Vibration analysis of hard-coating laminated plate considering strain-dependent characteristic

Abstract

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 + YSZ hard coating was chosen to demonstrate 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.

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. ¹ 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. ² They can also be smart materials^3,4 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. ⁵ Vibration control of such laminates becomes a more challenging task since the viscoelastic damping and active control techniques^6,7 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 damping coating^8,9 and the relative vibration reduction techniques have been applied to aero-engine compressor blades ¹⁰ and blisks. ¹¹ 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 ¹² analyzed the deformation and free vibration of rectangular laminated plates with classical lamination theory. Phan-Dao et al. ¹³ analyzed the free vibration and buckling behaviors of symmetric laminated plate. Mao et al. ¹⁴ 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. ¹⁵ analyzed nonlinear dynamic responses of cantilever rectangular laminates with external excitation. Hasheminejad and Keshavarzpour ¹⁶ created an exact 3D elasticity model of piezo-laminated composite circular plate based on the spatial state-space method and the Rayleigh integral formula. These aforementioned methods of dynamic modeling and analyzing can be used as references while studying the vibration characteristics of hard-coating laminates.

Hard-coating materials have strain-dependent characteristics,^17,18 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 + 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 of the laminates is $H_c$ . The composite laminate is considered to be symmetric before coating and the thickness is $H_s$ . After coating, the plate is still symmetric and the overall thickness is $H_s+2H_c$ . The xyz coordinate plane is set to locate at the neutral plane of the composite laminate. The definition of fiber orientation angle $\theta$ 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 $z_1,z_2,\dots ,z_{N+1}$ , $N$ is the number of layers.

Figure 1.

(a) Hard-coating cantilever laminated plate under base excitation, (b) representation of fiber orientation, and (c) cross-section of laminated plate.

The elastic modulus $E_{\text{c}}^{*}$ of strain-dependent hard-coating material can be expressed by polynomial as the function of equivalent strain ${\epsilon }_{\text{e}}$ as follows

$$E_{\text{c}}^{*}=E_{\text{c0}}^{\text{*}}+{\epsilon }_{\text{e}}{E}_{\text{c1}}^{\text{*}}+{\epsilon }_{\text{e}}^{\text{2}}{E}_{\text{c2}}^{\text{*}}+{\epsilon }_{\text{e}}^{\text{3}}{E}_{\text{c3}}^{\text{*}}+\cdots$$

(1)

where $E_{\text{c}j}^{\text{*}}$ can be expressed as

$$E_{\text{c}j}^{\text{*}}=E_{\text{cR}j}+\text{i}{E}_{\text{cI}j},j=0,1,2,\cdots$$

(2)

Here, $E_{\text{cR}j}$ and $E_{\text{cI}j}$ refer to the material coefficients relative to the storage modulus (Young’s modulus) and the loss modulus, respectively. Further, the elastic modulus of hard coating can also be expressed as

$$E_{\text{c}}^{*}=E_{\text{c,l}}^{*}+E_{\text{c,nl}}^{*}$$

(3)

where $E_{\text{c},1}^{*}=E_{\text{c0}}^{\text{*}}$ are defined as the linear part of the parameters of hard-coating materials and $E_{\text{c,nl}}^{*}={\epsilon }_{\text{e}}{E}_{\text{c1}}^{\text{*}}+{\epsilon }_{\text{e}}^2{E}_{\text{c2}}^{\text{*}}+{\epsilon }_{\text{e}}^{\text{3}}{E}_{\text{c3}}^{\text{*}}+\cdots$ are the nonlinear part.

Choose one layer arbitrarily in laminates matrix and the sequence number is $i$ , $i=2,3,\dots ,N-1$ . According to the classical lamination plate theory, the constitutive relation of stress and strain in this layer can be expressed as

$${\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}}^{\left(i\right)}={\left[\begin{array}{ccc}{\overline{Q}}_{11}& {\overline{Q}}_{12}& {\overline{Q}}_{16}\\ {\overline{Q}}_{21}& {\overline{Q}}_{22}& {\overline{Q}}_{26}\\ {\overline{Q}}_{61}& {\overline{Q}}_{62}& {\overline{Q}}_{66}\end{array}\right]}^{\left(i\right)}\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}{\}}^{\left(i\right)}$$

(4)

where ${\sigma }_x$ , ${\sigma }_y$ , ${\tau }_{xy}$ , ${\epsilon }_x$ , ${\epsilon }_y$ , ${\gamma }_{xy}$ are the stress and strain in the xy plane of the $i$ -th layer. ${\overline{Q}}_{jl}$ , $j$ , $l=1,2,6$ are the elements of elastic matrix ${\overline{Q}}^{(i)}$ . ${\overline{Q}}_{jl}$ can be solved according to the Young’s modulus, Poisson’s ratio, and fiber directional angle of the corresponding layer and the detailed expression can be referred to the Dozio’s study. ¹⁹

Since the hard coatings deposited on the surface of the laminate ( $i=1,N$ ) are isotropic materials, the constitutive relation of stress and strain can be expressed as

$${\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}}^{(i)}={\left[\begin{array}{ccc}\frac{E_{\text{c}}^{\text{*}}}{1-v_{\text{c}}^2}& \frac{v_{\text{c}}{E}_{\text{c}}^{\text{*}}}{1-v_{\text{c}}^2}& 0\\ \frac{v_{\text{c}}{E}_{\text{c}}^{\text{*}}}{1-v_{\text{c}}^2}& \frac{E_{\text{c}}^{\text{*}}}{1-v_{\text{c}}^2}& 0\\ 0& 0& \frac{E_{\text{c}}^{\text{*}}}{2\left(1+v_{\text{c}}\right)}\end{array}\right]}^{(i)}{\left\{\begin{array}{c}{\epsilon }_x\\ {\epsilon }_y\\ {\gamma }_{xy}\end{array}\right\}}^{(i)}$$

(5)

where $v_{\text{c}}$ is Poisson’s ratio of hard coating.

According to the mechanics of elasticity, the strain energy $U$ of the laminate can be expressed as

$$U=\frac{1}{2}{\displaystyle {\iiint }_V\left({\sigma }_x{\epsilon }_x+{\sigma }_y{\epsilon }_y+{\tau }_{xy}{\gamma }_{xy}\right)}\text{d}V=U_{\text{l}}+U_{\text{nl}}$$

(6)

where $V$ is the volume of hard-coating laminates, $U_1$ and $U_{\text{n}1}$ are the strain energy corresponding to the linear and 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

$$\begin{array}{l}{U}_{\text{l}}=\frac{1}{2}{{\displaystyle \iint [D_{11}\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}}^2+D_{22}{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2+2D_{12}\left(\frac{{\partial }^{2}w}{\partial x^2}\right)\left(\frac{{\partial }^{2}w}{\partial y^2}\right)+4D_{66}{\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2\\ +4D_{16}\left(\frac{{\partial }^{2}w}{\partial x^2}\right)\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)+4D_{26}\left(\frac{{\partial }^{2}w}{\partial y^2}\right)\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)]\text{d}x\text{d}y\end{array}$$

(7)

Here, the transverse motions of the hard-coating laminates are considered only, $w$ is the transverse displacement of neutral surface of composite laminates (completely expressed as $w(x,y,t)$ ), $D_{jl},j,l=1,2,6$ are bending 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

$$U_{\text{nl}}=\frac{1}{2}{\displaystyle \iint {\displaystyle \left(D_{\text{c1}\hspace{0.17em}}{f}_{\text{eq}}+D_{\text{c2}}{f}_{\text{eq}}^{\hspace{0.17em}2}+D_{\text{c1}}{f}_{\text{eq}}^{\hspace{0.17em}3}\right)g_{\text{nl}}\text{d}x\text{d}y}}$$

(8)

where

$$D_{\text{c1}}=\frac{E_{\text{c1}}^{\text{*}}}{4{\left(1-v_{\text{c}}^{\text{2}}\right)}^{\frac{3}{2}}}\left[{\left(H_{\text{c}}+\frac{H_{\text{s}}}{2}\right)}^4-{\left(\frac{H_{\text{s}}}{2}\right)}^4\right]$$

(9a)

$$D_{\text{c2}}=\frac{E_{\text{c2}}^{\text{*}}}{5{\left(1-v_{\text{c}}^{\text{2}}\right)}^2}\left[{\left(H_{\text{c}}+\frac{H_{\text{s}}}{2}\right)}^5-{\left(\frac{H_{\text{s}}}{2}\right)}^5\right]$$

(9b)

$$D_{\text{c3}}=\frac{E_{\text{c3}}^{\text{*}}}{6{\left(1-v_{\text{c}}^{\text{2}}\right)}^{\frac{5}{2}}}\left[{\left(H_{\text{c}}+\frac{H_{\text{s}}}{2}\right)}^6-{\left(\frac{H_{\text{s}}}{2}\right)}^6\right]$$

(9c)

$$\begin{array}{l}{f}_{\text{eq}}=\sqrt{{\mathrm{Re}}^2\left(\frac{{\partial }^{2}w}{\partial x^2}\right)+{\mathrm{Im}}^2\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}+\sqrt{{\mathrm{Re}}^2\left(\frac{{\partial }^{2}w}{\partial y^2}\right)+{\mathrm{Im}}^2\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}\\ +\sqrt{2\left(1-v_{\text{c}}\right)}\sqrt{{\mathrm{Re}}^2\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)+{\mathrm{Im}}^2\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}\end{array}$$

(9d)

$$g_{\text{nl}}={\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2+2v_{\text{c}}\frac{{\partial }^{2}w}{\partial x^2}\frac{{\partial }^{2}w}{\partial y^2}+2\left(1-v_{\text{c}}\right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2$$

(9e)

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

$$T=\frac{m}{2}{\displaystyle \iint {\dot{\lambda }}^2}\text{d}x\text{d}y=\frac{m}{2}{\displaystyle \iint {\left(\frac{\partial w}{\partial t}+{\dot{u}}_g\right)}^2}\text{d}x\text{d}y$$

(10)

where $m$ is the mass per unit area of the laminate, $\lambda$ is the total displacement at any point on the neutral surface of the laminate, and $u_g$ 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

$$w\left(x,y,t\right)={\displaystyle \sum _{j=1}^N{\phi }_j\left(x,y\right){\chi }_j\left(t\right)}$$

(11)

where ${\phi }_j(x,y)$ is the jth order modal shape, ${\chi }_j(t)$ is the jth order participation factors, and $N$ is the number 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

$$\frac{\text{d}}{\text{d}t}\left(\frac{\partial T}{\partial {\dot{\chi }}_k\left(t\right)}\right)+\frac{\partial U}{\partial {\chi }_k\left(t\right)}=0\left(k=1,2,\dots ,N\right)$$

(12)

Here, the base excitation is considered only, it can be assumed that

$${\chi }_j\left(t\right)={\chi }_j{e}^{\text{i}\omega t}$$

(13)

where $\omega$ is the excitation frequency. After calculations, the final expressions are

$$\left(K_{\text{ε}}^{\text{*}}-{\omega }^{2}M\right)\chi =F$$

(14)

which is a nonlinear algebraic equation since it considers the strain-dependent characteristics of hard-coating materials. Here, $K_{\text{ε}}^{\text{*}}$ is the $N\times N$ dimension complex stiffness matrix, $M$ is the $N\times N$ mass matrix, $\chi$ is the response vector which consists of ${\chi }_k$ ( $k$ = $1$ , $2$ , $\dots$ , $N$ ) and is independent of time, and $F$ is the excitation vector. Further, the complex stiffness matrix $K_{\text{ε}}^{\text{*}}$ can be expressed as

$$K_{\text{ε}}^{\ast }=K_{\text{l}}^{\text{*}}+K_{\text{nl}}^{\ast }$$

(15)

where $K_{\text{l}}^{\text{*}}$ is the complex stiffness matrix corresponding to the linear part of hard-coating materials and $K_{\text{nl}}^{\ast }$ is the complex stiffness matrix corresponding to the nonlinear part.

The element of the kth row and pth column of complex stiffness matrix $K_{\text{ε}}^{\ast }$ can be expressed as

$$\begin{array}{l}{K}_{\text{ε}}^{\ast }\left(k,p\right)={\displaystyle \iint [D_{11}}{\phi }_k^{xx}{\phi }_p^{xx}+D_{22}{\phi }_k^{yy}{\phi }_p^{yy}+4D_{66}{\phi }_k^{xy}{\phi }_p^{xy}+D_{12}\left({\phi }_k^{xx}{\phi }_p^{yy}+{\phi }_p^{xx}{\phi }_k^{yy}\right)\\ +2D_{16}\left({\phi }_k^{xx}{\phi }_p^{xy}+{\phi }_p^{xx}{\phi }_k^{xy}\right)+2D_{26}\left({\phi }_k^{yy}{\phi }_p^{xy}+{\phi }_p^{yy}{\phi }_k^{xy}\right)]\text{d}x\text{d}y\\ +{\displaystyle \iint \left[\left(D_{\text{c1}}+D_{\text{c2}}{f}_{\text{eq}}+D_{\text{c3}}{f}_{\text{eq}}^2\right)f_{\text{eq}}{h}_{\text{nl}}\right]}\text{d}x\text{d}y\end{array}$$

(16)

where

$$h_{\text{nl}}={\phi }_k^{xx}{\phi }_p^{xx}+{\phi }_k^{yy}{\phi }_p^{yy}+v_{\text{c}}\left({\phi }_k^{xx}{\phi }_p^{yy}+{\phi }_p^{xx}{\phi }_k^{yy}\right)+2\left(1-v_{\text{c}}\right){\phi }_k^{xy}{\phi }_p^{xy}$$

(17)

Here, ${\phi }^{xx}$ , ${\phi }^{yy}$ refer to the second partial derivatives of x and y, respectively, and ${\phi }^{xy}$ 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

$$M\left(k,p\right)=m{\displaystyle \iint {\phi }_k{\phi }_p\text{d}x\text{d}y}$$

(18)

The specific expression of the kth element of the excitation vector $F$ can be expressed as

$$F\left(k\right)=m{\omega }^2{U}_{\text{g}}{\displaystyle \iint {\phi }_k}\text{d}x\text{d}y$$

(19)

where $U_g$ 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 ${\epsilon }_e$ , 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

$${\epsilon }_{\text{e}}=\sqrt{\frac{z^2}{1-v_{\text{e}}^2}\left|{\left(\frac{{\partial }^{2}w}{\partial x^2}\right)}^2+{\left(\frac{{\partial }^{2}w}{\partial y^2}\right)}^2+2v_{\text{e}}\left(\frac{{\partial }^{2}w}{\partial x^2}\right)\left(\frac{{\partial }^{2}w}{\partial y^2}\right)+2\left(1-v_{\text{e}}\right){\left(\frac{{\partial }^{2}w}{\partial x\partial y}\right)}^2\right|}$$

(20)

where $v_e$ is the equivalent Poisson’s ratio of laminated plates and symbol $|\cdot |$ means solving the modulus of complex values. Considering that equation (20) is too complex, a relatively simple formula is given as follows

$${\epsilon }_{\text{e}}\hspace{0.17em}\approx \hspace{0.17em}{\epsilon }_{\text{e},\max }=\frac{z}{\sqrt{1-v_{\text{e}}^2}}\left(\left|\frac{{\partial }^{2}w}{\partial x^2}\right|+\left|\frac{{\partial }^{2}w}{\partial y^2}\right|+\sqrt{2\left(1-v_{\text{e}}\right)}\left|\frac{{\partial }^{2}w}{\partial x\partial y}\right|\right)$$

(21)

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

$$r\text{=}\left(K_{\text{ε}}^{\ast }-{\omega }^{2}M\right)\chi -F$$

(22)

where $r$ is the residual vector and its Jacobi matrix is

$$J=\left[\begin{array}{l}\text{Re}\left(\frac{\partial r}{\partial {\chi }_{\text{R}}}\right)\text{Re}\left(\frac{\partial r}{\partial {\chi }_{\text{I}}}\right)\\ \text{Im}\left(\frac{\partial r}{\partial {\chi }_{\text{R}}}\right)\text{Im}\left(\frac{\partial r}{\partial {\chi }_{\text{I}}}\right)\end{array}\right]$$

(23)

Here, ${\chi }_{\text{R}}$ and ${\chi }_{\text{I}}$ refer to the real part and imaginary part of response vector, respectively, $\partial r/\partial {\chi }_{\text{R}}$ and $\partial r/\partial {\chi }_{\text{I}}$ are both $N$ -order square matrix.

Extract the real and imaginary part of vector $r$ and $\chi$ , respectively, can form two new vectors

$$\widehat{r}={\left[\mathrm{Re}\left(r\right)\mathrm{Im}\left(r\right)\right]}^{\text{T}}$$

(24a)

$$\widehat{\chi }={\left[\mathrm{Re}\left(\chi \right)\mathrm{Im}\left(\chi \right)\right]}^{\text{T}}$$

(24b)

Then the iterative formulas of the Newton–Raphson method can be derived as

$${\widehat{r}}^{\left(n\right)}+J^{\left(n\right)}\cdot \Delta {\widehat{\chi }}^{\left(n\right)}=0$$

(25a)

$${\widehat{\chi }}^{\left(n+1\right)}={\widehat{\chi }}^{\left(n\right)}+\Delta {\widehat{\chi }}^{\left(n\right)}$$

(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.

Figure 2.

The procedure of solving vibration response of hard-coating laminated plate. TOL: is a setting value used to control calculation accuracy.

While calculating the iterative initial value ${\overline{\mathit{\chi }}}_0$ , the complex stiffness matrix $K_{\text{l}}^{\ast }$ corresponding to the linear part of hard-coating materials is considered only, and the solving equation is

$$\left(K_{\text{l}}^{\ast }-{\omega }^{2}M\right)\chi =F$$

(26)

The convergence condition is defined as the norm 2 of vector $r$ , that is stop iteration while $\left|\right|r|{|}_2\le TOL$ . When the terminal condition is satisfied, vector $\widehat{\mathit{\chi }}$ can be converted to complex vector $\chi$ . Then, the transverse displacement can be calculated with equation (11). Further, the total displacement of laminated plates under base excitation can be calculated with the following equation

$$\lambda \left(x,y,t\right)=w\left(x,y,t\right)+u_{\text{g}}\left(t\right)$$

(27)

Solution of resonant frequency

Obtaining the resonant response of hard-coating laminated plates is the first step of calculating resonant frequency, 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

$$\left(K_{\text{ε}}^{\text{*}}-{\omega }_j^{2}M\right)\chi =0$$

(28)

By calculating the complex stiffness matrix $K_{\text{ε}}^{*}$ 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.

Figure 3.

The procedure of solving resonant frequency of hard-coating laminated plate. TOL: ▪.

To obtain the initial value ${\overline{\omega }}_j$ of resonant frequency, the solution expression is shown in the following equation

$$\left(K_{\text{l}}^{\ast }-{\omega }^{2}M\right)\chi =0$$

(29)

Substituting the initial value of natural frequency into equation (26) can derive the resonant response vector $\overline{\mathit{\chi }}$ 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 ${\Vert \Delta {\omega }_n\Vert }_2\le TOL$ .

Study case

Problem description

In this section, a T300/QY8911 laminated plate with NiCoCrAlY + 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.

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.

	E1 (GPa)	E2 (GPa)		G12 (GPa)	ν 12	ρ (kg/m3)
Substrate	135.0	8.80		4.47	0.33	1380.0
	$E_{\text{cR0}}$ (GPa)		$E_{\text{cI0}}$ (GPa)		v c	ρc (kg/m3)
Hard coating	55.23		1.17		0.3	5600

It should be noticed that the material parameters given in Table 2 are the values corresponding to the linear part of NiCoCrAlY + YSZ hard-coating materials (which means the strain value is 0) and the complete expressions of cubic polynomial are

$$E_{\text{cR}}=\text{55}\text{.23437}-\text{0}\text{.02608}{\epsilon }_{\text{e}}+\text{6}\text{.89029}\times {\text{10}}^{-5}{\epsilon }_e^2+\text{1}\text{.09042}\times {\text{10}}^{-\text{14}}{\epsilon }_{\text{e}}^{\text{3}}$$

(30)

$$E_{\text{cI}}=\text{1}\text{.17448\hspace{0.17em}+\hspace{0.17em}0}\text{.00353}{\epsilon }_{\text{e}}-\text{1}\text{.00555}\times {\text{10}}^{-5}{\epsilon }_e^2\text{+1}\text{.49084}\times {\text{10}}^{-\text{9}}{\epsilon }_{\text{e}}^{\text{3}}$$

(31)

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 $E_{\text{c,nl}}^{*}$ of hard-coating materials is ignored) and the linear part $E_{\text{c,l}}^{*}=E_{\text{c0}}^{*}$ of hard-coating materials is 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.

Figure 4.

Finite element model of hard-coating cantilevered laminate plate.

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.

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).

Order	Analytic calculation ( $A_2$ )	ANSYS ( $F_2$ )	Difference $|A_2-F_2|/F_2$ (%)
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

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 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 convergence condition is defined as $TOL=0.001$ . 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.

Figure 6.

Frequency-domain responses of the hard-coating cantilever laminated plate near the fifth resonant region. (a) Linear calculation and (b) nonlinear calculation.

Table 4.

The fifth-order resonant frequency and resonant response under different excitation levels obtained by linear and nonlinear calculation.

Excitation level (g)	Calculation type	Resonant frequency (Hz)	Resonant response (mm)
1	Linear	244.9	4.13 × 10−3
	Nonlinear	244.8	4.11 × 10−3
3	Linear	244.9	1.24 × 10−2
	Nonlinear	244.6	1.22 × 10−2
5	Linear	244.9	2.07 × 10−2
	Nonlinear	244.4	2.01 × 10−2
7	Linear	244.9	2.89 × 10−2
	Nonlinear	244.2	2.78 × 10−2
9	Linear	244.9	3.72 × 10−2
	Nonlinear	244	3.55 × 10−2

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.

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 accomplished. 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.

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.

The vibration characteristic of a T300/QY89l1 laminated plate with NiCoCrAlY+YSZ 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 depositing coating.