5. Mechanical Design and Modelling
5.12 Mechanical Analysis of Drilling Subassembly
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5.12 Mechanical Analysis of Drilling Subassembly
In order that a static and dynamic analysis could be performed on the system, the vertical and horizontal stiffness‟s of individual modules have been modelled as springs in series, as illustrated in Figure 5.41.a. For simplicity the vertical and horizontal subsystems have been decoupled. This simplification will provide a reasonable approximation of the mechanical vibration except at frequencies close to resonance in each mode. The analysis considers the effect of a harmonic thrust force on the drilling head when it is vertically orientated and then horizontally orientated.
These scenarios are illustrated in Figures 5.41.b and 5.41.c respectively.
Table 5.17: Module Stiffness’s (Refer to Appendix C.5)
Range Extension
Arm (1) Cutting Head Rotary
Module (2) Column Module
KV (N.m-1) 1.592×106 95.908×106 1.949×10(3) 6
KH (N.m-1) 1.592×106 153.222×106 2.597×104
Table 5.17 contains the calculated stiffness of the related modules; these calculations are presented Appendix C.5. The stiffness of the column in particular, strictly corresponds to the column slide being in a position of 600 mm above its base. For springs in series the spring rates combine reciprocally; the total vertical and horizontal stiffness of the system are therefore:
𝐾𝑉 𝑡𝑜𝑡𝑎𝑙 = 1
1.592 × 106+ 1
95.908 × 106+ 1
1.949 × 106
−1
= 8.683 × 105 𝑁. 𝑚−1
𝐾𝐻 𝑡𝑜𝑡𝑎𝑙 = 1
1.592 × 106+ 1
153.222 × 106+ 1
2.594 × 104
−1
= 2.552 × 104 𝑁. 𝑚−1
Static Analysis
Figure 5.42: Graph of Static Deflection vs Drill Thrust Force for Vertical Drilling
Equation 5.55 presents Hooke‟s Law which relates the deflection of a spring to its stiffness and the force exerted on it. By manipulating Hooke‟s Law the deflection of the drilling head under the action of a static force may be obtained. The deflection of the drilling head is presented in Figures 5.42 and 5.43, for static vertical and horizontal forces.
𝛿 = 𝐹
𝐾 (5.55) 0.00
0.01 0.02 0.03 0.04 0.05 0.06 0.07
0 10 20 30 40 50
Static Deflection (mm)
Drlling Thrust Force (N)
Static Deflection vs Drill Thrust Force - Vertical Orientation
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5.12 Mechanical Analysis of Drilling Subassembly
Figure 5.43: Graph of Static Deflection vs Drill Thrust Force for Horizontal Drilling
A vertical thrust force of 50 N leads to a total deflection of 0.058 mm in the vertical plane. The drilling subassembly displayed less rigidity in the horizontal plane and a total deflection of 1.959 mm is expected for a drill thrust force of 50 N. The static deflections due to the thrust force largely affect the accuracy of the depth of holes drilled by the MRM. The reduced rigidity is due to 15 mm sliver steel guide rods that have been used as supporting members in the MRM column module, as opposed a more rigid yet costly dove tail slide system.
Dynamic Analysis at Multiple Frequencies a. b.
Figure 5.44: Spring-Mass-Damper Systems
a. Spring-Mass-Damper model of drilling assembly – vertical harmonic force b. Spring-Mass-Damper model of drilling assembly – horizontal harmonic force
The performance of the drilling subassembly under dynamic loading conditions was investigated.
The system was modelled as a Spring-Mass-Damper system, and separate investigations were conducted for vertical and horizontal harmonic loading; this is illustrated in Figures 5.44.a and 5.44.b respectively.
0.000 0.500 1.000 1.500 2.000 2.500
0 10 20 30 40 50 60
Static Deflection (mm)
Drilling Thrust Force (N)
Static Deflection vs Drill Thrust Force - Vertical Orientation
M
M
KV CV
KH
CH
FV(t)
FV(t)
xV+
xH+
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5.12 Mechanical Analysis of Drilling Subassembly
For a damped system under the action of a harmonic force, the motion is described by equation 5.56; where m is the mass of the drill module (kg), c is the damping coefficient (N.s/m), k is the stiffness of the system (N/m), F0 is the magnitude of the harmonic force (N) and ω is the frequency of the harmonic force (rad/s).
𝑚𝑥 + 𝑐𝑥 + 𝑘𝑥 = 𝐹0𝑐𝑜𝑠𝜔𝑡 (5.56) The complete solution to this differential equation for an underdamped system is given by equation 5.57, where X is the amplitude of the response (m), φ is the phase angle (deg), δ is the damping factor and ωd is the frequency of damped vibrations (rad/s).
𝑥 𝑡 = 𝑋0𝑒−𝜁𝜔𝑛𝑡𝑐𝑜𝑠 𝜔𝑑𝑡 − 𝜙0 + 𝑋𝑐𝑜𝑠 𝜔𝑡 − 𝜙 (5.57)
𝜔𝑑 = 1 − 𝜁2𝜔𝑛 (5.58)
𝜔𝑛 = 𝑘
𝑚 (5.59)
𝜁 = 𝐶
𝐶𝑐= 𝐶
2 𝑘𝑚 (5.60) 𝑟 = 𝜔
𝜔𝑛 (5.61) The relative performance of a system under a dynamic load with regard to its static loading characteristics is most effectively analysed by obtaining a ratio of the amplitude of the dynamic response to the amplitude of the static response of the system; under the action of force F0 at different frequencies. The amplitude ratio M is calculated by equation 5.62; where r is the frequency ratio.
𝑀 = 𝑋
𝛿𝑠𝑡 = 1
1 − 𝜔 𝜔𝑛
2 2
+ 2𝜁 𝜔 𝜔𝑛
2 0.5= 1
1 − 𝑟2 2+ 2𝜁𝑟 2 (5.62)
𝛿𝑠𝑡 =𝐹𝑜
𝑘 (5.63) The phase angle between the response and the excitation is calculated by equation 5.64.
𝜙 = 𝑡𝑎𝑛−1 2𝜁 𝜔 𝜔𝑛 1 − 𝜔
𝜔𝑛
2 = 𝑡𝑎𝑛−1 2𝜁𝑟
1 − 𝑟2 (5.64)
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5.12 Mechanical Analysis of Drilling Subassembly
Table 5.18: Physical Characteristics of Drilling Assembly in Horizontal and Vertical Directions
K (N/m) C (N.s/m) m (kg) ωn (rad/s) ζ ωd (rad/s)
Vertical 8.683×105 5 2.1 643.02 0.0019 643.02
Horizontal 2.552×104 5 2.1 110.24 0.0108 110.23
Table 5.18 presents physical characteristics of the drilling subassembly, where the stiffness‟s correspond to the column slide being in a position of 600 mm above its base. Information on the damping characteristics of drill subassembly was not available. The damping coefficients of the system have therefore been set at the conservatively small but finite value of 5 N.s.m-1 in both vertical and horizontal cases.
Figure 5.45: Graph of Amplitude Ratio vs Excitation Frequency – Vertical Excitation Force
Figure 5.46: Graph of Amplitude Ratio vs Excitation Frequency – Horizontal Excitation Force
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5.12 Mechanical Analysis of Drilling Subassembly
The physical characteristics of the system were used in conjunction with equation 5.62 to generate plots of the amplitude ratio at multiple frequencies, for both vertical and horizontal planes (see Figures 5.45 and 5.46). Equation 5.64 was used to generate plots of phase angle versus frequency for the subassembly (see Figures 5.47 and 5.48). The natural frequency of the drilling subassembly for a vertical excitation force is 102.23 Hz, while the natural frequency in the horizontal direction is 17.55 Hz. Excitation at the natural frequency is accompanied by a peak in amplitude ratio as seen in Figures 5.45 and 5.46. If the excitation were to occur twice per drill revolution, the respective resonant drilling speeds would be 3067 rev/min and 527 rev/min.
Figure 5.47: Graph of Phase Angle vs Excitation Frequency – Vertical Excitation Force
Figure 5.48: Graph of Phase Angle vs Excitation Frequency – Horizontal Excitation Force
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5.12 Mechanical Analysis of Drilling Subassembly
Figures 5.47 and 5.48 present the phase angle of the system response at various excitation frequencies. In both vertical and horizontal cases the excitation and the system response are approximately in phase for frequency ratios less than one. For frequencies greater than the natural frequencies of the system the response leads the excitation by approximately 180o. These characteristics are due to the (approximated) small damping capacity of the system.
Simulation Performed at 20 Hz
A simulation was performed to investigate the performance of the drilling subassembly under the action of a 50 N force at 20 Hz. The solving of equation 5.57 for these conditions is presented in Appendix C.6. Table 5.19 presents the results of the solution.
Table 5.19: Vibration Characteristics of Drilling Assembly in Horizontal and Vertical Directions
Xo (m) φo (deg) X (m) φ (deg) ωn (rad/s) ζ ω (rad/s) Vertical 8.360 ×10-5 0.1141 5.987×10-5 0.0442 643.02 0.0019 125.66 Horizontal 6.474×10-3 175.29 6.473×10-3 175.33 110.24 0.0108 125.66 Equation 5.57 was simulated for five seconds at a refresh rate of 5 ms; the results of the simulation are presented in Figures 5.49 and 5.50. The graphs display the displacement of the MRM cutting head with regard to time under the action of the excitation force. Theoretical displacements in excess of 10 mm were demonstrated for horizontal excitation (not practically verified). This is due to the excitation frequency being in close proximity to the natural frequency of the system. The dynamic deflections due to the harmonic excitation are expected to adversely affect the accuracy of the depth of holes drilled by the MRM.
Figure 5.49: Graph of Displacement vs Time – Vertical Excitation Force (50 N)
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