Finite element analysis (FEA)

Figure 4-9: Optimization process of cutting assembly

During design of the whole cutting assembly, a care was taken to increase dynamic stiffness of the system as much as possible and to avoid driving the ultrasonic head with a frequency causing a resonance of the cutting assembly.

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4.4.1.1 Introduction

A resonance in a system happens when a harmonic external driving force is applied to a system having a natural frequency equal or multiple of the driving force. In this case the resonance will cause large displacements, which might even result in the full structure to go over the elastic limit or even fail. In the simplified condition of absence of damping the harmonic driving force can be represented as:

, (4.3.1)

where, is the maximum magnitude of the force and is the frequency of the harmonic force in rad/sec.

If a system made of a simple mass attached to a spring in the absence of friction is considered [Figure 4-10]:

Figure 4-10:Schematic of simple mass attached to spring with coefficient k and applied harmonic force F(t)

it is possible to write the equation for the displacement x:

̈ . (4.3.2)

If each term of the equation is divided by the mass m, then:

̈ , (4.3.3)

where represents the natural frequency and is the acceleration of the system.

Solving this equation results in the total response of the system:

F(t) k

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In the absence of any damping the resonance condition brings ultimately large displacements that exceed any material mechanical characteristics. In the presence of damping, a viscous term does not allow for the displacement to grow infinitely but only up to a maximum limit yet negatively affecting the precision of the machining process. The ultrasonic transducer is the only element in the cutting head that is allowed to resonate; it is important that the whole system exhibits an excellent damping of vibrations and operates far from any resonance frequency.

4.4.1.2 FEA Model

For the finite element analysis the Simulia code Abaqus 6.11 Standard was used. It allowed calculating the approximate resonance frequencies and the associated resonance mode shapes of the complete cutting head. The driving frequency used in UAT, which is also the resonating frequency of the transducer complete with a cutting insert, was 17.8 kHz, this allowed reducing the input frequency range of the analysis, limited between 15 and 20 kHz, and, consequently, diminished the

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computational time. The system, being continuous, in fact, exhibits infinite resonance harmonics. In order to study the response of the system to low-frequency vibration, such as these causing chatter, the first 10 modes were also computed.

The developed model used quadratic tetrahedral elements with different dimensions;

the mesh was auto-generated with 26020 elements.

Figure 4-11: Model of cutting head with mesh

The bottom of the angle plate was restrained with all degrees of freedoms set to zero for all the analyses, to simulate it bolted to the cross-slide of the lathe.

4.4.1.3 Material properties

Most of the custom-machined parts of the cutting head were of mild steel, stainless steel was also the material of the dynamometer. The concentrator was the only

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element constructed in high-fatigue-resistant aluminium. For the sake of simplicity the material properties were considered to be uniform in the steel parts ignoring the differences between mild steel and stainless steel. The materials of this simulation were considered to behave as elastic isotropic materials. All the mechanical properties [Table 4-3] were obtained from the material supplier.

Table 4-3: Material properties

Component Material Supplier Mechanical properties

Concentrator High-fatigue

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Steel

N/A As above

Transducer holder

Steel East

Midlands alloy

As above

The contact surfaces between components and between the transducer holder and the concentrator were defined as surface-to-surface interactions with mechanical properties as ―rough‖. The friction was assumed to be unlimited so that the parts of the assembly would not move or rotate with regard to each other during the analysis.

This was necessary as all the parts of the assembly were bolted to each other not allowing relative movements during operations.

4.4.1.4 FEA analysis and results

The eigenvalues and natural frequencies in the frequency range around the resonating frequency of the transducer were obtained, all of them showing mixed vibrational modes. The effects affected predominantly only one or two of the system‘s components. An extra care was taken to check that the natural frequencies of the assembly were not coinciding with the resonating frequency of the ultrasonic transducer.

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Figure 4-12: Modal shape computed for frequency slightly below resonance frequency of transducer

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Figure 4-13: Modal shape computed for frequency slightly above resonance frequency of transducer

The colors represent the magnitude of the U3 eigenvalue (Z) direction, this orientation was selected because the transducer will effectively produce vibration along the Z direction of the model. The higher vibration amplitude appear to concentrate around zones of minor mechanical importance (around the edge of the flange), which are not directly involved in the machining process. This was observed for most of the natural frequencies around the resonating frequency [Figure 4-12][Figure 4-13] of the ultrasonic transducer with the exception of the eigenvalues correspondent to the frequency of 18111 Hz, this particular one showed heavy displacements [Figure 4-14].

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Figure 4-14: Modal shape for frequency of 18111 Hz showing large deformations of cutting assembly

Natural frequencies with mainly flexural modes of vibration of the cutting assembly were observed at 169, 298, 603 and 993 Hz (the first four modes). Largest deformations were observed in the first two modes, with the highest magnitude of the modal shape for the fourth mode [Figure 4-15]. The dominant effect observed was the flexural mode with some minor three-dimensional displacement effects.

An impact hammer test was used to calibrate the model using the resonating frequencies up to 6 kHz. Frequency response function analysis of the physical setup was valid only up to 9 kHz. Responses close or over to 9 kHz were showing reduced levels of coherence and were deemed unsafe to be used for calibration.

The low fundamental natural frequencies can be explained if the mass of the system

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is considered. The steel dynamometer and the transducer holder accounts for a mass of over 10 kg, to this it must be added the weight of the transducer (less than 5 kg) and of the steel block (around 7 kg).

Figure 4-15: Vibrational modes for first four natural frequencies

Summary

The impact hammer test and the FEA simulations allowed to identify that the implementation of a stiffening gusset at the back of the angle plate considerably reduced the flexural vibrations of the vertical slab. This additional element was preferred in the final stage III design.

In order to avoid oscillation of the system in the first modes due to periodic forces

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arousing from the machining process, such as the cutting force that a slightly offset work-piece with a rotational speed equal to any of the resonating frequencies could generate; it would be prudent to avoid them. The excellent damping properties of a lead sheet were used between the cutting head and the dynamometer; this was supposed to reduce the likelihood of exciting any of the vibrational modes.