88 W. Peng et al. Modal parameter indentification: m1, 1 Analytical Model updating: Initial Calibration: Strain-force transfer function: Displacement-force transfer function: Inverse Transfer Function: Strain Response: ε p Excitation Force: F (a) Excitation Force: Fo Strain Response: p Displacement: Wq Excitation Force Perdiction: Fq Displacement-force transfer function: Inverse Transfer Function: (b) Fig. 4 (a): initial calibration steps for analytical model. (b):A schematic diagram of the method for estimating the force acting on the tool tip using the strain response of the cutting tool. and poles can be expressed as: Rϵ pqr, Rϵ∗ pqr =∓j ϕϵ pr(ξp) · ϕqr(ξq) 2ωdrmr (2) λpqr,λ∗pqr =−ζrωr ±jωrp1−ζ 2 r =−ζrωr ±jωdr (3) ωr =(βrL) 2rEI mL3 (4) ϕqr(ξq)=C1 cos(βrξq)+C2 sin(βrξq)+C3 cosh(βrξq)+C4 sinh(βrξq) (5) ϕϵ pr(ξp)=δβ 2 r [C1 cos(βrξp)+C2 sin(βrξp) −C3 cosh(βrξp) −C4 sinh(βrξp)] (6) where ϕϵ pr and ϕqr refer to the individual mode shapes at the response and excitation points. ωdr refers to the damped resonance frequency of the mode r obtained from the poles and damping ratio ζr. ωr refers to the undamped resonance frequency andβr is the roots of the Euler-Bernoulli cantilever beam motion equation [11]. In equation 2, modal mass mr of the beam is a measure of amount of mass moving in the given mode r. The modal mass of the beam can be calculated by equation 7, as presented in [12]. mr =Z L 0 mρ · |ϕr(ξ)| 2dξ (7) The strain-force transfer function created in equation 1 need to be calibrated in order to accurately capture the boundary conditions in a real setup. The boundary conditions are typically more flexible than the idealized fixed conditions assumed in the analytical model. As a result, the resonance frequencies and peak values of the transfer function calculated from the model do not match those of the measured transfer function. To improve accuracy, the model is adapted to the cutting tool configuration based on the impulse strain response recorded from the strain sensor. This is achieved by applying a Fourier transform to this strain data allows the first resonance frequency ωm1 of the cutting tool to be obtained. To determine the damping ratioζr of the cutting tool, logarithmic decrement approach is utilized [13, 14]. Figure 5 illustrates that logarithmic decrement relates the damping ratio to the strain response peaks through equation 8. ζr(δN,N)= δN/(2πN) p1+(δN/(2πN)) 2 (8) where the logarithmic decrement is defined as: δN = ln µϵ1 µϵN (9)
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