APPLICATION OF THE METHOD OF SIMILARITY AND DIMENSIONS TO THE CALCULATION OF SOLENOID HAMMERS

P. M. Alabuzhev, N. P. Ryashentsev

The literature on the theory of similarity is extensive. Materials on the theory of similarity and modeling have entered the literature of both textbook \[3, 4, 5, 8, 13, 19, 20, 23, 25\] and reference \[15, 16\] character. The method of dynamic similarity as applied to the calculation of impact-action machines is considered in works \[2, 7, 22\]. In work \[2\] it is shown that similarity with respect to a number of dynamic characteristics also holds for forging hammers; the values of a number of similarity criteria for electropneumatic hammers are given, according to which, on the basis of experimental data, it is possible to select a series of values of the parameters of impact-action machines for an optimal operating mode.

In the present article it is shown that the method of similarity and dimensions can also be applied to the calculation of multi-coil solenoid hammers. According to literature data \[14, 24\], for the Sintron solenoid hammer taken as the model, the parameters determining the working process are the following values of the quantities: impact energy \(A_1 = 0.092\); impact frequency \(n_1 = 3000\) per minute; striker weight \(C_1 = 0.080\) kg; striker diameter \(d_1 = 12.5\) mm; striker travel between the floating bushing and the working tool \(l_1 = 12\) mm; number of coils \(Z_1 = 2\) pcs.

Taking this hammer as the model, it is required to determine the parameters of a new specimen of solenoid hammer with impact energy \(A = 3.00\) kgm and number of blows per minute \(n = 1500\).

We encountered such a problem when designing a more powerful solenoid hammer. Denoting the similarity coefficients by the same letter as the quantity of interest, but with the subscript “c” \[2\], we obtain:

\[ A_c = \frac{A_1}{A} = m_c v_c^2 = \frac{0.092}{3} = 0.031, \]

where

\[ m_c=\frac{m_1}{m}=\frac{G_1}{G} \]

is the ratio of the weights (masses) of the strikers;

\[ v_c=\frac{v_1}{v} \]

is the ratio of velocities, which we take, proceeding from the strength condition for the parts, to be equal to 0.8.

Then

\[ m_c=\frac{A_c}{v_c^2}=\frac{0.031}{0.8^2}=0.05, \]

whence the weight of the striker is

\[ G=\frac{G_1}{m_c}=\frac{0.08}{0.05}=1.6\ \text{kg}. \]

The time ratio is

\[ t_c=\frac{t_1}{t}=\frac{n}{n_1}=\frac{1500}{3000}=0.5. \]

From the relation \(v_c=\frac{l_c}{t_c}\) we determine the scale for the linear dimensions:

\[ l_c=v_c t_c=0.8\cdot 0.5=0.4. \]

Consequently, it should be expected that the stroke of the striker between the floating bushing and the working tool in the hammer being designed will be

\[ l=\frac{l_1}{l_c}=\frac{12.0}{0.4}=30\ \text{mm}, \]

and the diameter of the striker

\[ d=\frac{d_1}{l_c}=\frac{12.5}{0.4}=31\ \text{mm}. \]

In our works [1, 18] it is shown that it is possible to increase the tractive force of the hammer striker and the impact energy by means of the repeated application of electromagnetic forces to a striker consisting of magnetic and nonmagnetic links.

An increase in the tractive force with a smaller striker diameter, in turn, makes it possible to create a hammer with smaller transverse dimensions and to depart from complete geometric similarity.

On the basis of experimental investigations of tractive characteristics that have been carried out, we have established a dependence between the diameters of the strikers and the number of coils \(Z\):

\[ \left(\frac{d_1}{d}\right)^2=K^2\frac{Z}{Z_1}. \]

Here the empirical coefficient \(K=\frac{d_1}{d}\sqrt{\frac{Z_1}{Z}}\) makes it possible, in ...

in the second approximation, to determine more accurately the linear scale \(l_c\) in relation to the dimensions of the striker.

For the indicated parameters of the hammer, \(A = 3\ \text{kgm}\) and \(n = 1500\) blows/min. If, for design reasons, the number of coils is taken as \(z = 4\) pcs. and the striker diameter as \(d = 24.6\ \text{mm}\) (instead of \(d = 31\ \text{mm}\)), then

\[ l_c = K = \frac{d_1}{d}\sqrt{\frac{Z_1}{Z}} = \frac{12.5}{24.6}\sqrt{\frac{2}{4}} = 0.51 \cdot 0.71 = 0.36. \]

In the experimental model of the MS-3 hammer manufactured at TPI, with the indicated parameters it turned out that the striker stroke between the floating sleeve and the working tool was \(l = 40\ \text{mm}\), and the striker weight was \(G = 1.62\ \text{kg}\), and the number of striker blows was \(n = 1500\), which was achieved by means of a special sparkless switch [17].

Consequently, from the characteristic of the working process: the linear scale

\[ l_c = \frac{l_1}{l} = \frac{12}{40} = 0.3; \]

the scale for masses

\[ m_c = \frac{m_1}{m} = \frac{G_1}{G} = \frac{0.08}{1.62} = 0.049; \]

the scale for time

\[ t_c = \frac{1/n_1}{1/n} = \frac{n}{n_1} = \frac{1500}{3000} = 0.5. \]

Consequently, the ratio of works is

\[ A_c = m_c v_c^2 = m_c \left(\frac{l_c}{t_c}\right)^2 = 0.049 \frac{0.3^2}{0.5^2} = 0.0175 \]

then, according to the theory of similarity, the work on the striker should be

\[ A = \frac{A_1}{A_c} = \frac{0.080}{0.0175} = 4.5\ \text{kgm}. \]

For the MS-3 hammer, depending on the power-supply circuit of the coils in individual operating modes, the impact energy \(A\) was of the order of \(5\ \text{kgm}\). During stable operation, the average value was \(A = 3.7\ \text{kgm}\). A description of the experimental studies of the MS-3 hammer is given in the paper included in the present collection.

The scale for power

\[ N_c = m_c \frac{l_c^2}{t_c^3} = 0.049 \frac{0.3^2}{0.5^3} = 0.035 \]

In our case the power consumed from the mains is \(N_p = 2.0\ \text{kW}\) and the hammer efficiency is

\[ \eta = \frac{N_{\mathrm{eff}}}{N_p} = \frac{0.907}{2.00} = 0.457. \]

The efficiency of the Sintron hammer is of the order of \(0.32\).

The calculation of the hammer can be carried out on the basis of the method of dimensions for a number of quantities determining the operation of the hammer, proceeding from the following conditions.

1. With a proportional change of the geometrical dimensions of the hammer by a factor of \(K\), its other parameters will change by a factor of \(K^\alpha\). The exponent \(\alpha\) has, for each parameter, its own value, which is established on the basis of physical relations.

2. With a proportional change of the geometrical dimensions of the hammer, the induction of the steel of the magnetic circuit is preserved: \(B=\mathrm{const}\), which follows from the nonlinear character of the curve \(B=f(H)\) for the magnetization of the steel of the solenoid, where \(H\) is the magnetic-field strength.

On the basis of the formulated conditions, the values of the powers of the similarity coefficients will be:

1) for induction, \(K^0\), i.e. \(\alpha=0\);

2) for linear dimensions, \(K\), i.e. \(\alpha=1\) (except for the length in the presence of a multistage striker with a large number of coils \(Z\)).

Table 1 gives the values of the exponent \(\alpha\) for the principal parameters of the hammer when its linear dimensions are changed by a factor of \(K\)*.

Table 1

* The indicated system of units is not gravitational, since the dimension of force is \([l^2]\) and the dimension of mass is \([l^3]\).

Table 1 (continued)

Let us carry out, for example, a comparison of the calculation results obtained by the similarity method for the Cintron hammer, based on the test materials described in the literature [14], with the MS-3 hammer manufactured and tested by us at TPI. The characteristics of the hammers are given in Table 2.

Table 2

The value of the similarity coefficient $K$ is determined from the ratio of the work of the Cintron hammer to the work of the MS-3 hammer

\[ K^3=\frac{A_1}{A}=\frac{0.092}{3.00}=0.0307, \]

hence $K=0.31$.

Using the previously established relations of the similarity coefficient $K$ (Table 1), we determine the principal parameters of the Cintron hammer.

Let us find the striker weight

\[ A_c=\frac{A_1}{A}=m_c v_c^2=K^3=0.0307, \]

* The design of the MS-3 hammer is described in the article “Investigation of the Dynamics of Solenoid Hammers” in the present collection.

\[ v_c^2=\frac{v_1^2}{v^2}=\frac{2A_1g/G_1}{2Ag/G}=\frac{A_1G}{AG_1}=\frac{0.092\cdot 1.62}{3\cdot 0.08}=0.615. \]

Then the weight of the striker in the Sintron hammer will be

\[ G_1=m_cG=\frac{A_c}{v_c^2}\,G=\frac{0.0307}{0.615}\cdot 1.62=0.081\ \text{kg}. \]

We determine the diameter of the striker of the Sintron hammer

\[ d_1=dK\sqrt{\frac{z}{z_1}}=24.6\cdot 0.31\sqrt{\frac{4}{2}}=10.8\ \text{mm}. \]

The discrepancy between the values of the actual diameter of the striker \(d\) and the calculated \(d_1\) is explained by the fact that the supply circuit of the MS-3 hammer differs from the supply circuit of the Sintron hammer. The stroke of the striker between the floating bushing and the shank of the working tool is

\[ l_1=Kl=0.31\cdot 40=12.5\ \text{mm}. \]

Comparing the results of the calculation of the Sintron hammer obtained by the similarity method with its parameters given in Table 2, we come to the conclusion that the sought quantities obtained by the similarity method correspond sufficiently closely to practice. Consequently, having a manufactured and tested specimen of a solenoid hammer, it is easy to determine a series of parameters for new hammers operating on the same principle.

The particular dependences given on the basis of the method of similarity and dimensions are appealing in their simplicity. However, it must be borne in mind that these dependences can be used only in the case where, in the hammer being designed, the conditions of the process correspond to those that existed during the experiments and on the basis of which these dependences were obtained. Therefore, along with establishing the relationships given by the theory of similarity, it is necessary to study deeply the specific character of the phenomenon under concrete conditions.

The formulation and task of an experiment on the basis of the theory of dimensions and the theory of similarity is simplified and facilitated when there is a functional relationship between whole complexes of quantities determining the phenomenon, since in a number of cases there is no need to study the influence on the process of each factor separately, and it is possible to extend certain results of a single experiment to similar specimens [6, 9, 10, 11, 12, 19, 25].

One may arrive in various ways at establishing the relationship between similarity criteria: the most reliable solution and the obtaining of generalizing conclusions can be achieved by applying the theory of similarity on the basis of the differential equations describing

process and the conditions of uniqueness, by which the limits of applicability of a single experiment are established.

The motion of the striker of a solenoid hammer is described by the differential equation

\[ m \frac{d^{2}X}{dt^{2}} = F \pm F_{\mathrm{p}} - F_{\mathrm{c}} \pm F_{\mathrm{b}} + F_{\mathrm{y}}, \]

where

\[ \begin{aligned} m &= \frac{G}{g} &&\text{is the mass of the striker;}\\ X & &&\text{is the travel of the striker;}\\ F & &&\text{is the tractive force of the coil, which is determined}\\ & &&\text{from the expression } F = 5.1 \cdot 10^{-6}(iW)^2 \frac{d\mu}{dX};\\ (iW) & &&\text{is the number of ampere-turns of the coil (a variable}\\ & &&\text{quantity during the cycle);}\\ \mu & &&\text{is the magnetic permeability (a variable quantity}\\ & &&\text{during the cycle);}\\ F_{\mathrm{p}} = CX & &&\text{is the force acting from the spring during the}\\ & &&\text{return stroke of the striker;}\\ C & &&\text{is the stiffness coefficient of the spring;}\\ F_{\mathrm{c}} = fmg & &&\text{are the resistance forces (friction of the striker against the}\\ & &&\text{guides, resistance due to residual induction}\\ & &&\text{of the striker and poles);}\\ F_{\mathrm{b}} = P_0 \frac{\pi d^2}{4}\left[\left(\frac{l_0}{l}\right)^m - 1\right] & &&\text{is the resistance force of the air buffer,} \end{aligned} \]

where \(m = 1.32\) is the polytropic exponent (air leakage is neglected);

\[ F = \frac{m v_{\mathrm{y}}(1+e)}{\tau} \]

is the mean value of the reaction force of the tool during the impact time \(\tau\);

\[ e = \frac{v_{\mathrm{ot}}}{v_{\mathrm{y}}} \]

is the coefficient of restitution upon impact.

We assume the position of the hammer to be horizontal; we neglect the resistances due to residual induction of the striker and poles. From the condition of similarity the following relations follow:

\[ \frac{m_c v_c}{t_c} = i^2 W^2 \frac{\mu_c}{l_c} = C_c l_c = f_c m_c g_c = P_{0.c} d_c^2 = \frac{m_c v_c}{\tau_c}. \]

In addition, the boundary conditions must be satisfied: equality of the degree of compression \(\left(\frac{l_0}{l}\right)^m\) and equality of the value of the coefficient of restitution

\[ e = \frac{v_{\mathrm{ot}}}{v_{\mathrm{y}}}. \]

Hence follow the relations and the corresponding similarity criteria:

\[ \frac{m_c v_c}{t_c} = i_c^2 W_c^2 \frac{\mu_c}{l_c} \]

or

\[ 1.\quad \frac{i_c^2 W_c^2 \mu_c}{m_c v_c^2}=1,\quad \text{whence}\quad \Pi_1=\frac{i^2 W^2 \mu}{m v^2}=\mathrm{idem}; \]

\[ 2.\quad \frac{m_c v_c}{t_c}=C_c l_c,\quad \text{whence}\quad \Pi_3=\frac{m v^2}{l^2 C}=\mathrm{idem}; \]

\[ 3.\quad \frac{m_c v_c}{t_c}=f_c m_c g_c,\quad \text{whence}\quad \Pi_3=\frac{m v^2}{f l g m}=\mathrm{idem}; \]

\[ 4.\quad \frac{m_c v_c}{t_c}=P_{\mathrm{o.c}} d_c^2,\quad \text{whence}\quad \Pi_4=\frac{m v^2}{P l d^2}=\frac{m v^2}{P \nu}=\mathrm{idem}; \]

\[ 5.\quad \frac{m_c v_c}{t_c}=\frac{m_c v_c}{\tau_c},\quad \text{whence}\quad \tau_c=t_c. \]

The ratio of the impact time for similar systems must be equal to the time scale of the processes of the working cycle.

On the basis of the second similarity theorem (the \(\Pi\)-theorem), one may write the relation between the similarity criteria as

\[ \Phi(\Pi_1,\Pi_2,\Pi_3,\Pi_4)=0. \]

The form of the function is found from experiment. Partial dependences between individual similarity criteria may be found. These criteria could also be obtained in another way, on the basis of dimensional theory. For this it is necessary to know the list of all essential quantities determining the working process of the hammer and their dimensions.

Wishing to reduce the number of criteria, let us form:

\[ 1.\quad \Pi_3:\Pi_2=\frac{m v^2}{f m g l}:\frac{m v^2}{l^2 C}=\frac{C l}{f m g}=X_1, \]

where \(l\) is a characteristic linear dimension, for example, the maximum travel of the striker;

\[ 2.\quad \Pi_1:\Pi_4=\frac{i^2 W^2 \mu^2}{m v^2}\cdot\frac{m v^2}{P \nu}=\frac{i^2 W^2 \mu}{P \nu}=Y_1 \]

where \(i_1 W_1 \mu, P\) are the maximum or average values of the quantities over the cycle in the optimum operating mode of the hammer.

On the basis of experimental data we construct the graph of the function \(Y_1=Y_1(X_1)\). Each point of the graph, in dimensionless coordinates on the plane, relates 10 quantities to one another. In addition, a number of other dependences can also be constructed, for example: \(X_1=X_1(\Pi_1,\Pi_4)\) or \(X_1=X_1(\Pi_1,Y_1)\), etc.

From these graphs (nomograms), constructed from a comparatively small number of experimental data, it is easy to vary the choice of parameters for a newly designed machine. For this it is necessary to have data from several calculated and manufactured sam-

series of hammers (on the order of 5–6), or have an experimental installation on which, within comparatively wide limits, it would be possible to vary the parameters determining the working process of an impact-action machine built on the solenoid principle.

Experiment is one of the cornerstones of cognition, a research method by means of which the validity of hypotheses, conjectures, and models is tested and the values of a number of coefficients and indicators are established.

Academician M. V. Kirpichev showed in his works [9, 10, 11] that the theory of similarity is the theory (grammar) of experiment and modeling: it indicates how an experiment should be set up, how experimental data should be processed, and how the results obtained should be generalized and extended to new prototypes.

Having a well-functioning specimen of a successfully constructed machine, in which the experience of general human practice has already been embodied to a certain degree, on the basis of the theory of similarity it is possible, with sufficient confidence, comparatively easily and simply to produce (with a known approximation) a series of machines of the same type in accordance with the technical requirements and demands of industry.

LITERATURE

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6. L. I. Gutenmakher. Electrical Models. Publishing House of the Academy of Sciences of the USSR, 1949.

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16. A. V. Reznyakov. A Short Handbook on the Theory of Similarity. Izvestiya Kazakh Academy of Sciences of the USSR, 1950.

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18. N. P. Ryashentsev. On the question of the creation of electric drilling machines. Scientific Reports of Higher Education, No. 4, Sovetskaya Nauka, 1958.

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