one dimensional tion causes fatigue cracks to initiate and grow. This systems. Wahl and Schmidt [9] applied these discover- process is called high cycle fatigue (HCF) because of ies to experimental structural analysis. Jones and Tur- the tremendous number of cycles accumulated at the cotte [10], D’Ambrogio and Frengolent [11], Rade et high rotational rates of turbine engines. (HCF can also al. [12], and Lallement and Cogan [13] have applied be present in stationary airfoils due to rotating pressure antiresonance to finite element model updating with disturbances.) The U.S. Air Force spends a significant promising results. amount of money and effort inspecting and fixing HCF These papers analyzed antiresonance in single-input problems. single-output (SISO) transfer functions. Bladed disk HCF is exacerbated by mistuning, a breakdown in forced response requires a multi-input single-output the cyclic periodicity of bladed disks. A tuned bladed (MISO) method, because engine order excitation is ap- disk is a perfectly cyclic symmetric structure where all plied at all blades. Wang’s [14] formulation of an- blades are identical. Unfortunately, manufacturing tol- tiresonance can accommodate MISO problems and is erances will inevitably cause small differences in the applied to models of bladed disks in this paper. geometries, material properties, or interface boundary Although the transfer function forced response for- conditions of blades resulting in what is called a mis- mulation is mathematically equivalent to the modal for- tuned bladed disk. The maximum resonant response of mulation, it does provide a different perspective on mis- any blade on the disk, or maximum response for short, tuning phenomena. The insights that come from under- is usually greater for mistuned bladed disk than for a standing antiresonance may provide new approaches tuned bladed disk [1]. Therefore, mistuning shortens for reducing maximum response in bladed disks. component life, and negatively impacts the durability and reliability of engines. Most of the mistuning literature has relied on a modal 2. Theory approach when investigating bladed disk forced re- sponse. The modal approach relates natural frequen- 2.1. Forced response formulation cy and mode shape behavior to forced response using concepts such as eigenvalue veering and mode local- The undamped equations of motion (EOM) for a ization. Examples of the modal approach include pa- bladed disk assembly can be written as: pers by Ewins [2], Wei and Pierre [3,4], and Brewer et Mx¨ + Kx = F = Bu (1) al. [5] After surveying the mistuning literature, Slater et al. [6] concluded that amplified maximum response y = Cx (2) in mistuned bladed disks is due to light damping and mode localization. m11 m12 0 ··· m21 m21 m22 m12 0 ··· 1.2. Antiresonance M = ··· ··· ··· ··· ··· (3) ··· ··· ··· ··· ··· This paper investigates mistuning using a transfer m ··· m m function approach where the forced response is formu- 12 0 21 NN lated in terms of natural frequencies and antiresonant k11 k12 0 ··· k21 frequencies. This approach avoids using mode shapes, k21 k22 k12 0 ··· which become disordered in the presence of mistun- K = ··· ··· ··· ··· ··· (4) ing [3] making their effect on maximum response dif- ··· ··· ··· ··· ··· ficult to understand. Instead, the antiresonant frequen- k12 0 ··· k21 kNN cies incorporate the mode shape information while pro- viding a more direct and intuitive link to maximum where F is the vector of applied forces, mii and kii forced response levels. are the blade/sector mass and stiffnesses and N is the Antiresonance is an area of continuing research in number of blades/sectors. mii and kii are equal for the structural dynamics community. Miu [7] showed tuned bladed disks but differ for mistuned bladed disks. K. Jones and C. Cross / On antiresonance in the forced response of mistuned bladed disks 137 The EOM inputs (F ) and outputs (y) can be any lin- magnitude axis (see Fig. 1). To emphasize FRF an- ear combination of blade input forces (u) or respons- tiresonances, this paper defines the frequency response es (x) respectively. For the response of any particular function (FRF) to be the magnitude of the frequency blade i, C is a sparse row vector with Ci =1. response in decibels: The forcing function (F ) can be written as the prod- FRF =20log|h| (12) uct of an input shape matrix (B), which characterizes the spatial properties of the applied forces, and a vec- Damping was added to the transfer function FRFs tor (u), which characterizes the time properties of the to prevent infinite resonant peaks and antiresonant val- forces. Engine order excitation is assumed to be har- leys. Structural damping was incorporated by adding monic excitation which differs from blade to blade by an imaginary part to the undamped poles (p u): a constant interblade phase angle (θ): 2 2 2 p = pu +2ζpuj (13) j(i−1)θ Bi = e i =1, 2,...,N (5) Damping was assumed to have a similar effect on the jωt u = e (6) undamped zeros (zu): E z2 z2 ζz2j θ π = u +2 u (14) =2 N (7) To validate this approach, the transfer function FRFs E where is the engine order. were compared to the modal summation FRFs in equa- The solution to Eq. (1) can be written as the summa- tion (9) using the classic assumption of constant modal tion of modal responses in physical coordinates (The damping where: φT Mφ = I mode shapes are mass normalized so that ): 2 2 p = p +2ζpuωj (15) φ φT B φ φT B u y i1 1 i2 2 ··· i = 2 2 + 2 2 + A ζ of 0.001 was used for this comparison as well (p1 − ω ) (p2 − ω ) (8) φ φT B as the numerical examples in this paper. The transfer iN N u function and modal summation FRFs were virtually + p2 − ω2 ( N ) identical. where p is a system pole or natural frequency. The frequency response, h, is defined by y = hu so that: 2.2. Calculation of transfer function zeros φ φT B φ φT B h i1 1 i2 2 ··· Wang [14] showed that the undamped transfer func- i = p2 − ω2 + p2 − ω2 + ( 1 ) ( 2 ) tion zeros (zu) are the eigenvalues of reduced mass and T (9) φiN φ B stiffness matrices: + N (p2 − ω2) 2 N det K˜ − zuM˜ =0 (16) The frequency response can also be written as the ratio of two factored polynomials in w 2: where T M˜ = BN MCN (17) hi = (10) T 2 2 2 2 2 2 K˜ = BN KCN (18) G(z1 − w )(z2 − w ) ···(zN−1 − w ) 2 2 2 2 2 2 (p1 − w )(p2 − w ) ···(pN − w ) BN = nullspace(B) (19) where z is a transfer function zero and G is a constant CN = nullspace(C) (20) determined by the static deflection of the structure: M˜ and K˜ are non-symmetric matrices whose eigen- p2 p2 ··· p2 2 −1 ( 1)( 2) ( N ) values (zi ) can be positive, negative, real or complex G = K B 2 2 2 (11) (z1)(z2 ) ···(zN−1) numbers.
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