3.2 Beam
4.1.2 Physical Properties- Participation Factor
In a system with selected modes in consideration, the participation factor for the mode i is defined as:
Гi = Pi / mii , (4.5) where
{Pi} = []T [M] {1} / []T {M] [], (4.6)
where {P}is column vector of coefficient factor of considered modes, [] is mode shape, {1} is column vector of one, and mii is effective mass.
With a mass normalized mode shape, the effective mass matrix is a unit one:
[mii] =[]T[M][] = [I]. (4.7) Then (4.6) becomes:
{Pi} = []T [M] {1}. (4.8)
24
And the corresponding effective modal mass meff,i for mode i, that is the fraction of the total static mass associated with that mode, is given by:
mii = {Pi}2/ mii . (4.9) Consider a discrete spring-mass consists of four masses connected by five springs and fixed at two ends as shown in Figure 4.1.
Figure 4.1. 1D spring-mass system
Assuming k1= k2 = k3= k4 = 4 units, k5= 10 units, m1= m5 =1 unit and m2= m3 = 2 units, then by solving the equations of motion one obtains the system natural frequency with its corresponding mode shape:
Then from (4.6) and (4.8), the participation factors (equation 4.5) and the effective mass
The sum of the effective mass, Σmii, is just equal to the total mass of the system which is 6 units in this example. As observed, the first mass has the highest absolute participation factor and modal effective mass followed by the fourth mass. Thus, the modal participation factor and the modal effective mass provide a method for assessing the physical behavior of a vibration mode. In other words, it is a measure of the system response in a particular direction at a given natural frequency. A mode with relatively high effective mass has higher propensity to be excited by a forcing function than a mode with low effective mass.
However, up to this point one still does not have sufficient information to conclude that interference shown on the Campbell diagram would be a true indication of resonance.
Contrarily, suppose that resonance is encountered would it be tolerable? To answer these questions, one must examine the interaction between the structure and the forcing function.
26 4.2 Force Vibration
As stated above, the standard blade aeromechanical equation of motion is given by (4.1).
A study of the flow field to obtain the unsteady pressure distribution P(t), in the time domain, is not in the scope of this study. In essence, CFD simulations are used to determine the flow field characteristics. The starting point of the CFD analysis is the generation of the flow model of the domain to be analyzed. A finite number of points, connected by mesh are used to represent the continuous domain. At each of these points, a flow solver will solve the three basic governing conservations equations for mass, momentum and energy to determine the fluid flow field physical properties such as pressure, velocity, density and temperatures in the time domain as summarized in Figures 4.2 and 4.3.
Figure 4.2. Mach Number Contour.
Figure 4.3. Pressure Field
Through Fourier transform, the forcing function P(t) is decomposed into a sum of components each of which has a particular frequency ω, referred as Engine Orders. The subsequent step is to map the CFD pressure mesh onto the FE model. Each FE surface element is associated with a pressure node of the CFD mesh.
Figure 4.4. Blade Pressure Distribution in Frequency Domain
LE TE LE
28
To evaluate the response of the blade structure to the excitation source, a reduced modal model is needed. Consider x(t) as a linear combination of limited number of orthogonal mode shapes:
X =
k
k qk = [], (4.11)
where and q are normal modes and normal or modal coordinates, respectively.
Neglect the damping effect and substitute (4.11) into the equation of motion (4.1) and pre-multiply by T. The force vibration of the blade expressed in normal coordinates is:
[]T[M][]{q } + []T[K][]{q} = []T {p(ω)}, (4.12) where {p(ω)} is the Fourier transform of {p(t)}.
With a mass normalized mode shape:
[]T[M][] = [I],
and []T[K][] = [ω2]. (4.13) Equation [4.12] becomes:
[I]{q} + [ω2]{q} = []T {p(ω)}. (4.14) With this reduced decoupled system, the motion response of each blade mode to each component of the excitation sources can be evaluated independently. The term on the RHS of equation (4.14) represents the modal force. This is a measure of how much the physical loading is transformed into the modal coordinate. The blade steady state response could be obtained by any ODE solver.
Examining equation (4.14) one could deduce that an applied harmonic force at a particular frequency would induce maximum resonance vibration if it is acting in the physical coordinate corresponding to the blade mode with largest participation factor. It
would have no contribution to the mode vibration if it is aligned in the physical coordinate with a zero participation factor.
The level of interaction between fluid and blade could be measured from the energy transferring from fluid to blade. The aerodynamic work per cycle, W, of blade motion is taking as the time integration of the product of the pressure and the blade velocity (X ) in
one period of displacement over the blade area.
W 0 PnXdA]dt
T [ . (4.15) Representing the time dependence of oscillatory function by exp(iωt) and assume the motion is harmonic of Engine order ω, the modal displacement written in the complex form is:
X = Re{(X)(iωt) }. (4.16) With the complex velocity expressed as:
X = iω (X)(iωt) . (4.17) Similarly, the unsteady pressure is written as:
P = (Pr)(iωt+ φ) + i(Pim)(iωt+φ) . (4.18) The real part of pressure will be in phase with the displacement, and the blade stability is determined by the phase difference between the imaginary pressure, i(Pim)(iωt+φ), and the velocity, X .
When the work per cycle W is positive energy is transferred from fluid to blade, this eventually establishes conditions for the development and maintenance of resonance.
Conversely, an out of phase between the blade modal velocity and the imaginary part of the unsteady pressure implies a stable non-excited condition.
30
Figure 4.5. Physical Representation of the Harmonic Motion and Unsteady Pressure
(Pr + Pim)e(iωt+φ)
Xe(iωt)
4.3 Case Study
The aero-mechanical interaction is demonstrated in this case study. A blade natural frequency is excited if and only if there is positive energy transferred from fluid to the blade.
A blade shown in Figure 4.6, whose Campbell diagram illustrated in Figure 4.7, has mode N interfered with two Engine orders in the operating speed range. Engine order 13 corresponds to the immediate upstream stator while Engine order 16 is resulted from the immediate downstream nozzle. Modal analysis indicated that the nature of mode N is of stiff-wise bending (Figure 4.8) involving the dominant effective mass (close to 1.) in the axial direction (X axis) and rotation about Z axis. The coordinate system is shown in Figure 4.6.
The laser vibrometry test of mode N is shown in Figure 4.9, illustrating blade TE response. CFD analysis provides the un-steady pressure field. This pressure is mapped onto the FE grid for forced response analysis.
It was calculated that the center of the 13EO unsteady pressure is located on the pressure side of the blade profile about one quarter from the blade leading edge (LE) in the engine axis direction, while the one for the 16 EO from downstream effect is center about one fifth of the blade axial length measuring from the trailing edge (TE).
Forced response analysis indicated that the blade is not excited by the upstream stator (13EO). Instead, the blade is slightly sensitive to the unsteady pressure component derived from the downstream stator nozzle (16EO). From Figure 4.6, it is observed that the 13EO resultant driving force vector is close to normal to the blade mode. The work
32
per cycle blade motion is calculated and found close to zero for the 13EO excitation while a weak value is obtained for the 16EO.
An Engine Non-intrusive test measurement (NSMS) using fiber optics to measure blade vibrations was performed and the results confirmed the analytical prediction as shown in Figure 4.10. A small blade deflection was recorded on the 16EO line while no signal was measured at the expected location on the 13EO line. In this particular case the interferences of mode N shown in the Campbell diagram (Figure 4.7) are considered to be acceptable.
Figure 4.6. Blade and Velocity Diagram
LE
TE U
bladeV
1V
1relativeX
Y
Figure 4.7. Campbell Diagram
34
CHAPTER 5.
A. Eigenvalue Perturbation
5.1 The Perturbation of a Simple Eigenvalue Consider an eigenvalue problem
Au = λ u, (4.1) Where A is an (n x n) real matrix, λ is an eigenvalue with corresponding eigenvector, u.
The eignvalue is determined from the determinant of the characteristic equation
det(A – λI) = 0. (4.2) Consider the eigenvalue for the transposed matrix
ATv = λ v, (4.3) Since
det(AT– λI) = det(A – λI). (4.4) The eigenvalue value for the matrix AT can be written as
vTA = λvT, (4.5) where v is the left eigenvector corresponding to the eigenvalue λ.
Assume matrix A depends upon a vector of real parameter p = (p1,…,pn).
Then equation (4.2) gives
λ
det(A(p) – λI) ≠ 0, (4.6)
Applying implicit function theorem equation (4.6) can be expressed as
p λ
i
= - pi
det(A(p) – λI) / λ
det(A(p) – λI) , i= 1,...,n (4.7)
The derivative of the eigenvalue can be expressed in terms of the left hand vector v0.
The solution to the equation (4.9) exits if and only if vT0(
where vT0is the transposed of the left eigenvector corresponds to λ0. Therefore equation (4.10) gives The undamped, homogenous equation of motion is
[M]{ x } + [K]{x} = 0. (4.12) where [M] and [K] are system mass and stiffness matrices, {x} is displacement vector, and the dot denotes the derivative with respect to time.
Assume the structure stiffness and mass matrices are perturbed by [δK] and [δM], respectively, we have:
[K] = [Ko] + [δK] , (4.13)
36
[M] = [Mo] + [δM] . (4.14) The subscript (o) denotes to the unperturbed system and subscript (i) represents the ith
mode. For small perturbation (ie.[δK] « [K] and [δM] « [M]) the perturbed eigenvalue and eigenvector are expressed as
λi = λoi+ δλoi , (4.15) and
i = oi + δoi . (4.16) With the symmetric and positive definite of the mass and stiffness matrices, the orthogonal properties give:
oiT
[Mo] oi = 1, (4.17) Toi [Ko] oi = λoi = ωoi2
. (4.18) 1) Evaluation of λoi
To solve for
[K] i = λi [M] i (4.19)
Substitute (4.16), (4.17) and (4.18) into (4.19)
([Ko] + [δK])( oi + δoi ) = (λoi + δλoi)([Mo] + [δM])( oi + δoi ) , (4.20) Expanding the terms
[Ko] oi + [Ko] δoi + [δK] oi + [δK]δoi = λoi [Mo] oi + λoi [Mo] δoi + δλoi[Mo] oi + δλoi[δM]oi + δλoi[Mo] δoi + λoi[δM] δoi +
δλoi[Mo] δoi + δλoi [δM] δoi . (4.21)
Neglecting the higher order terms and simplifying, equation (4.21) yields
[Ko] δoi + [δK] oi = λoi [Mo] δoi + λoi [δM] oi + δλoi[Mo] oi (4.22)
One computational approximation method to calculate eigenvector derivative is to expand the derivative as a series of eigenvectors, thus for the ith eigenvector:
δi =
Applying the orthogonal properties by left multiplying byToi, equation (4.24) gives
38 But from equation (4.17)
Toi [Mo] oi = 1,
Therefore
δλi = Toi [δK] oi - λoi Toi [δM] oi . (4.29) 2) Evaluation of oi
Take the derivative of equation (4.17), Toi [Mo] oi = 1, with respect to the design variable (such as mass) if there is a constraint imposed on the eigenvector [16], the following is obtained.
(Toi)’ [Mo] oi + Toi [Mo]’ oi + Toi [Mo] i’= 0 , (4.30) where the prime denotes derivative, thus
Toi [Mo] oi’= -(1/2) Toi [Mo]’ oi . (4.31) To obtain ii substitute (4.23) into (4.31) and taking into account the normalization with respect to the mass matrix.
N 1
j εijToi [Mo]oj = -(1/2) Toi [Mo]’ I . (4.32)
For the case i = j
ii = -(1/2) Toi [Mo]’ oi . (4.33) For the case i ≠ j
Substitute (4.33) into the equation {([K] – λoi [M]) oi =0} after taking its derivative and then multiplying the resulting equation by Toj.
([K] – λoi [M]) oi’ + ([δK] – λoi[δM]) oi- δλoi [M]) oi = 0 , (4.34)
N 1
j εijToj ([K] – λoi [M]) oj + Toj([δK] - λoi [δM]) oi– δλoi Toj[M]) oi = 0 . (4.35) Due to M-orthogonal, the last term is equal to zero
with In summary , equations (4.15) and (4.16) are expressed in terms of the unperturbed system as mass and stiffness is perturbed.
λi = λoi + (Toi [δK] oi - λoi Toi [δM] oi) (4.37)
i = oi - (1/2) {Toi [δM] oi }oi + j1,N(ji) ( Toj{[δK] - λoi [δM]} oi) oj / (λoj - λoi) (4.38)
Through partial derivative of (4.37) and (4.38), the sensitivity of λi and i with respect to δ[K] and δ [M] perturbations is studied.
a.1) With the symmetric [K] matrix, change in kkl termwill change klk, thus
40 Similarly, for changes in the mass,
mkl
In terms of frequency, for a structural system with distinct eigenvalues in the range of interest, the shift of λi either in the direction of increasing or decreasing amplitude can be attained by perturbing only the elements of the [K] and [M] matrices associated with its
corresponding mode shape. The eigenvalue perturbation bounds are limited by the size and strength of its own mode shape as illustrated in equations (4.39) and (4.40).
While the perturbation of eigenvalue λi depends solely on the perturbation of [K] and [M]
matrices, the perturbation in the eigenvector i related to eigenvalue λi from the perturbation of [K] and [M] depends not only from the coefficient corresponding to λi but also from those of the other eigenvalues. As seen in equations (4.41) and (4.42), the sensitivity of eigenvector i is amplified if the difference between λi andλi+1 or λi-1 is small.
Numerically, the sensitivity stability of the eigenvalue matrix depends on the mathematical properties of the matrix itself. From Bauer-Fike theorem, the eigenvalues of a matrix A are very sensitive to perturbation if all of its eingenvalues are smaller than the largest element [11]. Fortunately, this is not the case of real and symmetric, or complex Hermitian of common structural application.
5.3 Nodes Antinodes and Maximum Frequency Shift
Neglecting the damping, the in-vacuo eigenvalues λi and eigenvectors i for a free vibration problem, are evaluated from the determinant of the homogeneous equation of motion
[M] { x } + [K] {x} = 0. (5.1) The eigenvalues gives the natural frequencies of vibration while its corresponding eigenvectors determine the shape of the vibrational modes. In other words, the eigenvectors define the displacement configurations of the various mode of the structure.
Each mode has a natural frequency associated with it. In a one-dimensional system at a
42
given mode the vibration will have nodes or places where the displacement is always equal to zero. The place where displacement is at maximum is called the primary node. The locations where displacements are relatively smaller than the primary anti-node are called the secondary anti-anti-nodes.
Consider the 1D beam of length L, whose mode shape is given by:
y = sin ( L
ix) (5.2)
where i denotes frequency index, i =1, 2, 3, etc…
For the first bending mode, the anti-node is located at the beam mid-span where the maximum beam deflection occurs. For the second bending mode, as shown below, a node is associated with the beam mid-span while the two anti-nodes of equal strength are located at one fourth and three fourth of the beam length from the left end.
Figure 5.1. 2nd Bending Mode.
In a two-dimensional system, the nodes are represented by lines where displacements are always equal to zero, while the primary anti-node becomes line where maximum displacement occurs.
In a three-dimensional system, the nodes become surfaces where displacement is always zero. In contrast, the primary anti-node is the surface where displacement is at maximum.
The perturbation of the ith eigenvalue, λi, is maximized as the elements of the matrices [K] and [M] are perturbed at the locations corresponding to the anti-nodes of its own mode shape, i. Equations (4.39) and (4.40) yield
kkk
i =
oi(k) oi(k) , (5.3)
mkk
i = oioi(k) oi(k) . (5.4)
B. Guided Tuning Turbine Blade Method (GTTB)
Based upon structural perturbations to the mass and stiffness at critical locations, as described previously from 5.1 to 5.3, the Guided Tuning of Turbine Blades method (GTTB) is developed.
The methodology is as follows:
(i) The various engine order excitations are identified.
(ii) Using a finite element description of the original blade geometry (pre-tuning), the free vibration eigenvalue problem is solved. This gives the natural frequencies and mode shapes.
(iii) A Campbell diagram is created, identifying all resonant conditions but especially the one(s) inside the operating speed range. The resonant conditions inside the operating range are referred to as the offending frequencies.
(iv) For all resonances inside the operating range, the maxima of their associated mode shape(s) are identified.
44
(v) A small amount of material is added to or removed from the blade at the maxima of the offending mode (see the previous step). This changes the mass and stiffness matrices and, hence, impacts the natural frequencies. But because it takes place at a modal peak, the change in the eigen-solution is most noticeable in the offending frequency/mode;
there is a smaller change in the other frequencies/modes. The process returns to step (ii) and the process continues until the offending frequency is pushed outside of the operating range.
The application of the GTTB method is illustrated in Paper 1 at the end of Chapter 6.
C. Summary
The perturbation of eigenvalue λn depends solely on the perturbation of [K] and [M]
matrices. The corresponding mode shape given by n consists of regions with zero displacements called nodes and regions with displacements called anti-nodes with the primary anti-node as the zone of highest deflections. Therefore, in order for the blade with distinct frequency λn with respect to its neighbor eigenvalues, to have maximum perturbation (frequency shift), the elements of the [δM] and [δK] matrices must contain non-zero terms at index locations corresponding to the anti-nodes terms of the eigenvector n. Thus, the maximum frequency shift of a distinct eigenvalue is bounded by its associated eigenvector.
CHAPTER 6.
Simulation and Validation
6.1 Finite Element Analysis
A simple turbine blade was modeled with tetrahedron elements to provide a better understanding on the perturbation tuning process as shown in Figure 6.2. Normally, the mechanical damping properties are small and have no effect on varying the blade resonance frequency. Therefore, the density and modulus of elasticity are the only two material properties of interest in this sensitivity study.
The studied blade has modulus of elasticity of 30E6psi and density of 0.283 lb/in3. A modal analysis was performed with the blade fixed in all degrees of freedom at the fir-trees (at the hub). The results illustrated in Figure 6.3 indicate that the structure has distinct eigenvalues. Moreover, one mode n interferes in the operating speed range.
Suppose it is required to shift this resonant frequency λn from thecurrent interference position (point A on Fig.6.4) to a position outside of the operating speed range (point B) with minimum perturbation on other eigenvalues such as frequency f1, shown in Figure 6.4, which is currently clear of resonance as indicated by point C. From perturbation theory, it is known that a Hermitian system of distinct eigenvalues (equations 4.39 &
4.40), δλn is linear proportional to the perturbation of [δK] and [δM].
The practical question is: at which location(s) on the structure do changes in the mass [M] with associated stiffness [K] produce the absolute maximum frequency shift? From a physical standpoint the eigenvector n is a matrix containing the 3DOF displacements of
46
each lumped mass representing the structure continuum. The mode shape given by n
consists of regions with zero displacements called nodes and regions with displacements called anti-nodes and the primary anti-node is the zone of highest deflections. Therefore, in order for the blade with distinct frequency λn with respect to its neighbor eigenvalues, to have maximum perturbation (frequency shift), the elements of the [δM] and [δK]
matrices must contain non-zero terms at index locations corresponding to the anti-nodes terms of the eigenvector n. Thus, the frequency shift of a distinct eigenvalue is bounded by its associated eigenvector. Practically, for minimum aerodynamic disturbance to the airfoil, the focus of perturbation is concentrated on the primary anti-node.
As in any FE application, to ensure the integrity of the analysis, the structural mesh must be maintained when a perturbation is introduced. This is approached by dividing the volume associated with the primary anti-node into sub-volumes such that number of nodes and elements remain unchanged while individual sub-volume material properties, density and modulus of elasticity vary.
From finite element determination, the element stiffness (k) and mass {m} are defined as:
{k} = ∫BTDBdv (6.1)
{m} = ∫ ρ[N]T[N]dv (6.2)
where dv is an element volume, ρ is the material density, [N] are the shape functions, B are the strain shape functions, and D is the elastic matrix.
From the baseline modal analysis, the eigenvalue λn and its corresponding mode shape n
are determined. This primary node corresponding to the mode shape n is found to be located on the blade trailing edge concentrated on a volume v consisting of three
sub-volumes v1,v2 and v3 with corresponding consistent mass matrices m1, m2 and m3 as shown in Figure 6.1. Numerically through FEA, the stability, the smoothness and the bounds of the perturbed solutions of equations (4.39) to (4.42) can be studied independently by either modifying only the mass matrix (through density property) or the stiffness matrix (through Young modulus, [D] matrix) at a time. Let’s consider the physical case in which the mass is reduced from the primary anti-node. The solutions of subsequent reduction simultaneously to zero of density and modulus of elasticity in v1, v2
and v3 demonstrate that:
1) The process of tuning a frequency out of the interference zone is feasible
2) The tuned eigenvalue and its corresponding eigenvector vary on the perturbed interval smoothly as indicated by variation of λn from point A to point B.
3) The frequency shift is bounded by the size of the anti-node of interest.
If we define ψ as the influence factor, the eigenvalue sensitivity is:
ψi = ( λj - λoi )/ λoi (6.3) The analysis indicates that, for a distinct eigenvalue system in the interested frequency domain, as used in this investigation, the eigenvalue sensitivity of the other modes in particular to those adjacent to the tuned frequency could be on the order of less than one
ψi = ( λj - λoi )/ λoi (6.3) The analysis indicates that, for a distinct eigenvalue system in the interested frequency domain, as used in this investigation, the eigenvalue sensitivity of the other modes in particular to those adjacent to the tuned frequency could be on the order of less than one