Mathematical Modeling of Certain Testing Conditions for Pressure Devices of Test Benches
Abstract Generated abstract
This paper develops mathematical models for comparing pressure devices used in bench vibration testing of hand-held pneumatic hammers with manual pressing under operating conditions. The authors formulate nonlinear differential equations for hammer-body motion with spring, pneumatic, weight, weight-spring, and hand pressing, incorporating viscous damping, approximate harmonic excitation, impact interaction with a clamped tool, and mechanical impedance of the hand-hammer system, then reduce the equations to electronic analog machine form. Modeling indicates that rebound displacement decreases with increasing stiffness or mechanical impedance, and suggests that weight-spring devices with stiffness of about 1 to 2 kg/cm and spring devices with stiffness of about 4 to 10 kg/cm can reproduce rebound displacements comparable to hand pressing.
Full Text
MATHEMATICAL MODELING OF CERTAIN TEST CONDITIONS FOR PRESSURE DEVICES OF TEST BENCHES
A. I. PASHNIN, A. V. TRIKHANOV
(Presented by the Department of Mining Machines, Mine Transport, and Mining Mechanics)
The nature of the vibration of pneumatic hammers, and its quantitative characteristics, depend, all other conditions being equal, on the type of pressure devices and absorbers of impact energy in test benches.
In bench tests, pressure applied to a pneumatic hammer is effected by various methods. Original designs of pressure devices have been proposed by many researchers. Thus, S. I. Dobroborskii [1] used a pneumatic actuator in the design of the ERD-3 work meter. G. S. Vilner [2] created a vibration-test bench in which a load–spring-type pressure device is used. A. M. Petreev [3] proposes a spring pressure device. A brief analysis of pressure devices (as well as impact-energy absorbers) without taking into account the collisions of the hammer body with the tool—
Fig. 1. Calculation scheme: \(x_k\) — displacement of the body, m; \(m_k\) — mass of the body, \(\mathrm{kg \cdot sec^2/m}\); \(m_{\mathrm{gr}}\) — mass of the load, \(\mathrm{kg \cdot sec^2/m}\); \(C_{\mathrm{ki}}\) — stiffness of the barrel–tool system during impact, \(\mathrm{kg/m}\); \(p_z, p_p\) — pressure in the rear and front cavities, respectively, \(\mathrm{kg/m^2}\); \(S_p\) — area of the actuator piston, \(\mathrm{m^2}\); \(p_i, V_i\) — pressure and volume of the air in the pneumatic actuator, \(\mathrm{kg/m^2}\), \(\mathrm{m^3}\); \(a\) — damping coefficient, \(\mathrm{kg \cdot sec/m}\); \(Q_0\) — magnitude of the preliminary pressure, kg.
...ment was carried out in [4]. A computational study of the effect of pressing devices on the vibration of a pneumatic hammer is associated with the need to introduce major assumptions, while an experimental study entails considerable expense for the manufacture of test benches. In this connection, the method of mathematical modeling was adopted for the investigations; it makes it possible to reflect, almost completely, all factors affecting the vibration of a pneumatic hammer.
The study is carried out for the limiting, basic operating mode of the hammer [3]. In the general case, the body of an operating hammer (Fig. 1) is acted upon by: air pressure, inertial force, pressing force, the weight of the hammer, friction, and the forces arising during impact between the body and the tool. In taking the forces into account, the following assumptions were adopted:
Fig. 2. Graph of the exciting force: 1 -- actual force; 2 -- calculated curve
1) only viscous friction, proportional to the velocity of motion of the hammer body, is taken into account;
2) the exciting force \(S_b(P_3-P_p)\), as a function of time, expanded in an infinite trigonometric series, is replaced by the first terms of the series (Fig. 2);
3) when studying the effect of pressing devices on the vibration of the hammer, the tool is assumed to be clamped, and the coefficient of restitution of the body velocity during its impact with the tool is, for simplicity, taken equal to 1.
The oscillations of the body of the pneumatic hammer under the action of various pressing devices are represented by a nonlinear second-order differential equation
\[ F_{\mathrm{in}}=F_{\mathrm{exc}}-F_{\mathrm{fr}}-F_{\mathrm{press}}-F_{\mathrm{imp}} \]
or
\[ m_k \frac{d^2 x_k}{dt^2} = A \sin \left(\omega t+\frac{\pi}{2}\right) +\Delta F -a\frac{d x_k}{dt} -Q(x_k) -\xi C_{\mathrm{ki}} x_k, \tag{1} \]
where
\[ \xi= \begin{cases} 0, & \text{when } x_k \ge 0,\\ 1, & \text{when } x_k < 0. \end{cases} \]
The pressing force \(Q(x_k)\), depending on the type of pressing device, is written as:
a) for pressing by a spring (Fig. 1, a),
\[ Q(x_k)=Cx_k+Q_0; \]
b) for a pneumatic pressing device, assuming that the process is adiabatic (Fig. 1, b),
\[ Q(x_k)=\frac{p_0 V_0^{\kappa} S_{\mathrm{p}}}{(V-S_{\mathrm{p}})}\,x_k+Q_0; \]
c) when pressing by a load,
\[ Q(x_k)=Q_0=\mathrm{const}. \]
In the latter case the expression for the inertial force takes the form
\[ (m_k+m_{\mathrm{gr}})\frac{d^2 x_k}{dt^2}, \]
where \(m_{\mathrm{gr}}\) is the mass of the load.
With load-spring pressing (Fig. 1, c), the equation of motion of the load is added to equation (1). We obtain the system
\[ m_k\frac{d^2 x_k}{dt^2}+\alpha\frac{dx_k}{dt}-C(x_k-x_{\mathrm{gr}})+Q_0+\xi C_{\mathrm{ki}}x_k =A\sin(\omega t+\varphi)+\Delta F, \]
\[ m_{\mathrm{gr}}\frac{d^2 x_{\mathrm{gr}}}{dt^2}+\alpha\frac{dx_{\mathrm{gr}}}{dt} +C(x_k-x_{\mathrm{gr}})+Q_0=0, \tag{2} \]
where
\[ \xi= \begin{cases} 0, & \text{for } x_k\ge 0,\\ 1, & \text{for } x_k<0. \end{cases} \]
In equations (1–2), the weight of the piston and rod of the pneumatic pressing device, the weight of the guides and of the spring in the load-spring and spring pressing devices are taken into account in the magnitude of the preliminary pressing force \(Q_0\).
The vibration of the hammer, depending on the type of pressing device during bench tests, must be compared with the vibration of the hammer under manual pressing, i.e., under actual operating conditions. Mathematically, it is most convenient to describe manual pressing by means of the mechanical impedance [3, 6], which represents the ratio of the harmonic force to the vibration velocity. It follows from this that, approximately, the action of the force of the hand may be regarded as a frictional resistance proportional to the velocity.
G. Scheffler [6] obtained values \(I=(10\text{--}30)\ \dfrac{\mathrm{kg}\cdot\mathrm{sec}}{\mathrm{m}}\), expressing the impedance of the human hand. However, for chipping hammers these impedance values cannot be used for the following reasons:
a) in G. Scheffler’s experiments the exciting force was specified by a harmonic (sinusoidal) law, whereas in actual hammers this force is not sinusoidal; replacing an infinite Fourier series by a single term leads to large measurement errors;
b) in experiments on a vibration stand it is possible to determine only the impedance of the human hand, whereas the mechanical losses (impedance) of the “human hand—pneumatic hammer” system can be determined under natural chipping conditions;
c) oscillograms show that the maxima of force and of vibration velocity, although they have the same frequency, do not coincide exactly in time; errors in the phase-shift angle and in determining the maximum amplitude may reach a large magnitude.
Fig. 3. Circuits of electronic analogs: a) for spring loading, b) for weight-and-spring loading, c) for manual loading.
To determine the value of the mechanical impedance of the “human hand—hammer” system, oscillograms of recoil and vibration velocity of pneumatic hammers were recorded under metal-cutting conditions, after which the impedance values were determined [4]. According to our calculations, these values proved to be equal to \(80—100\ \dfrac{kg\cdot sec}{m}\), which was subsequently confirmed in modeling (Fig. 4, \(a\)). Thus, the force for the case of manual pressing is
\[ Q(x_k)=I\frac{dx_k}{dt}=90\frac{dx_k}{dt}. \]
For mathematical modeling of mechanical systems, the differential equations (1) and (2) must be reduced to machine form, i.e., the differential equations of the realizable electronic analog must be obtained. Taking into account that the maximum voltage in the electronic model is equal to \(100\ v\), we choose the coefficients:
\[ a_x=\frac{x_k}{U_k};\qquad a_{F=}=\frac{F_=}{U_{F=}};\qquad a_{F\sim}=\frac{F_\sim}{U_{F\sim}}, \]
(Table 1). The time scale is chosen so that the machine coefficients (Table 2) do not exceed unity:
\[ A_i=\frac{aF_\sim}{m_k};\qquad \mathrm{Б}_i=\frac{aF_=}{m_k};\qquad B_i=\frac{Ca^2\tau}{m_k};\qquad \Gamma_i=\frac{a\cdot a\tau}{m_k};\qquad D_i=\frac{C_{ki}a^2\tau}{m_k}. \]
Table 1
Scale coefficients
| Name | Mathematical expression | Dimension | Values |
|---|---|---|---|
| Scale of body displacement | \(a_x\) | \(m/v\) | \(2\cdot10^{-5}\) |
| Time scale | \(a_\tau\) | \(4{,}2\cdot10^{-3}\) | |
| Scale of constant forces | \(a_{F=}\) | \(kg/v\) | \(0{,}12\) |
| Scale of variable forces | \(a_{F\sim}\) | \(kg/v\) | \(0{,}3\) |
Table 2
Machine coefficients of the equations
| \(i\) | \(A_i\) | \(\mathrm{Б}_i\) | \(B_i\) | \(\Gamma_i\) | \(D_i\) |
|---|---|---|---|---|---|
| 1 | \(0{,}9\) | \(0{,}36\) | \(0{,}004—0{,}48\) | \(0{,}18\) | \(100\) |
| 2 | \(0{,}9\) | \(0{,}36\) | — | \(0{,}83—0{,}18\) | \(100\) |
| 3 | \(0{,}9\) | \(0{,}36\) | \(0{,}004—0{,}48\) | \(100\) |
Thus, equation (1) is reduced to machine form
\[ \ddot U_k= \frac{a_\tau^2 a_{F\sim}}{m_k}U_{F\sim} - \frac{a_\tau^2 a_{F=}}{m_k}U_{F=} - \frac{Ca_\tau^2}{m_k}U_k - \frac{a\cdot a_\tau}{m_k}\dot U_k \]
\[ {}-\xi\,\frac{C_{ki}a_\tau^2}{m_k}U_k; \tag{3} \]
for the spring pressing device and the pneumatic feeder
\[ \ddot U_{\mathrm{k}}=A_1U_{F\sim}-B_1U_{F=}-B_1U_{\mathrm{k}}-\Gamma_1\dot U_{\mathrm{k}}-\xi D_1U_{\mathrm{k}}, \tag{4} \]
for the weight-spring device
\[ \left\{ \begin{aligned} \ddot U_{\mathrm{k}}&=A_2U_{F\sim}-B_2U_{F=}-B_2(U_{\mathrm{k}}-U_{\mathrm{gr}})-\Gamma_2\dot U_{\mathrm{k}}-\xi D_2U_{\mathrm{k}},\\ \ddot U_{\mathrm{gr}}&=-B'_2U_{F=}+B'_2(U_{\mathrm{k}}-U_{\mathrm{gr}})-\Gamma'_2\dot U_{\mathrm{gr}}, \end{aligned} \right. \tag{5} \]
for pressing by a weight and for hand pressing
\[ \ddot U_{\mathrm{k}}=A_3U_{F=}-B_3U_{F=}-\Gamma_3\dot U_{\mathrm{k}}-\xi D_3U_{\mathrm{k}}. \tag{6} \]
In equations (3–6)
\[ \xi= \begin{cases} 0, & \text{for } U_{\mathrm{k}}\ge 0,\\ 1, & \text{for } U_{\mathrm{k}}<0. \end{cases} \]
On the basis of these equations, structural diagrams of the mathematical models were constructed (Fig. 3).
Fig. 4. Dependence of the rebound displacement of the pneumatic hammer body on the stiffness of the pressing device: a) hand pressing, b) weight-spring pressing, c) spring pressing, d) installation of a weight on the hammer
The results of solving the equations are presented in Fig. 4, from which it is evident that the amplitude of rebound displacement under weight-spring and spring pressing decreases as stiffness increases (curves b, c). A decrease in rebound displacement is also evident with an increase in mechanical impedance (curve a). Depending on the volume of air in the pneumatic feeder, its “stiffness” changes (the zone of hammer rebound displacements, depending on the pneumatic feeders used, is shown in Fig. 4 by the lower hatched region).
As a result of the modeling it was found that, in comparison with hand pressing, identical rebound displacements of the pneumatic hammer body are given by weight-spring pressing devices with a spring stiffness coefficient of 1–2 kg/cm, and by a spring pressing device with a coefficient
stiffness of 4–10 kg/cm. These clamping devices should be recommended to designers when developing test benches for vibration testing of hand-held pneumatic hammers.
References
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