Substructure analysis of vibration-assisted drilling systems

Abstract

Designing vibration-assisted drilling (VAD) toolholders requires accurately modeling the dynamics of the mechanical structure and its electromechanical coupling with the piezoelectric actuator. The overall dynamics of the VAD toolholder depends on the dynamics of its individual components (e.g., concentrator, piezoelectric disk, drill bit) and the interactions between them. Modelling by substructure analysis therefore provides an efficient framework for studying the effect of each component on the overall dynamics of the VAD system. Nonetheless, the existing substructure analysis methods are only applicable to holders with basic concentrator geometry, which limits their application in designing high-performance holders.

In this paper, Receptance Coupling Substructure Analysis (RCSA) is used to combine the dynamic models of the toolholder’s components to determine its vibration response to the electrical excitation of the piezoelectric actuator. This method allows for combining the 3D finite element models of the concentrator with the simplified and analytical models of the rest of the VAD components, enabling the inclusion of complex concentrator designs in modeling without imposing a large computational load. Furthermore, because this method couples the mechanical model of the structure with the electrical model of the piezo-actuator’s impedance, it can be used as a design tool to modify the toolholder’s natural frequencies by tuning the electrical impedance of its piezoelectric components. This possibility effectively converts the piezoelectric components to mechanical members with adjustable structural dynamics, while they also actuate vibrations. Both axial and axial-torsional VAD toolholders are studied in this work, and the accuracy of the presented modelling approach is verified experimentally.

Appendix : 1: Modeling vibrations of VAD components for RCSA

In this section, the analytical mode shapes of rods with uniform cross-section, the finite element model of rod vibrations, and the finite element model of twisted rod vibrations are explained.

1.1 1.1 Uniform cylindrical rod

Consider an elastic rod with length l, Young’s modulus E, density ρ, and cross-section area A. Assuming that the rod is rigid in lateral directions, and its torsional and axial deformations are uncoupled, the natural frequencies (ω_n) and mass-normalized mode shapes (U_n) of its axial vibrations in free-free end condition are obtained as follows [26]:

$$ \begin{array}{@{}rcl@{}} {\omega_{n}} &= &\frac{{n\pi }}{l}\sqrt {\frac{E}{\rho}} ;\\ {U_{0}}(x) &=& \frac{1}{{\sqrt {\rho Al} }};{U_{n}}(x) = \sqrt {\frac{2}{{\rho Al}}} \cos \frac{{n\pi x}}{l} \end{array} $$

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where n = 0, 1, 2,…, and x is the distance from a free end of the rod. Using these mode shapes, the axial FRF between two points at x_i and x_j is synthesized as follows:

$$ h_{ij}^{xf}\left( {x_{i}},{x_{j}},\omega\right) =\frac{x_{i}}{f_{j}}= \sum\limits_{n = 0}^{N_{m}} {\frac{{{U_{n}}\left( {x_{i}}\right){U_{n}}\left( {x_{j}}\right)}}{{ - {\omega^{2}} + {\omega_{n}^{2}}}}} $$

(25)

where N_m is the number of modes considered in synthesizing the FRF. Now, consider the same elastic rod with modulus of rigidity of G and polar moment of inertia of J. The natural frequencies (ω_n) and mass-normalized mode shapes (Θ_n) of its torsional vibrations are obtained as follows [26]:

$$ \begin{array}{@{}rcl@{}} {\omega_{n}} &= &\frac{{n\pi }}{l}\sqrt {\frac{G}{\rho }};\\ {{\varTheta}_{0}}(x) &=& \frac{1}{{\sqrt {\rho Jl} }};{{\varTheta}_{n}}(x) = \sqrt {\frac{2}{{\rho Jl}}} \cos \left( \frac{{\pi nx}}{l}\right) \end{array} $$

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where n = 0, 1, 2,…. Similar to Eq. 25, the torsional FRF between points at x_i and x_j is also synthesized using the torsional mode shapes:

$$ h_{ij}^{\theta t}\left( {x_{i}},{x_{j}},\omega\right) =\frac{\theta_{i}}{t_{j}}= \sum\limits_{n = 0}^{N_{m}} {\frac{{{{\varTheta}_{n}}\left( {x_{i}}\right){{\varTheta}_{n}}\left( {x_{j}}\right)}}{{ - {\omega^{2}} + {\omega_{n}^{2}}}}} $$

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1.2 1.2 Rod finite element model

Finite element method is used to model the vibrations of rods with non-uniform cross-section—for example, the tapered part of back mass in Section 2.1. A rod element with cross-section area A, length l, polar moment of inertia per unit length I_p, density ρ, modulus of elasticity E, modulus of rigidity G, and polar moment of area J is considered. Assuming uncoupled axial and torsional deformations, the mass and stiffness matrices of a 2-nod, 4 DOF, rod element with DOF vector $\textbf {u}=\left [x_{1},\theta _{1},x_{2},\theta _{2}\right ]^{T}$, are obtained as follows [26]:

$$ \begin{array}{l} {\textbf{m}} = \frac{{\rho l}}{6}\left[ {\begin{array}{*{20}{c}} {2A}&0&A&0\\ 0&{2{I_{p}}}&0&{{I_{p}}}\\ A&0&{2A}&0\\ 0&{{I_{p}}}&0&{2{I_{p}}} \end{array}} \right]\\ {\textbf{k}} = \frac{1}{l}\left[ {\begin{array}{*{20}{c}} {EA}&0&{ - EA}&0\\ 0&{GJ}&0&{ - GJ}\\ { - EA}&0&{EA}&0\\ 0&{ - GJ}&0&{GJ} \end{array}} \right] \end{array} $$

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The global mass and stiffness matrices of the non-uniform rod are obtained by assembling its element matrices; the desired FRFs are subsequently obtained based on the resulting global mass and stiffness matrices [26].

1.3 1.3 Coupled axial-torsional finite element to model the drill bit

Due to the pre-twisted geometry of the drill bit, its axial and torsional deflections are coupled. This coupling is considered in the presented work by using Rosen’s model [21] of the coupled non-linear deflection of pre-twisted rods. The drill bit is divided into finite rod elements with pre-twisted geometry as shown in Fig. 11. As shown in this figure, when an axial force (f ) is applied to the pre-twisted rod element, it simultaneously deflects in axial direction by Δx₁ and twists through by Δ𝜃₁. Similarly, when the element is subjected to torque t, it deflects axially by Δx₂ and twists through by Δ𝜃₂.

Fig. 11

Deformations of a drill bit element under a an axial force and b a torsional torque

Rosen [21] presented a non-linear model to describe the mechanics of the torsional-axial deflections in pre-twisted rods. Nonetheless, Jin and Koya [27] showed that, for typical deflections during drilling, the values of the non-linear terms in Rosen’s model are less than 1% of the linear terms. Therefore, the effect of non-linearities can be neglected. Based on the uniform axial strain and twist per unit length assumptions in Rosen’s work [21], the following equation is developed to describe the coupling between the axial and torsional loads (f and t) and the resulting deflections (Δx and Δ𝜃) in the pre-twisted rod element shown in Fig. 11:

$$ f = k^{xf}{\varDelta} x + k^{\theta f}{\varDelta} \theta ; t = k^{xt}{\varDelta} x + k^{\theta t}{\varDelta} \theta $$

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where the stiffness coefficients are expressed as follows:

$$ \begin{array}{l} {k^{xf}} = {EA}/{l};{k^{\theta f}} = {ES_{s}}/{l};\\ {k^{xt}} = {ES_{s}}/{l};{k^{\theta t}} = \left( {G{J_{s}} + EK_{s}}\right)/{l} \end{array} $$

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and E, G, A, and l being the modulus of elasticity, shear modulus, cross-section area, and length of the element. The parameters S_s, J_s, K_s, and F_s are section integral parameters [21] and depend on the geometrical dimensions of the cross-section. Various methods are available in the literature to obtain the section integral parameters. For example, a numerical method based on a 2D FE solution was used in [27]. In this work, assuming a standard twist drill cross-section geometry, a curve-fitting method is used to obtain these parameters as a function of drill diameter, d. A pre-twisted rod element is modelled in the commercial finite element software COMSOL Multiphysics, and its stiffness coefficients are obtained by applying axial force and torsional torque to the element and measuring the deformations, as shown in Fig. 11. This process is repeated for rods with various diameters. Subsequently, a cubic polynomial is fitted on the resulting stiffness coefficients, shown below:

$$ \begin{array}{@{}rcl@{}} {k^{xf}}& =& 765.6{d^{3}} + 1.125 \times {10^{5}}{d^{2}} + {1.49910^{5}}d\\ &&- 3.089 \times {10^{5}}\\ {k^{\theta f}} &=& 13.38{d^{3}} + 0.8516{d^{2}} - 0.6822d - 4.749\\ {k^{\theta t}} &= &0.1071{d^{3}} - 1.209{d^{2}} + 5.622d - 8.954\\ {k^{\theta t}} &=& 0.1071{d^{3}} - 1.209{d^{2}} + 5.622d - 8.954 \end{array} $$

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where d is rod diameter in mm.

The material considered for obtaining the cubic polynomials is tungsten carbide with modulus of elasticity of E_WC and for a unit length of the element. Since the stiffness coefficients are proportional to the modulus of elasticity E and inversely proportional to the element length l, the stiffness matrix for a material with the modulus of elasticity of E and the element length of l, is expressed as follows:

$$ \begin{array}{l} {\mathbf{m}} = \left[ {\begin{array}{*{20}{c}} {\frac{m}{3}}&0&{\frac{m}{6}}&0\\ 0&{\frac{I}{3}}&0&{\frac{I}{6}}\\ {\frac{m}{6}}&0&{\frac{m}{3}}&0\\ 0&{\frac{I}{6}}&0&{\frac{I}{3}} \end{array}} \right];\\ {\mathbf{k}} = \frac{{ E}}{{l{E_{WC}}}}\left[ {\begin{array}{*{20}{c}} {{k^{xf}}}&{{k^{\theta f}}}&{ - {k^{xf}}}&{ - {k^{\theta f}}}\\ {{k^{xt}}}&{{k^{\theta t}}}&{ - {k^{xt}}}&{ - {k^{\theta t}}}\\ { - {k^{xf}}}&{ - {k^{\theta f}}}&{{k^{xf}}}&{{k^{\theta f}}}\\ { - {k^{xt}}}&{ - {k^{\theta t}}}&{{k^{xt}}}&{{k^{\theta t}}} \end{array}} \right] \end{array} $$

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where m and I are mass and moment of inertia of the element, respectively.