2.1. Calculation Principle
The schematic of a hard-coating composite structure is shown in
Figure 1 and it consists of a hard-coating layer and a metal substrate. Symbols
Hc,
Hs, and
Ec*,
Es* are the thickness and the complex modulus of the hard coating and the metal substrate, respectively.
Ec* and
Es* can be expressed as
where, * represents a complex number,
i =
, and
Ec,
Es, η
c, η
s are the Young’s modulus and loss factor of the hard coating and the metal substrate, respectively.
Since the elastic modulus of the hard coating and the metal substrate are expressed by complex modulus, the finite element dynamic equation of the system in frequency domain becomes
where,
M is the mass matrix of the composite structure,
KR is the real part of the complex stiffness matrix,
KI is the imaginary part of the complex stiffness matrix,
F is the exciting force vector, and
X is the displacement vector.
The real and imaginary parts of the complex stiffness matrix can be further expressed as
where,
KRS,
KRC and
KIS,
KIC represent the contributions of the metal substrate and the hard coating to the real and imaginary parts of the complex stiffness matrix, respectively.
For the CMSE method, usually, only the real part
KR of the complex stiffness matrix is adopted to construct characteristic equation, which is used to solve the real mode and can be expressed as
where, ω
r, φ
r are the
r-th order natural frequency and modal shape, respectively. Further, the loss factor of composite structure can be solved using the obtained real modal shape and the solving equation is
where, ηr is the r-th order modal loss factor of composite structure. In addition, Equation (6) only considers the contribution of hard coating to the loss factor of the whole system, so the CMSE method cannot be used to predict the damping characteristics directly.
Then, the CMSE method is modified in this study. Two modified methods are proposed here and the main difference between the two methods is the pattern of constructing the characteristic equation. For the method of MMSE 1, the characteristic equation of solving mode shape can be written as
where,
are the
r-th order natural frequency and modal shape respectively obtained by the method of MMSE, and â is a modified coefficient which describes the contribution of the imaginary part of the complex stiffness matrix to the solution of mode shapes. The value of â can be calculated by [
17]
where “trace ()” is a function to gain the trace of a matrix, namely, the sum of matrix diagonal elements.
Compared with the method of MMSE 1, the characteristic equation of the method of MMSE 2 is expressed as
It can be seen from Equation (9) that the modified coefficient, â is replaced by â
r in the method of MMSE 2, which means that different order will have different modified coefficient. The solving formula for â
r is shown as follows
It should be noted from the Equation (10) that the modal shape vector corresponding to the CMSE method (shown in Equation (5)) needs to be used during the solution of modified coefficient âr for the method of MMSE 2.
According to the MMSE method, the prediction formula of damping characteristics for the hard-coating composite structure can be finally determined as
It can be seen from the prediction formula that the loss factor of both the metal substrate and the hard coating are considered and the contribution of the imaginary part of the complex stiffness matrix to the modal shapes are also introduced. So the calculation accuracy of damping characteristics should be higher than the CMSE method.
2.2. Calculation Procedure of MMSE Method
According to the aforementioned calculation principles, the calculation procedure of the two MMSE methods can be described according to
Figure 2. It mainly includes five key steps: Solving the mass matrix and complex stiffness matrix, decomposing complex stiffness matrix, computing the modified coefficient, calculating the modified modal shape, and predicting the damping characteristics.
It should be noted that, for the method of MMSE 1, the modified coefficient â can be obtained by the direct utilization of real and imaginary parts of the complex stiffness matrix without using the calculation steps of CMSE method. However, for the method of MMSE 2, the CMSE method should be adopted to obtain the modal shape φr, and then âr—corresponding to different modal order—can be calculated. Obviously, the computational efficiency of MMSE 1 should be higher than that of MMSE 2. However, due to the fact that the influence of different modal order has been taken into account in MMSE 2, as a general rule, the computational accuracy of MMSE 2 should be higher than that of MMSE 1.