Methodology for Calculating the Vibration-Isolating Element of Pneumatic Hammers with Elastic Handles
Abstract Generated abstract
This paper presents a calculation procedure for the vibration-isolating elastic link in pneumatic hammers equipped with elastic handles, addressing the reduction of harmful recoil and vibration transmitted to workers. Based on a two-mass model of the hammer body, spring, and handle under disturbing, constraint, and feed forces, it derives criteria for selecting the rational stiffness of the spring link as a function of hammer body weight, handle weight, blow frequency, and dimensionless frequency. The methodology specifies steps for determining isolated masses, choosing frequency parameters to avoid resonance and improve isolation, calculating spring stiffness and working forces, and sizing helical springs for riveting, chipping, breaker, and drilling hammers.
Full Text
Methodology for Calculating the Vibration-Isolating Link of Pneumatic Hammers with Elastic Handles
V. F. Gorgunov, V. I. Kopytov, M. P. Shumskii
(Presented by the Department of Mining Machines and Mine Transport)
On the basis of theoretical works on the analysis of vibrations of mechanical systems of the pneumatic-hammer type, carried out by Prof. P. M. Alabuzhev and Candidate of Technical Sciences V. I. Kopytov, an effective principle for limiting the harmful influence of recoil and vibration of hammers on the health of workers is the separation of the handle from the vibrating body of the hammer by means of springs.
From consideration of the oscillatory system “hammer body—spring—handle” under the action of a disturbing force (air pressure), a constraint (the tool collar), and the pressing force of the worker’s hand, differential equations describing the behavior of a hammer with an elastic handle were derived and solved [2, 6, 7]. As a result it was possible to obtain equations
Fig. 1. Mechanical model of a hammer with an elastic handle:
$m_1$—mass of the body; $m_2$—mass of the handle; $C$—elastic link; $Q$—pressing force on the handle; $H\cos \omega t$—disturbing force (air pressure); $x_1$ and $x_2$—current coordinates of the masses $m_1$ and $m_2$.
for determining the rational stiffness of the vibration-isolating link as a function of the ratio of the weights of the body and handle and of the acting forces (hammer parameters).
For a system of hammers with an elastic handle (Fig. 1), the equation for
for determining the stiffness coefficient of the link at which the most favorable conditions for vibration isolation are created has the form
\[ C=\frac{4\cdot \pi G_1 \nu^2}{g p^2},\ \text{kg/cm}, \tag{1} \]
where
\[ \begin{aligned} C &\text{— stiffness coefficient of the vibration-isolating link, kg/cm;}\\ G_1 &\text{— weight of the hammer body, kg;}\\ g &\text{— acceleration of free fall of bodies }(981\ \text{cm/sec}^2);\\ \nu &\text{— frequency of piston blows, blows/sec;}\\ p&=(0.5+n)\sqrt{1+\frac{1}{\mu}}\ \text{— dimensionless frequency;}\\ \mu &\text{— ratio of the weight of the carriage (handle) to the weight of the hammer body;}\\ n &\text{— values of the order of the subharmonics }(n=2,3,4\ldots). \end{aligned} \]
The results obtained from theoretical work, taking into account practical and design factors, make it possible to present the procedure for calculating vibration-isolating devices in the following order.
1. Determination of the weights of the hammer body and of the isolated handle
This question is of a purely design nature. Depending on the design of the hammer for which the vibration-isolating device is being designed, it is determined which parts (devices) of the hammer can be attached to the body and which to the handle. As a special case of such a design solution, the path may be chosen of supplementing the hammer design with a separate vibration-damping device (the VK-1 carriage for the PR-24l hammer of the Pneumatika plant). However, despite the simplicity of solving the problem and its obvious expediency from a technological point of view, this design cannot be considered rational, since the weight of the drilling hammer with the vibration-damping device is significantly increased in comparison with a hammer of ordinary design.
Therefore, when selecting a design, one should strive to preserve the total weight of the hammer by including individual parts and mechanisms of the hammer in the isolated device.
For example, when developing a chipping hammer with an elastic handle, the designers of the Tomsk Electromechanical Plant include in the isolated device: a) the actual handle of the hammer; b) the starting device; c) the “false barrel”—a metal casing that covers (isolates) the hammer barrel and protects the worker’s left hand from vibration of the body.
In riveting hammers of the Moscow City Council of National Economy of the KMP type and in Leont’ev’s hammer (Kharkov Turbine Plant), a similar separation of devices was carried out for the purpose of vibration isolation of the workers’ hands.
It follows from the laws of vibration damping [2, 5—7] that the mass of the handle (carriage) should be as large as possible. In this case the vibration-isolation effect increases. However, this recommendation is not entirely decisive, since the necessary (acceptable) vibration-isolation effect can also be achieved with a comparatively “light” handle.
In designing hand-held drilling hammers with elastic handles, in our opinion, one should consider the possibility of making even the air-distribution device part of the isolated link. This will inevitably complicate the hammer design, but will make it possible to substantially improve vibration isolation while preserving the weight of the hammer.
2. Determination of the dimensionless frequency \(p\), which determines the minimum values of the amplitudes of handle oscillations
As studies [6] have shown, the “hammer body with handle” system may have very large oscillation amplitudes in the event of resonance phenomena under the influence of a periodically varying disturbing force. According to preliminary calculations and experiments, the value of the frequency \(p\) at which the amplitudes of handle oscillations will be minimal can be calculated by the formula
\[ p=(0.5+n)\sqrt{1+\frac{1}{\mu}}, \tag{2} \]
where
\(n\) — order of the subharmonic \((2, 3, 4\ldots)\);
\(\mu\) — ratio of the weight of the isolated member (handle, carriage) to the weight of the hammer body.
By assigning values to the parameter \(n\), with a known ratio of the oscillating masses, one can determine the specific value of \(p\). In doing so, it should be borne in mind that the value of \(p\) must be no less than 5. The reason is that at smaller values the vibration-isolating effect for handle oscillations with a frequency equal to the frequency of the disturbing force will drop sharply—the elastic connection will be “rigid.”
Values of \(p\) greater than 10–12 are also undesirable, since the elastic connection becomes too compliant and, in order to balance the internal forces acting in the system, it is necessary to make the elastic member long, which is not always easy to accomplish structurally.
In the general case, it is recommended to select the parameter \(n\) so that the dimensionless frequency \(p\) lies within the range 5–12. As a rule, when \(\mu\) is close to unity, the parameter \(n\) should be taken equal to 4.
If, structurally, it is possible to install a long elastic member in the system, the frequency \(p\) may be taken even greater than 12. The vibration-isolating effect will thereby be improved.
3. Determination of the stiffness coefficient of the elastic member
As the vibration-isolating element of hammers with elastic handles, ordinary helical cylindrical springs may be recommended. An undoubted advantage of such springs is their linearity, i.e., the rectilinear dependence of the spring resistance force on deformation under compression or tension (Fig. 2), and their durability.
As shown above, the stiffness coefficient of the connecting spring is determined by formula (1), which includes the principal parameters of the hammer.
To facilitate calculations of the spring stiffness coefficient by formula (1) and to eliminate possible arithmetical errors, nomograms may be drawn up. For chipping and riveting hammers we have drawn up a nomogram (Fig. 3) based on calculation of the variation of the hammer blow frequency within the range 600–1920 blows/min. For drilling hammers, which have a high blow frequency, similar nomograms are not difficult to construct.
Depending on the blow frequency of the hammer, the nomograms make it easy to determine the ratio of the stiffness coefficient to the weight of the hammer body, taking into account the reduced masses,
\[ g=\frac{C}{G_1},\quad \frac{\mathrm{kg}/\mathrm{cm}}{\mathrm{kg}}. \]
The availability of such nomograms will enable the designer quickly to analyze a number of variants of the elastic linkage and to select from them the one most suitable for particular specific conditions.
4. Determination of the Spring Dimensions and Its Force Characteristic
Proceeding from the operating conditions of a pneumatic hammer with a vibration-isolating handle (carriage), it is necessary to determine the range of feed forces at which the elastic linkage (spring) will operate.
Fig. 2. “Force—deformation” graph for the spring of an elastic handle.
For hand-held breaking, scaling, and chipping hammers these forces may be limited to the range of 10–30 kg. Rock drills require the creation of optimum feed forces from 30 to 120.0 kg. The required feed force for rock drills may be determined from formula [3]
\[ Q_{\mathrm{opt}} = (1.45 \div 1.68)\,\nu\,\sqrt{2Am} + C_1 + G_0 \sin \alpha , \tag{3} \]
where
\[ \begin{aligned} A &\text{ — impact energy, kgm;}\\ m &\text{ — mass of the striker piston, kg sec}^2/\text{m;}\\ C_1 &\text{ — constant force taking into account hose resistance, the reaction of the exhaust air, friction in the feeding device, etc.}\\ &\quad \text{(in practice } C_1 = 5\text{–}6 \text{ kg);}\\ G_0 &\text{ — total weight of the hammer, kg;}\\ \alpha &\text{ — angle of inclination of the drill steel to the horizontal, degrees.} \end{aligned} \]
Operation at the optimum feed force is highly desirable; however, in practice there will almost always be deviations from it. Therefore the elastic element must be designed for operation with a varying force from \(Q_{\min}\) to \(Q_{\max}\). These limits must be specified for the particular hammer design and are the initial data in calculating the springs.
In the case where, between the handle and the body, not only ...
Fig. 3. Nomogram for determining the ratio of the spring stiffness coefficient to the weight of the hammer body at \(i = 4\).
Horizontal axis: Number of blows per minute
Scale: 600, 720, 840, 960, 1080, 1200, 1320, 1440, 1560, 1680, 1800, 1920.
Vertical axis: Ratio of the stiffness coefficient to the weight of the hammer body, \(\mathrm{kg/cm}/\mathrm{kg}\)
Scale: 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6.
Family of curves: Ratio of the weight of the body to the weight of the handle
Curve labels: 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0.
one, but several springs are in parallel, then their stiffness coefficients can be determined as
\[ C_i=\frac{C}{i}, \tag{4} \]
where \(i\) is the number of springs installed in parallel.
The working force of the spring is determined by the formula
\[ Q \geq \frac{Q_{\max}}{i} + C_i\Delta,\ \text{kg}, \tag{5} \]
where \(\Delta\) is the maximum amplitude of vibration of the hammer body, cm.
Formula (5) expresses the following condition: the working force of the spring (Fig. 2) must not be less than the greatest force arising during operation with the maximum feed force and at the maximum amplitude of vibrations.
In order to make the design compact, it is advisable to precompress the springs by a certain amount. In this case, however, the condition must be observed
\[ Q_0 \leq \frac{Q_{\min}}{i} - C_i\Delta,\ \text{kg}, \tag{6} \]
where \(Q_0\) is the precompression force of one spring, kg.
The precompression force \(Q_0\) (Fig. 2) must not be greater than the smallest force arising in the spring during operation with the minimum feed force and at the maximum amplitude of vibrations.
Further calculation of the springs is recommended to be carried out according to Table 9 given in work [4].
For a specified working force and spring stiffness coefficient, the table makes it possible to determine its diameter \(D_{\text{н}}\), wire cross-section \(d\), and set \(f\).
Having determined \(D_{\text{н}}\), \(d\), and \(f\), we calculate the number of active coils of the spring
\[ z=\frac{Q}{C_i f}, \tag{7} \]
where
\[ \begin{aligned} Q &\text{— working force, kg;}\\ C_i &\text{— stiffness coefficient, kg/cm;}\\ f &\text{— set, cm.} \end{aligned} \]
The pitch of springs for \(d<5\) mm is calculated by the formula
\[ h \geq d+f+0.05,\ \text{cm}. \tag{8} \]
The length of the springs in the free state is
\[ l=zh+d,\ \text{cm}. \tag{9} \]
The length of the spring in the assembled state is
\[ l_0=zh+d-\frac{Q_0}{C_i},\ \text{cm}. \tag{10} \]
During operation of the hammer the springs will be compressed to the value
\[ l_1=zh+d-\frac{Q_{\max}}{C_i}-\Delta,\ \text{cm}. \tag{11} \]
The design of the hammer must provide for the possibility of bring-
of the handle (carriage) and the housing from the position they occupy after assembly by the amount
\[ l_{2}=l_{0}-l_{1}=\frac{Q_{\max}-Q_{\min}}{C_{i}}+\Delta,\ \text{cm}. \tag{12} \]
This methodology can be used in calculations of the elastic elements of pneumatic riveting, chipping, breaker, and drilling hammers with vibration-isolating devices.
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