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An analytical modeling method of hard-coating laminated plate under base excitation was studied considering strain-dependent characteristic of coating material (i.e. a kind of material nonlinear behavior). For convenience, the strain-dependent characteristic of hard-coating material was characterized by polynomial, and the material parameters were divided into two parts: linearity and nonlinearity. Hard coating was regarded as a special layer in the analysis and Lagrange’s equation was used to acquire nonlinear equation of motion of the hard-coating laminated plate. Based on Newton–Raphson method, the procedure of solving resonant response and resonant frequency of composite plate was presented. Finally, a T300/QY89l1 laminated plate with NiCoCrAlY þ YSZ hard coating was chosen to dem- onstrate the proposed method, the linear and nonlinear vibrations of the composite plate were solved, and only the linear results were validated by ANSYS software. The results reveal that there is a big difference between the calculation results considering the nonlinearity of coating material and the linear results, which means the laminated plate displays soft nonlinear phenomenon because of depositing coating. Keywords Strain-dependent characteristic, hard coating, laminated plate, vibration characteristics, analytical analysis Introduction Composite laminated plates have been widely used in aeronautic, astronautic, and other fields because of their superior mechanical properties such as high stiffness-to-weight and strength-to-weight ratios. The vibration control of laminated structure has been increasingly concerned to ensure that the components made of laminated plates can work stably and reliably under multifield coupling loads. The main method of vibration control of laminated plates is adding several layers of damping materials to the external surfaces or interior of the plates. These can be damping materials which implement passive vibration control with their inner damping. They can 3,4 also be smart materials which realize vibration control with active methods. Some composite laminates need to work in extreme environments such as high stress, high temperature, or high corrosion, for example, titanium matrix composites and fiber reinforced plastic composite components in aircraft engines. Vibration control of such laminates becomes a more challenging task since the viscoelastic damping 6,7 and active control techniques which need additional energies are difficult to work in such extreme conditions. The hard coating is a kind of coating material which is made of metals, ceramics, or their mixtures and can be used as the thermal coating, antifriction and antierosion coating. In recent years, it has also been found to be used as 8,9 damping coating and the relative vibration reduction techniques have been applied to aero-engine compressor School of Mechanical Engineering & Automation, Northeastern University, Shenyang, China Key Laboratory of Vibration and Control of Aero-Propulsion System, Ministry of Education of China, Northeastern University, Shenyang, China Corresponding author: Wei Sun, Northeastern University, Shenyang 110819, China. Email: weisun@mail.neu.edu.cn Creative Commons CC BY: This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://www. creativecommons.org/licenses/by/4.0/) which permits any use, reproduction and distribution of the work without further permission provided the original work is attributed as specified on the SAGE and Open Access pages (https://us.sagepub.com/en-us/nam/open-access-at-sage). 790 Journal of Low Frequency Noise, Vibration and Active Control 37(4) 10 11 blades and blisks. The biggest technical advantage is that the hard coating can maintain its damping capacity in high temperature or high corrosion environment. Thus, the vibration reduction problems can be solved by depositing hard coatings on the outer surface of laminated plates. To implement damping treatments efficiently, dynamic analysis model of hard-coating laminated composite structures is needed. There have been many researches on the dynamic modeling and analysis of laminated composite structures presently. For example, Kabir analyzed the deformation and free vibration of rectangular laminated plates with classical lamination theory. Phan-Dao et al. analyzed the free vibration and buckling behaviors of symmetric laminated plate. Mao et al. approximated the displacement function of constrained damping of laminated cantilever plates with beam function; they established the dynamic model of plates and obtained the transient response. Zhang et al. analyzed nonlinear dynamic responses of cantilever rectangular laminates with external excitation. Hasheminejad and Keshavarzpour created an exact 3D elasticity model of piezo-laminated composite circular plate based on the spatial state-space method and the Rayleigh integral for- mula. These aforementioned methods of dynamic modeling and analyzing can be used as references while study- ing the vibration characteristics of hard-coating laminates. 17,18 Hard-coating materials have strain-dependent characteristics, which means their storage modulus and dissipation modulus (or loss factors) change with the strain response amplitude of composite structure and this is a unique material nonlinear behavior. Therefore, it becomes a challenging task to study the dynamic modeling method of hard-coating laminates under the consideration of strain-dependent characteristic. In this paper, an analytical modeling method of hard-coating laminated plate under base excitation was studied considering strain-dependent characteristic of coating material. In the next section, the hard coating was regarded as a special layer of plate in analysis and the equations of motion of hard-coating laminates were derived based on classical lamination theory. In “Solution of vibration response considering the strain-dependent characteristic of hard-coating laminates” section, the procedure of solving resonant response and resonant frequency of composite plate was presented based on Newton–Raphson method. In “Study case” section, a T300/QY89l1 laminated plate with NiCoCrAlY þ YSZ hard coating was chosen to demonstrate the proposed method, the linear and nonlinear vibrations of the composite plate were solved, and the linear results were validated by ANSYS software. Some relevant conclusions were listed in the final section. Equations of motion of hard-coating laminates under base excitation The hard-coating cantilever laminated plate structure is shown in Figure 1(a). The length and width of the plate are a and b, respectively, and the thickness of hard coatings which are deposited on the upper and lower surfaces (a) Hard coating ut () Laminated plate (b) (c) y Hard coating Hard coating Hard coating z 1 N+1 Ө x H +2H s c Figure 1. (a) Hard-coating cantilever laminated plate under base excitation, (b) representation of fiber orientation, and (c) cross- section of laminated plate. Sun et al. 791 of the laminates is H . The composite laminate is considered to be symmetric before coating and the thickness is H . After coating, the plate is still symmetric and the overall thickness is H þ 2H . The xyz coordinate plane is set s s c to locate at the neutral plane of the composite laminate. The definition of fiber orientation angle h is shown in Figure 1(b). Number 1 donates the fiber direction and number 2 donates the direction perpendicular to the fiber direction (also called fiber transverse direction). As is shown in Figure 1(c), if the hard coating is regarded as a special layer of composite laminate, then the coordinate in thickness direction from the lower surface to upper surface can be represented by z ; z ; ... ; z , N is the number of layers. 1 2 Nþ1 The elastic modulus E of strain-dependent hard-coating material can be expressed by polynomial as the function of equivalent strain e as follows 2 3 E ¼ E þ e E þ e E þ e E þ (1) c c0 c1 e c2 e c3 where E can be expressed as cj E ¼ E þ iE ; j ¼ 0; 1; 2; (2) cRj cIj cj Here, E and E refer to the material coefficients relative to the storage modulus (Young’s modulus) and the cRj cIj loss modulus, respectively. Further, the elastic modulus of hard coating can also be expressed as E ¼ E þ E (3) c c;l c;nl where E ¼ E are defined as the linear part of the parameters of hard-coating materials and E ¼ e E þ e c;1 c0 c;nl c1 e E þ e E þ are the nonlinear part. c2 e c3 Choose one layer arbitrarily in laminates matrix and the sequence number is i, i ¼ 2; 3; .. . ; N 1. According to the classical lamination plate theory, the constitutive relation of stress and strain in this layer can be expressed as 8 9 2 3 8 9 ðÞ i ðÞ i ðÞ i > > r e > > Q Q Q > x > > > 11 12 16 > > > > 6 7 > > < = < = 6 7 6 7 r e ¼ Q Q Q y (4) 6 21 22 26 7 > > > > > > 4 5 > > > > > > : ; > > : ; s c Q Q Q xy xy 61 62 66 where r , r , s , e , e , c are the stress and strain in the xy plane of the i-th layer. Q , j, l ¼ 1; 2; 6 are the x y xy x y xy jl ðiÞ elements of elastic matrix Q . Q can be solved according to the Young’s modulus, Poisson’s ratio, and fiber jl directional angle of the corresponding layer and the detailed expression can be referred to the Dozio’s study. Since the hard coatings deposited on the surface of the laminate (i ¼ 1; N) are isotropic materials, the consti- tutive relation of stress and strain can be expressed as 2 3 ðiÞ E v E c c 6 7 8 9 8 9 2 2 ðiÞ ðiÞ 6 1 v 1 v 7 c c > > r e > > 6 7 > x > > > > > > > 6 7 > > < = < = 6 7 v E E c c 6 7 r ¼ 0 e (5) 6 2 2 7 > > 1 v 1 v > > > > 6 c c 7 > > > > > > : ; > > 6 7 : ; s c 6 7 xy xy 4 5 21ðÞ þ v where v is Poisson’s ratio of hard coating. According to the mechanics of elasticity, the strain energy U of the laminate can be expressed as ZZZ U ¼ r e þ r e þ s c dV ¼ U þ U (6) ðÞ x x y y xy l nl xy V 792 Journal of Low Frequency Noise, Vibration and Active Control 37(4) where V is the volume of hard-coating laminates, U and U are the strain energy corresponding to the linear and 1 n1 nonlinear parts of the hard-coating materials, which can be defined as linear strain energy and nonlinear strain energy, respectively. Substituting equations (4) and (5) into equation (6) can yield the expression of linear and nonlinear strain energy, of which the linear one is ! ! ! 2 2 ZZ 2 2 2 2 2 2 1 @ w @ w @ w @ w @ w U ¼ D þ D þ 2D þ 4D l 11 22 12 66 2 2 2 2 2 @x @y @x @y @x@y (7) ! ! !# 2 2 2 2 @ w @ w @ w @ w þ 4D þ 4D dxdy 16 26 2 2 @x @x@y @y @x@y Here, the transverse motions of the hard-coating laminates are considered only, w is the transverse displace- ment of neutral surface of composite laminates (completely expressed as wðx; y; tÞ), D ; j; l ¼ 1; 2; 6 are bending jl stiffness coefficients of laminates. If the polynomial in equation (1) adopts to its third only, then the expression of nonlinear strain energy can be shown as ZZ 2 3 U ¼ D f þ D f þ D f g dxdy (8) nl c1 eq c2 c1 nl eq eq where "# 4 4 E H H s s c1 D ¼ H þ (9a) c1 c 2 2 2 2 41 v "# 5 5 E H H s s c2 D ¼ H þ (9b) c2 c 2 2 51 v "# 6 6 E H H s s c3 D ¼ H þ (9c) c3 c 2 2 2 61 v vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 2 2 2 2 @ w @ w @ w @ w 2 2 t 2 2 f ¼ Re þ Im þ Re þ Im eq 2 2 2 2 @x @x @y @y (9d) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! ! 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ w @ w 2 2 ðÞ Re þ Im þ 21 v @x@y @x@y ! ! 2 2 2 2 2 2 2 @ w @ w @ w @ w @ w g ¼ þ þ 2v þ21ðÞ v (9e) nl c c 2 2 2 2 @x @y @x @y @x@y Here, Re () and Im () refer to the real part and imaginary part of the value of quantities, respectively. The kinetic energy of hard-coating laminates can be expressed as ZZ ZZ m m @w _ _ T ¼ k dxdy ¼ þ u dxdy (10) 2 2 @t Sun et al. 793 where m is the mass per unit area of the laminate, k is the total displacement at any point on the neutral surface of the laminate, and u is the displacement of base excitation. Here, the steady-state response of hard-coating laminated plates is considered only and the Galerkin discrete method is used to reduce the order of this nonlinear problem. Assuming that the transverse displacement in natural surface is ðÞ wxðÞ ; y; t ¼ / x; y v t (11) ðÞ j j j¼1 where / ðx; yÞ is the jth order modal shape, v ðtÞ is the jth order participation factors, and N is the number j j of orders. Substituting equation (11) into equations (7), (8), and (10) can yield the expressions of linear strain energy, nonlinear strain energy, and kinetic energy. To obtain the equations of motion of the laminated plates, the derived kinetic energy expression and strain energy expression are substituted into the following Lagrange’s equation d @T @U ðÞ t þ ¼ 0ðÞ k ¼ 1; 2; ... ; N (12) dt @v @v ðÞ t Here, the base excitation is considered only, it can be assumed that ixt v ðÞ t ¼ v e (13) j j where x is the excitation frequency. After calculations, the final expressions are K x M v ¼ F (14) which is a nonlinear algebraic equation since it considers the strain-dependent characteristics of hard-coating materials. Here, K is the N N dimension complex stiffness matrix, M is the N N mass matrix, v is the response vector which consists of v (k=1,2,...,N) and is independent of time, and F is the excitation vector. Further, the complex stiffness matrix K can be expressed as K ¼ K þ K (15) e l nl where K is the complex stiffness matrix corresponding to the linear part of hard-coating materials and K is the l nl complex stiffness matrix corresponding to the nonlinear part. The element of the kth row and pth column of complex stiffness matrix K can be expressed as ZZ yy xy yy xx xx yy xy xx yy xx KðÞ k; p ¼ D / / þ D / / þ 4D / / þ D / / þ / / 11 22 66 12 e k p k p k p k p p k xy yy xy xx xy xx xy yy (16) þ 2D / / þ / / þ 2D / / þ / / dxdy 16 26 k p p k k p p k ZZ hi þ D þ D f þ D f f h dxdy c1 c2 eq c3 eq nl eq where xx xx yy yy xx yy xx yy xy xy ðÞ h ¼ / / þ / / þ v / / þ / / þ21 v / / (17) nl c c k p p k p p p k k k 794 Journal of Low Frequency Noise, Vibration and Active Control 37(4) xx yy xy Here, / , / refer to the second partial derivatives of x and y, respectively, and / represents the partial derivative of x and y successively. The element of the kth row and pth column in mass matrix can be expressed as ZZ MðÞ k; p ¼ m / / dxdy (18) k p The specific expression of the kth element of the excitation vector F can be expressed as ZZ ðÞ F k ¼ mx U / dxdy (19) where U is the basic excitation amplitude. Solution of vibration response considering the strain-dependent characteristic of hard-coating laminates The nonlinear algebraic equation shown in equation (14) can be solved with the nonlinear iterative algorithm. In the solving process, it is necessary to modify the material parameters of hard coatings with the equivalent strain e , so obtaining the equivalent strain of hard-coating laminates is the first step of the whole iterative computation. According to the principle of equal strain energy density of composite laminates, the solving equations can be derived as follows sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 2 2 2 2 z @ w @ w @ w @ w @ w e ¼ þ þ 2v þ21ðÞ v (20) e e e 2 2 2 2 2 1 v @x @y @x @y @x@y where v is the equivalent Poisson’s ratio of laminated plates and symbol jj means solving the modulus of complex values. Considering that equation (20) is too complex, a relatively simple formula is given as follows 2 2 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z @ w @ w @ w e e ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ þ 21ðÞ v (21) e e;max e 2 2 @x @y @x@y 1 v Practice shows that this simplification does not introduce large errors but has significantly improved the computational efficiency. Solution of vibration response The Newton–Raphson method is adopted to solve the vibration response of hard-coating laminates, and equation (14) can be transformed as r¼ K x M v F (22) where r is the residual vector and its Jacobi matrix is 2 3 @r @r Re Re 6 7 6 @v @v 7 R I 6 7 J ¼ (23) 6 7 @r @r 4 5 Im Im @v @v R I Here, v and v refer to the real part and imaginary part of response vector, respectively, @r=@v and @r=@v R I R I are both N-order square matrix. Sun et al. 795 Extract the real and imaginary part of vector r and v, respectively, can form two new vectors br ¼½ ReðÞ r ImðÞ r (24a) bv ¼ Re v Im v (24b) ðÞ ðÞ Then the iterative formulas of the Newton–Raphson method can be derived as ðÞ n ðÞ n ðÞ n br þ J Dbv ¼ 0 (25a) ðÞ nþ1 ðÞ n ðÞ n bv ¼ bv þ Dbv (25b) where n is the iteration number. The specific procedure of solving the vibration response of hard-coating laminated plate is shown in Figure 2 and some key steps are explained as follows. While calculating the iterative initial value v , the complex stiffness matrix K corresponding to the linear part 0 l of hard-coating materials is considered only, and the solving equation is K x M v ¼ F (26) The convergence condition is defined as the norm 2 of vector r, that is stop iteration while jjrjj TOL. When the terminal condition is satisfied, vector b v can be converted to complex vector v. Then, the transverse displace- ment can be calculated with equation (11). Further, the total displacement of laminated plates under base exci- tation can be calculated with the following equation ðÞ kðÞ x; y; t ¼ wxðÞ ; y; t þ u t (27) Step1,calculating the iterative initial value under a certain excitation frequency Step 2,calculating the complex stiffness matrix K Step 3,calculating the residual vector r and transforming it into r ˆ Step 4,calculating the Jacobi matrix J ˆ ˆ Step 5,calculating χˈ χ ˆ χ Step 6, transforming into χ Step 7, recalculating the residual vector r No Step 8,verifying r TOL Yes Outputting wx,,y t , x,y,t Figure 2. The procedure of solving vibration response of hard-coating laminated plate. TOL: is a setting value used to control calculation accuracy. 796 Journal of Low Frequency Noise, Vibration and Active Control 37(4) Step 1, calculating the iterative initial value of Step 2, calculating the j-order resonant responseχ Step 3, confirming the equivalent strain Step 4, calculating the complex stiffness matrix K Step 5, calculating the resonant frequency according to Eq.(36) No Step 6, verifying TOL n 2 Yes Outputting Figure 3. The procedure of solving resonant frequency of hard-coating laminated plate. TOL: . Solution of resonant frequency Obtaining the resonant response of hard-coating laminated plates is the first step of calculating resonant frequen- cy, so in this section, the calculation method of vibration response is described first. Neglecting the excitation force items in equation (14), the characteristic equation of hard-coating laminated plates considering strain-dependent characteristic can be expressed as K x M v ¼ 0 (28) e j By calculating the complex stiffness matrix K corresponding to different strain response amplitudes, the resonant frequency of hard-coating laminated plates can be derived with the characteristic equations expressed in equation (28). The procedure is shown in Figure 3 and some key steps are explained as follows. To obtain the initial value x of resonant frequency, the solution expression is shown in the following equation K x M v ¼ 0 (29) Substituting the initial value of natural frequency into equation (26) can derive the resonant response vector v which is needed in the subsequent calculations. Similar to the response calculation, the convergence condition is defined as the norm 2 of the difference between two adjacent natural frequencies and the expression is kDx k TOL. Study case Problem description In this section, a T300/QY8911 laminated plate with NiCoCrAlY þ YSZ hard coating was chosen to demonstrate the proposed method assuming that the total number of layers after coating is 6. The upper and lower surface are hard coating, so the number of layers of substrate is 4 and the fiber orientation angles from upper surface to lower surface are 0 ,30 , 30 , 30 ,30 ,0 , respectively. The relative dimension of the laminated plate is shown in Table 1 and the material parameters of substrate and hard coatings are shown in Table 2. It should be noticed that the material parameters given in Table 2 are the values corresponding to the linear part of NiCoCrAlY þ YSZ hard-coating materials (which means the strain value is 0) and the complete expres- sions of cubic polynomial are 5 2 14 3 E ¼ 55:23437 0:02608e þ 6:89029 10 e þ 1:09042 10 e (30) cR e e e Sun et al. 797 Table 1. Geometry parameters of composite plate (mm). Material type Length Width Thickness (per layer) Substrate 100 100 0.06 Hard coating 100 100 0.06 Table 2. Material parameters of composite plate. E (GPa) E (GPa) G (GPa) q (kg/m ) 1 2 12 12 Substrate 135.0 8.80 4.47 0.33 1380.0 E (GPa) E (GPa) v q (kg/m ) cR0 cI0 c c Hard coating 55.23 1.17 0.3 5600 Hard coating Vibration picking point Laminated plate Figure 4. Finite element model of hard-coating cantilevered laminate plate. 5 2 9 3 E ¼ 1:17448 þ 0:00353e 1:00555 10 e þ1:49084 10 e (31) cI e e e Then the following parameters can be calculated with the proposed procedure: (1) the linear resonant frequency and frequency response of this laminated plate in the 250 Hz frequency range under 1 g excitation level; (2) the fifth-order nonlinear resonant frequency and resonant region response under different excitation levels (1, 3, 5, 7, 9 g). Analysis of linear vibration characteristic The linear vibration analysis mentioned here has neglected the strain-dependent characteristic (which means the nonlinear part E of hard-coating materials is ignored) and the linear part E ¼ E of hard-coating materials is c;nl c;l c0 considered only. Then the derived equation (14) becomes a linear system and the direct frequency method can be adopted to obtain the frequency responses of laminated plates. Further, using the characteristic equation described by equation (28) can obtain the resonant frequency of the laminated plates. In order to demonstrate the rationality of the proposed method, the finite element software ANSYS is used to do the same calculations. SHELL281 element is adopted to simulate the hard-coating laminated plates and the finite element model is shown in Figure 4. This model contains 1200 elements and 1281 nodes in total. The section/ shell command is used to set the parameters of laminated plates, such as the values of fiber angle and thickness. The DMPRAT command is used to input the values of material damping into the analytical model. The ACEL command is used to exert 1 g acceleration excitation to the finite element modal. For the model created above, the Block Lanczos method is adopted to calculate the natural frequency and the mode superposition method is adopted to calculate the harmonic response of laminated plates, the resonant frequency and frequency response can be obtained at the same time. All the results of linear calculation are listed in Figure 5 and Table 3. It can be seen from Figure 5 that the linear frequency responses obtained by the analytic algorithm proposed in this study are basically the same as the results calculated by ANSYS software. Meanwhile, it can be seen in Table 3 that the maximum difference between analytic calculation and ANSYS method is 0.45%.Then, the rationality of analytic algorithm proposed in this study can be demonstrated. 798 Journal of Low Frequency Noise, Vibration and Active Control 37(4) ANSYS 10 Analytical method -1 -2 -3 -4 0 25 50 75 100 125 150 175 200 225 250 Excitation frequency/Hz Figure 5. Frequency-domain responses of the hard-coating cantilever laminated plate obtained by analytical method and ANSYS software. Table 3. Resonant frequencies of the hard-coating cantilever laminated plate obtained by analytical method and ANSYS software (Hz). Analytic Difference Order calculation (A ) ANSYS (F ) jA F j=F (%) 2 2 2 2 2 1 27.7 27.6 0.36 2 67.2 66.9 0.45 3 166.9 166.2 0.42 4 197.4 197.2 0.1 5 244.9 244.3 0.25 (b) (a) -2 -2 4×10 4×10 1g 1g 3g 3g -2 -2 3.5×10 3.5×10 5g 5g -2 7g -2 7g 3×10 3×10 9g 9g -2 -2 2.5×10 2.5×10 -2 -2 2×10 2×10 -2 -2 1.5×10 1.5×10 -2 -2 1×10 1×10 -3 -3 5×10 5×10 240 242 244 246 248 250 240 242 244 246 248 250 Frequency(Hz) Frequency(Hz) Figure 6. Frequency-domain responses of the hard-coating cantilever laminated plate near the fifth resonant region. (a) Linear calculation and (b) nonlinear calculation. Analysis of nonlinear vibration characteristic If the linear and nonlinear part of hard-coating materials are considered simultaneously, then equation (14) becomes a nonlinear equation and all these analyses are nonlinear vibration analysis. ANSYS software cannot consider the strain-dependent characteristic of hard-coating materials, so the proposed method is adopted to Response/mm Response/mm Response Sun et al. 799 Table 4. The fifth-order resonant frequency and resonant response under dierent excitation levels obtained by linear and nonlinear calculation. Excitation Calculation Resonant Resonant level (g) type frequency (Hz) response (mm) 1 Linear 244.9 4.13 10 Nonlinear 244.8 4.11 10 3 Linear 244.9 1.24 10 Nonlinear 244.6 1.22 10 5 Linear 244.9 2.07 10 Nonlinear 244.4 2.01 10 7 Linear 244.9 2.89 10 Nonlinear 244.2 2.78 10 9 Linear 244.9 3.72 10 Nonlinear 244 3.55 10 calculate the nonlinear resonant frequency and resonant region response under different excitation levels, and the fifth-order frequency and response are chosen as examples. The nonlinear iterative computations are performed according to Figures 2 and 3, the vibration picking point is the same as that in linear calculation and the con- vergence condition is defined as TOL ¼ 0:001. The results of the fifth-order nonlinear resonant frequency and resonant region response under different excitation levels are shown in Figure 6 and Table 4, respectively. For comparison, the corresponding results of linear calculation are listed in Table 4 as well. It can be seen in Figure 6 that the peak value of frequency response obtained by nonlinear calculation is less than that of the linear one under the same excitation level. And with the increase of excitation amplitude, the magnitude of attenuation increases. It can be seen in Table 4 that for the same order, the resonant frequency obtained by nonlinear calculation is less than that of the linear one under the same excitation level and with the increase of excitation amplitude, the magnitude of attenuation increases. Figure 6 shows that the resonant peak shifts to the left of horizontal axis, and the offset distance increases with excitation amplitude, this phenomenon also indicates the decrement of nonlinear resonant frequency. In conclusion, the laminated plate displays a soft nonlinear phenomenon because of depositing hard coating. Conclusions In this paper, the dynamic model of hard-coating laminated plates is proposed and the vibration characteristic is analyzed considering the strain-dependent characteristic of hard-coating materials. The conclusions are listed as follows. 1. The hard coating is regarded as a special layer in the laminated plate, using polynomials to express the material parameters of hard coatings with strain-dependent characteristics, then the modeling of plates can be accom- plished. In practical computations, the parameters of strain-dependent hard-coating materials can be divided into linear part and nonlinear part to conduct linear and nonlinear calculations, respectively. 2. After confirming the proper equivalent strain, the vibration responses and resonant frequencies of laminated plates can be calculated with Newton–Raphson method, and the detailed procedure is given in this study. 3. The vibration characteristic of a T300/QY89l1 laminated plate with NiCoCrAlYþYSZ hard coating was calculated with the proposed method and the linear results were validated by ANSYS software. The results also reveal that the resonant frequency of the laminated plate shifts to the left with the increase of excitation amplitude, which means the laminated plate displays soft nonlinear phenomenon because of deposit- ing coating. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. 800 Journal of Low Frequency Noise, Vibration and Active Control 37(4) Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 51375079) and the Fundamental Research Funds for the Central Universities of China (Grant No. N170308028). References 1. Zhang YX and Yang CH. Recent developments in finite element analysis for laminated composite plates. Compos Struct 2009; 88: 147–157. 2. Hu BG and Dokainish MA. Damped vibrations of laminated composite plates – modeling and finite element analysis. Finite Elem Anal Des 1993; 15: 103–124. 3. Hoseinzadeh M and Rezaeepazhand J. 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"Journal of Low Frequency Noise, Vibration and Active Control" – SAGE
Published: Jun 9, 2018
Keywords: Strain-dependent characteristic; hard coating; laminated plate; vibration characteristics; analytical analysis
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