Calculation of Parameters of Pneumatic Impact Units and the Amplitude of Compressive Stresses in Drill Rods under Longitudinal Impact Using Nomograms
Abstract Generated abstract
This paper presents a nomographic method for selecting design parameters of pneumatic impact units and estimating compressive stress amplitudes in drill rods during longitudinal impact. Using established relations for impact energy, striker mass and velocity, barrel cavity dimensions, and stress wave formation, the authors construct aligned-point nomograms for determining cylinder diameter, striker mass, and maximum compressive stress for cylindrical and two-stage strikers. The approach incorporates assumed operating constants for air pressure, energy losses, sound speed, and elastic modulus, and uses preferred ranges of impact energy, striker velocity, and stroke. The resulting nomograms are intended to enable rapid preliminary design of new pneumatic impact units, including units for fatigue testing of rods, while accounting for stress amplitudes generated in the rod.
Full Text
CALCULATION OF PARAMETERS OF PNEUMATIC IMPACT UNITS AND THE AMPLITUDE OF COMPRESSIVE STRESSES IN DRILL RODS UNDER LONGITUDINAL IMPACT USING NOMOGRAMS
V. F. GORBUNOV, L. A. SARUEV, A. S. SERDECHNY, L. A. PARSHIKOVA
Works [1–2] are devoted to the calculation, using nomograms, of the principal parameters of pneumatic drilling, scaling, and riveting hammers.
In the present work an attempt is made to select the parameters of new designs of pneumatic impact units, as well as designs of impact units intended for testing rods for fatigue life, and to calculate the amplitude of compressive stresses in drill rods using nomograms.
It is proposed to construct the calculation of the parameters on the basis of the known relations [3–5]:
\[ A_y = K_1 P_0 F_y S_k = \frac{m_b V_b^2}{2}; \tag{1} \]
\[ \sigma_{\max}=\frac{1}{2}\left[1+g_1+(1-g_1^2)g_2\right]\frac{V_b}{a_{\text{rod}}}E_{\text{rod}}; \tag{2} \]
\[ \sigma_{\max}=\frac{1}{2}\left[1+g_1\right]\frac{V_b}{a_{\text{rod}}}E_{\text{rod}}; \tag{3} \]
where \(P_0\) is the pressure of compressed air in the network, kg/cm\(^2\);
\(F_y\) is the cross section of the barrel cavity of the impact unit, cm\(^2\);
\(S_k\) is the stroke of the striker, representing the difference between the length of the cavity (minus the length of the tool shank and the length of the striker), m;
\(m_b\) is the mass of the striker, kg·s\(^2\)/m;
\(K_1\) is the energy-loss coefficient, taking into account the degree of filling with compressed air, mechanical losses, and also the change in temperature of the compressed air under optimal operating modes of the pneumatic impact unit; for approximate calculations it may be taken equal to 0.5–0.55 [3];
\(\sigma_{\max}\) is the maximum amplitude of stress in the rod, kgf/cm\(^2\);
\(V_b\) is the pre-impact velocity of the striker, m/s;
\(a_{\text{rod}}\) is the speed of sound in the rod material, m/s;
\(E_{\text{rod}}\) is the modulus of elasticity, kgf/cm\(^2\).
\[ q_1=\frac{F_{1b}-F_{\text{rod}}}{F_{1b}+F_{\text{rod}}}, \tag{2,a} \]
where \(F_{1b}\) is the cross section of the first stage of the striker, \(\text{cm}^2\);
\(F_{\text{rod}}\) is the cross section of the rod, \(\text{cm}^2\).
\[ q_2=\frac{F_{2b}-F_{1b}}{F_{2b}+F_{1b}}, \tag{2,b} \]
\(F_{2b}\) is the cross section of the second stage of the striker.
In constructing the nomograms it was assumed that:
\[ P_0=5\ \text{atm};\quad K=0.5;\quad a_{\text{rod}}=5200\ \text{m/sec};\quad E_{\text{rod}}=2.1\cdot 10^6\ \text{kg}/\text{cm}^2. \]
The limits of variation of the impact energy (\(A_y\) from 4 to 31.5 kgf·m), as well as of the striker velocity (\(V_b\) from 5 to 20 m/sec), were selected according to the series of preferred numbers [6]. The intervals of values of the constructive stroke of the striker were chosen on the basis of existing impact units in drilling machines [7].
For the formulas presented, which are comparatively simple, it is convenient to use nomograms of aligned points with logarithmic scales [8]. The nomogram (Fig. 1) is constructed according to formula (1). For
Fig. 1. Nomogram for determining the diameter of the barrel cavity of the impact unit
ease of use of the nomogram, in formula (1) we replace the cross section of the barrel cavity of the impact unit by the diameter and, after transformation, obtain
\[ D_c=\sqrt{\frac{0.51A_y}{S_k}}. \tag{4} \]
Substituting the adopted values of impact energy and constructive stroke, we obtain the calculated values of the cylinder diameter of the impact units. The second nomogram (Fig. 2) is also constructed according to formula [1], represented in the form
\[ m_b=\frac{2A_y}{V_b^2}. \tag{5} \]
From the specified parameters of impact energy and pre-impact velocity we find the mass of the striker.
The maximum amplitude of the compressive stress in the rod under a longitudinal impact for strikers of two-stage shape is determined from the nomogram (Fig. 3), constructed according to the formula
\[ \sigma_{\max}=\bar{\varepsilon}_{1}\frac{V_{\mathrm{b}}}{a_{\mathrm{rod}}}E_{\mathrm{rod}}, \tag{6} \]
where \(\bar{\varepsilon}_{1}\) is the maximum longitudinal strain for strikers of two-stage shape, equal to
\[ \bar{\varepsilon}_{1}=\frac{1}{2}\left[1+g_{1}+(1-g_{1}^{2})g_{2}\right]. \tag{7} \]
Fig. 2. Nomogram for determining the striker mass of the impact unit
In formulas (2, a and 2, b), for ease of calculation using the adopted parameters, we replace the cross section \(F\) by the diameter:
\[ g_{1}=\frac{d_{1\mathrm{b}}^{2}-d_{\mathrm{rod}}^{2}}{d_{1\mathrm{b}}^{2}+d_{\mathrm{rod}}^{2}}, \tag{7,a} \]
\[ g_{2}=\frac{d_{2\mathrm{b}}^{2}-d_{1\mathrm{b}}^{2}}{d_{2\mathrm{b}}^{2}+d_{1\mathrm{b}}^{2}}. \tag{7,b} \]
The maximum amplitude of the compressive stress in the rod under a longitudinal impact for strikers of cylindrical shape is determined from the nomogram (Fig. 4), constructed according to the formula
\[ \sigma_{\max}=\bar{\varepsilon}_{1}\frac{V_{\mathrm{b}}}{a_{\mathrm{rod}}}, \tag{8} \]
\[ \bar{\varepsilon}_{1}=\frac{1}{2}(1+g_{1}). \tag{9} \]
Fig. 3. Nomogram for determining the maximum amplitude of compressive stresses in the rod under longitudinal impact for two-stage strikers
Fig. 4. Nomogram for determining the maximum amplitude of compressive stress in the rod under longitudinal impact for cylindrical strikers
It should be noted that in the nomograms (Fig. 3 and Fig. 4) a scale of bore diameters has been introduced to determine the maximum longitudinal deformation. Knowing the longitudinal deformation from the bore diameter and the striker velocity, it is easy to find the maximum stress amplitude in the rod.
Thus, the constructed nomograms make it possible to quickly select the parameters of new impact units, taking into account the amplitudes of compressive stresses in the rod under longitudinal impact.
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