Budapest University of Technology and Economics Faculty of Mechanical Engineering
Department of Applied Mechanics
Booklet of thesis statements
Unexplored parameter variations in cutting processes for chatter suppression
Author:
Bence Beri
Supervisor:
GaborStepan, DSc professor
Submitted to
Géza Pattantyús-Ábrahám Doctoral School of Mechanical Engineering Sciences, Budapest University of Technology and Economics
Budapest, 2022
Overview of the dissertation
In the manufacturing industry, there is a continuous driving force to increase efficiency by achieving high material removal rate, high qual- ity and environmentally friendly production at the same time. It is difficult to balance between the significant decrease of the energy con- sumption of machines and the continuously ascending demands of the customers, which have to be fulfilled. There is a need to reduce further the material loss during machining operations. While this requirement pushes the industry to strict limitations, the cutting processes such as turning, milling and drilling still provide high production efficiency and excellent accuracy. There is a critical phenomenon that interfere with meeting these high precision standards, which is the possible machine tool vibrations.
In machining, the surface regeneration effect is often responsible for the destabilisation of the underlying material removal operation. It leads to self-excited oscillations that are often called chatter in man- ufacturing. These harmful vibrations are difficult to be eliminated.
They may result in increased noise levels of the machine tool, in deteri- orated surface topology and/or in decreased lifespan of the cutting tool.
In material removal operations, the specified phenomena are in connec- tion with the flexibility of the tool and/or the workpiece, which are also related to the compliance of the whole machine tool structure. For this reason, the deformation of cutting tools and thin-walled or beam-like slender workpieces has an important role in the fundamental dynamics of the corresponding system. The flexible nature induces the variation of the chip thickness due to the relative vibration between the cutting tool and the workpiece.
The mathematical description for surface regeneration is given by means of delay differential equations where the cutting tool meets the surface of the workpiece that was already machined in a preceding cut.
If the cutting tool is assumingly compliant and the workpiece is rigid, the instantaneous chip thickness can be defined by the present and the previous positions of the tool. The time period between two succeed- ing cuts corresponds to the arising time delay in the process. Based
on these properties, a lobe structure may exist with stable pockets in the technological parameter space, by which stable machining can be explored at high depth of cut values. While these pockets are highly sensitive on the dynamic characteristics of the machine tool, they can result in the increased efficiency of the material removal processes.
The stability charts provide a guidance for the machinist to reach the highest possible material removal rate still without facing chatter.
While all the operations are generally subjected to strong nonlinearities, the conventional charts show the boundaries of the linearly stable pa- rameter regions only, which are in agreement with the system properties in a linear subspace. Based on chatter tests, it is also possible to detect abrupt vibrations near the stability boundaries within the linearly sta- ble domains. These are commonly originated in the nonlinear character of the arising cutting force or in the asymmetries of the machine tool structure. To forecast their appearance, the sensitivity of the machine tool dynamics on the initial perturbations has to be investigated.
Although, the theoretical stability charts present increasingly reli- able chatter predictions as the underlying mechanical models improve, the experimental identification of the stability limits is still a demand- ing task. The reliability of chatter predictions is low in real industrial applications. To bypass these difficulties, three ways can be followed.
One of these is the expensive laboratory tests carried out in an environ- ment where cutting force sensors and vibration measurement devices support the online prediction and detection of chatter.
The second solution is the establishment of a computational envi- ronment for virtual machining, which has the capability to simulate any material removal process via the implemented tool geometries and the underlying mechanical models of the machine tool structure. To further improve the credibility of the digital twin, the analysis of the machine tool can be supported by the modal testing of the real machine tool structure. However, the reliability and the efficiency of this virtual ap- proach are limited by the precision of the embedded cutting conditions and the models of the machine tool elements.
A third possibility is to remain as close as possible to the real dy- namics of the machine tool by investigating the cutting operation in
an accurately developed computer controlled environment. It can be achieved in the framework of the hardware-in-the-loop (HIL) method- ology (also called substructuring), which provides an intermediate tool between theoretical modelling and experimental work by carrying out semi-virtual cutting processes.
This thesis work aims at the mechanical modelling of the dynam- ics of turning and milling processes where the stability properties can change due to the varying system parameters. Two essential cases are investigated for the intricate parameter variation. First, the cutting tool and the workpiece interaction results in different directional cut- ting force components that may have non-negligible linear and nonlin- ear effects on the cutting processes. During the theoretical analysis, the lateral deformation of the beam-like segments of the machine tool structure such as workpieces and cutting tools is investigated in the presence of axial cutting forces, which affect the lateral stiffness of the corresponding beam. As a second aspect, the identification of the ma- chine tool dynamics is a crucial part of the accurate determination of the cutting process stability that is very sensitive to tiny internal and external perturbations. Experimental study on the varying stability properties is carried out in HIL environment by means of a real ma- chine tool spindle and emulated cutting force characteristics.
The thesis expands the related literature by both qualitative and quantitative results on the stability properties of machining processes.
Since one of the most important goals of the current research is to reach the highest material removal rates, the theoretical outcome of the dissertation helps to exploit formerly unexplored stable parame- ter combinations. The development of semi-virtual cutting processes and their accurate emulation in the novel HIL setup contributes to the improving credibility of chatter predictions in industrial environments.
Modelling approaches
The first topic focuses on the dynamic behaviour of material re- moval processes. It presents the fundamental mechanical modelling of
the beam-like parts of the machine tools. Cantilever beams subjected to axial and lateral forces are analysed based on the Euler-Bernoulli theory. The nonlinear relationship between the small axial and lateral displacements of the free end of the beam is derived analytically, by which the corresponding lateral stiffness variation of the beam can also be identified. Considering torsional moment acting on the beam addi- tionally, the structural stability of rotating beams is presented (Thesis statement 1).
The second topic investigates the dynamics of turning in distinct cutting configurations. First, the effect of the axial cutting force com- ponent acting on the cutting tool is taken into account in the stability calculation (Thesis statement 2). In the second arrangement, a time- periodic external axial excitation is considered, by which the relevant stable parameter domain of the process is improved (Thesis statement 4). In the meantime, a qualitative study is performed on the effect of intermittent cutting that is approximated by a reduced order turning model (Thesis statement 3).
As the next result, the stability properties of milling processes are determined when the axial component of the cutting force appears as compression. On the one hand, the dynamic effect of the compressive force is considered through the lateral stiffness variation of the bearing support of the main spindle. This results in stable and unstable islands within the unstable and stable parameter domains. The relevance of these islands in real cutting environment is analysed by means of local bifurcation theory. On the other hand, the dynamics of a reformulated milling process is investigated in the presence of a slender cutting tool subjected to compressive cutting force (Thesis statement 5).
Finally, an experimental HIL environment is presented that is ca- pable of reconstructing the stability of machining processes in a wide range of spindle speeds. The structure of the measurement setup is de- scribed in details and the chatter prediction process is summarised. For the experimental stability analysis, three case studies are discussed via emulated turning processes. Each case one-by-one reveals the effect of complex parameter variations on the corresponding stability properties (Thesis statement 6).
Thesis statements
To identify the dynamic parameters of slender cutting tools and/or slender, cylindrical workpieces involved in material removal processes, I have analysed the deformation of cantilever beams based on the Euler- Berno-ulli theory. The beams are subjected to axial and lateral forces at their free end. In case of no axial load, a numerical estimation is available in the literature for the relationship between the axial and the lateral displacements of the free end of the beam. As an extension of this result, I have determined the relation between these displacements analytically in closed form while also taking into account the axial load.
The results can be summarised as follows.
Thesis statement 1
Consider an elastic, homogeneous, isotropic, inextensible and prismatic cantilever beam of length L and bending stiff- ness IE, which is subjected to an axial force Fa and a lateral force at its free end. Assuming the conditions of the Euler- Bernoulli theorem for small displacements, the resulting axial displacement δx and lateral displacement δy at the free end have a closed form nonlinear relationship:
δx =κδ2y, where κ= 2(2 + cos(2√
γL))γL−3√
γsin(2√ γL) 8(√
γLcos(√
γL)−sin(√ γL))2 and γ = Fa
IE. Related publications: [1,2]
I have analysed the stability of the turning process when the cutting force has a component that provides relevant axial load on the tool.
When the cutting tool is modelled by a cantilever beam based on the Euler-Bernoulli theory, the effect of the axial force can be considered by means of the lateral stiffness variation at the tool tip. This way, the governing equation of the relevant dynamics of the system assumes the
form
m¨y(t) +cyy(t) +˙ ky(Fz(t))y(t) =Fy(t)
with lateral displacement y of the cutting tool; modal mass m, modal damping cy and modal (lateral) stiffness ky = ky,0 +ky,1Fz(t). The cutting force components are indicated by Fy and Fz in the lateral y and axialzdirections, respectively. Assume the power law cutting force characteristics asFz(t) =Fy(t)/σandFy(t) =Kywhν(t)whereσis an empirical ratio,Ky is the cutting force coefficient,wis the depth of cut, his the instantaneous chip thickness andνis the cutting exponent. As a new result, the effect of the axial component of the regenerative cutting force was revealed on the stable domain of the technological parameters, and the limit was determined where the static loss of stability of the cutting tool occurs. The results are summarised as follows.
Thesis statement 2
Consider the single degree-of-freedom orthogonal cutting model of a turning process with flexible cutting tool and rigid workpiece, where the cutting force satisfies the power-law and its typically compressive component Fz(t) in the axial direc- tion of the tool decreases the tool’s lateral stiffness: ky = ky,0−ky,1Fz(t) at the tool tip. This leads to the reduction of the stable technological parameter domain in the plane of the depth of cutw and the spindle speedΩ: the so-called stability lobes are shifted in the direction of lower depth of cuts and they are also bent in the direction of low spindle speeds as w increases. The system undergoes a static loss of stability and the cutting tool buckles when the dimensionless critical depth of cut
¯
wc,cr= 5Lνσ 6h0
is reached where L is the length of the tool, ν is the cut- ting force exponent, σ is the empirical ratio of the cutting force components, h0 is the theoretical chip thickness; w¯c =
Kywνhν−10 /ky,0 holds for dimensioning with cutting force coef- ficient Ky.
Related publications: [1,2,3,4]
These phenomena become relevant in case of slender cutting tools like the ones used for cutting inside lengthy tubes.
I have analysed the global dynamics of the turning process qualita- tively when the so-called fly-over effect is taken into account, that is, the cutting tool may leave the surface of the workpiece during machin- ing. Consider the low-order Taylor expansion of the infinite dimensional governing equations with respect to the regenerative delayτ in the form η00(t)+2ζη0(t)+η(t) =
( −w¯cτ η0(t)− 12τ η00(t) +16τ2η000(t)
h(t)>0
0 h(t)≤0
where the instantaneous chip thickness is h(t) =h0−τ η0(t) +1
2τ2η00(t)
with small perturbationηaround the steady-state solution, with damp- ing ratioζ, specific cutting coefficientw¯cand theoretical chip thickness h0. I have found that the chaotic behaviour of turning can be identified in the three-dimensional phase space of the reduced system. The re- duced non-smooth dynamic model of turning was investigated by means of discrete iterated diagrams that identify the characteristic parameters of the chaotic processes. The results lead to the following conclusion.
Thesis statement 3
Consider the reduced form of the single degree-of-freedom dynamical model of turning, which includes the fly-over ef- fect. The reduction is achieved by means of the second de- gree Taylor-expansion with respect to the regenerative delay.
The three-dimensional subspace approximating the essential part of the original system’s spectrum describes the possi- ble chaotic behaviour of the turning process qualitatively by means of the geometric representation of the non-smooth dy- namics. This description is acceptable when the time delay is an order of magnitude smaller than the time period of the sys- tem’s natural oscillation. In case of unstable parameter com- binations, the behaviour of the reduced system is presented in the form of discrete iterated diagrams, which also provide the characteristic parameters of the chaotic process.
Related publications: [5,6]
I have analysed the multi-parameter stable domain of turning in a novel configuration where the workpiece is excited by a time-periodic axial force Fa(t) at the tailstock. The dynamic parameters of the clamped and pinned beam-like workpiece were fitted at the tool tip by means of the dominant bending vibration and the deformation of the corresponding Euler-Bernoulli beam. The one degree-of-freedom mechanical model linearised around the steady-state solution for the small perturbationη assumes the form
m(t)¨η(t) +cy(t) ˙η(t) +ky(lp;w, Fa(t))η(t) =Fy(t)
with time-periodic coefficients where Fy is the corresponding regener- ative cutting force component, m , cy and ky are the modal mass, the modal damping and the lateral stiffness, respectively,lpis the tool posi- tion andwis the depth of cut. In case of distinct excitation waveforms and frequencies, I showed that the stable parameter domain of the ma- terial removal process increases significantly in the relevant low spindle speed domain even though the mean lateral stiffness of the workpiece decreased. This allows the use of higher attainable depth of cuts. The results are summarised below.
Thesis statement 4
In case of orthogonal turning of periodically compressed slender workpieces, consider the single degree-of-freedom me- chanical model where the workpiece is assumed to be flexible and the cutting tool to be rigid. Due to the effect of the pe- riodic axial force Fa(t) = Fa0+Fa1S(t), the lateral stiffness of the workpiece and the corresponding proportional damping also change periodically. For sinusoidal, triangular and rect- angular waveforms S(t), and for excitation frequencies either dependent or independent of the regenerative delay, the stable parameter domain of turning improves essentially in case of realistic force amplitudesFa1 in the relevant low spindle speed domain depending on the cutting tool position along the work- piece. In contrast, constant compression definitely decreases the stable parameter domain.
Related publications: [1,2,7,8]
I have analysed the stability of milling and turning operations when time-periodic axial compressive force is acting on the milling tool or on the slender workpiece in case of turning. This axial compression is originated either in the dynamics of the material removal process itself or in some external excitation. The effect of the periodic axial forcing is taken into account in the lateral stiffness variation of the system at the tool tip. As a new result, I have identified that essential changes occur in the stable domains of the above processes and formerly un- explored stable parameter combinations appear. Based on the results, the following conclusions are reached.
Thesis statement 5
Consider the single degree-of-freedom mechanical model of milling or turning processes where the cutting tool or the slender workpiece, respectively, is subjected to a time-periodic
axial compressive force. The time-periodic compression mod- ifies the lateral stiffness of the corresponding system and it perturbs the surface regeneration effect in a way that the sta- ble parameter domain of the cutting process changes essen- tially compared to the case of constant axial force. Due to the parametric excitation induced by the periodic axial force, the originally vertical asymptotes of the stability lobes be- come bent and stable/unstable islands may appear inside the originally unstable/stable regions. By means of the stable is- lands, the material removal rate can be increased further and chatter-free machining may also be assured as confirmed by the nonlinear analysis.
Related publications: [1,2,4,7,9,10]
I was a member of the team that developed a semi-virtual hardware- in-the-loop (HIL) experimental setup, which investigates the dynamics of cutting processes in a way that in case of a given machine tool, the cutting forces are generated with electromagnetic actuators and re- sponse signals of contactless sensors. I carried out the identification of the spindle speed dependent dynamics of the machine tool structure, by which the relevant modal parameters of the system were determined.
The experimental setup is eligible to determine the stability of machin- ing processes without damaging the machine/workpiece structure even in unstable cutting conditions. The process stability/instability can be reconstructed in a wide range of spindle speeds. Based on the dis- placement signal of the rotating workpiece, the stability of the turning process is identified with the help of the dominant vibration frequency.
According to the experimental observations, the results and the conclu- sions are summarised as follows.
Thesis statement 6
By means of the developed hardware-in-the-loop experi- mental setup, the stability of turning processes is given with
high accuracy by the presence of the dominant regenerative vibration frequency in the spectrum of the response signal of the workpiece. During experimental stability analysis, the frequency identification of the arising regenerative vibrations in the unstable parameter domain determines the nature and the order of magnitude of the nonlinearities and also their dependence on the spindle speed.
Related publications: [11,12,13]
Bibliography
[1] B. Beri, G. Stepan, and S. J. Hogan. “Effect of Potential Energy Variation on the Natural Frequency of an Euler-Bernoulli Can- tilever Beam Under Lateral Force and Compression”. In:Journal of Applied Mechanics 84 (2017), p. 051002.
[2] B. Beri, G. Stepan, and S. J. Hogan. “Structural stability of a light rotating beam under combined loads”. In: Acta Mechanica 228 (2017), pp. 3735–3740.
[3] B. Beri and G. Stepan. “Stability of turning process with tool subjected to compression”. In:Procedia CIRP 77 (2018), pp. 179–
182.
[4] B. Beri and G. Stepan. “Axiális és torziós terhelés hatása fúrási folyamatok stabilitására”. In: XIII. Magyar Mechanikai Konfer- encia (MaMek), Augusztus 27-29. Magyarország, Miskolc, 2019, pp. 1–6.
[5] B. Beri and G. Stepan. “Essential chaotic dynamics of chatter in turning processes”. In:Chaos 30 (2020), p. 053108.
[6] B. Beri and G. Stepan. “Approximated Dynamics of Chatter in Turning Processes”. In: Nonlinear Dynamics of Structures, Sys- tems and Devices (2020), pp. 463–470.
[7] B. Beri, G. Meszaros, and G. Stepan. “Machining of slender work- pieces subjected to time-periodic axial force: stability and chat- ter suppression”. In: Journal of Sound and Vibration 504 (2021), p. 116114.
[8] B. Beri, G. Meszaros, and G. Stepan. “Stability of turning process on workpiece subjected to time-periodic compression”. In: XXV.
ICTAM, 22-27 August. Italy, Milano, 2021, pp. 1–2.
[9] B. Beri and G. Stepan. “Effect of axial force on the stability of milling: Local bifurcations around stable islands”. In: Journal of Vibration and Control ? (2021), ?
[10] B. Beri and G. Stepan. “Local bifurcations around a stable island in the parameter plane of milling processes”. In: ENOC, 17-22 July. France, Lyon, 2022, pp. 1–2.
[11] G. Stepan, B. Beri, A. Miklos, R. Wohlfart, D. Bachrathy, G.
Porempovics, A. Toth, and D. Takacs. “On stability of emulated turning processes in HIL environment”. In:CIRP Annals68 (2019), pp. 405–408.
[12] B. Beri, A. Miklos, D. Takacs, and G. Stepan. “Nonlinearities of hardware-in-the-loop environment affecting turning process emu- lation”. In:International Journal of Machine Tools and Manufac- ture 157 (2020), p. 103611.
[13] D. Bachrathy, H. Sykora, D. Hajdu, B. Beri, and G. Stepan. “Why is it hard to identify the onset of chatter? A stochastic resonance perspective”. In:CIRP Annals 70 (2021), pp. 329–332.
Categorisation of scientific papers:
Journal papers: [1,2,5,7,9,11,12,13]
International conference papers: [3,6]
National conference papers: [4]
International conference abstracts: [8,10]
