On the Problem of Energy Transfer During Impact in Drilling and Chipping Hammers
Abstract Generated abstract
This paper examines how kinetic energy is transferred during impact between the striker and drill rod in drilling and demolition hammers. It compares two wave-theory treatments: one accounting for wave propagation in both striker and rod, following Belyaev, and another deriving a transfer coefficient while treating the striker as a rigid body and considering waves only in the drill rod. Calculated coefficients from both formulas are tested against experiments using a shot-type apparatus and rebound-height measurements. The results indicate close agreement between theory and experiment over much of the parameter range, suggesting that modeling the striker as rigid is often adequate for estimating the impact-energy transfer coefficient, while more detailed wave analysis remains useful for studying stress-wave propagation and energy losses in long drill rods.
Full Text
ON THE QUESTION OF ENERGY TRANSFER DURING IMPACT IN DRILLING AND DEMOLITION HAMMERS
A. G. TSUKANOV
(Presented by the Department of Theoretical Mechanics)
The article considers mathematical relationships for the coefficient of transfer of kinetic energy when the striker and the drill rod collide. A comparison is made of the calculated values of this coefficient under two assumptions adopted in using the wave theory of impact.
First, wave processes are considered both in the drill rod and in the striker. The mathematical formulation of the problem and the relationship for the coefficient of energy transfer in this case were obtained by Yu. V. Belyaev [1]. In the second case, the propagation of waves only in the drill rod is taken into account, assuming that the striker is an absolutely rigid body. The calculated values of the coefficient are compared with experimental data. Yu. V. Belyaev, generalizing the known solution on the collision of two elastic rods to the case when they have different diameters, considers the wave equation with the corresponding initial and boundary conditions:
\[ \frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2}, \tag{1} \]
where \(u\) is the displacement of some transverse section of the rod;
\(x\) is the coordinate of the section under consideration;
\(a\) is the velocity of wave propagation in the material of the rods.
Having determined the relative deformation in the contact section of the rods, the contact time, and the impact impulse, the velocity of the center of inertia of the second rod (the drill) and the kinetic energy corresponding to this velocity are determined by the theorem of momentum. Then, for the coefficient of transfer of kinetic energy in the collision of two elastic rods (striker–drill), Yu. V. Belyaev obtains the formula
\[ \eta=\frac{m_1}{m_2}\left[1-f^p+\alpha(1-f)f^p\right]^2 . \tag{2} \]
Here
\[ f=\frac{a_2-\kappa a_1}{a_2+\kappa a_1}, \qquad \kappa=\frac{E_2F_2}{E_1F_1}, \]
\(a_1, a_2\) are, respectively, the velocities of propagation of sound waves in the material of the rods;
\(p\) is an integer in the expression
\[ \frac{T_2}{T_1}=p+a \qquad (0<a<1), \]
\(T_1, T_2\) are, respectively, the times of motion of the wave along the rods during impact.
We now obtain a dependence for the coefficient of transfer of kinetic energy in the collision of an elastic rod (drill) and a striker, under the assumption that the latter is an absolutely rigid body. The problem reduces to solving the wave equation
\[ \frac{\partial^2 u}{\partial t^2}=a^2\frac{\partial^2 u}{\partial x^2} \]
with the initial conditions (Fig. 1):
\[ \begin{gathered} \text{for } t=0 \quad u=0,\\ \frac{\partial u}{\partial t}=0 \qquad (0<x<l),\\ \frac{\partial u}{\partial t}=-v_0 \qquad (x=l); \end{gathered} \]
with the boundary conditions:
\[ \begin{gathered} \text{for } x=0 \quad \frac{\partial u}{\partial x}=0,\\ x=l \quad bl\frac{\partial^2 u}{\partial t^2}=-a^2\frac{\partial u}{\partial x}, \end{gathered} \]
where \(b\) is the ratio of the weight of the striker \(p_1\) to the weight of the rod \(P_2=\gamma Fl\). The solution of the wave equation under the indicated initial and boundary conditions can be found in [2]. We shall use this solution and obtain a new dependence for the coefficient of transfer of impact energy.
Fig. 1.
The velocity of the striker, considered as a rigid body, at the end of the collision phase is represented by the following expression:
\[ v_1=\left(\frac{\partial u}{\partial t}\right)_{x=l}=-v_0 e^{-\frac{2}{b}}. \]
To determine the velocity of the rod (drill) after the collision, we use the momentum theorem. Taking into account that the impulse of the external forces during the collision in the system (striker–drill) is equal to zero, we obtain the following expression for the velocity of the center of mass of the drill at the end of the impact phase:
\[ v_2=-bv_0\left(1-e^{-\frac{2}{b}}\right). \]
Now we are able to determine the kinetic energy of the drill, received from the striker upon collision,
\[ A_2=\frac{m_2 v_2^2}{2}. \]
Knowing the kinetic energy of the striker at the beginning of the impact phase,
\[ A_1=\frac{m_1 v_1^2}{2}, \]
we determine the coefficient of transfer of impact energy
\[ \eta=\frac{m_1}{m_2}\left(1-e^{-\frac{2m_2}{m_1}}\right)^2 . \tag{3} \]
Thus, the coefficient of transfer of kinetic energy during the collision of the striker and the drill rod is represented by two formulas (2) and (3), whose derivation is based on the premises of the theory of wave propagation. The calculated data obtained from the indicated formulas were compared with experimental data.
In conducting the experiment, the “shot” method was used.
Figure 2 shows a diagram of the experimental setup. It consists of a mechanism for accelerating the striker and a guiding device. The mechanism for accelerating the striker consists of a cylindrical housing 1, rigidly fixed on a massive base of a copra; a steel spring 2; two cylindrical stops 3, 4, intended for compressing the spring and the subsequent ejection of striker 5.
The striker had a weight of 3 kg, diameter \(d_1=38\) mm, was made of 40KhN steel and hardened to \(H=45\text{–}50\). The end surface of the striker was flat. As rods 6 receiving the impact, ordinary drill rods of diameter \(d_2=25\) mm with hardened shanks were used. The efficiency, or coefficient of energy transfer during collision of the striker and the drill, was determined from the relation
\[ \eta=\frac{A_2}{A_1}, \]
where \(A_1\) is the kinetic energy of the striker at the beginning of the collision phase;
\(A_2\) is the kinetic energy of the drill rod received from the striker at the end of the impact event.
Fig. 2. Diagram of the experimental setup.
To determine \(A_1\), free flight of the striker was carried out without colli-
with the drill rod, and the height of this rebound was recorded. In determining \(A_2\), the height of the drill-rod rebound was recorded after it had received a blow from striker 5.
From the rebound height of the drill rod, the kinetic energy corresponding to the displacement of the drill rod at the velocity of the center of mass was determined, as was assumed in the theoretical solution of the problem. It should be noted that, although the coefficient \(\eta\) does not take into account the wave component of the energy, it is an important characteristic of the energetics of impact. The impact energy corresponding to the motion of bodies at the velocity of the center of mass is of predominant importance in many problems.
Figure 3 presents curves characterizing the variation of the coefficient \(\eta\).
Analysis of the calculated values of the coefficient \(\eta\), obtained from formulas (2) and (3), shows that in the region \(1<\kappa<2\) these values practically coincide, while for \(\kappa>2\) they are of the same order of accuracy.
Fig. 3. Experimental and theoretical values of the impact-energy transfer coefficient.
\(\triangle\) — experimental curve;
\(\square\) — curve corresponding to formula (2);
\(\circ\) — curve corresponding to formula (3).
The experimental values of the coefficient \(\eta\) agree sufficiently well with the calculated data, and for \(\kappa>2\) the curves corresponding to formulas (2) and (3) are identical to the experimental curve. Both the calculated and the experimental curves have a maximum different from unity. At the same time, the maximum of the experimental curve and of the curve corresponding to formula (3) is shifted toward values \(\kappa<1\). The difference in the values of \(\eta\) occurs in the region \(\kappa<1\), where obtaining sufficiently accurate values of \(\eta\) experimentally with flat end faces is practically impossible, and the experimental values of the coefficient are underestimated. The nature of the variation of the experimental curve gives grounds to suppose that, in the region \(\kappa<1\), other theoretical assumptions, close to the classical theory of impact, should be adopted for obtaining the coefficient \(\eta\). Thus, consideration of wave processes in the striker on the basis of the assumptions of the one-dimensional theory of wave propagation does not introduce substantial refinements into the determination of the impact-energy transfer coefficient. Treating the striker as a rigid body, as is done in a number of studies [3; 4], is entirely possible; in this case, greater attention should be paid to ensuring the most favorable conditions of impact contact between the striker and the drill shank. The possibilities of the wave theory of impact are very promising in an in-depth study of the propagation
stress waves and the analysis of energy losses in drill rods, whose length may be quite large.
References
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Yu. V. Belyaev. On estimating the degree of utilization of impact energy in percussion machines. VNIIsstroydormash, issue X, 1955.
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N. A. Kilchevsky. Theory of Collisions of Solid Bodies. Gostekhizdat, 1949.
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V. M. Mostkov. Fundamentals of the Theory of Pneumatic Drilling. Ugletekhizdat, 1952.
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Mechanisms and Machines of Percussive, Rotary, and Rotary-Percussive Action. Interuniversity collection of papers, issue 1, Novosibirsk, 1963.