Calculation and Selection of Solenoid Hammer Parameters

N. P. Ryashentsev

Symbols

$A$ — impact energy of the hammer, kg·m;
$f$ — current frequency, Hz;
$n$ — number of blows per minute;
$m$ — number of hammer coils;
$S$ — striker stroke, cm;
$d_{\mathrm{b}}$ — striker diameter, mm;
$r_{\mathrm{b}}$ — striker radius, mm;
$l_{\mathrm{b}}$ — total length of the striker, mm;
$g$ — wall thickness of the guide tube, mm;
$d_2$ — outside diameter of the magnetic pole of the coil, mm;
$G_{\mathrm{b}}$ — weight of the striker, kg;
$\rho_1$ — average specific weight of steels 40KhN and 1Kh18N9T, kg/dm³;
$\rho_2$ — specific weight of copper, kg/dm³;
$F_{\mathrm{avg}}$ — average traction force of the striker, kg;
$\eta$ — efficiency of the hammer, %;
$\eta_{\mathrm{m}}$ — magnetic efficiency of the hammer, %;
$\cos\varphi$ — power factor of the hammer;
$A_{\mathrm{m}}$ — electromagnetic energy of the solenoids over the full cycle of the striker stroke, kg·m;
$g_{\mathrm{b}}$ — cross-sectional area of the striker, cm²;
$I$ — effective value of the solenoid current, A;
$U$ — line voltage, V;
$W$ — number of turns of the solenoid;
$IW$ — number of ampere-turns of the solenoid, A·turns;
$P$ — active power of the hammer, kW;
$g_{\mathrm{w}}$ — cross-section of the solenoid winding wire, mm²;
$i$ — current density, A/mm²;
$d$ — diameter of the winding wire, mm;
$K_1$ — amplitude coefficient of the current curve;
$R$ — ohmic resistance of the coil, ohm;
$l_{\mathrm{k}}$ — length of the coil wire, m;
$\rho_{\mathrm{m}}$ — electrical resistivity of copper, ohm·mm²/m;
$Q$ — window area of the solenoid coil, mm²;
$d_{\mathrm{o}}$ — outside diameter of the coil, cm;
$d_{\mathrm{i}}$ — inside diameter of the coil, cm;
$l_{\mathrm{v}}$ — mean length of a coil turn, cm;
$G_{\mathrm{m}}$ — weight of the solenoid copper, kg;

$l_1$ and $l_2$ — height of the pole in the regions of parasitic gaps, cm;
$\Phi_p$ — calculated magnetic flux of the gap, Mx;
$B_{\max.p.}$ — maximum induction of the solenoid pole, G/cm²;
$l_x$ — height of the coil pole at its radius $r_x$, cm;
$h$ — height of the solenoid coil, cm.

Calculation of the Electrical Part of a Multi-Coil Hammer

The existing methods for calculating electromagnets do not always correspond sufficiently well to practice. These calculation methods do not take into account the shape of the current diagram feeding the solenoid, do not determine rational coil parameters, and do not reveal the picture of the dynamic process of the hammer.

Theoretical studies of the dynamics of an electromagnet with direct and alternating sinusoidal current are presented in works [1—6]. However, the authors of these works point out that theoretical investigations of electromagnet dynamics are very difficult (complex), and at present theoretical studies of electromagnet dynamics have been insufficiently solved. The indicated calculation methods make it possible only to estimate approximately the quantities characterizing the working process. And there is no calculation at all for solenoid hammers operating according to the scheme shown in Fig. 1.

Before describing the dynamics of the solenoid hammer, it is necessary to make a preliminary selection of its basic parameters.

When the hammer coils are supplied according to the circuit shown in Fig. 1, a, the current oscillogram is shown in Fig. 1, b. From the supply circuit and the current oscillogram it is evident that the current flowing through the coils is nonsinusoidal owing to the inclusion of nonlinear inductance. The frequency of the current feeding the coil is 25 cps.

The work of the electromagnetic forces acting on the striker is determined by the active power of the hammer, which it must deliver to the working tool.

It is known [5] that, with a rational design of the magnetic system of the Cintron hammer, the magnetic efficiency $\eta_m$ is not more than 50%.

The calculated value of ampere-turns is determined from the expression

$$ \Phi_p I W \eta_m = A \cdot 10^9 . \tag{1} $$

According to the technical specifications, the dimensions of the striker should be adopted—the diameter $d_b$ and the length $l_b$.

Then the weight of the striker will be

$$ G_b = \frac{\pi d_b^2}{4} l_b \rho_1, \tag{2} $$

where $\rho_1$ is the averaged specific weight of steels 40KhN and 1Kh18N9T, kg/dm³;

\[ \rho_1=\frac{\rho' V' + \rho'' V''}{V' + V''}, \]

where \(\rho'\) and \(\rho''\) are, respectively, the specific weights of steels 40KhN and 1Kh18N9T, \(\text{kg}/\text{dm}^3\);

\(V'\) and \(V''\) are, respectively, the volumes of steels 40KhN and 1Kh18N9T, \(\text{dm}^3\).

For a specified impact energy, the velocity of the striker before the moment of impact is determined from the formula

\[ v=\sqrt{\frac{2A}{m}}=\sqrt{\frac{2Ag}{G_b}}. \tag{3} \]

Taking into account losses due to friction of the striker against the walls of the guide tube, the efficiency of spring recoil, and the impact of the striker against the spring [5], amounting to 10% of the impact energy, we determine the required electromagnetic energy of the solenoids over the complete cycle of striker travel:

\[ A_m=\frac{A}{0.9}=\frac{m v^2}{2\cdot 0.9}. \tag{4} \]

Let us determine the number of ampere-turns. At the maximum magnetic induction in the striker, \(B_{\max}=16000\ \text{Gs}/\text{cm}^2\), the calculated flux of one coil will be

\[ \Phi_p=B_{\max}s_b=B_{\max}\frac{\pi d_b^2}{4}. \tag{5} \]

The ampere-turns of one solenoid are determined from the expression [5, p. 63]

\[ A=\frac{1}{2}\,m I W \Phi_p\,10^{-9}, \]

whence

\[ I W=\frac{2A}{m\Phi_p 10^{-9}}. \tag{6} \]

In accordance with works [5 and 6], the efficiency of the solenoid is \(\eta=40\%\), and \(\cos\varphi=0.6\).

The effective value of the current in the coil is determined as

\[ I=\frac{P}{mU\eta\cos\varphi}, \tag{7} \]

where

\[ P=\frac{An}{60\cdot 102}. \tag{8} \]

The amplitude value of the current will be

\[ I_{\max}=K_1 I, \tag{9} \]

where \(K_1\) is the amplitude coefficient of the current curve, which is equal to the ratio of the amplitude value of the current to the effective value,

\[ K_1=\frac{I_{\max}}{I}. \tag{10} \]

When the hammer coil is supplied according to the circuit shown in Fig. 1, \(K_1 = 4.2\) (the value of \(K_1\) was calculated from the oscillogram of the coil current recorded during operation of the solenoid hammer).

The number of turns in the solenoid is

\[ W = \frac{I W}{I_{\max}} . \tag{11} \]

The cross section of the solenoid winding wire is

\[ g_{\mathrm{o}} = \frac{I}{i}, \tag{12} \]

where \(i\) is the current density; for the MS-3 hammer with water cooling and winding wire with heat-resistant insulation, we take \(i = 5\ \mathrm{A/mm^2}\).

Fig. 1. Circuit for switching on the coils of the MS-3 hammer \((a)\) and current oscillogram \((b)\).

The wire diameter is

\[ d = \sqrt{\frac{4g}{\pi}} . \tag{13} \]

The total cross section of the solenoid coil window, with filling factor \(K_{\mathrm{m}}\), will be

\[ Q = \frac{gW}{K_{\mathrm{m}}}. \tag{14} \]

The inner diameter of the coil is determined by the equation

\[ d_{\mathrm{v}} = d_{\mathrm{b}} + 2g . \]

With the outer diameter of the coil \(d_{\mathrm{n}}\), its height will be

\[ h = \frac{2Q}{d_{\mathrm{n}} - d_{\mathrm{v}}}. \tag{15} \]

The average length of a turn in the drawing is

\[ l_{\mathrm{v}}=\pi\frac{d_{\mathrm{v}}+d_{\mathrm{n}}}{2}. \tag{16} \]

The ohmic resistance of the coil is

\[ R=\rho_{\mathrm{m}}\frac{l_k\,10^{-2}}{g} =\rho_{\mathrm{m}}\frac{\pi(d_{\mathrm{v}}+d_{\mathrm{n}})W}{2g}. \tag{16a} \]

Example. Calculate a solenoid hammer satisfying the following technical specifications:
1. Impact energy \(A=3\) kgm.
2. Impact frequency \(n=1500\) blows/min.
3. Efficiency \(\eta=40\%\).
4. Allowable heating temperature of the coil \(t^0=90^\circ\mathrm{C}\).
5. Water cooling.
6. Line voltage \(U=127\) V.
7. Replacement of worn parts must be simple and quick.

We adopt a four-coil hammer design and the dimensions of the striker—diameter \(d_{\mathrm{b}}=26\) mm and length \(l_{\mathrm{b}}=380\) mm; then the weight of the striker will be

\[ G_{\mathrm{b}}=\frac{3.14\cdot0.26}{4}\cdot3.8\cdot8.0=1.62\ \text{kg}. \]

The velocity of the striker just before the instant of impact is

\[ v=\sqrt{\frac{2\cdot3\cdot9.81}{1.62}}=6.0\ \text{m/sec}. \]

The electromagnetic energy of the four solenoids of the hammer is

\[ A_{\mathrm{m}}=\frac{3.0}{0.9}=3.34\ \text{kgm}. \]

The magnetic flux of one solenoid is

\[ \Phi_{\mathrm{r}}=16000\frac{3.14\cdot2.5^2}{4}=85000\ \text{Mx}. \]

The required number of ampere-turns of one solenoid is

\[ IW=\frac{2\cdot3.34\cdot10^9}{4\cdot85\cdot10^3}=19640\ \text{A-turns}. \]

The effective value of the coil current is

\[ I=\frac{3\cdot1500\cdot10^3}{60\cdot102\cdot4\cdot127\cdot0.40\cdot0.60}=6.28\ \text{A}. \]

The amplitude value of the current is

\[ I_{\max}=4.2\cdot6.28=26.4\ \text{A}. \]

The number of turns of the coil is

\[ W=\frac{19640}{26.4}=744\ \text{turns}. \]

The cross section of the coil winding wire is

\[ g_{\mathrm{p}}=\frac{6.28}{5}=1.26\ \text{mm}^2. \]

Diameter of the coil winding wire:

\[ d_{\text{w}}=\sqrt{\frac{4\cdot 1.26}{3.14}}=1.26\ \text{mm}. \]

We adopt, according to GOST, PSD-1, 20 wire; then the coil fill factor is \(K_m=0.54\). The cross section of the coil window will be

\[ Q=\frac{1.26\cdot 800}{0.54}=1870\ \text{mm}^2. \]

With an internal coil diameter \(d_{\text{in}}=30\ \text{mm}\) and an external diameter \(d_{\text{out}}=93\ \text{mm}\), the height of the coil will be

\[ h=\frac{1870\cdot 2}{93-30}=60\ \text{mm}. \]

The mean length of a turn is

\[ l_{\text{t}}=\frac{3.14(30+93)}{2}=194\ \text{mm}. \]

We shall check the calculated number of coil turns by the formula

\[ W=\frac{UK_2}{4.44\Phi_{\text{av.t}}f} -\frac{(IW)R}{2U} \left( 2\sin\theta+ \frac{4.44\Phi_{\text{av.t}}fRIW} {U^2-8.88\Phi_{\text{av.t}}fR(IW)\sin\theta} \right), \tag{17} \]

given in the literature [6],

where \(K_2\) is the amplitude coefficient of the current curve; when the coil is supplied according to the circuit shown in Fig. 1, \(K_2=0.45\cdot 1.57=0.71\) [8];

\(\Phi_{\text{av.t}}\) is the amplitude of the mean flux of the coil, Wb; \(\Phi_{\text{av.t}}=\Phi_{\text{p}}\);

\(IW\) is the rms value of the coil magnetomotive force,

\[ IW=\frac{IW_{\max}}{K_1}; \]

\(R\) is the coil resistance,

\[ R=\rho\,\frac{l_k}{g}\,10^{-2}\ \text{ohm}; \tag{18} \]

\(l_k\) is the length of the coil wire,

\[ l_k=\pi\left(\frac{d_{\text{e}}+d_{\text{i}}}{2}\right)W=l_{\text{t}}W\ \text{m}; \]

\(f\) is the frequency of the current supplying the coil, \(f=25\ \text{Hz}\);

\(\theta\) is the phase-shift angle between \(\Phi_{mn}\) and \(IW\); for magnetic circuits without short-circuited turns and with negligibly small losses in the steel, \(\theta=0\), and then

\[ W=\frac{UK_2}{4.44\Phi_{\text{av.t}}f} -\frac{4.44\Phi_{\text{av.t}}f(IWR)^2}{2U^3}. \tag{19} \]

For the specified technical conditions of the example, we calculate the number of coil turns by formula (19).

The rms value of the coil magnetomotive force is

\[ IW=\frac{19640}{4.2}=4680\ \text{A-turns}. \]

Ohmic resistance of the coil

\[ R = 1.725 \cdot 10^{-5}\frac{3.14(30+93)800}{2 \cdot 1.26}=2.05\ \text{ohm}. \]

Number of turns of the coil

\[ W = \frac{127 \cdot 0.71}{4.44 \cdot 0.00085 \cdot 25} - \frac{4.44 \cdot 0.00085 \cdot 25(4680 \cdot 2.05)^3}{2 \cdot 127^3} = \]

\[ =956-2.0=954\ \text{turns}. \]

From the calculation it is evident that, for \(R \ll \dfrac{U}{R}\), the second component of the right-hand side of the equation may be set equal to zero, and the number of turns of the coil determined from the formula

\[ W=\frac{U}{4.44\Phi_{\mathrm{av.}n}f}. \tag{20} \]

From the check calculation it follows that the number of turns of the coil, calculated by formula (17), is greater by 210 turns than that determined by formula (11). Taking into account the experience of manufacturing solenoid hammers and the results of their testing, we take the number of coil turns as \(W=800\).

We choose the shape of the coil poles from the condition of the smallest possible parasitic gaps between the inner diameter of the pole and the striker, and also between the outer diameter of the pole and the hammer body. Based on the foregoing, we take the height of the pole in the places of parasitic gaps as \(l_1=20\ \text{mm}\) and \(l_2=10\ \text{mm}\), and determine the intermediate dimensions of the pole height from the condition of obtaining a constant induction value over the pole cross section. Specifying the maximum induction \(B_{\max.\mathrm{p}}=13000\ \text{Gs}/\text{cm}^2\) and the pole radius, we determine its height by the formula

\[ l_x=\frac{\Phi_p}{2\pi r_x B_{\max.\mathrm{p}}} = \frac{85000}{2 \cdot 3.14 \cdot 12000 r_x} = \frac{1.15}{r_x}, \tag{21} \]

where \(r_x\) is the pole radius, cm.

It should be noted that, in the calculation methods given above, the dimensions of the striker are taken from design considerations. In doing so, the geometrical dimensions of the coils and the nonsinusoidal character of the current, due to the inclusion of the nonlinear inductance [5], which depends on the coil supply circuit and on the magnitude of the striker stroke, are not taken into account. Therefore, below we give a calculation taking into account the principal dimensions of the coils.

Calculation of the electrical part of a solenoid hammer with allowance for the shape of the coils

According to the specified technical conditions of the example given above, we choose the optimum dimensions of the coil and striker from the condition of minimum losses with the following physical parameters:

thermal-conductivity coefficient \(K_t=4.4 \cdot 10^3\ \text{W}/\text{cm}^\circ\text{C}\);

heat-transfer coefficient \(h=1.7 \cdot 10^{-3}\ \text{W}/\text{cm}^2{}^\circ\text{C}\);

maximum heating temperature of the coil \(t_{\max}=90^\circ\text{C}\);

specific weight of the winding wire \(\rho_2=8.8\ \text{g}/\text{cm}^3\);

coil filling factor \(K_n=0.56\);

specific resistance of copper \(\rho_m=1.725 \cdot 10^{-2}\ \text{ohm}\,\text{mm}^2/\text{m}\).

We determine the ampere-turns per unit length of the coil from the formula given in [1],

\[ \frac{IW}{2C}= \frac{ 200\sqrt{\dfrac{K_{\mathrm{m}}Kt}{\rho_{\mathrm{m}}}\,(r_2-r_1)} }{ \sqrt{ \dfrac{p-g}{m-n} \left( \ln \dfrac{p-g}{m-n}-1 \right) + 2\,\dfrac{gm-pn}{m-n} } }, \tag{22} \]

where \(2C\) is the length of the coil;
\(r_1\) and \(r_2\) are, respectively, the outer and inner radius of the coil;

[Graph: vertical axis—“Ampere-turns per 1 cm of coil length”; horizontal axis—“Width of the coil window, cm”; curves labeled \(r_1=1.0\), \(r_1=1.4\), \(r_1=1.6\), \(r_1=1.8\).]

Fig. 2. Dependence of the width of the coil window on the specific values of ampere-turns per unit length of the coil.

\[ \delta=r_2-r_1 \]

is the width of the coil window;

\[ p=\frac{r_2^2}{2}+\frac{K_{\mathrm{T}}r_2}{h}; \qquad m=\ln r_2+\frac{K}{hr_2}; \]

\[ g=\frac{r_1^2}{2}-\frac{Kr_1}{h}; \qquad n=\ln r_1-\frac{K}{hr_1}. \]

Assigning values of \(\delta=r_2-r_1\) (Fig. 3) in the interval from 0.3 to 2.1 and keeping the other coil dimensions unchanged, from equation (23) we find the dependence between the ampere-turns expended for driving the magnetic flux through \(1\ \mathrm{cm}\) of coil length,

\[ \frac{IW}{2C}=aW, \]

and the width of the coil window \(\delta\) (lying within certain limits) for a given coil length.

If the magnetic flux, which is determined by formula (6), is known, then each selected value of \(r_1\) corresponds to a definite induction and, consequently, to the ampere-turns per \(1\ \mathrm{cm}\) of coil length required for the working magnetic flux. Then, from the formula

\[ \frac{IW}{2C}=f(\delta) \]

(Fig. 2), we determine the most advantageous radius for various values of ampere-turns per unit length of the coil

\[ IW_0=IW-IW_{\mathrm{t}}=\mathrm{const}, \]

where \(IW_{\mathrm{t}}=2CaW_{\mathrm{t}}\) are the ampere-turns expended in conducting the magnetic flux through the striker.

The results of calculating

\[ \frac{AW}{2C}=f(\delta) \]

for specified values of \(r_1\) and \(r_2\) are summarized in Table 1.

Table 1

Let us take \(r_1=1.4\ \mathrm{cm}\); then, allowing for the coil insulation thickness of \(0.05\ \mathrm{cm}\) and the guide tube thickness of \(0.12\ \mathrm{cm}\), the striker radius will be \(r_{\mathrm{b}}=1.25\ \mathrm{cm}\). With a magnetic flux \(\Phi_{\mathrm{p}}=85\,000\ \mathrm{Mx}\), the induction in the striker is determined from the expression

\[ B_{\mathrm{b}}=\frac{\Phi_{\mathrm{p}}}{\pi r_{\mathrm{b}}^2} =\frac{85\,000}{4.75}=17\,800\ \mathrm{G}. \]

From the magnetization curve, for this induction \(B_{\mathrm{b}}=17\,800\ \mathrm{G}\), we find the value of the ampere-turns per unit length of the striker,

\[ \frac{IW_{\mathrm{t}}}{2C}=aw_{\mathrm{t}} \]

for the specified values

\[ \frac{IW_0}{2C}, \]

which we conventionally take to be \(200,\ 300,\ 400\ \mathrm{A/cm}\). These ampere-turns are necessary for pushing through-

of the magnetic flux per 1 cm of coil length, which are determined from the equation

\[ \frac{IW}{2C}=\frac{IW_0}{2C}+\frac{IW_{\tau}}{C}. \tag{23} \]

From the found value of ampere-turns, we determine from the graph (Fig. 2) the width of the coil window \(\delta\). Then the outer radius of the coil will be \(r_2 = r_1 + \delta\).

The weight of 1 cm of coil length is determined by the formula

\[ \frac{G}{2C}=\rho_2 K_\tau \pi (r_2^2-r_1^2). \tag{24} \]

Using equation (24), we determine the weight of 1 cm of coil length for the ratio \(\dfrac{IW_0}{2C}\), equal to 200, 300, and 400 av/cm. The results of the calculation are shown in Fig. 3, from which it is seen that the most advantageous value of the radius \(r_1\) for the specified parameters lies within the range 1.6–2.0 and depends little on the magnitude of \(\dfrac{IW}{2C}\).

Fig. 3. Dependence of the weight of a unit length of the coil on its inner radius.

At \(r_1 = 1.80\) cm, the radius of the striker is \(r_b = 1.63\) cm, and the induction in the striker is \(B_b = 10\,200\) gs. Taking into account that the hammer is a fast-acting electromagnet and the desire that the striker operate close to the saturated regime, we take \(r_1 = 1.60\) cm; then we obtain \(r_b = 1.43\) cm and \(B_b = 13\,900\) gs.

After determining the inner radius of the coil, it is necessary to determine its rational length and its outer radius, i.e., to find such a coil length which, at minimum weight, would have minimum electrical losses. To this end, we find the relationship between the minimum losses in the magne-

in the magnetic core and coil at different lengths. The electrical losses in the coil with steel are determined by the expressions:

\[ P_{\mathrm{v}}=\sigma_{\mathrm{v}}\left(\frac{f}{100}\cdot\frac{B_{\max}}{1000}\right)^2 G_{\mathrm{c}}; \tag{25} \]

\[ P_{\mathrm{g}}=\sigma_{\mathrm{g}}\frac{f}{100}\left(\frac{B_{\max}}{10\,000}\right)^2 G_{\mathrm{c}}; \tag{26} \]

\[ P_{\mathrm{m}}=I^2R, \tag{27} \]

where \(P_{\mathrm{v}}\) and \(P_{\mathrm{g}}\) are the losses due to eddy currents and hysteresis per \(1\ \mathrm{kg}\) of steel;

\[ \begin{aligned} P_{\mathrm{m}}&\text{ is the ohmic loss in the coil;}\\ f&\text{ is the current frequency;}\\ B_{\max}&\text{ is the maximum magnetic induction;}\\ G_{\mathrm{c}}&\text{ is the weight of the magnetic core;}\\ \sigma_{\mathrm{v}}\ \text{and}\ \sigma_{\mathrm{g}}&\text{ are coefficients depending on the grade of steel and the}\\ &\text{geometric dimensions of the magnetic core.} \end{aligned} \]

The magnetic core of the coil is made of pressed powder materials with transverse slots to eliminate short-circuited turns, and operates at a maximum induction \(B_{\max}=10\text{--}12\cdot10^3\) (except for the striker). Taking into account also that the magnetic core must not become overmagnetized, we take the specific hysteresis and eddy-current losses for the adopted magnetic-core material to be the same as for high-quality structural annealed steel, \(P_{\mathrm{c}}=6\ \mathrm{W/kg}\) [4]. Taking into account that the specific losses in the magnetic core are 2–3 times greater than the specific losses in the steels listed in the table, we take the specific losses in the steel to be \(P_{\mathrm{c}}=18\ \mathrm{W/kg}\) [4].

Then

\[ P_{\mathrm{c}}=P_{\mathrm{v}}+P_{\mathrm{g}}=18G_{\mathrm{c}}. \]

Before determining the minimum losses in the steel of the coil, it is necessary to calculate its minimum weight. The window area of the coil is determined from expression (14) and is equal to \(Q=1870\ \mathrm{mm}^2\). At the maximum induction in the coil magnetic core \(B_{\max}=10\,000\ \mathrm{G}\), the weight of the pole as a function of \(r_2\) is determined from the equation

\[ G_{\mathrm{p}}=\frac{\Phi_{\max}}{B_{\max}}\rho_{\mathrm{st}}K_{\mathrm{p}}(r_{\mathrm{p}2}-r_{\mathrm{p}1}) =2\pi r l\,\rho_{\mathrm{st}}-K_{\mathrm{p}}(r_{\mathrm{p}2}-r_{\mathrm{p}1}), \tag{28} \]

where

\[ \begin{aligned} r&\text{ is the pole radius, calculated by formula (21);}\\ r_{\mathrm{p}1}\ \text{and}\ r_{\mathrm{p}2}&\text{ are the inner and outer radii of the pole;}\\ \rho_{\mathrm{st}}&\text{ is the specific weight of structural steel, } \rho_{\mathrm{st}}=7.85\ \mathrm{kg/dm^3};\\ K_{\mathrm{p}}&\text{ is the magnetic-core filling factor, } K_{\mathrm{p}}=0.8. \end{aligned} \]

Then

\[ G_{\mathrm{n}}=\frac{85000}{16000}\,8.75\cdot 0.8(r_2-0.16)=(59.5r_2-95.2)\ \text{g}. \]

The weight of the coil yoke is determined from the equation

\[ G_{\mathrm{y}}=\frac{\Phi_{\max}}{B_{\max}}\rho_{\mathrm{st}}K_{\mathrm{n}}2(C+h_2) =\rho_{\mathrm{st}}K_{\mathrm{n}}2(C+h_2)(r_{\mathrm{y}2}-r_{\mathrm{y}1}), \tag{29} \]

where \(r_{\mathrm{y}1}\) and \(r_{\mathrm{y}2}\) are the inner and outer radii of the yoke;
\(h_2\) is the pole height at \(r_{\mathrm{n}2}\).

Substituting into equation (29) the numerical values of the known quantities, the weight of the yoke is determined from the expression

\[ G_{\mathrm{y}}=\frac{85000}{10000}\cdot 8.75\cdot 0.8\cdot 2(C+h_2) =119(C+h_2)\ \text{g}. \]

Weight of the striker

\[ G_{\mathrm{b}}=\pi r_{\mathrm{b}}^2\rho_{\mathrm{st}}2(C+h_1), \tag{30} \]

where \(h_1\) is the pole height at \(r_{\mathrm{n}1}\); then

\[ G_{\mathrm{b}}=3.14\cdot 1.43^2\cdot 8.75\cdot 2(C+h_1) =113(C+h_1)\ \text{g}. \]

For \(r_{\mathrm{n}1}=1.6\ \text{cm}\) and \(h_1=0.73\ \text{mm}\), the weight of the striker will be \(G_{\mathrm{b}}=113C+83\ \text{g}\).

Assigning a value to the coil length \(2C\), we find the dependence \(P_{\mathrm{c}}=18G_{\mathrm{c}}=f_1(2C)\), which is shown in Fig. 4 (the dependence of the steel weight \(G_{\mathrm{c}}\) on the coil length is given in Fig. 5).

The ohmic losses in the coil are calculated from the following conditions: the operating temperature of the coil is \(t^\circ=90^\circ\text{C}\), the conductor is copper, the wire diameter is \(b_{\mathrm{n}}=1.3\ \text{mm}\), the current is \(I=6.28\ \text{A}\), and the coil window area is \(Q=1870\ \text{mm}^2\).

The resistance of the coil is determined from the equation

\[ R=\rho_{\mathrm{m}}\frac{l_{\mathrm{av}}}{g}W\cdot 10^{-3}, \tag{31} \]

\[ \rho_{\mathrm{m}}=\rho_0(1+\alpha\Delta t) \]
— the resistivity of copper;

\[ \rho_0=0.01754 \]
— the resistivity of copper at \(t^\circ=20^\circ\text{C}\);

\[ \alpha=0.0393 \]
— the temperature coefficient of copper;

\[ \Delta t=90^\circ-20^\circ=70^\circ\text{C}; \]

\[ l_{\mathrm{av}}=\frac{\pi(D_{\mathrm{n}1}+D_{\mathrm{n}2})}{2} \]
— the mean turn length, mm;

\(D_{\mathrm{n}1}\) and \(D_{\mathrm{n}2}\) are the outer and inner diameters of the coil, mm;

\[ g=\frac{\pi d^2}{4} \]
— cross section of the bare wire, \(\text{mm}^2\).

Substituting the values of the quantities entering into equation (32), we obtain

\[ R=\rho_0(1+a\Delta t)\frac{2(D_{\mathrm{p}1}+D_{\mathrm{p}2})}{d^2}\,W\cdot 10^{-3}, \tag{33} \]

or

\[ R=0.01754(1+0.00393\cdot 70)\frac{2(16+D_{\mathrm{p}2})}{1.30^2}\cdot 800\cdot 10^{-3} =0.021(32+D_{\mathrm{p}2})\ \Omega, \]

or

\[ R=0.042(r_1+r_2)=0.67+0.042r_2\ \Omega. \]

Assigning values to the coil length \(2C\), we find the dependence \(P_{\mathrm{m}}=I^2R=f(2C)\), which is shown in Fig. 4. From Fig. 4

Fig. 4. Dependence of the coil losses on its length.

it is seen that at \(r_1=1.6\ \mathrm{cm}\) the minimum electrical losses for the given hammer are obtained at \(2C=11\text{–}13\ \mathrm{cm}\), and \(r=3.04\text{–}3.27\ \mathrm{cm}\).

In addition to the hammer having a high efficiency, it must also be light. To find the minimum weight of the striker, we determine the weight of the coil \(G\) as a function of its length \(2C\) and find the minimum of this function

\[ G=G_{\mathrm{c}}+G_{\mathrm{m}}=f(2C), \]

where \(G_{\mathrm{c}}=2G_{\mathrm{p}}+G_{\mathrm{y}}+G_{\mathrm{b}}\) is the weight of the steel of the coil, which is determined by equations (28, 29, and 30);

\[ G_{\mathrm{m}}=\rho_2\frac{\pi d^2}{4}l_{\mathrm{av}}W \]
is the weight of the copper of the coil.

Substituting into the formula for \(G_{\mathrm{m}}\) the value of \(l_{\mathrm{av}}\), we obtain

\[ G_{\mathrm{m}}=\rho_2\frac{(\pi d)^2(D_{\mathrm{p}1}+D_{\mathrm{p}2})}{8}\,W \tag{34} \]

where \(\rho_2 = 8.3\ \mathrm{kg/dm^3}\) is the specific weight of copper, \(d = 1.3\ \mathrm{mm}\) is the diameter of the bare wire; \(d_{n1}=1.6\ \mathrm{cm}\).

Substituting the values of the quantities entering into equation (34), we obtain

\[ G_m=\frac{8.8(3.14+0.013)^2(1.60+D_{n3})800}{8} =234.5+146.5D_{n3}\ \mathrm{g}, \]

or

\[ G_m=(234.5+293r_2)\ \mathrm{g}. \]

From the results of the calculation we plot the dependence of the weight of the coil on its length (Fig. 5), from which it is seen that the minimum weight of the coil is obtained at \(2C=7\text{–}9\ \mathrm{cm}\). At \(2C>8\ \mathrm{cm}\) the electrical losses decrease only slightly (Fig. 4).

Fig. 5. Dependence of the solenoid weight on the coil length.

On the basis of the calculation presented, the optimal parameters of the coil for the given winding are given in Table 2.

Using the optimal parameters of the coil and the average tractive force of the striker, we determine the number of turns of the coil.

To obtain an impact energy equal to \(3\ \mathrm{kgm}\) with a striker travel \(S=4\ \mathrm{cm}\), the necessary average force (average tractive force of the striker) per coil is

\[ F_m=\frac{A}{Sm}=\frac{3}{0.04\cdot4}=18.8\ \mathrm{kg}. \tag{35} \]

Table 2

To calculate the total electromagnetic tractive force of the striker, it is necessary to determine the tractive force separately for each coil of the winding in accordance with the change in current at various posi-

positions of the striker. Then these forces are to be added, taking into account the time of their action and the position of the striker at that moment. The analytical expression for these forces is very complicated, and they have to be found by a graphical-analytical method, which requires considerable time. Therefore we carry out the calculation of the hammer in a simplified manner, assuming that the average tractive force of the striker is obtained over its travel equal to \(S : 2 = 2\) cm.

Fig. 6. Diagram of the coil of a solenoid hammer.

The number of turns of the coil [1] is determined by the following equation:

\[ W = \frac{ \sqrt{\frac{0.1\,(K_2 U \cdot 10^4)^2 P}{f_m}} }{f}, \tag{36} \]

where \(K_2\) is the voltage waveform coefficient, which depends on the hammer supply circuit; when the hammer coil is supplied according to the circuit shown in Fig. 1, \(K_2 = 0.45 \cdot 1.57 = 0.71\) [8];

\(U = 127\) V — mains voltage;

\(f = 50\) Hz — frequency;

\(P\) is the coil-geometry coefficient

\[ P = \frac{ \frac{\pi r_b^2}{S^2} + g\left(\frac{z}{l}\right)^2 }{ \left[ \frac{\pi r_b^2}{S} + g\,\frac{z^3 + n^3}{3l^2} \right] }, \tag{37} \]

the designations in formula (37) are shown in Fig. 6;

\[ g = \frac{2\pi}{\ln \frac{r_b}{r_1}} \]

is the leakage permeance per unit length of the core.

Substituting the adopted coil parameters into formulas (37) and (36), we obtain:

\[ g=\frac{2\cdot 3,14}{\ln \frac{3,92}{1,43}}=6,2; \]

\[ P= \frac{ \frac{3,14+1,43^2}{2^2} + 6,2\left(\frac{4}{8}\right)^2 }{ \left[ \frac{3,14+1,43^2}{2} + 6,2\frac{4^3+2^3}{3\cdot 8^2} \right] } =0,093; \]

\[ W= \frac{ \sqrt{\frac{0,1(0,71\cdot 127\cdot 10^4)^2\cdot 0,093}{18}} }{25} =815. \]

Comparing the results of coil calculations by formulas (11), (17), and (36) with manufactured and tested specimens of solenoid hammers [7], we have established that the calculation results obtained by formula (36) correspond most closely to practice. Moreover, formula (36) takes into account the selection of optimal coil parameters, which is very important for calculating a light hand tool with minimal electrical losses.

REFERENCES

1. A. Ya. Builov. Fundamentals of Electrical Apparatus Engineering. Gosenergoizdat, 1946.
2. A. I. Moskvitin. The attraction work of a permanent magnet. Elektrichestvo, No. 1, 1939.
3. Roters. Electromagnetic Mechanisms. Gosenergoizdat, 1951.
4. F. A. Stupel. Calculation and Design of Electromagnetic Relays. Gosenergoizdat, 1950.
5. N. A. Shmarov. An Electric Demolition Hammer for the Stone-Coal Industry. ONTI NKTP, 1935.
6. Electrical Engineering Handbook. Gosenergoizdat, 1952.
7. N. P. Ryashentsev. On the question of creating electric drilling machines. Scientific Reports of Higher School, Mining Series, No. 3, 1958.
8. Academy of Sciences of the USSR. Semiconductors in Science and Technology, Vol. 1, Publishing House of the Academy of Sciences of the USSR, 1957.

Noted Misprints

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