AMP Advances in Mathematical Physics 1687-9139 1687-9120 Hindawi Publishing Corporation 549430 10.1155/2014/549430 549430 Research Article Fractal Theory and Contact Dynamics Modeling Vibration Characteristics of Damping Blade Yuan Ruishan Zhou Qin Zhang Qiang http://orcid.org/0000-0002-9364-327X Xie Yonghui Yang Xiao-Jun School of Energy and Power Engineering Xi'an Jiaotong University Xi'an 710049 China xjtu.edu.cn 2014 25 3 2014 2014 27 02 2014 16 03 2014 7 4 2014 2014 Copyright © 2014 Ruishan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The contact surface structure of dry friction damper is complicate, irregular, and self-similar. In this paper, contact surface structure is described with the fractal theory and damping blade is simplified as 2-DOF cantilever beam model with lumped masses. By changing the position of the damper, lacing and shroud structure are separately simulated to study vibration absorption effect of damping blade. The results show that both shroud structure and lacing could not only dissipate energy but also change stiffness of blade. Under the same condition of normal pressure and contact surface, the damping effect of lacing is stronger than that of shroud structure. Meanwhile, the effect on changing blade stiffness of shroud structure is stronger than that of lacing. This paper proposed that there is at least one position of the blade, at which the damper dissipates the most vibration energy during a vibration cycle.

1. Introduction

While forced by the exciting force, blades vibrate and the relative displacements among adjacent blades are formed. The dry friction structures of lacing and shroud, on the one hand, dissipate vibration energy. On the other hand, the structure not only changes the mass of the blade but also changes the stiffness of the blade so that the inherent frequency and the exciting force frequency of the blade could avoid each other well. Thus the purpose of reducing the vibration level of the blade is achieved.

The contact surface structure of dry friction damper can be described with the fractal theory accurately. Since the fractal theory was proposed by Benoit B. Mandelbrot in 1975, it has been widely applied. Based on fractal dimension, Ritchie and Olff  proposed biodiversity calculation model by researching the relationship between space scaling relation and biodiversity. Taylor et al.  studied Pollock’s watercolors by using the fractal theory.

Using the dry friction model to inhibit vibration is a very effective method. The famous scholars Den Hartog  proposed the Coulomb model. After fully developing the Coulomb model and intensively studying dry friction, Meng et al. did further research on two-dimensional sliding movement on the basis of the microscopic sliding study in 1991 [4, 5]. Muszynska et al.  simplified the blade and damper structure using the centralized quality model of multidegrees of freedom. Csaba  then combined the one-bar microslip model with the centralized quality model of two degrees of freedom to study the blade with a damper. Lui et al.  and others proposed a new model to research the dry friction under different roughness and positive pressure condition.

In order to accurately describe contact interface and reveal the vibration reduction mechanisms of lacing and shroud structure, a centralized quality cantilever model of two degrees of freedom is built using the fractal theory in this paper. By placing the damper on different masses, the vibration reduction effect of lacing and shroud structure is simulated. By using the one-bar microslip model, the constitutive equation of the force among friction surface is established. The time-frequency domain interactive method is adopted in the solving process of lacing and shroud structure. Therefore, the corresponding results at the steady state are obtained.

2. The One-Bar Microslip Model

For damping blade, the damper structure and boundary conditions are very complex. The contact interface, especially, consists of a series of concave and convex structure in different sizes. It has high self-similarity and it is difficult to describe the structure using the same size. In order to develop a model as simple as possible but also complete enough to show the most important properties of friction interface, the contact interface is described using fractal dimension and fractal length in this paper. Fractal dimension can be calculated by a variety of methods. And similarity dimension method is adopted here.

Assuming objects or geometric shapes can be divided into N parts, each part is similar to the whole object with a likelihood ratio β. The fractal dimension can be expressed as (1)D=lnNln(1/β)=-lnNlnβ.

The fractal dimension D does not have to be an integer.

The fractal curve length estimation model given by Mandelbrot is expressed as (2)L=L0ε1-D, where L is the fractal curve’s Euclidean length; L0 is the initial operation length of fractal curve; ε is the scale of fractal curve; D is the fractal dimension.

The damper is modelled by a rectangular bar pressed against a rigid surface with a normal load q and subjected to a force F, shown in Figure 1. The normal load on the bar is assumed to be constant over the width of the bar and defined by a quadratic normal load function in the lengthwise direction: (3)q(x)=q0+q24(xl-x2)l2.

Microslip model for the friction interface.

The force F can be expressed as (4)F=Fasinωt. The bar has a modulus of elasticity E and a cross section area A. The length of the bar is l, and the coefficient of friction μ is assumed to be constant across the contact zone and independent whether the bar is sliding or not.

The bar starts to slip, as the force is applied. Then the bar may be divided into two zones, one that is slipping and one that is stuck. The length of the slip zone is defined as the slip length δ and the slip length corresponding to the maximum force Famp is δa, shown in Figure 2(a).

Microslip model for damper.

Illustrative plot of force and displacement when F is decreasing

Illustrative plot of force and displacement when F is increasing

The zone that is slipping is stretched after the initial loading with F=Famp. As the force decreases from Famp to -Famp, the bar may be divided into three zones in this analysis, which is shown in Figure 2(b).

Zone A is stuck and has zero strain.

Zone B is stuck and stretched.

Zone C is slipping and compressed.

The length of the compression zone is denoted as the slip length δd. Zone C will increase and zone B will decrease as F decreases. This will continue until F=-Famp. Zone B is then eliminated and δd equals δa.

Here we have the opposite situation as in the previous part. As the force increases from -Famp to Famp, the bar may be divided into three zones in this analysis, which is shown in Figure 2(c).

Zone A is stuck and has zero strain.

Zone B is stuck and compressed.

Zone C is slipping and stretched.

The length of stretched zone is denoted as the slip length δi. Zone C will increase and zone B will decrease as F increases. This will continue until F=Famp. Zone B is then eliminated and δd equals δa.

The length of the slip zone is defined as the slip length δ. The slip length corresponding to the maximum force Famp is δa. The displacement of the bar is defined as u. According to the mechanics analysis, it can be seen that force F and displacement u are the function of slip length δ.

As the force is applied, the force at the right bar end F and displacement can be expressed as (5)F(δ)=0δFf(x)dx=μq0δ+2μq23l2(3δ2l-2δ3),u(δ)=μq0δ22EA+2μq23EAl2δ3-μq23EAl23δ4. As the force decreases from Famp to -Famp, the force at the right bar end F and displacement can be expressed as (6)Fd(δa,δd)=μq0(δa-2δd)+2μq23l2(3lδa2-2δa3+4δd3-6lδd2),ud(δa,δd)=μEA(q0δa2-2δd22+q2(4(δa3-2δd3)3l-δa4-2δd4l4)). As the force decreases from -Famp to Famp, the force at the right bar end F and displacement can be expressed as (7)Fi(δa,δi)=μq0(2δi-δa)+2μq23l2(2δa3-3lδa2+6lδi2-4δi3),ui(δa,δd)=μEA(q02δi2-δd22+q2(4(2δi3-δa3)3l-2δi4-δa4l2)). Tow linearization techniques are discussed to transform the nonlinear properties of friction into equivalent damping and stiffness. Two criteria are established in Lazan’s linearization method, for equivalence of the two loops.

The same value of loop area is the same as damping energy per cycle W.

The amplitude of force Famp and displacement uamp should be the same.

The first criterion gives the equivalent viscous damping ceq, while the second criterion together with ceq gives the equivalent stiffness keq: (8)w(δa)=2μ2δa33EA[q02+(4δal-14δa25l2)q0q2+(8δa47l4-4δa3l3+16δa25l2)q22],(9)ceq=2μ2δa33EAπωuamp2[q02+(4δal-14δa25l2)q0q2+(8δa47l4-4δa3l3+16δa25l2)q22],(10)keq=(Fampuamp)2-(ωceq)2.

3. The Forced Vibration Response Analysis of Damped Blade

A two-degree freedom system simplifying the blade is used for forced vibration response analysis in this paper. The 2-DOF system is built up of a massless beam, two concentrated masses, and the damper. Compared with the previous study, this paper simulates lacing and shroud structure by locating damper on m1 and m2, respectively, and the structure is shown in Figures 3 and 4. The damper is described by a viscous damper and spring, ceq1 and keq1 and ceq2 and keq2, respectively. Displacements where the damper is attached and the force is applied are x1 and x2, respectively.

Simplified damping blade with lacing structure.

Simplified damping blade with shroud structure.

The following parameters of the whole system are as follows, assuming that Q0=μq0l/Pa:

beam system k1=1.0×107 N/m, k2=1.0×107 N/m, m1=0.05 kg, and m2=0.05 kg;

damper system E1A1=40000 N, l=0.2 m, and Q2=Q0/2.

The equations of motion for the system are (11)m1x1(t)+ceq1x1(t)+(k1+k2+keq1)x1(t)-k2x2(t)=0,m2x2(t)+k2x2(t)-k2x1(t)=Paeiωt.

Assuming that harmonic motion yields (12)x1(t)=x1aeiωtx2(t)=x2aeiωt, we define the complex stiffness as (13)K1a=keq1+iωceq1.

Solving equations for x1 and x2 yields (14)x1=Pak2(k2-m2ω2)(k1+k2+K1a-m1ω2)-k22,x2=Pa(k2-m2ω2)+((Pak22)×((k2-m2ω2))-1((k2-m2ω2)(k1+k2+K1a-m1ω2)-k22)×(k2-m2ω2))-1).

The equations of motion for the system are (15)m1x1(t)+(k1+k2)x1(t)-k2x2(t)=0,m2x2(t)+ceq2x2(t)+(k2+keq2)x2(t)-k2x1(t)=Paeiωt.

Assuming that harmonic motion yields (16)x1(t)=x1aeiωtx2(t)=x2aeiωt, we define the complex stiffness as (17)K1a=keq2+iωceq2.

Solving equations for x1 and x2 yields (18)x1=Pak2(k2+K2a-m2ω2)(k1+k2-m1ω2)-k22,x2=Pa(k2+K2a-m2ω2)+((k2+K2a-m2ω2))-1(Pak22)×(((k2+K2a-m2ω2)(k1+k2-m1ω2)-k22)×(k2+K2a-m2ω2))-1).

4. Results and Discussion

Through blades vibration analysis with lacing in Figure 3, amplitude-frequency response curves of m1 and m2 in the range of 0~3000 Hz are achieved. The results are shown in Figures 5 and 6. Different positive pressure and damper parameters under different resonance states are shown in Table 1.

Resonance parameters of lacing damper.

Positive pressure Q0 u amp (μm) F amp (N) Resonance frequency (Hz) Equivalent viscous damping (N·s/m) Equivalent stiffness (N/s)
5.6 6.23854 4.217 1420 176 676039
16 2.55285 4.309 1465 453 1687830
80 0.75866 5.022 1630 1689 6619180
160 0.52390 5.858 1740 2696 11180741
320 0.40169 7.223 1855 4085 17980921
800 0.32137 10.183 1985 6750 31686540

Vibration response of dry friction blade m1 with lacing.

Vibration response of dry friction blade m2 with lacing.

From Figures 5 and 6, it is obvious that, when the damper is placed on the mass m1 to simulate lacing structure, the resonance amplitude of m1 decreases with the increase of the positive pressure. On the contrary, the resonance frequency of m1 increases with the increase of positive pressure. It could also be seen that, with the increase of positive pressure, the equivalent stiffness increases. From the vibration mechanics, the resonant frequency is positively correlated with the system stiffness as the resonance amplitudes negatively correlated with the stiffness. As a result, the resonance amplitude of m1 decreases and the resonant frequency increases.

The resonance amplitude of m2 decreases first and then increases along with the increase of positive pressure. Moreover, the changing trend of the resonance amplitude in Figure 6 is consistent with the conclusion as above. While the positive pressure Q0 equals 320, the resonance amplitude reaches the minimum. This indicates that there exists optimal positive pressure which makes blade resonance response reach minimum for a blade with lacing structure. As the same as mass m1, the resonance frequency of m2 increases with the increase of positive pressure.

Comparing the amplitudes of the mass m1 and m2, the amplitude of m2 is found greater than that of m1 in calculation of all frequencies. In this case, the vibration inhibition effect towards m1 that damping structure brought is much greater than the inhibition effect towards m2.

The damper is placed on the mass m2 to simulate shroud structure. Figures 7 and 8 present the amplitude-frequency response curves in the frequency range of 0~3200 Hz of m1 and m2 with shroud structure. Table 2 shows the resonance parameters of the damper under various positive pressures.

Resonance parameters of shroud structure damper.

Positive pressure Q0 u amp (μm) F amp (N) Resonance frequency (Hz) Equivalent viscous damping (N·s/m) Equivalent stiffness (N/s)
5.6 2.37847 2.527 1525 265 1062646
16 0.86828 2.445 1715 659 2815978
80 0.17324 2.377 2435 2369 13719903
160 0.08665 2.366 2795 4129 27308918
320 0.04329 2.360 3010 7670 54505874
800 0.01733 2.357 3120 18506 136020960

Vibration response of dry friction blade m1 with shroud structure.

Vibration response of dry friction blade m2 with shroud structure.

As shown in Figures 7 and 8, when the damper is placed on the mass m2, the resonance amplitude of m1 decreases first and then increases along with the increase of positive pressure. While the positive pressure Q0 equals 160, the resonance amplitude reaches the minimum. And the resonance frequency of m1 increases gradually with the increase of positive pressure. On the other hand, the resonance amplitude of m2 decreases with the increase of positive pressure. The resonance frequency of m2 increases along with the increase of positive pressure.

Comparing the amplitude of m1 and m2, it is found that, when the positive pressure is smaller than or equal to 80, the amplitude of m2 is greater than the amplitude of m1 and the difference between the two amplitudes is smaller than the difference obtained when the damper is placed on m1. When the positive pressure is greater than 80, the amplitude of m1 is greater than that of m2 while the frequency is near the resonance frequency. However, the amplitude of m1 remains smaller than that of m2 when the frequency is beyond this range. In this case, the vibration inhibition effect towards m2 that damping structure brought is much greater than the inhibition effect towards m1. When the positive pressure is relatively small, the inhibition effect could be small accordingly. But if the positive pressure is relatively large, particularly under the resonance state, the equivalent stiffness of the damper is very large and the inhibition effect towards m2 would be great as well. This results in the fact that the amplitude of m2 stays smaller than that of m1 within certain range around the resonance frequency.

Comparing the results obtained when the damper is placed separately on m1 and m2, it is found that the resonance amplitude changing trend of m1 in the first condition is the same as the changing trend of m2 in the second condition. And the resonance amplitude changing trend of m2 is the same as the changing trend of m1 in the second condition. But, under the same positive pressure, the resonance frequency of m2 with the damper is higher than that of m1 with the damper. By the contrast of Tables 1 and 2, it is obvious that the equivalent stiffness of m2 with the damper is much greater than that of m1 with the damper. So, in the same situation, the resonance frequency of m2 increases drastically.

Figures 9 and 10 show hysteresis loops in an excitation cycle of the damper at resonance state. Table 3 gives energy dissipation in a vibration cycle of the damper at resonance state.

Energy dissipation of the damper at resonance state.

Pressure Q0 (N) Damper located on m1 Damper located on m2
Resonance frequency (Hz) Sliding part length δa (m) The dissipation of energy ×10-6 (w) Resonance frequency (Hz) Sliding part length δa (m) The dissipation of energy ×10-6 (w)
5.6 1420 0.1115447 30.4667 1525 0.0710186 7.1782
16 1465 0.0451878 13.5863 1715 0.0271995 2.6746
80 1630 0.0118769 4.9762 2435 0.0057784 0.5436
160 1740 0.0070775 4.0425 2795 0.0029157 0.2720
320 1855 0.0044180 3.8393 3010 0.0014642 0.1359
800 1985 0.0025145 4.3451 3120 0.0005877 0.0544

Resonance hysteresis loops of dry friction blade with lacing wire.

Resonance hysteresis loops of dry friction blade with shroud structure.

The area surrounded by a hysteresis loop represents the energy dissipation amount in a vibration cycle of the damper. From Figures 9 and 10 and Table 3, it could be seen that, whether the damper is placed on m1 or m2, the energy dissipation amount of the damper decreases overall along with the increase of positive pressure. Based on formula (8), the energy dissipation amount in a vibration cycle of the damper is relevant to the positive pressure and the sliding part length δa. The energy dissipation amount is positively correlated with the positive pressure and negatively correlated with the sliding part length.

Under the same condition, the consumed energy of vibration when the damper is placed on lacing is greater than that when the damper is placed on shroud structure. Comparing Tables 1 and 2 and Figures 9 and 10, it is found that, when the damper is placed on lacing structure, the maximum excitation force on the damper presents linear variation with the increase of pressure. And the excitation force on the damper increases as the positive pressure increases. When damper is placed on shroud structure, the maximum excitation force on the damper presents linear variation with the increase of pressure. The variation amplitude is pretty small and decreases as the positive pressure increases.

Assuming the placing of the damper on the blade root, as the vibration displacement amplitude at the root is close to zero, the consumed energy in a vibration cycle is less than the consumed energy at the lacing structure. Meanwhile, from Figures 9 and 10 and Table 3, it could be seen that, under the same positive pressure, the consumed energy when the damper is placed on lacing wire is greater than that while the damper is placed on shroud structure. Combining mathematical knowledge, it could be obtained that there existed one position so that, when the damper is placed on this position, the consumed energy in a vibration cycle reaches maximum.

5. Conclusions

Because of the complexity and self-similarity of contact surface geometry, contact surface can be described accurately using fractal dimension and fractal length.

Whether shroud or lacing structure, it not only has the effect of dissipating energy but also changes blade stiffness. Meanwhile, in the same condition of pressure and contact area, the damping effect of lacing structure is stronger than that of shroud structure, but the stiffness effect is opposite.

When damper is located on mass m1, as pressure increases, the resonance frequencies of masses m1 and m2 increase. The resonance amplitude of mass m1 decreases and the resonance amplitude of mass m2 decreases first and then increases. It is obvious that there is an optimal pressure, making resonance amplitude minimum. Moreover, the resonance amplitude of m1 is less than that of m2.

When damper is located on mass m2, as pressure increases, the resonance frequencies of masses m1 and m2 increase. The resonance amplitude of mass m1 decreases first and then increases while the resonance amplitude of mass m2 decreases. There is an optimal pressure that makes resonance amplitude of mass m1 minimum. When pressure is relatively small, the resonance amplitude of m1 is smaller than that of m2. When pressure is relatively large, near the resonance frequency, there is a frequency range that makes resonance amplitude of m1 bigger than that of m2.

Under the same circumstance, resonance frequency is lower when damper is located on m1 than when it is located on m2, but the vibration amplitude is opposite.

The damper consumes more energy when it is located on mass m1 than the situation when it is located on m2. No matter which mass damper is located, the energy consumed in a vibration cycle declined as the pressure increases.

Under the same circumstance, there is at least one position on the blade, making damper energy consumption maximum in one vibration cycle.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Ritchie M. E. Olff H. Spatial scaling laws yield a synthetic theory of biodiversity Nature 1999 400 6744 557 560 2-s2.0-0033527041 10.1038/23010 Taylor R. P. Micolich A. P. Jonas D. Fractal analysis of Pollock's drip paintings Nature 1999 399 6735 422 2-s2.0-0033519642 Den Hartog J. P. Forced vibrations with combined Coulomb and viscous friction Transactions of the American Society of Mechanical Engineers 1931 53 9 107 115 Meng C.-H. Bielak J. Griffin J. H. The influence of microslip on vibratory response, part I: a new microslip model Journal of Sound and Vibration 1986 107 2 279 293 2-s2.0-0023044770 Meng C.-H. Chidamparam P. Friction damping of two-dimensional motion and its application in vibration control Journal of Sound and Vibration 1991 144 3 427 447 2-s2.0-0026419464 Muszyńska A. Jones D. I. G. On tuned bladed disk dynamics: some aspects of friction related mistuning Journal of Sound and Vibration 1983 86 1 107 128 2-s2.0-0021097285 ZBL0507.73054 Muszynska A. Jones D. I. G. Lagnese T. Whitford L. On nonlinear response of multiple blade systems Shock and Vibration Bulletin 1981 51 3 89 110 2-s2.0-0019010086 Jones D. I. G. Muszynska A. Design of turbine blades for effective slip damping at high rotational speeds Shock and Vibration Bulletin 1979 49 87 96 Csaba G. Modelling Microslip Friction Damping and Its Influence on Turbine Blade Vibrations 1998 Division of Machine Design Department of Mechanical Engineering, Linköping University Liu Y. Shangguan B. Xu Z. A friction contact stiffness model of fractal geometry in forced response analysis of a shrouded blade Nonlinear Dynamics 2012 70 3 2247 2257 10.1007/s11071-012-0615-8