### 2.1 Dominated Dynamics Based Cutting Stage Division

Machining system composes of machine tool and part, and its dynamics is influenced by two sub-systems together.

$$[{\Delta }^{m} ] + [{\Delta }^{p} ] = [{\Delta }],$$

(1)

where [Δ^{m}], [Δ^{p}], and [Δ] represent the dynamics of machine tool, part, and machining system respectively. In three-axis milling with unchangeable tool and its overhang length, [Δ^{p}] changes with the material removal in cutting process, while [Δ^{m}] keeps constant.

The whole cutting process can be divided into three stages based on the relative dynamics between [Δp] and [Δ^{m}]. At the beginning (stage 1), as shown in Figure 1(a), the part is thick and can be regarded as rigid. With the material removal, the thickness and stiffness of part decrease, and the dynamics of two subsystems affects the machining process together (stage 2) as depicted in Figure 1(b). At the last stage (stage 3), the dynamics is dominated by part as illustrated in Figure 1(c). The relative dynamic characteristic varies with cutting process for thin-walled parts.

In whole cutting process, the dynamics of machining system is rewritten as a piecewise function based on the thickness of part, expressed as

$$[{\Delta }] = \left\{ {\begin{array}{*{20}l} {[{\Delta }_{{}}^{m} ],} \hfill \\ {[{\Delta }_{{}}^{m} ] + [{\Delta }^{p} (t)],} \hfill \\ {[{\Delta }^{p} (t)],} \hfill \\ \end{array} \begin{array}{*{20}c} {} & {\begin{array}{*{20}l} {t > t_{1} ,} \hfill \\ {t_{1} \ge t \ge t_{2} ,} \hfill \\ {t_{2} > t,} \hfill \\ \end{array} } \\ \end{array} } \right.$$

(2)

where *t* represents the thickness of part; *t*_{1} and *t*_{2} are the critical thicknesses to divide stages; [Δ^{m}] can be obtained by the impact testing or RCSA method. However, [Δ^{p}] changes with the thickness.

### 2.2 Modelling of Thickness-dependent Dynamics of Thin-walled Parts

The thickness of part decreases with the material removal. When the dynamics of part influences the machining process, the aspect ratio of the length and height to the thickness is large enough so that the part can be modelled by using Kirchhoff’s thin plate theory, in which stress changes in the thickness direction is ignored. The thickness-dependent dynamics model is based on the assumption that the wall of part is reduced from both side simultaneously to remain the neutral plane unchanged during machining. The error of the assumption on the dynamics prediction is negligible [18], and four nodes quadrilateral element is used to mesh the neutral plane.

Based on the vibration of part and the Kirchhoff’s thin plate theory, the degree-of-freedom (DOF) of a node in the element can be reduced from 6 to 3, and the shape function of an element can be expressed as

$$\begin{array}{*{20}c} {{\varvec{N}} = \left[ {N_{{n_{1} }} } \right.} & {N_{{xn_{1} }} } & {N_{{yn_{1} }} } & {} & {} & {} \\ {} & {N_{{n_{2} }} } & {N_{{xn_{2} }} } & {N_{{yn_{2} }} } & {} & {} \\ {} & {} & {N_{{n_{3} }} } & {N_{{xn_{3} }} } & {N_{{yn_{3} }} } & {} \\ {} & {} & {} & {N_{{n_{4} }} } & {N_{{xn_{4} }} } & {\left. {N_{{yn_{4} }} } \right]} \\ \end{array} ,$$

(3)

$$\begin{gathered} \left\{ {\begin{array}{*{20}c} {N_{i} = \frac{1}{8}\left( {1 + \frac{x}{{x_{i} }}} \right)\left( {1 + \frac{y}{{y_{i} }}} \right)\left[ {2 + \frac{x}{{x_{i} }}\left( {1 - \frac{x}{{x_{i} }}} \right) + \frac{y}{{y_{i} }}\left( {1 - \frac{y}{{y_{i} }}} \right)} \right],} \\ {N_{xi} = - \frac{1}{8}y_{r} \left( {1 + \frac{x}{{x_{i} }}} \right)\left( {1 + \frac{y}{{y_{i} }}} \right)^{2} \left( {1 - \frac{y}{{y_{i} }}} \right),} \\ {N_{yi} = - \frac{1}{8}x_{r} \left( {1 + \frac{x}{{x_{i} }}} \right)^{2} \left( {1 + \frac{y}{{y_{i} }}} \right)\left( {1 - \frac{x}{{x_{i} }}} \right),} \\ \end{array} } \right. \hfill \\ \begin{array}{*{20}c} {(i = n_{1} ,n_{2} ,n_{3} ,n_{4} ),} & {} \\ \end{array} \hfill \\ \end{gathered}$$

(4)

where *x*_{i} and *y*_{i} (*i*=*n*_{1}*, n*_{2}*, n*_{3}*, n*_{4}) are the coordinates of node in the element coordinate system. *B* is the independent variable of the stiffness matrix, and is derived from Eqs. (3) and (4):

$${\varvec{B}} = \left[ {\begin{array}{*{20}c} {{\varvec{B}}_{{n_{1} }} } & {{\varvec{B}}_{{n_{2} }} } & {{\varvec{B}}_{{n_{3} }} } & {{\varvec{B}}_{{n_{4} }} } \\ \end{array} } \right],$$

(5)

$$\begin{array}{*{20}c} {} & \begin{gathered} {\varvec{B}}_{r} = - \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} N_{r} }}{{\partial x^{2} }}} & {\frac{{\partial^{2} N_{xr} }}{{\partial x^{2} }}} & {\frac{{\partial^{2} N_{yr} }}{{\partial x^{2} }}} \\ {\frac{{\partial^{2} N_{r} }}{{\partial y^{2} }}} & {\frac{{\partial^{2} N_{xr} }}{{\partial y^{2} }}} & {\frac{{\partial^{2} N_{yr} }}{{\partial y^{2} }}} \\ {2\frac{{\partial^{2} N_{r} }}{\partial xy}} & {2\frac{{\partial^{2} N_{xr} }}{\partial xy}} & {2\frac{{\partial^{2} N_{yr} }}{\partial xy}} \\ \end{array} } \right], \hfill \\ (r = n_{1} ,n_{2} ,n_{3} ,n_{4} ). \hfill \\ \end{gathered} \\ \end{array}$$

(6)

By using the virtual work principle, the stiffness matrix *K*^{e} and mass matrix *M*^{e} of element are expressed as the function of thickness.

$${\varvec{K}}^{e} (t) = \int {_{{V^{e} }} t^{3} {\varvec{B}}^\text{T} {\varvec{DB}}{\text{d}}V} ,$$

(7)

where *D* is the modified bending modulus, and is calculated as

$${\varvec{D}} = \frac{E}{{12(1 - \upsilon^{2} )}}\left[ {\begin{array}{*{20}c} 1 & \upsilon & 0 \\ \upsilon & 1 & 0 \\ 0 & 0 & {{{(1 - \upsilon )} \mathord{\left/ {\vphantom {{(1 - \upsilon )} 2}} \right. \kern-\nulldelimiterspace} 2}} \\ \end{array} } \right],$$

(8)

where *E* is the elasticity modulus, and *υ* is the Poisson ratio of materials.

$${\varvec{M}}^{e} (t) = e_{w} e_{l} \rho t\int {_{{S^{e} }} {\varvec{N}}^{{\text{T}}} {\varvec{N}}{\text{d}S}} ,$$

(9)

where *e*_{w} and *e*_{l} are the width and length of element; *ρ* is the density of material.

The damping matrix can be represented in terms of mass and stiffness matrices. For highly computational efficiency, Rayleigh damping is chosen to establish the damping matrix, which can be formulated as

$${\varvec{C}}^{e} (t) = \alpha {\varvec{M}}^{e} (t) + \beta {\varvec{K}}^{e} (t),$$

(10)

where *α* and *β* are damping coefficients identified from experiments.

To develop the matrices of part, the stiffness matrix, mass matrix and damping matrix of elements need to be assembled in nodes by using matrix displacement method. The dynamics model of the thickness-dependent dynamics of thin-walled part is

$${\varvec{M}}(t)\ddot{x} + {\varvec{C}}(t)\dot{x} + {\varvec{K}}(t){\varvec{x}} = {\varvec{F}}.$$

(11)

[Δ^{p}(*t*)] representing the direct FRFs of the cutting point is solved from the dynamics model, as expressed by

$$[\Delta^{p} (t)] = \sum\limits_{i = 1}^{n} {\frac{{{{\varvec{\uppsi}}}_{iq} (t){{\varvec{\uppsi}}}_{iq}^{\rm T} (t)}}{{(\omega_{i}^{2} (t) - \omega^{2} ) + 2j\zeta_{i} (t)\omega_{i} (t)\omega }}} ,$$

(12)

where *q* represents the DOF of lateral deflection in the weakest stiffness point. **Ψ**_{iq} and *ω*_{i} are the eigenvector and eigenvalue of the dynamics model respectively, *ζ*_{i} is the modal damping ratio, and *n* is the number of natural modes considered in synthesizing the FRF.