MODELING OF DISTURBING FORCES

A. V. TRIKHANOV, V. M. RAZIN

(Presented by the scientific seminar of the Faculty of Automation and Computer Engineering)

Analysis of the interrelation between internal disturbing forces and the motion of the body, striker, and also the tool of a pneumatic hammer shows that, in those cases where the working tool may be considered immobile, when studying the vibration of the body there is no particular need to consider the motion of the striker.

For many designs in these cases, the disturbing forces acting on the hammer body can be specified in the form of variable periodic functions that do not depend on the motion of the body.

Studies of the motion of the body under various forms of disturbing forces are very important.

When solving the nonlinear differential equation of the motion of the body by the method of mathematical modeling, it becomes necessary to reproduce various forms of disturbing forces. Certain limited possibilities in this direction are provided by a low-frequency generator of periodic oscillations. For example, a generator of the NGPK-3 type [1] serves to create electrical oscillations only of the following forms: rectangular (with a duty factor equal to \( \frac{1}{2} \)), sawtooth, triangular, and also sinusoidal.

In investigations it is required to realize a broader range of different forms of disturbing forces.

For these purposes we have proposed using variable-coefficient blocks, which are usually included in electronic modeling installations. One period of the disturbing force is reproduced by multiplying a constant voltage by a variable coefficient whose graph coincides with the form of the disturbing force. As is known, the graph of a variable coefficient is approximated by means of a stepped curve and is set according to the reference points of a setting grid on a plane whose coordinate axes are the time axis and the axis of the values of the variable coefficient. The setting grid is obtained by dividing the working range into 100 equal intervals \(T_0\) along the time axis and into 200 equal intervals (from \(-1\) to \(+1\)) along the axis of the magnitude of the variable coefficient [2].

The motion of the body must be traced under the action of the disturbing force over the course of several periods. It would be possible to set these periods during the time of a complete working cycle of the block, corresponding to 100 intervals \(T_0\).

Obviously, the more periods of the force action are sampled, the less accurately the stepwise approximation of an individual period is carried out. The greatest accuracy will be achieved if one period of the force is sampled over the complete operating cycle of the unit.

However, no serially produced electronic modeling installation provides an operating mode in which, in solving, it would be possible to use the variable-coefficient unit for several complete cycles of its operation. Very close to the indicated mode is the repeat-solution mode provided in serially produced models. In this mode, at the end of a complete operating cycle of the variable-coefficient unit, the capacitors in the negative-feedback circuits of the integrating amplifiers are automatically discharged and the initial conditions of the problem being solved are restored; that is, the circuit of the modeling installation returns to its initial state.

Apparently, to implement the operating mode we propose, one can use the repeat-solution mode if the return of the installation circuit to the initial state is blocked. Let us consider the circuit of an operational-amplifier unit of type BOU-2 in the integration mode (Fig. 1). The discharge of capacitor \(C_1\) at the 101st step of the selector of the variable-coefficient unit is carried out through the resistance \(R_4\), which is connected to it by means of the normally closed contacts \(1P_{\mathrm{d}}\), \(2P_{\mathrm{y}}\) and the normally open contact \(2P_{\mathrm{k}}\). In this case the relay \(P_{\mathrm{k}}\) is energized, while relays \(P_{\mathrm{y}}\) and \(P_{\mathrm{d}}\) are de-energized. At the 102nd step the relay \(P_{\mathrm{k}}\) is de-energized. Through the normally closed contact \(2P_{\mathrm{k}}\), the circuit for setting the initial conditions is closed. By applying the voltage \(U_{\mathrm{ny}}\), the initial condition is set at the output of the unit.

Fig. 1. Circuit of BOU-2

At the first step the relay \(P_{\mathrm{y}}\) is energized; the input circuit, through the normally open contact \(1P_{\mathrm{y}}\) and the normally closed contact \(1P_{\mathrm{d}}\), is connected to the input of the dc amplifier UPT-4. Integration of the input voltage \(U_{\mathrm{in}}\) is performed.

Let us consider the operation of the control panel in the repeat-solution mode. A simplified circuit of the control panel of the IDPT-5 model is shown in Fig. 2, and its operating diagram in Fig. 3.

When the model is started, starting relay $P_{\mathrm{p}}$ operates and is locked through its own contact $4P_{\mathrm{p}}$. Contact $2P_{\mathrm{p}}$ switches on relay $P_0$ and prepares the circuit for the operation of relay $P_1$. Relay $P_0$ repeats the pulses of the interrupter. Through its contact $2P_0$, relay $P_1$ is energized, and it repeats the operation of relay $P_0$. Through contact $1P_0$, $+26\ \mathrm{V}$ is applied to contactor $K$.

During a pulse, relay $P_1$ operates and, by contact $1P_1$, switches on relay $P_2$, which is locked by its contact $1P_2$ for the entire period of operation of the circuit. At the beginning of the next pause, when relay $P_1$ releases its armature, relay $P_3$ operates and is locked through its contact $2P_3$.

At the same time, at the beginning of the pause, all control relays $P_{\mathrm{u}}$ in the operational-amplifier units operate. Through contact $1P_3$, $-26\ \mathrm{V}$ is applied to the contactor, which, with the appearance of the next pulse,

Fig. 3. Diagram of operation of the control panel

comes into operation and, through its contact $1K$, applies pulses to all seekers of the variable-coefficient blocks of the model (this, in particular, ensures the synchronism of their operation).

At the 10th step of the seeker in the BOU-2 block (to the winding of relay $P_{\mathrm{k}}$), $-26\ \mathrm{V}$ is applied, which de-energizes relay $P_{\mathrm{k}}$, which, by its contact $1P_{\mathrm{k}}$, energizes relay $P_{\mathrm{u}}$. Relays $P_{\mathrm{k}}$ and $P_{\mathrm{u}}$ close the discharge circuit of capacitor $C_1$ through a small resistance.

At the 100th step of the seeker, relay $P_{\mathrm{k}}$ is de-energized. In BOU-2, the circuit for setting the initial conditions is switched on. At the zeroth step of the seeker, $U_0$ operates relay $P_{\mathrm{k}_1}$ of the control block and, by its contact $1P_{\mathrm{k}_1}$, opens the $-26\ \mathrm{V}$ circuit, thereby de-energizing contactor $K$ and relay $P_{\mathrm{p}}$,

Fig. 2. Simplified diagram of the control panel of the IPT-5 model.

Visible inscriptions in the diagram: “from the interrupter”; \(+26\,\mathrm{V}\); \(-26\,\mathrm{V}\); \(+26\,\mathrm{V}\) to \(P_k\); \(-26\,\mathrm{V}\) to \(P_k\); \(+26\,\mathrm{V}\) to \(I_k\); \(-26\,\mathrm{V}\) to \(I_k\); \(P_0\), \(P_6\), \(P_k\), \(K\), \(P_1\), \(P_2\), \(P_3\), \(P_n\), \(D_k\), \(D_{k2}\), \(D_3\), \(D_n\), \(U_0\); \(1P_k\), \(2P_k\), \(3P_3\), \(4P_3\), \(1P_1\), \(2P_2\), \(2P_6\), \(1P_3\), \(1K\); \(1RU_0\), \(5RU_0\), \(8RU_0\), \(7RU_0\); terminals \(10\), \(11\), \(12\), \(122\), \(123\), \(124\), \(125\), \(126\).

Through contact \(2P_{\mathrm{k}}\), relay \(P_{\mathrm{b}}\) operates; with its contact \(3P_{\mathrm{b}}\) it prepares relay \(P_{\mathrm{p}}\) for operation.

As soon as relay \(P_{\mathrm{p}}\) releases, the relays will begin to release sequentially: \(P_1\), \(P_2\), \(P_3\), \(P_{\mathrm{k}_2}\), and, finally, relay \(P_{\mathrm{k}_1}\), which with its contact \(1P_{\mathrm{k}}\) will close the circuit \(-26\ \mathrm{V}\), after which relay \(P_{\mathrm{p}}\) operates and the process begins anew.

From consideration of the BOU-2 circuit and analysis of the operation of the control panel of the IPT-5 model, it follows that the return of the circuit to the initial state can be blocked by switching on relay \(P_{\mathrm{d}}\) (stop relay) at steps 101, 102, 103, and 1. For this it is necessary to make a slight change in the circuit of the model control panel, namely to add toggle switch \(T\) and to use a separate row \((7PU_0)\) of the step selector \(U_0\).

The circuit is transferred to the proposed mode by setting toggle switch \(T\) to the “On” position.

From the 2nd step through the 100th step, the panel circuit operates exactly as in the solution-repetition mode. Beginning with step 101, by means of row \(7PU_0\) of the step selector \(U_0\), \(+26\ \mathrm{V}\) is applied through toggle switch \(T\) to relay \(P_{\mathrm{d}}\) (in BOU-2); \(-26\ \mathrm{V}\) is continuously applied to relay \(P_{\mathrm{d}}\) by means of toggle switch \(T\).

Relay \(P_{\mathrm{d}}\) is switched on, as was noted, at the same time as relay \(P_{\mathrm{k}}\) is switched on.

With its normally closed contact \(1P_{\mathrm{k}}\), it opens the circuit through which the voltage \(+26\ \mathrm{V}\) is applied to relay \(P_{\mathrm{y}}\). With a certain delay relative to the switching-on of relay \(P_{\mathrm{k}}\), and consequently also relay \(P_{\mathrm{d}}\), relay \(P_{\mathrm{y}}\) is switched off. This delay ensures sufficient reliability of the operation of BOU-2 at the beginning of step 101.

Relay \(P_{\mathrm{d}}\) must remain switched on until relay \(P_{\mathrm{y}}\) is switched on for a new working cycle of the variable-coefficient unit.

As can be seen from the diagram (Fig. 3), relay \(P_{\mathrm{d}}\) must not be de-energized when the step selector is switched from the 103rd panel to the zero panel, since for some time relay \(P_{\mathrm{y}}\) is in the switched-on state.

Consequently, relay \(P_{\mathrm{d}}\) can be de-energized at the beginning of the second step of the next complete working cycle of the variable-coefficient unit. Thus, the first step of the variable-coefficient unit should not be used in the operating mode under consideration; the force period must be set starting from the 2nd step of the selector. Obviously, this cannot lead to any appreciable increase in the error of setting the force curve.

In the solution-repetition mode, toggle switch \(T\) must be in the “Off” position.

As a result, relay \(P_{\mathrm{d}}\) will not be switched on, which is shown in Fig. 3 by hatching. In this case, integration will begin when the step selector is positioned on the zero panel at the moment relay \(P_{\mathrm{y}}\) is switched on.

REFERENCES

1. Low-frequency generator of periodic oscillations, type NGPK-3. Technical passport PR 3.268.001T.P.

2. IPT-5 electronic model. Technical description and operating instructions M006-00-000TO.