I have investigated a human balancing model in the frontal plane with mediolateral balance control.
The robust stability boundaries associated with unstructured complex-valued perturbations can be found in the available literature. As an extension, I have determined the more realistic real structured stability radius, and showed that the most robust control gains differ from the ones obtained by the previous method. The results are summarized as follows.
3.4 Main results 31
Thesis statement 2
Let the linearized equation of motion around the equilibrium position of the single-degree-of-freedom model of the mediolateral balance control of a human balancing model be written as
Ix(t) +¨ Gx(t) =−kpx(t−τ)−kdx(t˙ −τ),
wherekpandkdare control gains of the proportional-derivative controller,I >0is the equivalent inertia, G < 0is the equivalent stiffness,x(t)is the generalized coordinate, andτ is the reaction delay. The domain of the robust stable control gains shrinks with increasing uncertainty in the system parametersIandG. The most robust stable control gains are located in the left bottom part of the stable region in the plane(kp, kd), i.e., whenkp&−Gandkd &−Gτ.
Related publications: [83].
CHAPTER 4
Turning operations
Metal cutting is one of the most widely used manufacturing process for the final shaping of products.
Cutting processes have evolved significantly over the past century, when numerical control (NC) and computer numerical control (CNC) have appeared in the industry. Nowadays these CNC machines provide high accuracy, flexibility and computational capability while the manufacturing time has been reduced remarkably. Despite the advanced technological improvements, the increase of the machining speed is still limited by the harmful self-excited vibrations that arise with the change of technological parameters. This phenomenon is called machine tool vibration ormachine tool chatter, which produces noise, limits the productivity, increases toolwear, and spoils the surface quality. Detection and avoidance are therefore highly important tasks in machining [84].
Researchers have found strong potential in optimization of machining operations and several numerical methods have been developed to predict chatter effectively. Stability lobe diagramshave been introduced to separate the domains on the plane of technological parameters, such as the spindle speed and the depth of cut, where chatter can or cannot arise. These diagrams provide a guide to the machinists to select optimal machining parameters and to avoid undesired vibrations.
However, in practice, predicted stability diagrams often does not meet experimental cutting tests due to uncertainties an inaccuracies in the mechanical model. In order to overcome this problem,robust stability lobe diagramscan be used that are capable to include the effects of model uncertainties.
There are many attempts to solve the problem in the literature by including uncertainties in the governing equations. Local sensitivity techniques, such as the local partial derivatives [85, 86] or first-order second moment methods and their extensions [87, 88, 89] are used to characterize the probability of stability. Neural networks have been used by [90], while [91, 92]
presented approximating numerical methods for milling operations, which provide confidence levels of stability boundaries for high number of uncertain parameters. Robust chatter prediction method (RCPM) has been introduced by [93], that is based on the discretization of the probability density function (PDF) and evaluation of stability on a discrete grid in the space of uncertain system parameters. A different solution is used by [94] and [95] by introducing Fuzzy arithmetics.
Meantime, [96] used the dimension reduction method combined with saddlepoint approximation.
While statistical methods are based on probability distributions, robust techniques provide conservative bounds on the stability within typically shorter calculation time. Only a few works have been published related to these types of techniques. For example, the edge theorem combined with the zero exclusion method is presented in [97, 98] and compared to the results obtained by linear matrix inequalities in [99].
In this chapter we use the stability radius method and introduce a frequency-domain technique to obtain the robust stability lobe diagrams. Utilization of frequency response functions (FRFs) does not require modal parameter estimation and therefore gives opportunity to evaluate robustness based on the measured FRFs directly. The structure of the chapter is as follows. Section 4.1 introduces the widely used mechanical and mathematical model of turning operations. Robust stability analysis is conducted in Sec. 4.2 using the stability radius method and in Sec. 4.3 using a
33
34 CHAPTER 4 Turning operations frequency-domain solution. A case study is presented in Sec. 4.4 and main results are concluded in Sec. 4.5.
4.1 Model of turning operations
The first mathematical models dealing with the self-excited vibrations in machining operations appeared in the work of Tobias [100] and Tlusty [101] in the 1950s and 1960s. After their pioneering research, the so-called regenerative effect became the most commonly accepted explanation for machine tool chatter. During the cutting process the vibrating tool leaves a wavy surface behind, which affects the chip thickness and induces variation in the cutting-force one revolution later.
The chip formation process in case of an ideal rigid turning tool and compliant tool is presented in Fig. 4.1(a-b). From the dynamic system’s point of view, chatter is associated with the loss of stability of the stationary (chatter-free) machining process followed by a large amplitude self-excited vibration between the tool and the workpiece.
The dynamical model of an orthogonal cutting operation involving multiple vibration modes is shown in Fig. 4.1(c). The modes are projected to the directionxof the surface regeneration. Note that vibrations in the y-direction do not affect linear stability properties [102]. It is assumed that the cutting forceFqdepends linearly on the chip widthw, but it is not necessarily a linear function of the chip thicknessh, i.e.,
Fq(h) =
(wfq(h), if h≥0,
0 if h <0, (4.1)
wherefq(h)is the specific cutting force andq = r,t,a, . . . denote the standard radial, tangential, and axial direction. There exists several models in the literature to approximate the characteristics, such as
fqL(h) =Kq,ch, (4.2a)
fqSL(h) =Kq,e+Kq,ch, (4.2b)
fqP(h) =Kq,αhαq, 0< αq <1 (4.2c) fqE(h) =Kq,ch+Kq,e 1−e−Eqh
, (4.2d)
fqC(h) =ρq,0+ρq,1h+ρq,2h2+ρq,3h3, (4.2e) which are linear (L), shifted linear (SL), power (P) exponential (E), and polynomial (C-cubic) functions ofh, respectively,and the parameters are determined from experiments, see [14, 103].
The equation of motion of then-degree-of-freedom system is written as M¨q(t) +Cq(t) +˙ Kq(t) = F h(t)
, (4.3)
where q(t) = [q1(t), q2(t), . . . , qn(t)]> ∈ Rn is the generalized coordinate vector, M ∈ Rn×n is the mass matrix, C ∈ Rn×n is the damping matrix, K ∈ Rn×n is the stiffness matrix and F h(t)
= [wfq h(t)
,0, . . . ,0]> ∈ Rn is the forcing vector. Due to the vibrations of the tool, the instantaneous chip thicknessh(t) is determined by the feed, the current tool position and the previous position of the tool one revolution before, i.e.,
h(t) =h0+q1(t−τ)−q1(t), (4.4) whereh0 =vfτ is the prescribed chip thickness,vf is the feed velocity,q1(t)indicates the position of the tool tip in directionxat time instantt,q1(t−τ)is the position one revolution before andτ
4.1 Model of turning operations 35
(a) Rigid tool (b) Compliant tool
tool (c) Mechanical model of turning
x
x
x y
y
y z
z
z
R Ω
w
0
0
Fq
vf
vc
vc
q1(t) q1(t−τ) h(t)
h(t)
h0 q1
q1
m1
k1
c1
Actual cut Previous cut
Fig. 4.1. Surface regeneration in an orthogonal cutting process: (a) Chip formation in case of a rigid tool; (b) Chip formation in case of a compliant tool (1stvibration mode); (c) Mechanical model of turning operations with multiple vibration modes.
is the regenerative time delay. For constant spindle speeds, the time delay can be given explicitly asτ = 60/Ω, whereΩis the workpiece revolution given in rpm.
Local stability of the steady-state (chatter-free) motion can be analyzed by means of the lin-earized system. The general solution can be given asq(t) = qst+x(t), whereqst ∈Rnis related to the static deformation and x(t) ∈Rn is a small perturbation around the equilibriumqst. Note that the unique equilibrium can be calculated fromKqst =F(h0).
After the linearization, the variational system is given in the form M¨x(t) +Cx(t) +˙ Kx(t) =κ x(t−τ)−x(t)
, (4.5)
and
κ=
κ 0 . . . 0 0 0 . . . 0 ... ... ...
0 0 . . . 0
∈Rn×n, κ=wfq0(h0), (4.6)
withκbeing the specific cutting-force coefficient. Note thatκis linearly proportional to the depth of cutw, which is one of the main technological parameters.
Stability analysis
There exists several methods to construct the stability lobe diagrams corresponding to system (4.5).
The root-crossing boundaries where a pair of a complex eigenvalues cross the imaginary axis can be determined by the D-subdivision method. The characteristic equation is written as
D(λ) = det D(λ)
= 0, (4.7)
where
D(λ) =λ2M+λC+K+κ 1−e−λτ
(4.8)
36 CHAPTER 4 Turning operations is the characteristic matrix. The solution of (4.7) forλ= iωafter separating to real and imaginary parts gives the stability boundaries. Utilization of a bisection algorithm provides a fast solution for the problem, cf. [104].
Alternatively, other numerical methods can be used which are based on the approximation of the largest characteristic exponents. For example, the semi-discretization method [1], the full-discretization method [105], the integration method [106], the Chebyshev collocation method [107, 108] and the spectral element method [15, 109] are often used numerical techniques, just to mention a few. Calculated stability lobe diagrams are given in the following subsection.