On the Theory of a Solenoid Hammer with Free Run-Out of the Striker
Abstract Generated abstract
This paper develops a theoretical description of a solenoid hammer with free runout of the striker, focusing on the timing conditions needed to avoid electromagnetic braking and improve efficiency. Under simplifying assumptions about the magnetic circuit, air gap permeance, and losses, the authors derive coupled equations for coil tractive force, current, inductance, and striker motion from energy balance, circuit equilibrium, and Newtonian dynamics. The analysis yields integrodifferential equations for sinusoidal and constant voltage supply, verifies limiting locked-striker cases against known force expressions, and reformulates the model for motion with spring and friction resistance. The work identifies the variation of the solenoid electrical parameters with air gap as the key factor in calculating hammer operation.
Full Text
On the Theory of a Solenoid Hammer with Free Runout of the Striker
N. P. Ryashentsev, E. M. Timoshenko
(Submitted by the Department of Mining Machines and Mine Transport)
The design and power-supply circuit of a solenoid hammer [1] with free runout of the striker are simple. The same cannot be said of the dynamics and the processes occurring in the hammer during its operation. A necessary condition for obtaining the maximum value of the hammer efficiency is the absence of electromagnetic braking of the striker as it leaves the position of magnetic equilibrium. Electromagnetic braking of the striker will be absent in the case when the flow of current in the coil ceases before the moment it leaves the neutral position. On the other hand, if the current in the coil ceases long before the striker leaves the neutral position, then the useful energy delivered by the coil will be reduced, since, with a small air gap, there is a larger value of the conductance and of the tractive force of the coil at a smaller value of the magnetizing force.
The best operating regime of the hammer, from the point of view of obtaining the greatest efficiency, is observed when the exit of the striker from the neutral position corresponds to the instant of time lying between the cessation of current in the coil and the instant of collapse of the magnetic field of the solenoid, i.e., the instant when the magnetic induction \(B = 0\), or is equal to the residual induction.
The time between the instant at which the current in the coil ceases and the instant of collapse of the magnetic field (magnetic lag) is determined by the properties of the magnetic circuit and by the structural design of the hammer. The magnitude and character of the change in the current in the coil is determined by the transient process, the course of which depends on the electromagnetic properties of the magnetic circuit, the electrical parameters of the circuit (active resistance \(R\), inductance \(L\)), and the rate of their change during operation of the hammer. The rate of change of the electrical parameters of the circuit is determined by the velocity of motion of the striker and, consequently, by the mass of the striker and the magnitude of the driving and braking forces.
Analytical determination of the optimal parameters [1] is possible only by solving the equation of motion of the striker in general form, which presents great difficulties, since the dynamics of the hammer is characterized by a system of nonlinear differential equations.
Below is given the derivation of an equation describing the dynamics of the hammer under the following assumptions: a) the magnetic resistance of the mag-
of the magnetic circuit is absent; b) the magnetic resistance of the main air gap is proportional to its length; c) eddy-current and hysteresis losses are absent.
Starting from the energy balance of the solenoid, the instantaneous value of the tractive force of the coil is determined by the expression
\[ F=\frac{w}{2}\left(i\frac{d\Phi}{d\delta}-\Phi\frac{di}{d\delta}\right). \tag{1} \]
Without taking into account the resistance of the magnetic circuit, the instantaneous value of the flux will be
\[ \Phi=4\pi\cdot10^{-7}wiG, \]
whence
\[ d\Phi=4\pi\cdot10^{-7}w(i\,dG+G\,di). \]
Substituting the values of \(\Phi\) and \(d\Phi\) into expression (1), we obtain
\[ F=2\pi\cdot10^{-7}w^{2}i^{2}\frac{dG}{d\delta}, \tag{2} \]
where \(G\) is the permeance of the air gap. Neglecting leakage fluxes according to [2], the permeance \(G\) is determined by the equation
\[ G=\frac{S_{\mathrm{в}}}{\delta_{0}-\displaystyle\int_{0}^{t} V\,dt}, \tag{3} \]
where
\(S_{\mathrm{в}}\) — area of the air gap;
\(\delta_{0}\) — magnitude of the initial air gap;
\(V\) — velocity of motion of the striker.
From the fundamental equation of motion
\[ F-F_{m}=m\frac{dV}{dt} \]
the velocity of the striker is
\[ V=\frac{d\delta}{dt}=\frac{1}{m}\int_{0}^{t}(F-F_{m})\,dt, \tag{4} \]
where
\(m\) — mass of the striker;
\(F\) — tractive force developed by the coil;
\(F_{m}\) — sum of the forces resisting the motion of the striker;
\(\delta\) — magnitude of the displacement of the striker.
If the obtained value of \(V\) is substituted into expression (3), the permeance of the air gap will be
\[ G=\frac{S_{\mathrm{в}}}{\delta_{0}-\displaystyle\frac{1}{m}\int_{0}^{t}dt\int_{0}^{t}(F-F_{m})\,dt}. \tag{5} \]
Then the time derivative of the permeance is expressed as
\[ \frac{dG}{dt} = \frac{ S_{\mathrm{в}}\displaystyle\int_{0}^{t}(F-F_m)\,dt }{ m\left[ \delta_0-\frac{1}{m}\displaystyle\int_{0}^{t}dt\int_{0}^{t}(F-F_m)\,dt \right]^2 }. \tag{6} \]
Multiplying and dividing equation (2) by \(dt\) and substituting (4) and (6) into the resulting expression, the tractive force of the coil is determined as
\[ F=2\pi\cdot 10^{-7}w^2 i^2\, \frac{S_{\mathrm{в}}}{ \left[ \delta_0-\frac{1}{m}\displaystyle\int_{0}^{t}dt\int_{0}^{t}(F-F_m)\,dt \right]^2 }. \tag{7} \]
The variation of the current \(i\) with time is governed by the equation of electrical equilibrium, which, for a moving striker, has the form
\[ u=iR+L\frac{di}{dt}+i\frac{dL}{dt}. \tag{8} \]
In order to determine the character of the variation of the coil tractive force with time, it is necessary to solve equations (7) and (8) jointly. To this end we replace the quantities entering equation (8) by their values.
The inductance of the coil is
\[ L=4\pi w^2\cdot 10^{-7}G \]
or, substituting the value of the permeance of the air gap, we obtain
\[ L= \frac{ 4\pi w^2\cdot 10^{-7}S_{\mathrm{в}} }{ \delta_0-\frac{1}{m}\displaystyle\int_{0}^{t}dt\int_{0}^{t}(F-F_m)\,dt }. \tag{9} \]
The time derivative of the coil inductance is determined by the expression
\[ \frac{dL}{dt} = \frac{ 4\pi w^2\cdot 10^{-7}S_{\mathrm{в}}\displaystyle\int_{0}^{t}(F-F_m)\,dt }{ m\left[ \delta_0-\frac{1}{m}\displaystyle\int_{0}^{t}dt\int_{0}^{t}(F-F_m)\,dt \right]^2 }. \tag{10} \]
From equation (7), the current flowing in the coil is
\[ i= \frac{ \sqrt{F}\left[ \delta_0-\frac{1}{m}\displaystyle\int_{0}^{t}dt\int_{0}^{t}(F-F_m)\,dt \right] }{ \sqrt{2\pi w^2\cdot 10^{-7}S_{\mathrm{в}}} }. \tag{11} \]
and the derivative of the current with respect to time
\[ \frac{di}{dt} = \frac{ \frac{dF}{dt}\left[ \delta_0 - \frac{1}{m}\int_0^t dt \int_0^t (F-F_m)\,dt \right] -\frac{2F}{m}\int_0^t (F-F_m)\,dt }{ \sqrt{F}\cdot 2\sqrt{2\pi\omega^2\cdot 10^{-7}S_{\mathrm{v}}} }. \tag{12} \]
From the oscillogram (Fig. 1) it is seen that in the time interval from \(t=17.5\) to \(t=20\) msec and from \(t=37.5\) to \(t=40\) msec, owing to the large reverse resistance of the valve, no current flows in the coil and the e.m.f. induced in the coil is zero, i.e., the magnetic field of the coil has time to collapse before the arrival of the next half-wave of voltage.
Fig. 1. Oscillogram of the working process of a solenoid hammer with free runout of the striker (\(S\)—curve of striker travel, current curve \(i\), and voltage \(u\)).
The state of the circuit during this interval is equivalent to switching off the coil. Therefore, characteristic of the hammer operation is the fact that the magnitude of the current and the nature of its variation are established immediately from the moment the hammer is switched on (if the magnetic state of the magnetic circuit at the switching-on of the hammer corresponds to the magnetic state during its operation at the moment when current begins to flow in the coil). We assume that the rebound phenomenon is absent and that, at the moment the hammer is switched on, the striker is at the shank of the working tool. This makes it possible, by introducing boundary conditions, to solve the equation of motion of the striker piecewise, while taking
\[ u = U_m \sin \omega t. \tag{13} \]
Substituting expressions (9–13) into equation (8), after certain transformations we obtain the integrodifferential equation
\[ 4\pi\omega^2\cdot 10^{-7}S_{\mathrm{v}}\frac{dF}{dt} -2U_m\sin\omega t\sqrt{2\pi\omega^2\cdot 10^{-7}S_{\mathrm{v}}}\cdot \sqrt{F} +2R\delta_0F \]
\[ = \frac{2R}{m}F\int_0^t dt\int_0^t (F-F_m)\,dt. \tag{14} \]
To simplify the equation obtained, we replace the constant quantities entering into it, which are determined by the parameters of the hammer, by the following coefficients:
\[ c_1=4\pi\omega^3\cdot 10^{-7}S_{\mathrm{v}};\qquad c_2=2U_m\sqrt{2\pi\omega^2\cdot 10^{-7}S_{\mathrm{v}}}; \]
\[ c_3=2R\delta_0;\qquad c_4=\frac{2R}{m}. \]
Then equation (14) is written as
\[ c_1\frac{dF}{dt}-c_2\sin\omega t\sqrt{F}+c_3F = c_4F\int_0^t dt\int_0^t(F-F_m)\,dt. \tag{15} \]
When the coil is switched on to a constant voltage, by analogy, the change of tractive force with time will be described by the equation
\[ c_1\frac{dF}{dt}-c_2\sqrt{F}+c_3F = c_4F\int_0^t dt\int_0^t(F-F_m)\,dt. \tag{16} \]
If the only driving forces are those developed by the hammer coil, then, before the beginning of motion of the striker, the equality \(F_m=F\) holds; therefore, for a motionless striker, the expression on the right-hand side of equations (15), (16) is equal to zero,
\[ c_4F\int_0^t dt\int_0^t(F-F_m)\,dt=0. \]
For the case of a locked striker, expressions (15), (16) will respectively have the form
\[ c_1\frac{dF}{dt}-c_2\sin\omega t\sqrt{F}+c_3F=0 \tag{17} \]
\[ c_1\frac{dF}{dt}-c_2\sqrt{F}+c_3F=0. \tag{18} \]
Solving equation (17), we obtain
\[ F= \left[ \frac{ c_2c_3\sin\omega t-2c_1c_2\cos\omega t+2c_1c_2\omega e^{-\frac{c_3t}{2c_1}} }{ c_3^2+4c_1^2\omega^2 } \right]^2. \]
Substituting the values of the coefficients \(c_1\), \(c_2\), and \(c_3\) and carrying out certain transformations, we have
\[ F=\frac{L_0}{2\delta_0}I_m^2 \left[ \sin(\omega t-\varphi)+\sin\varphi\,e^{-\frac{Rt}{L_0}} \right]^2. \tag{19} \]
For the case when the coil is switched on to a constant voltage with the striker locked, solving equation (18), we obtain
\[ F=\frac{c_2^2}{c_3^2} \left( 1-e^{-\frac{c_3t}{2c_1}} \right)^2, \]
or, substituting the values of \(c_1\), \(c_2\), and \(c_3\), we have
\[ F=\frac{L_0}{2\delta_0}I_y^2 \left( 1-e^{-\frac{Rt}{L_0}} \right)^2. \tag{20} \]
In expressions (19), (20)
\(L_0\) is the inductance of the coil at the air gap \(\delta_0\);
\(R\) — active resistance of the winding;
\(\varphi = \operatorname{arctg}\dfrac{\omega L_0}{R}\) — phase angle between current and voltage;
\(I_m = \dfrac{U_m}{Z}\) — amplitude value of the steady-state current;
\(Z = \sqrt{R^2 + (\omega L_0)^2}\) — total impedance of the coil;
\(I_y = \dfrac{U}{R}\) — steady-state current of the coil.
Formulas (19, 20), for determining the change in the tractive force of the coil when the latter is connected to sinusoidal and constant voltage, are analogous to those found in the literature [2, 3]. This testifies to the correctness of the approach to deriving equations (15, 16).
The sum of the resistance forces \(F_m\) to the motion of the striker may be a function of the path. Thus, in the presence of braking forces of the spring and assuming the friction forces to be constant over the entire path of motion, we have
\[ F_m = F_{cm} + c_5 \delta, \]
where \(c_5\) is the spring stiffness coefficient.
In this case, proceeding from the fundamental equation of motion of the striker,
\[ F - (F_{cm} + c_5\delta) = m \frac{d^2\delta}{dt^2} \tag{21} \]
in equations (15, 16) the variable \(F\) should be replaced by the variable \(\delta\). Then, after transformations, we obtain
\[ c_1 m \frac{d^3\delta}{dt^3} + c_1 c_5 \frac{d\delta}{dt} - c_2 \sin \omega t \sqrt{ m \frac{d^2\delta}{dt^2} + c_5\delta + F_{cm} } + \]
\[ + (c_3 - m c_4 \dot{\delta}) \left( m \frac{d^2\delta}{dt^2} + c_5\delta + F_{cm} \right) = 0. \tag{22} \]
And when the coil is connected to constant voltage, we have
\[ c_1 m \frac{d^3\delta}{dt^3} + c_1 c_5 \frac{d\delta}{dt} - c_2 \sqrt{ m \frac{d^2\delta}{dt^2} + c_5\delta + F_{cm} } + \]
\[ + (c_3 - c_4 m \dot{\delta}) \left( m \frac{d^2\delta}{dt^2} + c_5\delta + F_{cm} \right) = 0 \tag{23} \]
Equations (22) and (23), under the adopted assumptions, describe the operating processes of the hammer for the cases when the hammer coil is connected to sinusoidal (22) and constant voltage (23). The solution of equations (22) and (23), comparison of the calculated data with experimental data, and also the admissibility of the adopted assumptions for each particular case will be published later.
The determining factor in calculating a solenoid hammer is the value and the character of the variation of the electrical parameters of the solenoid with the change in the air gap.
REFERENCES
- Ryashentsev N. P., Timoshenko E. M., Frolov A. V., Tronov A. P., On the parameters of a solenoid hammer with free runout of the striker. The article is published in the present collection.
- Builov A. Ya., Fundamentals of Electrical Apparatus Construction. Gosenergoizdat, 1946.
- Babikov M. A., Electrical Apparatus. Gosenergoizdat, 1951.