Vibration suppression and coupled interaction study in milling of thin wall casings in the presence of tuned mass dampers

Damping of machining vibrations in thin-wall structures is an important area of research due to the ever-increasing use of lightweight structures such as jet engine casings. Published literature has focussed on passive/active damping solutions for open geometry structure (e.g. cantilever thin wall), whereas more challenging situations such as closed geometry structures (e.g. thin wall ring-type casings) were not taken into consideration. In this study, a passive damping solution in the form of tuned viscoelastic dampers is studied to minimise the vibration of thin wall casings while focussing on the change in coupled interaction between tool and workpiece due to added tuned dampers. Finite element simulation was carried out to evaluate the effectiveness of tuned dampers in single impact excitation, and this was further validated experimentally through modal impact testing. A reduction in root mean square value, with tuned dampers, of about 2.5 and 4 times is noted at higher and lower depths of cut, respectively, indicating a moderate dependency on depth of cut. A change in coupled interaction of workpiece with tool’s torsional mode (in undamped state) to that of tool’s bending mode (with tuned dampers) was also noted. Variation in machined wall thickness of the order of 6 µm is noted due to the change in coupled interaction from torsional mode to bending mode of tool.

chain. Calculation of stable machining parameters must take into account one or more of the weak links of this system. The stability lobe algorithms, available through commercially available software [6,7], are currently being used mostly for machines, tools, and fixtures and in general are utilised for simple workpiece geometries [8][9][10]. Workpiece vibration when machining complex components could be more practically addressed through fixturing solutions [11]. This is mainly due to the complexity involved in obtaining a workpiece frequency response and also due to the fact that it changes with removal of material during machining. Hence, the way to suppress vibration is through appropriate fixture designs. Moreover, the research in suppressing vibration through fixturing solutions was not only driven by the need to overcome the complexity of the component's geometry, but also, indirectly, by the use of difficult-to-machine workpiece materials so commonly employed in high value-added products such as gas turbine engines. In these situations, due to process damping effects, the stability lobe algorithms are not particularly useful at the low cutting speeds at which these materials are machined [12].
Typical fixturing solutions researched for improving thin wall machining stability are either standard mechanical fixtures or damping solutions. Research has been carried out in both passive and active damping treatments with a view to suppressing vibrations during machining. However, these solutions have sought to dampen vibrations in toolholders (boring, turning) [13][14][15] and machine tool structures [16]; only a few researchers have investigated solutions to dampen vibrations arising from the workpiece.
Sims et al. [17] reported mitigation of workpiece chatter during milling using granular particle dampers to provide energy dissipation through friction. Using this technique the depth of cut was able to exceed the previous limit by an order of magnitude. Zhang et al. [18] reported on workpiece chatter avoidance in milling using piezoelectric active damping mounted directly on the workpiece.
Although this is more difficult to implement in real industrial environments, by using this approach a seven-fold improvement in the limiting depth of cut has been obtained. However, this has been done on simple geometry parts such as a cantilever plate. Rashid et al [19] proposed an active control of workpiece vibrations in milling through piezo-actuators embedded in work holding systems. However, the part on which the demonstration was done was of simple geometry (rectangular blocks) and was dynamically stiff. Nevertheless, this was directed to improve the dynamics of a production workholding system, (i.e. a pallet) by generating a secondary controlled signal (i.e. vibration) that cancels the primary disturbing signal generated by the cutting process (i.e. milling); improvements in surface finish and tool life were reported.
In addition to damping through piezo-actuators, passive dampers such as tuned mass dampers were also employed to mitigate workpiece vibration. Rashid et al. [20] proposed the use of tuned mass dampers mounted on a stiff workpiece, a rigid steel block. They presented an experimental validation of tuned viscoelastic dampers for damping a targeted mode of a solid workpiece during a milling operation, in which a reduction in vibration acceleration by 20dB for the targeted mode was reported. The vibration absorbers for structural dynamics applications are usually tuned based on Den Hartog's method which gives two peaks of equal magnitude in the damped frequency response function. However, considering the special nature of machining chatter problems, where the limiting depth of cut in machining is inversely proportional to the negative real part of the transfer function, Sims [21] proposed a novel tuning methodology for vibration absorbers in machining application.
This consisted of a tuning methodology with equal troughs of the real part of frequency response function instead of the conventional equal-peaks of amplitude method.
The aforementioned research validates both the passive and active damping concepts on simple geometries which can be represented by a few degrees of freedom. Hence, the solutions that were proposed up to now can be analytically designed and evaluated. However, on actual industrial applications which have a much more complex response with multiple modes, no work has been reported to demonstrate these solutions. Of course, in such situations there is a significant difficulty in simultaneously supressing various modes that also might change their values while the workpiece is machined. Also, the clamping setups/forces may have some inherent variations.
Moreover, damping of machining vibrations of thin walled cylindrical structures has not been reported. Chang et al [22] studied the chatter behaviour of a thin wall cylindrical workpiece while turning, in which they showed that the chatter phenomenon is determined by the ratio of the internal diameter of the casing to the wall thickness. With an increase in this ratio, the compliance of shell mode increases and hence is easily excited when compared to beam mode. Lai et al [23] studied the stability characteristics while turning a thin walled cylindrical workpiece clamped by a three jaw chuck. They reported that the vibration properties such as stiffness coefficient and vibration direction angle of beam and shell modes change with the relative position of the cutting tool to the chucking jaw. Mehdi et al [24][25] also studied the dynamic behaviour of three jaw clamped thin walled cylindrical workpieces during turning and a proposed response to a Dirac excitation in the form of Nyquist curves to characterise the stability of the turning process -a negative real abscissa value less than -1 indicating an unstable process. They also reported a reduction in vibrations when a supplementary damper, an arbitrarily selected rubber tube, was attached to the workpiece system. However, there was no mention of the scientific basis of damper selection and also no attempt was made to validate the experimental results using numerical simulations. Also the variation in the dynamic behaviour of the workpiece with added supplementary damping was not presented. To address this need, in this work a finite element simulation was carried out to design and validate the tuned viscoelastic dampers for a thin walled cylindrical component during milling operations, and also to study the behaviour of the workpiece with dampers mounted on it. The advantage of such a simulation is the ability to evaluate the effectiveness of the dampers on actual components during setting up of the fixture, so that vibration during milling is effectively suppressed. Using the proposed simulation methodology, design parameters such as the number and location of the dampers as well as the mass ratio can also be optimised. In this paper, firstly, the dynamic response analysis of the un-damped workpiece (both through finite element and experiment) is discussed.
Then, the following sections present the details of the design of the Tuned Viscoelastic Dampers (TVDs) based on the analysis of un-damped workpiece response supported by numerical and experimental validation of TVDs.

Finite element analysis
An understanding of the dynamic response of the workpiece is essential before designing the tuned dampers. This section presents numerical and experimental evaluations of the workpiece response.
The workpiece chosen was a generic component to represent a practical range of thin walled casings typically used in aerospace structures. The component is made of a Nickel based superalloy (Waspaloy®) and has a thin wall of 2.5mm thickness, a height of 95 mm, and an inner diameter of 360mm. As per industrial specifications, a peripheral milling operation needs to be performed on the thin wall to generate specific features (pockets/bosses). As shown in Figure 1 [26]. This high number of modes was chosen considering the fact that most of the modes have symmetric counterparts. Generally, for most of the structures the criteria to decide the number of modes to be extracted depends on the total effective mass in each direction; as this should represent a significant fraction, e.g. 85% [27], of the total mass of the structure. However, for casings such as the one presented here, it is not practical to decide the number of modes based on such criteria due to the fact that part of the structure (the thin wall) has very low stiffness when compared to the remaining part (e.g. the flange). For example, the total casing weights 80 kg and the thin wall section amounts to only 4 kg. Due to this reason, most of the modes extracted are local modes of the thin wall and the effective mass in each direction accounts for only 70% of the total structural mass even after the extraction of 500 modes (highest frequency corresponding to 20,000 Hz).
Harmonic analysis was carried out to study the workpiece response at all the extracted natural frequencies in which the casing thin wall participates. The harmonic response was computed using a linear perturbation step in Abaqus®, steady state dynamics, and direct integration. Although this step is computationally expensive, it was chosen as it gives accurate results in the presence of material properties that depend on frequency -a characteristic of viscoelastic materials, the analysis of which is presented in the next section. To maintain a direct comparison between damped and undamped casings, the same analysis step was chosen for the un-damped casing. As the amplitude of the harmonic response of the un-damped casing will be unrealistically high, a preliminary experimental modal analysis was carried out to find out the order of structural damping. This was found to be of the order of 0.1% after curve-fitting the frequency responses. The same damping factor was input for the finite element simulations. The harmonic response of the un-damped casing, as shown in Figure 2, is computed using fifteen frequency points, around each resonance peak, with frequency spacing of 0.2Hz. Such close frequency spacing was chosen to capture the resonance peak considering the un-damped nature of the frequency response. However, considering the high computational cost associated with using direct integration, the number of points is kept to a minimum, and hence the discrete nature of the spectrum in Figure 2.

Machining experiments on un-damped casing
To study the behaviour of the dominance of a group of modes in the dynamic response as observed in FE analysis, machining trials were carried out on the casing with two accelerometers mounted on it within the machining zone. Peripheral milling cuts were taken over one full quarter of the casing using a Ø16mm tool with two inserts. An axial and radial depths of cut of 2mm and 1mm along with a feed rate of 0.1mm/tooth and cutting speed of 40 m/min are employed. The frequency spectrum (FFT) of the machining acceleration signal was studied at various instances of the entire machining sequence. Figure 5 shows one such analysis of frequency spectra for a time period of one tooth contact, which can be considered to be an impact excitation of the structure. Note that Figure 5 shows the acceleration signal for one complete revolution of the tool having 2 inserts. It can be seen that 4869Hz and 4767Hz and their harmonics are quite dominant in the whole spectrum. In fact, this behaviour was observed, within a variation of only a few Hz, without fail in the whole acceleration signal acquired over a machining period for a quarter part of the casing.

Modal testing of un-damped casing
The variation between the finite element prediction of 4724.7Hz and the experimental observation of 4869Hz in frequency spectra of the machining signal is analysed. Experimental modal analysis was carried out on the casing in the actual machining set up, as shown in Figure 6, considering the sensitivity of modes to the boundary conditions. As the thin walled section of the casing is of primary interest, the test was carried out only on this area. The grid size for modal testing was decided after studying the mode shapes in the finite element analysis. As the height of the thin wall of the casing is small when compared to the diameter, it has a relatively easier tendency to vibrate at higher order circumferential modes as compared to axial direction. This can be observed from the fact that within the studied frequency band (0-6000 Hz) seventeen circumferential and second axial modal orders are noticed. Hence the thin wall was meshed into 36 sections circumferentially and 4 sections axially.
Considering that such a symmetrical structure has repeating modes, two reference accelerometers are used to capture the symmetric modes. The mode shape analysis revealed that the m=2 and n=1 mode (terminology as explained in section 2.1) occurs at 4874.7Hz as shown in Figure 7, the mode shape of which compares with that of the FE mode 4724.7Hz, see Figure 4(a), and thus confirming the FE prediction. Similarly, m=2 and n=2 mode occurs at 4760.41Hz which corresponds to the FE mode of 4618.3Hz. The slight mismatch in frequencies could be due to non-updating of the finite element model. A finite element model with bolted joints modelled using spring dashpot elements, as reported in [28] and updating their stiffness and damping using experimental results will predict the correct natural frequencies; however in the present study this error of 3% is considered acceptable for further analysis.

Finite element modelling of TVDs
Finite element analysis of viscoelastic damping is a popular approach for dealing with constrained layer damping of automotive panel vibrations. Research has been carried out on various aspects such as developing methodologies for the faster prediction of the damped response of structures [29][30], optimisation of damper location [31][32], and development of new damping polymers to suit different automotive components [33][34]. A detailed review of modelling and finite element implementation of viscoelastic damping is given in Vasques et al [35].
A viscoelastic material is represented using a complex modulus in the frequency domain: * = ′ + ′′ = ′(1 + ) Where G' and G'' represent storage and loss modulus respectively, and η = G''/G' represents the loss factor of material. Manufacturers' data sheets for the damping polymers generally provide the storage modulus and loss factor in the form of nomographs showing their frequency and temperature dependence. In this research, 3M® ISD112 viscoelastic tape is used as it has a high loss factor and a moderate modulus value, and with a little variation of these values within the expected operational temperature range. The frequency domain viscoelastic data for 3M® ISD112 provided in [36] which was extracted from manufacturer's data sheet is used in this simulation and is reproduced in the Appendix. The guidelines for static and dynamic analysis of viscoelastic materials using finite elements are summarised in [37].
Considering the experimental dynamic response of the un-damped casing as presented in Figure 5(b) and the experimental modal analysis results presented in Figure 7, (2)  For the casing under study, the 2.5mm thin wall is considered to be the main vibrating mass (4.47kg).
With the above calculations, for a mild steel damper block with a mass equal to 5% of vibrating mass and dimensions of 40x40x20mm, the thickness of viscoelastic tape was found out to be 0.163mm. A finite element model, shown in Figure 8, was created with these parameters in which a single damper block is modelled (attached) inside the periphery of the casing. The location of the damper block is arranged at the maximum amplitude location of 4724.7Hz mode of FE, mode shape as shown in Figure 4(a). Though this mode also has an identical symmetric mode shape at 90 degrees, only one damper block is modelled initially to study the effect of a single damper block. The damper block was meshed using C3D20R elements, similar to the casing, and the viscoelastic layer was meshed using hybrid continuum elements C3D20RH which will give accurate results while simulating incompressible rubber-like materials [37], such as the 3M ISD112 viscoelastic material used in this study. The nodes of the viscoelastic layer are tied to those of the casing and damper block on either side. To minimise the distortion due to the automatic re-adjustment of the nodes by Abaqus® before the analysis, a uniform mesh was designed for the viscoelastic and damper blocks where nodes of these parts fall close to each other. During initial trials, it was noticed that this step is essential as non-matching nodes can result in distorted elements leading to aborting of the simulation.
After completion of the initial frequency extraction step, a harmonic analysis was run using Steady State Dynamics, Direct step. As explained in the previous section, this allows viscoelastic material properties, which vary with frequency, to be considered, albeit at additional computational burden.
It should be noted that while extracting natural frequencies, Abaqus® gives the option of considering the frequency dependent properties at only one frequency point, and in this case 4725Hz was considered for that option as the damper is tuned for this frequency. Also, out of all the frequencies extracted, some of the modes correspond to the rigid body motion of the damper block, viscoelastic layer and the bending modes of the fixture plate without affecting the casing. These frequencies are discarded while evaluating the harmonic response, retaining only those modes where the casing participated. The drive point harmonic response (Figure 9) clearly shows that the targeted mode of 4725Hz is damped about 7 times, while decreasing the overall response acceleration magnitude. A close analysis of mode shapes with respect to the response reveals that out of the two symmetric modes of 4725Hz, the mode shape which is targeted for damping has shifted to 4707.7Hz with mode shape as shown in Figure 10(a) and the other mode remained at similar frequency of 4725.2Hz with mode shape as shown in Figure 10(b). However, the response magnitude in both the cases has reduced to approximately 600 m/s 2 from an un-damped response of 4992 m/s 2 (at the same location as shown in Figure 2). This not only shows the importance of choosing the placement location of the damper but also its efficacy on damping the targeted mode. local tuned viscoelastic dampers attached, such an assumption is not valid [39]. However, in this work, modal parameter estimation routines using proportional damping only are used. Figure 13. Frequency spectrum of acceleration signal acquired inbetween two damper blocks for one tooth contact Figure 14 shows the mode shape of the casing with tuned dampers at 4490.79Hz; the mode shape correlates well with that of Figure 11(b) which shows that the FE predicted mode of 4255Hz corresponds to 4490Hz in actual casing. While it is difficult to appreciate the exact similarity between two mode shapes, some of the points used to observe the phase similarity are shown in Figure 11(b) and can be matched to identify the mode similarity. Also, the top profile of the casing of FE mode shape is overlaid in Figure 14.

Conclusions
Damping of machining vibrations is crucial in achieving a better surface quality of the product and also to improve productivity by utilising aggressive process parameters. Thin-walled ring type structures are prone to workpiece vibrations during machining, and passive damping treatments are potential solutions to address this problem. In this work, a novel approach into modelling and utilisation of passive damping, i.e. tuned viscoelastic dampers, is devised to minimise vibrations on a thin walled casing. The paper brings into attention the following key aspects:  No previous research has been reported in the literature on the dynamic behaviour of thin walled casings in highly dynamic processes such as milling. It has been shown that the casing studied in this work has a dynamic behaviour consisting of a dominant response from few modes at higher frequency prompting the usage of tuned dampers to minimise machining vibrations. This behaviour was validated using spectral analysis of the acceleration signal acquired during machining tests and also through experimental mode shape analysis. The variation in predicted and experimental natural frequencies of interest for un-damped casing is only 3%, and considering that this deviation is achieved without any model updating demonstrates the accuracy of simulation.
 Finite element simulation of the tuned viscoelastic damper was not previously reported for manufacturing applications. In the present work this is performed with a view to implement and validate a methodology for designing passive damping solutions as fixturing concepts for large industrial thin walled structures. Frequency response predictions using FE analysis are validated through spectral analysis of machining acceleration signal and experimental mode shapes; predictions are made with an error of 5%. Considering that the un-damped casing predictions itself differ by 3%, such a variation of 5% for response simulation with dampers is considered to be very good.
 The benefits of tuned dampers in minimising the machining vibration are evaluated through the root mean square (RMS) value of the acceleration signal acquired during machining; a reduction in RMS value of nearly 4 times is observed when using the proposed passive damping solution.
 The damped dynamic response of the casing also showed a dominant mode, though with reduced acceleration, that is related to the vibration of the casing inbetween the damper blocks. It is likely that even a fully damped casing (continuous damping treatment) will have a dominant mode; this will be further investigated in future.
Overall, this work reports the dynamic response of thin walled casings with and without tuned dampers, and also throws some light on the behaviour of the structure when shell vibrations alone are treated with discrete damping solution. The accuracy with which the results are obtained and validated proves the benefits of using FE analysis for designing and validating the damping solutions in manufacturing applications, thereby achieving important benefits such as better surface finish and higher productivity while machining thin walled structures.