4.1.1. Establishment of the Vibration Model of Milling Cutter
In the modeling process of the vibration of the milling cutter, the effect of tool runout on the machining process is ignored. The milling cutter is assumed to be a uniform continuous system (uniform simply supported beam of equal cross-section), which means that the density, cross-sectional area, and stiffness of the milling cutter do not vary with position. During the milling process, the milling cutter is held by the knife clip at one end and is in contact with the workpiece at the other end. Therefore, the vibration model of the milling cutter can be simplified to a fixed-hinged beam system subjected to concentrated forces. The force model of the milling cutter is shown in
Figure 22.
Let the intrinsic frequency at a given boundary condition be
and the corresponding formation function be
. For a homogeneous beam of equal cross-section, the intrinsic frequency
can be expressed as [
37,
38]:
where
E is the modulus of elasticity,
J is the moment of inertia of the section,
A is the cross-sectional area, and
ρ is the mass per unit volume.
L is the overhanging length of the milling cutter, and
is the characteristic root of Equation (2). For the fixed-hinged beam system, the characteristic roots of Equation (2) can be expressed as [
39]:
where
, denotes the order. Based on the above assumptions about the vibration model, the formation function of the milling cutter can be expressed as:
where
denotes the hyperbolic function and
is a function related to the intrinsic frequency and material properties:
Since the stiffness of the milling cutter in the axial direction is much greater than that in the radial direction, the milling machining system can be assumed as a two-degree-of-freedom system (
x and
y directions). As the solution process of the vibration model in the
x and
y directions is the same, for the sake of simplicity, only the solution process in the
x-direction is discussed. Assume that the force applied to the milling cutter in the
x-direction is the concentrated force
, which can be expressed as:
where
denote the
δ function. For a homogeneous beam of equal cross-section, the vibration differential equation can be expressed as [
40]:
Introducing the regular coordinates
, based on the formation superposition method, the solution
for a given boundary condition can be expressed as [
41]:
The vibration differential equation (Equation (7)) can be expressed as:
The boundary conditions of the fixed-hinged beam system can be expressed as [
42]:
Regularize both sides of Equation (9) while integrating over the entire interval
. An independent system of ordinary differential equations is obtained as follows [
43]:
where
is the generalized force corresponding to the canonical coordinate
. When
acts on the end of the milling cutter, the generalized force
can be expressed as:
Combined with the previous studies, the response of Equation (12) is [
44,
45]:
Substituting
into the canonical equation (Equation (12)):
For the fixed-hinged beam system,
. Ultimately, the vibration response at the end of the milling cutter in the
x-axis direction is:
During the milling process, the milling force is a force that varies from moment to moment. Milling force and vibration displacement exist in a mutually coupled relationship. To find the milling cutter displacement in the vibration state, the time
is discretized into multiple tiny moments. For the period
, the milling force in the
x-direction is approximated by considering it as a constant force (see
Figure 23):
For the convenience of calculation, the formation function is only taken to the first three orders. The discretized equation can be expressed as:
Similarly, the vibration response of the milling cutter in the
x-direction can be found. Therefore, the vibration response of the end of the milling cutter during milling can be expressed in vector form as:
It is worth noting that the coordinate systems of the dynamometer and the milling cutter are different. Regarding the x-direction, the directions specified in both coordinate systems are the same. Regarding the y- and z-directions, they are opposite in both coordinate systems.
4.1.2. Surface Roughness Model Considering Milling Cutter Vibration
In previous studies, the trajectory of the milling cutter was usually generated by the radial runout and tilt runout of the milling cutter [
46,
47,
48]. As the milling cutter rotates, the position and trajectory of each point on the cutting edge of the milling cutter will deviate from the ideal value. In this study, the radial runout of the milling cutter is ignored since the milling cutter moves in the
x-direction as a feed. For the tilt runout, it is attributed to the inclination between the milling cutter axis and the machine spindle due to the vibration of the milling cutter.
Figure 24 shows the creation of surface topography.
Before the creation of the surface topography, the tool coordinate system (O
c-x
c-y
c-z
c) and the workpiece coordinate system (O
w-x
w-y
w-z
w) are established. The origin of the tool coordinate system is set to the projection of the bottom center of the milling cutter on the workpiece. For easy calculation, it is necessary to discretize the milling cutter along the axial direction. Under ideal conditions, when
, the coordinate values of the cutting edge at the moment
can be expressed as [
49]:
where
j denotes the
jth flute,
k denotes the
kth unit in the axial direction,
denotes the milling speed, and
denotes the spindle speed.
, is the number of teeth of the milling cutter.
R = 6 mm is the radius of the milling cutter.
denotes the thickness of the axial unit.
is the helix angle of the flathead milling cutter, and
is the axial milling depth. For the axial coordinate values,
is written as
for a uniform expression form.
When the vibration responses of the milling cutter in the
x- and
y-directions are
and
, respectively, the tilt angle
τ of the milling cutter can be expressed as:
In this study, the position angle
of the tilt angle is defined as the angle between the tilt direction of the milling cutter and the
x-direction in the
x–o–
y plane. According to the direction of the force on the milling cutter, the position angle
takes a range of
. The position angle
can be expressed as:
Once the tilt angle
and the position angle
of the milling cutter are determined, the coordinates of the cutting edge of the milling cutter under vibration response can be obtained by rotating (around the
y-axis and
z-axis) and translating the coordinate values in the ideal condition. The real coordinates of the cutting edge can be expressed as:
For simplicity of representation, the coordinate values of the cutting edge can be expressed as:
Combined with the previous studies, the undeformed cutting thickness can be derived from two adjacent cutting-edge trajectories (see
Figure 25) [
36,
49]. Suppose there exist two points P
and Q
. The point P is any point on the
j-th flute trajectory. The point Q is the intersection of the
j-th flute trajectory and the line from the tool center O to the point P.
denote the time corresponding to the cutting edge passing through the points P and Q, respectively.
The undeformed cutting thickness considering the vibration of the milling cutter during milling can be expressed as:
The coordinates of the point P can be expressed using Equation (26):
For the point Q, the coordinates can be expressed as:
Since the time
corresponding to point P is the current time, the coordinates corresponding to point P can be obtained by Equation (26). For the coordinates of point Q, they can be calculated by the position relationship between point
, point P, and point Q. Since point
, point P, and point Q are located in a straight line, the coordinate relationship between the three points can be expressed as:
By iterating the Equation (28), the coordinates of point Q can be obtained where
denotes the contact angle of the
j-th flute of the milling cutter, which can be expressed as [
36]:
The minimum cutting thickness in the milling process is an important factor affecting the surface roughness. Based on the geometric parameters of the surface roughness and the minimum cutting thickness, the surface roughness model considering the vibration of the milling cutter is established. When the undeformed cutting thickness is less than the minimum cutting thickness , the material deformation of the workpiece surface is dominated by the plowing effect under the extrusion of the cutting edge of the milling cutter. When the undeformed cutting thickness gradually increases to the minimum cutting thickness, the milling process is dominated by the shear effect and the elastic response of the workpiece is negligible.
According to the radius of the cutting edge of the milling cutter and the material properties, the minimum cutting thickness is expressed as [
36]:
where,
is the minimum cutting thickness proportionality constant, and
is the radius of the cutting edge. Based on the relationship between the minimum cutting thickness
and the undeformed cutting thickness
on the milling mechanism, the actual undeformed cutting thickness can be expressed as [
49]:
where
denotes the elastic recovery rate.
Combining the above derivation, the actual cutting-edge trajectory considering the vibration response of the milling cutter and the minimum cutting thickness of the workpiece can be derived. When
, the actual cutting-edge trajectory can be expressed as:
When
, the actual cutting-edge trajectory can be expressed as:
In this study, the evaluation index of surface roughness is the arithmetic mean height of the surface . The object of evaluation is the milling side surface, which is . From the relationship between the workpiece coordinate system and the milling cutter coordinate system, when , the coordinates of any point in the workpiece coordinate system can be expressed as:
where
denotes the radial cutting depth of the milling cutter. Eventually, based on the actual trajectory of the milling cutter, the surface morphology model considering the vibration of the milling cutter can be obtained. Based on the sampled points on the sub-oscillation trajectory of the cutting edge, the longitudinal coordinate values at the same position are compared. The point set consisting of the minimum longitudinal coordinate values constitutes the final surface topography of the workpiece generated by the milling process.
Figure 26 shows the modeling process of surface roughness.