On the Modeling of Impacts and Contacts in Impact-Action Machines
Abstract Generated abstract
This paper examines how elastic impacts and contacts should be represented in electronic analog models of impact-action machines, with emphasis on hand-held pneumatic hammers. It reviews existing approaches based on either instantaneous velocity changes or explicit modeling of collision duration, noting limitations in complexity, applicability, and treatment of restitution coefficients. The authors describe a simpler analog circuit for modeling elastic impact with an adjustable coefficient of restitution from zero to unity, extend the idea to central impact of two moving masses, and discuss required experimental determination of rebound parameters. They propose practical modeling choices for striker-tool, tool-material, housing-tool, and emergency housing impacts, reporting that their model was used for pneumatic hammer studies with results in good agreement with experimental data.
Full Text
ON THE PROBLEM OF MODELING IMPACTS AND CONTACTS IN IMPACT-ACTION MACHINES
V. F. GORBUNOV, V. M. RAZIN, A. V. TRIKHANOV
(Presented by the seminar of the Faculty of Automation and Computer Engineering)
In studying machines of impact and vibratory action, it becomes necessary to take elastic impacts and contacts into account.
Recently, for such studies, the means of computer technology have begun to be used—namely, electronic modeling installations [1, 2, 3, 6, 9]. In connection with this, a number of electronic models constructed with allowance for impacts have been proposed. We have applied the method of electronic modeling to the study of processes of isolation and damping of vibration in hand-held pneumatic impact machines. During their operation the following impacts and contacts occur.
The striker of a pneumatic hammer periodically delivers blows to the shank of the tool. The energy transmitted by the striker to the tool is expended on the destruction of the material being worked, also by impact. Before being struck by the striker, the tool—depending on the force with which the worker presses on the hammer body—may either be in contact with the material being worked or at some distance from it. The hammer body receives return impacts from the tool as the latter rebounds from the material being worked, and it also comes to rest on the collar of the tool during the reverse stroke of the striker. When the body is displaced significantly away from the face, the striker delivers blows not to the shank of the tool, but to the front wall of the body.
Thus, during operation of the hammer, the striker delivers blows to the tool; the body and the tool may either be in contact or collide with the striker and with one another. In addition, under certain conditions other impacts also occur in the hammer—in particular, “hard” impacts of the striker against the rear wall of the hammer body, impacts on the body caused by oscillations in the tool, and so on. However, these impacts are not considered by us at present because of their insignificant influence on the operation of the hammer.
In the electronic modeling of the vibration of pneumatic hammers, it is necessary to realize the impacts and contacts noted above. Let us briefly analyze the works known to us in this direction.
All the works proceed from the fact that different interpretations of the impact process are possible:
a) the colliding bodies may be regarded as perfectly rigid, and the impact process itself as instantaneous (the classical theory of collision of rigid bodies);
b) in analyzing the process, the time of interaction of the colliding bodies can be taken into account.
The different interpretations of the impact process are determined mainly by the data of direct experiment, which, in the general case, should precede the formulation of problems for theoretical investigation.
The classical theory of collision of rigid bodies regards the colliding bodies as absolutely rigid, bearing in mind that the impact occurs practically instantaneously and is characterized by the coefficient of restitution of velocity.
However, experimental data [1] show that in a number of cases the duration of impact is commensurate with the time of motion of the system in the interval between impacts. This emphasizes the need to obtain data characterizing the processes that occur in the system directly during the period of collision. In our opinion, this is true for cases when the period between impacts is so small that it proves to be commensurate with the duration of impact. Depending on the interpretation of impacts, electronic models of two types have been described in the literature. In models of one type an instantaneous change of velocity is modeled [2, 4, 7, 8, 9, 10], while in others the impact process itself is modeled [1, 5].
In [2] it is proposed to model the rebound of the striker after its impact on the end of the armature in a solenoid hammer. This is carried out in the following way. In the feedback circuit of an integrating amplifier, at the output of which there is a voltage corresponding to the velocity of motion of the striker, two capacitors, \(C_1\) and \(C_2\), are connected in parallel (Fig. 1, a). When the displacement coordinate of the striker reaches its limiting values, a relay interchanges the contacts of capacitor \(C_2\), thereby changing the voltage at the amplifier output that models the velocity of displacement of the striker.
As calculation shows, the ratio of the velocities before and after switching of the rebound coefficient is equal to
\[ R=\frac{C_1-C_2}{C_1+C_2}. \]
Simultaneously with the switching of the capacitor contacts, the voltages corresponding to the attained limiting values of the displacement coordinate are changed.
As soon as the coordinate reaches the second limiting value, the switching is repeated, and so on.
In our opinion, the proposed method of reproducing the rebounds of the striker in a solenoid hammer has certain shortcomings. First of all, it provides a limited set of values of the rebound coefficient. For each new value a new capacitance is required.
In addition, this method requires that impacts be repeated when the displacement coordinate of the striker reaches its limiting values. Otherwise, all subsequent rebounds will be reproduced incorrectly. It follows from this that the proposed method has a comparatively narrow field of application.
In [3] a model is described for an elastic impact with a coefficient of restitution of velocity equal to unity. In this case the elastic stop against which the mass strikes is represented in the form of a spring with a stiffness coefficient corresponding to the modulus of elasticity of the material being worked.
However, the coefficient of restitution of velocity \(R\) in real machines may vary within the limits from zero to unity.
In [7] a model of a vibration impact device is proposed for the case when the coefficient of restitution of velocity is equal to zero (the case of inelastic ...).
impact). The modeling idea is based on the fact that, when the impact condition is satisfied, a normally open relay is actuated and instantaneously discharges the capacitor in the negative-feedback circuit of the integrator. The output voltage of the integrator corresponds to the velocity of the striker.
In [8] the following procedure is described for realizing impact in a vibratory hammer. The machine operates in a “stop” mode; that is, integration proceeds until the difference between the coordinates of the impact part of the vibratory hammer and the pile is positive. When the coordinates become equal, the stop relay is actuated and integration ceases. The values of displacements and velocities at the instant of stopping are the sought values for the instant before impact. Then, by means of simple calculations, the velocities of the impact part and of the pile after impact are found; these are set on the model as initial conditions for studying the next cycle of motion.
The velocities are recalculated on the basis of the coefficient of restitution of velocity in the collision of two bodies.
In [9] two methods are proposed for modeling an impact with a coefficient of restitution of velocity \(R\), different from unity. The essential feature of these methods is the use of a tracking system of the “inertial element” type for continuously duplicating, on an additional integrator, the velocity of the vibratory striker according to its value on the main integrator. With each impact the roles of the main and additional integrators are interchanged. This is done by means of sign blocks. After each impact, on the next main integrator the post-impact velocity is realized as the initial condition.
However, these models for \(R < 1\) are very complex.
We have developed a simple model of an elastic impact, constructed under the assumption that an instantaneous change of velocity occurs during impact [4]. It is assumed that some moving mass strikes an immovable body. In this case the relation between the velocities before and after impact may be represented as follows:
\[ \dot{x} = -R\dot{x}_{0}, \]
where \(\dot{x}_{0}\) is the velocity before impact;
\(\dot{x}\) is the velocity after impact;
\(R\) is the coefficient of restitution of velocity.
The model makes it possible to assign any values of \(R\) in the range from zero to unity.
The block diagram of the model (Fig. 1, b) includes: operational amplifier \(1\), operating in the integration mode; a block of constant coefficients \(R\); operational amplifier \(2\), operating in the inversion mode, to the output of which is connected a capacitor \(C_{2}\), equal to the capacitor \(C_{1}\) in the feedback circuit of the integrator; and a switching device. Amplifier \(1\) integrates the second derivative with respect to time of the displacement of the body producing the impact. At the output of the integrator the first derivative is obtained, that is, the velocity. Through the block of constant coefficients, on which the coefficient of restitution of velocity is set, the voltage is fed to the input of the inverting amplifier \(2\). At its output, consequently, the velocity is continuously duplicated in magnitude as at the output of the integrator.
At the instant of impact, the relay of the switching device interchanges capacitors \(C_{1}\) and \(C_{2}\) (Fig. 1, b). After each impact, at the output of integrator \(1\), the post-impact velocity \(-R\dot{x}_{0}\) is realized as the initial condition.
To weaken the effect of nonsimultaneous operation of relay contacts on the setting of the coefficient \(R\), a capacitor is connected to the output of amplifier 2 in series with a resistor \(r\).
The moment of impact is determined by a switching device [10]. To determine the moment of impact, the condition is introduced that the coordinate of the moving body reaches the value corresponding to the position of the stationary body.
Comparison of the described impact-modeling circuit with similar ones [8] shows that, when it is used, savings are achieved in operational amplifiers (half as many amplifiers are required) and in the number of elements included in the commonly used signature blocks.
Fig. 1. Structural diagrams of impact models:
a) with a set of capacitors in the feedback of integrator 2; b) with an additional block of constant coefficients; c) the same for two free masses
The proposed model has certain shortcomings. First of all, it is not universal. Furthermore, it requires minor changes in the operational-amplifier circuit that make it possible, first, to switch the capacitance and, second, to increase the accuracy of reproducing the coefficient of restitution of velocity.
The point is that, when the capacitance is switched in a standard dc amplifier of the UPT-4 type, a decrease in the value of \(R\) by 5–10% compared with the specified value is observed. This change is explained by the presence of local negative feedback in the output stage of the UPT-4, which transmits voltage changes at the anode of the output-stage tube (when the capacitances are switched) to the grid of the output tube. This leads to a considerable recharge of the capacitance newly connected into the feedback circuit of the integrator. We moved the local negative feedback from the output stage to the intermediate stage of the UPT-4.
Studies of the impact model with changes in the UPT circuit confirmed sufficient relative accuracy (1–2%) in obtaining the coefficient \(R\).
In view of the features indicated above, it is advisable to use the model described when employing analog machines that do not have standard signature blocks (for example, IPT-5, MPT-9, etc.).
The impact model we have proposed was used in studying pneumatic hammers on an electronic analog modeling installation of the IIPT-5 type. The results agree well with experimental data obtained in studies of real hammers.
An interesting method for modeling the rebound of the striker from a stationary tool is described in [9]. It has something in common with the methods proposed by Bebikhov [8]. The circuit for modeling the rebound is shown in Fig. 2. It operates as follows.
When the electronic model is started, integration of the second derivative of the striker displacement, $\ddot{x}$, is performed by operational amplifier 1. In this case amplifiers 2 and 3 operate alternately in the mode of scale amplification,
Fig. 2. Impact model with two units operating alternately in the mode of scale amplification
charging capacitor $C_2$ or $C_3$ to a voltage corresponding to the striker velocity $\dot{x}$. The alternating operation of amplifiers 1 and 2 is caused by the jump-like change in velocity at the instant of impact. At this instant (it is determined by means of a comparison circuit and a step finder), relay $P_2$ is switched on, which performs switching in the feedback of amplifier 2.
Amplifier 1 is disconnected, and amplifier 2 begins to operate in the integration mode throughout the entire subsequent cycle, having as its initial condition the value of the velocity $\dot{x}$. At the instant of the next impact, relay $P_1$ is switched on and relay $P_2$ is switched off. Owing to this, amplifier 2 returns to the scale-amplifier mode, and amplifier 3 is switched on in the integrator mode with an initial condition proportional to the final velocity in the preceding period. The proportionality coefficient (the coefficient of restitution of velocity $R$) is set by means of a block of constant coefficients.
Comparison of this circuit with the one we have proposed shows that it requires 2.5 times more amplifiers and relays and, evidently, is inferior in speed, since its operation presupposes sequential switching of relays.
The impact model we proposed (Fig. 1, b) can be made more universal by posing the problem of studying the process of central impact of two bodies of different masses, moving before impact with different velocities.
In this case the post-impact velocities \(\dot U_1\) and \(\dot U_2\) of the masses \(m_1\) and \(m_2\), on the basis of the velocities of the masses before impact \(v_1\) and \(v_2\), are determined by the formulas:
\[ U_1 = a_{11}V_1 + a_{12}V_2, \]
\[ U_2 = a_{21}V_2 + a_{22}V_1, \]
where \(a_{11}, a_{12}, a_{21}, a_{22}\) are constant coefficients.
For real conditions these coefficients are determined experimentally. In accordance with these expressions, the block for specifying the post-impact velocity for the mass \(m_1\) may be represented as shown in Fig. 1, c.
In the general model scheme for two masses, two modeling schemes for post-impact velocities of the indicated type (Fig. 1, c) must be provided, with one common control scheme (Fig. 1, b), to whose input in this case the voltage difference modeling the coordinates of the moving colliding masses must be applied.
Above, models were considered that proceed from the classical formulation of the impact problem. However, in a number of cases the impact time is commensurable with the time of motion of the system in the period between impacts [1, 5].
For these cases it is necessary to obtain data on the processes that occur in the system directly during the period of collision.
If, in analyzing the system, one uses the classical theory of collision of rigid bodies, then it should be assumed that during impact, at the point of contact of the colliding bodies, infinitely large forces must arise in magnitude, but acting instantaneously, leading to a finite change in the quantity of motion.
However, on a system of material points, along with instantaneous forces arising only during the period of collision, there act forces finite in magnitude (for example, the force of gravity, forces of dry and viscous friction, etc.). The impulses of these forces over the infinitely small time of impact will be small and, in the presence of finite impulses of instantaneous forces, may be neglected.
In connection with the foregoing, it seems expedient in the general case to divide the entire process into the period between impacts and the period of impact. For each of these periods its own mathematical description is adopted. The process between impacts is considered on one time scale, and the impact itself on another.
To pass from the system of differential equations describing the motion in the period between impacts to the system of equations for the period of impact, the corresponding matching condition is introduced.
It is precisely in this way that modeling of the process of vibrational immersion is considered in [5]. This mechanical system is in a certain sense analogous to a pneumatic hammer.
The method considered in [5] takes account of the collision time regardless of whether it is small or commensurable with the cycle time. Evidently, such a method is expedient to apply in cases where the total collision time is fixed or varies within certain insignificant limits. It is on this basis that the time scale is chosen, which may be taken as common for the impact and non-impact periods.
In [1] a comparison is given of various methods for solving the problem of impact in a vibratory mill.
The results of the comparison show agreement (within the error of the machine solution of the equations) of the parameters characterizing the behavior of the impact part of the system.
On the basis of the foregoing analysis of the impact-modeling techniques known to us, it is advisable to model impacts and contacts during the operation of a pneumatic hammer in the following manner.
The post-impact velocities of the striker and the tool in their collision are specified by means of the block described by the authors in [10].
The rebound coefficients of the striker and the tool, used as initial data for the simulation, must be determined experimentally, since impact theory has no information on this matter.
The impact of the striker against the front wall of the housing, which occurs in emergency operating modes of the hammer, can be modeled by means of a spring with a stiffness coefficient corresponding to the modulus of elasticity of steel. In this case the velocity recovery coefficient will be equal to unity (there are no losses in the impact).
The elastic properties of the material being worked are likewise modeled by a spring with a stiffness corresponding to the modulus of elasticity of the material. From experimental studies it is known that this coefficient of rebound of the tool from the material being worked is less than unity and is determined by the specific conditions and by the tool parameters. The modeling can be simplified if the coefficient of rebound of the tool when it strikes the material being worked is attributed to the rebound of the tool when it is struck by the striker. In this case the initial velocity of motion of the tool after it is struck by the striker is specified with allowance for the coefficient of rebound of the tool from the material being worked.
The contact between the housing and the tool is modeled by us by means of a spring having a stiffness corresponding to the modulus of elasticity of steel. The post-impact velocity for the housing when it collides with the tool is likewise specified by means of the impact block [10]. The coefficients affecting the rebound of the housing are determined experimentally.
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