Investigation on chatter stability of thin-walled parts in high-speed milling based on relative transfer functions

2.1. Relative transfer function of tool-workpiece system model

Machine tools have multiple degrees of freedom (DOF) in various directions. The vibrations between the cutting tool and generated workpiece surface are the main interest. From the Newton’s second law, the equation of motion for the cutting tool or workpiece can be written in matrix form yields as:

Because the modes are orthogonal to each other, they have the following properties:

where $m_{q1}$ is the modal mass associated with the first mode. When the orthogonality pinciple is applied aimilaily to the remaining mode shapes, the local mass and stiffness mateixes are the transformed into modal coordinates:

The resulting modal mass ([$M_q$]) and modal stiffness ([$K_q$]) matrixes are diagonal, and each diagonal element represents the modal mass or modal stiffness associated with a mode. The transformed modal damping matrix [$C_q$] is also disgnal.

By applying modal transformations to Eq. (1), we can obtain the equations of motion in modal coordinates:

By applying the same modal transformations to both sides of the equations of motion for forced vibrations Eq. (1) the force vector $\left\{F\right\}$ can be transformed to modal coordinates:

The forced vibration expressions in modal coordinates become:

The modal displacements can be expressed as matrix form:

where ${\mathrm{\Phi }}_q$ is the diagnal modal transfer function matrix. Substitutiong $\left\{R\right\}=\left[P{]}^T\right\{F\}$ and $\left\{x\right\}=\left[P\right]\left\{q\right\}$ into Eq. (7), we obtain the vibrations in local coordinates:

or:

where ${\left\{P\right\}}_k$ is the eigenvector for mode k, and n is the number of degrees of freedom. Thus the harmonically forced vibrations for each coordinate can be calculated using the mode shaps ${\left\{P\right\}}_k$, mode transfer functions ${\mathrm{\Phi }}_{q,k}$, and the external force vector $\left\{F\right\}$.

The frequency response function is expressed for workpiece – tool system in one degree of freedom:

where $k$ is stiffness (N/mm²), is $\zeta$ damp ratio, is $\omega$ frequency (Hz)

The frequency response function is expressed for tool subsystem in multi-degrees of freedom as follows:

Similarly, the workpiece subsystem frequency response function is given:

The relative displacement between the tool and workpiece is ($x_t-x_w$), where $x_t$ and $x_w$ are displacements of the tool and workpiece, respectively. The cutting forces acting on the tool ($F_t$) and the workpiece ($F_w$) have the same opposing directions (i.e., $F_t=-F_w$). The displacement and force vectors for an $n$-DOF system can be expressed in local coordinates as:

The equation of motion for the system is:

where the local mass [$M_x$], damping [$C_x$], and stiffness [$K_x$] matrices are square with dimentions [$n\times n$], and the force vector $\left\{F\right\}$ has a dimension of [$n\times 1$]. The solution of the eigenvalue problem leads to a modal matrix [$P$] with a dimention of [$n\times n$]:

or:

where each column represents a mode shape $\left\{P\right\}$ of the n-DOF system structure. From the modal coordinate transformation equation ($x=\left[P\right]\left\{q\right\}$), the and the workpiece displacements can be calculated using the associated rows t and w of the above modal matrix:

Substituting the modal force $\text{(}\left\{R\right\}=\left[P\right]T\left\{F\right\}\text{)}$ into the modal displacement vector $\text{(}\left\{q\right\}=\left[{\mathrm{\Phi }}_q\right]\left\{R\right\}\text{)}$, and remembering that all elements of the force vector {$F$} are zero except the two corresponding to coordinates $x_t$ and $x_w$, yields:

Substitution of these modal displacements into Eq. (17) gives the local displacements of the tool and workpiece, respectively:

When the machine tool structure is excited by a harmonically varying force at the cutting point, the relative transfer function between the tool and workpiece becomes:

2.2. The calculation model of chatter stability

After the development of this relative transfer function, the calculation model of the stability lobe diagram analytically is the next step. The resulting chip thickness consists of the static part $\left(f_z\mathrm{s}\mathrm{i}\mathrm{n}{\varphi }_j\right)$, attributed to rigid body motion of the cutter, and a dynamic component caused by the vibrations of the tool at the present and previous tooth periods. Since the chip thickness is measured in the radial direction, the general dynamic chip thickness can be expressed as follows:

where, $f_z$ is feed rate per tooth, $\left[x_t\right(t),y_t(t){]}^T$ is the dynamic displacement vector of tool at the present time $t$. $\left[x_w\right(t),y_w(t){]}^T$ is displacement vector of workpiece at the present time $t$. ${\varphi }_j\left(t\right)$ is the instantaneous angular immersion of tooth $j$ as following:

where, $N$ is number of cutter, $\mathrm{\Omega }$ is rotational speed.

The function $g\left({\varphi }_j\right)$ is a unit step function that determines whether the tooth is in or out of cut, that is:

where, ${\varphi }_{st}$ and ${\varphi }_{ex}$ are that start and exit immersion angles of the cutter to and from the cut, respectively.

The governing equation of milling is a delayed differential equation with the tooth-passing period $T$ as delay. The dynamic equation of “cutter-workpiece” is expressed as follows:

where, $s{\varphi }_j=\mathrm{s}\mathrm{i}\mathrm{n}\left({\varphi }_j\right(t\left)\right)$, $c{\varphi }_j=\mathrm{c}\mathrm{o}\mathrm{s}\left({\varphi }_j\right(t\left)\right)$.

Using $q\left(t\right)=M\dot{p}\left(t\right)+Cp\left(t\right)/2$ to the Eq. (12), which may be rewritten as [19]:

Discretization is introduced using a time interval $[t_i,t_{i+1}]$ with $t_{i+1}-t_i=\mathrm{\Delta }t$. The delayed state $q(t-T)$ is approximated by:

The solution of milling dynamic equation is expressed as follows:

There are two possible instabilities can be observed: the critical eigenvalue is greater than 1. This case corresponds to the Hopf bifurcation causing the quasiperiodic chatter. The critical eigenvalue is real and its value is smaller than –1. This case corresponds to the period doubling or flip bifurcation, which causes the periodic chatter.

3.1. Modal tests of both cutter and workpiece

The difference between the transfer function of the relative movement with respect to the treatment of the tool’s and workpiece’s transfer functions as two independent entities seems obvious. For the transfer function of tool and workpiece, the modal test is conducted in 5-axis machining center (DMU50). The diameter of cutter is 16 mm. There are some equipment as follows: 1) modal testing and analysis software Cut/Pro; 2). USB Carrier I/O-9233; Accelerometer sensor8778a500 (Sensitivity 10.00 mV/g); Hammer type (B&K8206-002) 9722a500 (Sensitivity 10.00 mV/g); Sampling Rate (Hz):50000; Frequency Range (Hz): 50-5000; Transfer functions: Displacement-Force.

Fig. 2The modal test of cutter and thin-wall part

a) Accelerometer sensor on cutter tip

b) Modal tests of thin-wall part

c) Modal tests of cutter

The frequency response functions (real part and imagine part) of tool point are measured in $x$ and $y$ direction, as shown in Fig. 3. The modal parameters are identified based the transfer function, as shown in Table 1. The frequency response functions (real part and imagine part) of thin-walled part are measured in $x$ and $y$ direction, as shown in Fig. 4. The multi-modes relative transfer function of tool and workpiece are shown in Fig. 5. It can be shown that the transfer function of the structure of a processing system depends on the relative relationship between the two subsystems. A first-order natural frequency equal to the natural frequency between the lowest among all the natural frequency, the last-order natural frequency is equal to the natural frequency between the highest natural frequency, that is the natural frequency of the system is relatively low to the high order by sequentially increased number order.

Fig. 3Two orthogonal direction FRFs measured at the tool tip

Fig. 4Two orthogonal direction FRFs measured at thin-walled part

The frequency response function (FRF) matrix of the machine-tool structure at the tool tip (tip-tip FRFs) was determined by an impact test procedure. In order to check for the presence of mode coupling, the tool response was measured in $x$ and $y$ directions at both locations. The measured FRFs were curve fit by commercial modal analysis software to identify the modal parameters and calculate the tip-tip FRFs. The obtained diagonal elements of the system for the tip-tip FRFs are listed in Table 1.

Fig. 5Relative transfer function of multi-modes in x direction

According to the analyzed first order and second order frequency for cutters and workpieces and the relative transfer function theory, the transfer function of the structure of the processing system depends on the relative relationship between the tool and workpiece subsystem. The first order frequency equals the lower natural frequency of tools and workpieces; the second order frequency equals the higher natural frequency of tools and workpieces.

3.2. Measurement of milling force coefficient

3.2.1. Experiment setup

The milling tests were a CNC vertical machining center VMG850, equipped with variable spindle speed from 100 to 8000 rpm. A three-dimensional Kislter9265B dynamometer was mount on machine table to measure the cutting forces, and the instantaneous cutting force were recorded in $x$, $y$ and $z$ direction by a type Dyno Ware signal analyzer software after amplification using a type 5070A multi-channel charge amplifier. The workpiece was mounted on the dynamometer through a specially designed fixture. The cutting tool (one insert used) in the experiment was a 12-mm diameter end mill (catalog number: EGO EAP300R 12D120L GC-0109) with a tool holder. The insert is APMT 1135 PDER DP5320. The milling tests were conducted on a high speed milling center using a cylindrical end mill with a single cutting edge ($N=$1). A relatively large overhang was used to assure a single domain vibration mode of the tool, whereas a single edged cutter was used to avoid the disturbances due to tool runout. The setup of the machining experiment is shown in Fig. 6.

The experimental materials are thin-walled Ti-6Al-4V alloy workpiece (200×150×5 mm), which has a series of advantages of good corrosion resistance, smaller density, higher strength and better toughness and weldability, etc. It is used widely in the aerospace, automotive and biomedical industries etc. The chemical and physical properties of Ti-6Al-4V can be seen in Table 2 and Table 3 separately.

Table 2Chemical composition of titanium TC4

Alloying element	Al	V	Fe	C	N	H	O	Ti
Content (%)	6.1	4.1	0.15	0.02	0.02	0.006	0.12	Margin

Table 3Main physical properties of titanium TC4

Density kg/m3	Elasticity modulus GPa	Poisson ratio	Tensile strength MPa	Yield strength MPa	Elongation %	Thermal conductivity W/(m·K)	Hardness HB
4420	110	0.31	960-1270	820	≥8	7.955	195

Fig. 6Measurement of milling force coefficients

a) Before machining

b) After machining in one parameter

3.3. Predicted milling forces

The total instantaneous cutting forces are evaluated by integration of the contributions of all the differential edge segments of the tool. Furthermore, the total cutting forces can be derived from the three orthogonal force components in Cartesian coordinates in $x$, $y$ and $z$ directions as follows:

where, $\varphi$ is the tool’s rotational angle; the start angle ${\varphi }_{st}=0$ and the exit angle ${\varphi }_{ex}=a\mathrm{c}\mathrm{o}\mathrm{s}\left(r-a_e/r\right)$. $A_1={\int }_{z_1}^{z_2}dz$, $A_2={\int }_{z_1}^{z_2}\mathrm{s}\mathrm{i}\mathrm{n}\kappa \left(z\right)dz$, $A_3={\int }_{z_1}^{z_2}\mathrm{c}\mathrm{o}\mathrm{s}\kappa \left(z\right)dz$, $B_1={\int }_{z_1}^{z_2}dS\left(z\right)$, $B_2={\int }_{z_1}^{z_2}\mathrm{s}\mathrm{i}\mathrm{n}\kappa \left(z\right)dS\left(z\right)$, $B_3={\int }_{z_1}^{z_2}\mathrm{c}\mathrm{o}\mathrm{s}\kappa \left(z\right)dS\left(z\right)$. $z_1$, $z_2$ are integral up limit and down limit.

The milling parameters TC4 Thin-walled part are listed as follows: the depth of cut: 2 mm; feed per tooth: 0.05 mm/z; tool diameter: 12 mm, the number of tooth: 1; the radial depth of cut: $ae=$0.25$D$ and 0.5$D$. The milling forces of thin-walled part are simulated in different parameters, as shown in Fig. 7. It can be noticed that the milling forces value is not change with the increase of radial depth of cut, but the exit angle increase.

In the experiment of milling force coefficients measurement, the instantaneous cutting forces in three directions are measured by dynamometer and the data has been saved in the computer. Then the data is calculated to obtain the milling force coefficients in regression method. With the milling force coefficients, the stability lobes can be elaborated after getting the modal parameters in modal tests.

Fig. 7Milling forces in x, y and z directions

a)$ae=$0.25$D$

b)$ae=$0.5$D$