Wang et al 2018a - Revision history

Rimni at 13:15, 30 August 2021

2021-08-30T13:15:32Z

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Rimni: /* 3. Modal and harmonic response analysis of blades */

2021-08-30T12:51:02Z

‎3. Modal and harmonic response analysis of blades

Rimni: /* 2.2 Parameters of viscoelastic damping material */

2021-08-30T12:46:27Z

‎2.2 Parameters of viscoelastic damping material

Rimni at 13:51, 6 May 2020

2020-05-06T13:51:09Z

Rimni at 13:49, 6 May 2020

2020-05-06T13:49:31Z

Rimni at 12:40, 6 May 2020

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Rimni at 12:39, 6 May 2020

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Rimni at 12:34, 6 May 2020

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Rimni at 12:33, 6 May 2020

2020-05-06T12:33:41Z

Rimni at 12:24, 6 May 2020

2020-05-06T12:24:23Z

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 ===3.1 Blade modeling and mesh division===
-Effect of viscoelastic material on blade vibration and its damping performance is mainly researched. A simplified model of blade is adopted to draw a direct conclusion. The results can also be applied to actual aero-engine blades. According to the discussion in Chapter 2 , Solid186 element is adopted for straight mortise blade and volume sweep method is selected to divide mesh. Finite element model after mesh division is shown in Figure 2. Element size is set to 1.5mm. In Figure 2 (a), there are 5778 elements in blade and 6075 elements in mortise. In Figure 2 (b), there are 4428 elements in blade and 2187 elements in mortise.
+Effect of viscoelastic material on blade vibration and its damping performance is mainly researched. A simplified model of blade is adopted to draw a direct conclusion. The results can also be applied to actual aero-engine blades. According to the discussion in Chapter 2, Solid186 element is adopted for straight mortise blade and volume sweep method is selected to divide mesh. Finite element model after mesh division is shown in [[#img-2|Figure 2]]. Element size is set to 1.5mm. In [[#img-2|Figure 2]](a), there are 5778 elements in blade and 6075 elements in mortise. In [[#img-2|Figure 2]](b), there are 4428 elements in blade and 2187 elements in mortise.
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
@@ Line 261: / Line 261: @@
-For blade without viscoelastic block shown in Figure 2 (b), there is a contact between blade and mortise side. Surface contact pairs are needed to make blade and mortise connect better. For blade that viscoelastic damping block is added to mortise sides shown in Figure 4 (a), viscoelastic block is in contact with mortise sides. In order to connect two parts of different materials, volumes glue command in ANSYS software is required. Force will pass through blades for the combined structure that viscoelastic block has been added. Two surface contact pairs are established. For blades that viscoelastic damping block is added between blade bottom and mortise shown in Figure 4 (b), there is a contact between viscoelastic damping block and mortise, which should be coupled through degree of freedom of overlapping joint.
+For blade without viscoelastic block shown in [[#img-2|Figure 2]](b), there is a contact between blade and mortise side. Surface contact pairs are needed to make blade and mortise connect better. For blade that viscoelastic damping block is added to mortise sides shown in [[#img-4|Figure 4]](a), viscoelastic block is in contact with mortise sides. In order to connect two parts of different materials, volumes glue command in ANSYS software is required. Force will pass through blades for the combined structure that viscoelastic block has been added. Two surface contact pairs are established. For blades that viscoelastic damping block is added between blade bottom and mortise shown in [[#img-4|Figure 4]](b), there is a contact between viscoelastic damping block and mortise, which should be coupled through degree of freedom of overlapping joint.
 ===3.2 Modal and harmonic response analysis===
@@ Line 314: / Line 314: @@
-Analysis shows that calculation of natural frequency comes down to an eigenvalue problem. Eigenvalues <math>{\omega }^2_i</math> can be obtained by solving equation (17). The corresponding eigenvectors are <math>U_i</math>. <math>{\omega }_i</math> is natural circular frequency of i order, where i ranges from 1 to number of degrees of freedom. So natural frequency is <math>f_i={\omega }_i/(2\pi )</math>. In order to represent dynamic response of the structure, many orders’ data are needed. Here, suppression effect of viscoelastic block on blade vibration is mainly discussed, in which high order’s data have little effect. So, only ten orders’ data will be extracted. Modal analysis of blades is solved by Block Lanczos method and natural frequency from 1 to 10 orders is obtained for blades with and without viscoelastic block. And viscoelastic block is added on mortise sides. Natural frequency of blades is given in Table 3.
+Analysis shows that calculation of natural frequency comes down to an eigenvalue problem. Eigenvalues <math>{\omega }^2_i</math> can be obtained by solving equation (17). The corresponding eigenvectors are <math>U_i</math>. <math>{\omega }_i</math> is natural circular frequency of i order, where i ranges from 1 to number of degrees of freedom. So natural frequency is <math>f_i={\omega }_i/(2\pi )</math>. In order to represent dynamic response of the structure, many orders’ data are needed. Here, suppression effect of viscoelastic block on blade vibration is mainly discussed, in which high order’s data have little effect. So, only ten orders’ data will be extracted. Modal analysis of blades is solved by Block Lanczos method and natural frequency from 1 to 10 orders is obtained for blades with and without viscoelastic block. And viscoelastic block is added on mortise sides. Natural frequency of blades is given in [[#tab-3|Table 3]].
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 3'''. Natural frequency of blades with and without viscoelastic block/Hz</div>
+<div id='tab-3'></div>
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:70%;"
 |-

@@ Line 137: / Line 137: @@
 ===2.2 Parameters of viscoelastic damping material ===
-Material of blade plate is plain carbon structural steel Q235A. Its geometrical size is shown in Figure 1. Material of viscoelastic damping block is selected as ZN-33 rubber and its parameters are given in Table 1.
+Material of blade plate is plain carbon structural steel Q235A. Its geometrical size is shown in [[#img-1|Figure 1]]. Material of viscoelastic damping block is selected as ZN-33 rubber and its parameters are given in [[#tab-1|Table 1]].
 {| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
 |-
@@ Line 151: / Line 152: @@
 | colspan="1" style="padding:10px;"| '''Figure 1'''. Geometrical size and blade model
 |}
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 1'''. Material parameters of blade and viscoelastic block</div>
+<div id='tab-1'></div>
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:45%;"
 |-
@@ Line 208: / Line 208: @@
-According to equation (8) and (10), relative modulus  <math>{\alpha }_i^G</math> and  <math>{\alpha }_i^K</math> can be available. The calculated values of parameters are given in Table 2.
+According to equation (8) and (10), relative modulus  <math>{\alpha }_i^G</math> and  <math>{\alpha }_i^K</math> can be available. The calculated values of parameters are given in [[#tab-2|Table 2]].
 <div class="center" style="width: auto; margin-left: auto; margin-right: auto;font-size:75%;">
 '''Table 2'''. Parameters of Prony series</div>
+<div id='tab-2'></div>
 {| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:45%;"
 |-

@@ Line 1,039: / Line 1,039: @@
 [17] Lu C.H., Bai H.B.  Study on constitutive model of viscoelastic material. Polymer Materials Science & Engineering, 23(6):28-31+35, 2007.
-[18] Gamonpilas C., Mccuiston R.,  Mccuiston R.  A non-linear viscoelastic material constitutive model for polyurea. Polymer, 53(17):3655-3658, 2012.
+[18] Gamonpilas C., McCuiston R.  A non-linear viscoelastic material constitutive model for polyurea. Polymer, 53(17):3655-3658, 2012.
 [19] Park S.W., Schapery R.A.  Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. International Journal of Solids & Structures, 36(11):1653-1675, 1999.

@@ Line 1,005: / Line 1,005: @@
 <div class="auto" style="width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-[1] Huang YX and Chen GX (2011) The structure crack detection of aero-engine compressor based on modal analysis. ''Science & Technology Information'' (23): 83-84(in Chinese).
+[1] Huang Y.X., Chen G.X. The structure crack detection of aero-engine compressor based on modal analysis. Science & Technology Information, (23):83-84, 2011 (in Chinese).
-[2] Zhang W, Feng ZQ and Guo XY (2013) Nonlinear vibration of aero-engine compressor blades.'' SCIENCE CHINA Physics, Mechanics & Astronomy'' (4): 345-362.
+[2] Zhang W., Feng Z.Q.,  Guo X.Y.  Nonlinear vibration of aero-engine compressor blades. Science China Physics, Mechanics & Astronomy, (4):345-362, 2013.
-[3] Zhao YF (2001) Properties and Applications of ZN Series Viscoelastic Damping Materials. ''Aerospace Materials & Technology'' (31):19-23.
+[3] Zhao Y.F. Properties and applications of ZN series viscoelastic damping materials. Aerospace Materials & Technology, (31):19-23, 2001.
-[4] Williams ML (2012) Structural analysis of viscoelastic materials. ''Aiaa Journal'' 2(5):785-808.
+[4] Williams M.L.  Structural analysis of viscoelastic materials. AIAA Journal, 2(5):785-808, 2012.
-[5] Zhang F, Guo M, Xu K et al (2014) Multilayered damping composites with damping layer/constraining layer prepared by a novel method. ''Composites Science & Technology'' 101(101):167-172.
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-[6] Shokrieh M M and Najafi A (2007) Damping Characterization and Viscoelastic Behavior of Laminated Polymer Matrix Composites Using a Modified Classical Lamination Theory. ''Experimental Mechanics'' 47(6):831-839.
+[6] Shokrieh M.M., Najafi A. Damping characterization and viscoelastic behavior of laminated polymer matrix composites using a modified classical lamination theory. Experimental Mechanics, 47(6):831-839, 2007.
-[7] Fan RP (2010) Experimental study of viscoelastic damping materials for vibration and noise control within railway vehicles. ''Journal of Mechanical Strength'' 32(2): 188-195(in Chinese).
+[7] Fan R.P.  Experimental study of viscoelastic damping materials for vibration and noise control within railway vehicles. Journal of Mechanical Strength, 32(2):188-195, 2010 (in Chinese).
-[8] Yang JS (2013) Vibration and damping characteristics of hybrid carbon fiber
+[8] Yang J.S.  Vibration and damping characteristics of hybrid carbon fiber composite pyramidal truss sandwich panels with viscoelastic layers. Composite Structures, (106):570-580, 2013.
-composite pyramidal truss sandwich panels with viscoelastic layers. ''Composite ''Structures'' (106): 570-580.''
+[9] Robinson M.J., Kosmatka J.B.  Embedding viscoelastic damping materials in low-cost VARTM composite structures. Proceedings of SPIE-The International Society for Optical Engineering, 5760:349-360, 2005.
-[9] Robinson MJ and Kosmatka JB (2005) Embedding viscoelastic damping materials in low-cost VARTM composite structures. ''Proceedings of SPIE-The International Society for Optical Engineering'' 5760:349-360.
+[10] Zhou X.Q.  Simplified-super-element-method for analyzing free flexural vibration characteristics of periodically stiffened-thin-plate filled with viscoelastic damping material. Thin-Walled Structures, (94):234-252, 2015.
-[10] Zhou XQ (2015) Simplified-super-element-method for analyzing free flexural vibration characteristics of periodically stiffened-thin-plate filled with
+[11] Martinez M.A., Elejabarrieta M.J. Dynamic characterization of high damping viscoelastic materials from vibration test data. Journal of Sound & Vibration, 330(16):3930-3943, 2011.
-viscoelastic damping material. ''Thin-Walled Structures'' (94): 234-252.
+[12] Trindade M.A. Optimization of passive constrained layer damping treatments applied to composite beams. Latin American Journal of Solids & Structures, 4(1):19-38, 2007.
-[11] Martinez MA and Elejabarrieta MJ (2011) Dynamic characterization of high damping viscoelastic materials from vibration test data. ''Journal of Sound & Vibration'' 330(16):3930-3943.
+[13] Wang H.J., Chen L.W.  Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer. Composite Structures, 58(4):563-570, 2002.
-[12] Trindade M A (2007) Optimization of passive constrained layer damping treatments applied to composite beams. ''Latin American Journal of Solids & Structures'' 4(1):19-38.
+[14] Montemurro M., Vincenti A., Yao K., et al.  A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches. Journal of Vibration & Control, 21(9):1778-1800, 2014.
-[13] Wang HJ and Chen LW (2002) Vibration and damping analysis of a three-layered composite annular plate with a viscoelastic mid-layer. ''Composite Structures'', 58(4):563-570.
+[15] Tan D.H., Yu H.  Viscoelastic Materials Used for Vibration and Noise Control. Marine Electric & Electronic Engineering, (2):63-66, 2015.
-[14] Montemurro M, Vincenti A, Yao K, et al (2014) A two-level procedure for the global optimization of the damping behavior of composite laminated plates with elastomer patches. ''Journal of Vibration & Control'' 21(9):1778-1800.
+[16] Zhou S.C., Liu J.H.  Dynamic modeling and engineering application of viscoelastic damping elements. Journal of Astronautics, (4):1346-1350, 2009.
-[15] Tan DH and Yu H (2015) Viscoelastic Materials Used for Vibration and Noise Control. ''Marine Electric & Electronic Engineering'' (2):63-66.
+[17] Lu C.H., Bai H.B.  Study on constitutive model of viscoelastic material. Polymer Materials Science & Engineering, 23(6):28-31+35, 2007.
-[16] Zhou SC and Liu JH (2009) Dynamic modeling and engineering application of viscoelastic damping elements. ''Journal of Astronautics'' (4):1346-1350.
+[18] Gamonpilas C., Mccuiston R.,  Mccuiston R.  A non-linear viscoelastic material constitutive model for polyurea. Polymer, 53(17):3655-3658, 2012.
-[17] Lu CH and Bai HB (2007) Study on Constitutive Model of Viscoelastic Material. ''Polymer Materials Science & Engineering'' 23(6):28-31+35.
+[19] Park S.W., Schapery R.A.  Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. International Journal of Solids & Structures, 36(11):1653-1675, 1999.
-[18] Gamonpilas C, Mccuiston R and Mccuiston R (2012) A non-linear viscoelastic material constitutive model for polyurea.'' Polymer'' 53(17):3655-3658.
+[20] Zhang H.L.,  Zhou J.P.  Viscoelastic stochastic finite element simulation of solid propellant grain with random Poisson’s ratio. Journal of Propulsion Technology, 22(3):245-249, 2001.
-[19] Park SW and Schapery RA (1999) Methods of interconversion between linear viscoelastic material functions. Part I—a numerical method based on Prony series. ''International Journal of Solids & Structures'' 36(11):1653-1675.
+[21] Xiao S.Y., Wang H., Ruan Z.Q.  Test Assessment Method of Damping Property of Viscoelastic Damping Materials. Chinese Journal of Ship Research, 9(4):84-87, 2014.
-−
-[20] Zhang HL and Zhou JP (2001) Viscoelastic stochastic finite element simulation of solid propellant grain with random Poisson’s ratio. ''Journal of Propulsion Technology'' 22(3):245-249.
-−
-[21] Xiao SY, Wang H and Ruan ZQ (2014) Test Assessment Method of Damping Property of Viscoelastic Damping Materials.'' Chinese Journal of Ship Research'' 9(4):84-87.
 </div>

@@ Line 689: / Line 689: @@
 <div class="center" style="font-size: 75%;">'''Table 7'''. Natural frequency of blades with different thickness of viscoelastic block/Hz</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:65%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:80%;"
 |-
 |  colspan='2'  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Thickness/mm

@@ Line 689: / Line 689: @@
 <div class="center" style="font-size: 75%;">'''Table 7'''. Natural frequency of blades with different thickness of viscoelastic block/Hz</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:45%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:65%;"
 |-
 |  colspan='2'  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|Thickness/mm
@@ Line 697: / Line 697: @@
 |  colspan='2'  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|4<sup>th</sup> order
 |  colspan='2'  style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;"|5<sup>th</sup> order
 |-
 |  colspan='2'  style="border-top: 2pt solid black;text-align: center;"|No block is added
@@ Line 704: / Line 709: @@
 |  colspan='2'  style="border-top: 2pt solid black;text-align: center;"|4353.5
 |  colspan='2'  style="border-top: 2pt solid black;text-align: center;"|4797.4
 |-
 |  colspan='2'  style="text-align: center;"|0.2
@@ Line 711: / Line 721: @@
 |  colspan='2'  style="text-align: center;"|3962.5
 |  colspan='2'  style="text-align: center;"|4343.0
 |-
 |  colspan='2'  style="text-align: center;"|0.4
@@ Line 718: / Line 733: @@
 |  colspan='2'  style="text-align: center;"|3504.2
 |  colspan='2'  style="text-align: center;"|4336.7
 |-
 |  colspan='2'  style="text-align: center;"|0.6
@@ Line 725: / Line 745: @@
 |  colspan='2'  style="text-align: center;"|3182.9
 |  colspan='2'  style="text-align: center;"|4331.3
 |-
 |  colspan='2'  style="text-align: center;"|0.8
@@ Line 732: / Line 757: @@
 |  colspan='2'  style="text-align: center;"|2923.6
 |  colspan='2'  style="text-align: center;"|4327.0
 |-
 |  colspan='2'  style="border-bottom: 2pt solid black;text-align: center;"|1.0
@@ Line 739: / Line 769: @@
 |  colspan='2'  style="border-bottom: 2pt solid black;text-align: center;"|2703.8
 |  colspan='2'  style="border-bottom: 2pt solid black;text-align: center;"|4323.0
 |  colspan='2'  style="border-bottom: 2pt solid black;text-align: center;"|5798.0
 |  colspan='2'  style="border-bottom: 2pt solid black;text-align: center;"|7390.1

@@ Line 925: / Line 925: @@
 <div class="center" style="font-size: 75%;">'''Table 10'''. The maximum displacement of blades with different loss factor and elasticity modulus of the VDM/mm</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:60%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:55%;"
 |-
 | style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;" | Loss factor

@@ Line 984: / Line 984: @@
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 45%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 55%;"
 |-
 |  style="text-align: center;padding:10px;"| [[File:Review_Wang_et_al_2018a_9300_Fig_15(a).png|centre]]

@@ Line 652: / Line 652: @@
 Frequency-displacement curves of blade with viscoelastic block added to mortise sides are obtained in 0～12000 Hz ranges by harmonic response analysis as shown in Figure 7. It can be seen that the maximum displacement of blade decreases about 15.6% when viscoelastic block is added. The larger vibration displacements are obtained by extracting nodes data of the 1<sup>st</sup>, 3<sup>rd</sup>, 6<sup>th</sup> and10<sup>th</sup> orders at the tip of blades.
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 60%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
 |style="padding:10px;"| [[File:Review_Wang_et_al_2018a_8127_Fig_7.png|centre]]
@@ Line 879: / Line 879: @@
 Frequency-displacement curves in 0-12000Hz ranges with different elastic moduli are obtained by harmonic response analysis as shown in Figure 10 (a). It can be seen that the maximum displacement happens in the 1<sup>st</sup> order of blade. Displacement data in the first order of natural frequency are extracted and Origin curves are shown in Figure 10 (b). Comparing with the blade that elastic modulus is 2.0 GPa, the maximum displacement in the 1<sup>st</sup> order decreases about 16% when elastic modulus is 0.5 GPa.
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
 |-
 |  style="text-align: center;padding:10px;"| [[File:Review_Wang_et_al_2018a_8431_Fig_9(a).png|centre]]
@@ Line 925: / Line 925: @@
 <div class="center" style="font-size: 75%;">'''Table 10'''. The maximum displacement of blades with different loss factor and elasticity modulus of the VDM/mm</div>
-{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:40%;"
+{| style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:60%;"
 |-
 | style="border-top: 2pt solid black;border-bottom: 2pt solid black;text-align: center;" | Loss factor
@@ Line 984: / Line 984: @@
-{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 50%;"
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 45%;"
 |-
 |  style="text-align: center;padding:10px;"| [[File:Review_Wang_et_al_2018a_9300_Fig_15(a).png|centre]]
@@ Line 992: / Line 992: @@
 |  style="text-align: center;padding:10px;"|[[File:Review_Wang_et_al_2018a_2077_Fig_15(b).png|centre]]
 |-
-|  style="text-align: center;font-size: 75%;"|(b) The maximum displacement of blades with and without viscoelastic block added to mortise sides
+|  style="text-align: center;font-size: 75%;padding:10px;"|(b) The maximum displacement of blades with and without viscoelastic block added to mortise sides
 |- style="text-align: center; font-size: 75%;"
 | colspan="2" style="padding:10px;"| '''Figure 15'''. Frequency-displacement curves when centrifugal force is considered