On the Dynamic Coefficient under Impact Damping

V. F. Gorbunov, A. I. Pashnin, S. S. Deroberti, I. G. Reznikov

As is known [1], the dynamic coefficient, or amplification factor, is the ratio of the amplitude of forced oscillations \(y\) to the static displacement \(S_a\), i.e.,

\[ A^\circ = \frac{y}{S_a} = \frac{1}{ \sqrt{ \left(1-\frac{\Omega^2}{\omega_0^2}\right)^2 + \frac{4\nu^2\Omega^2}{\omega_0^4} } }, \tag{1} \]

where \(\Omega\) is the circular frequency of the forced oscillations of the system;
\(\omega_0\) is the circular frequency of the free oscillations of the system;
\(\nu\) is the damping coefficient.

The dynamic coefficient is the most important characteristic of vibration damping. Its value makes it possible to judge how effective one or another type of damping is. Therefore, the authors set themselves the task of obtaining the dependence of the dynamic coefficient on the geometric parameters of an impact damper and on the principal characteristics of the disturbing force.

The scheme of an elastic system with an impact damper is shown in Fig. 1. Here \(M\) and \(K\) are, respectively, the mass and stiffness of the system being damped; \(m\) is the mass of the damping element, placed with a clearance \(\delta\) in the cavity of the body of mass \(M\).

The most dangerous operating regimes of machines are resonant ones; therefore we shall confine ourselves to considering a system with an impact damper in the resonance region. For this case \(\omega_0 = \Omega\), and therefore the expression for the dynamic coefficient takes the following form:

\[ A^\circ = \frac{\omega_0}{2\nu}. \tag{2} \]

In this expression, \(\nu\) can be expressed through the damping or resistance coefficient \(h_{\mathrm{eqv}}\), which for a system with an impact damper, under the assumption that the action of the impact damper is equivalent to the action of a viscous-friction force, may be represented [2] as

\[ h_{\mathrm{eqv}} = \frac{m\Omega}{\pi}(1-\cos\varphi)^2. \tag{3} \]

where \(m\) is the mass of the damping element;
\(\varphi\) is the phase of impacts.

In view of the fact that the value \(\nu\) is determined by the expression \(\dfrac{h}{2M}\) [1], from (2) and (3) we obtain

\[ A^\circ=\frac{\pi\cdot\mu}{(1-\cos\varphi)^2}, \tag{4} \]

where \(\mu\) is the ratio of the colliding masses \(\dfrac{M}{m}\) (Fig. 1).

Fig. 1. Diagram of an impact damper

The collision phase (3), depending on the ratio of the gap \(\delta\) to the amplitude of the forced vibrations of the system \(y\), according to [2], is determined by the expression

\[ \frac{\delta}{y}=\varphi-\sin\varphi, \tag{5} \]

where \(\varphi=\Omega\cdot t_{\mathrm{ud}}\) is the collision phase, and \(t_{\mathrm{ud}}\) is the time from impact to impact.

Solving equations (4) and (5) jointly and taking \(y=A^\circ\cdot S_a\), we obtain the expression for the dynamic coefficient for a system with an impact damper

\[ A^\circ= \frac{1}{\Omega\cdot t_{\mathrm{ud}}\cdot S_a} \left[ \delta+ \frac{\pi\cdot\mu\cdot S_a} {\tan \dfrac{\Omega\cdot t_{\mathrm{ud}}}{2}\left(1-\cos\Omega\cdot t_{\mathrm{ud}}\right)} \right]. \tag{6} \]

It follows from equation (6) that the dynamic coefficient depends primarily on the gap \(\delta\) and the collision phase \(\varphi=\Omega\cdot t_{\mathrm{ud}}\). The collision phase is an important characteristic of impact damping, since any change in the amplitude of the excitation, the gap, the material, and the shape of the colliding elements entails its change. According to dependence (6), the dynamic coefficient decreases somewhat with an increase in the amplitude of the exciting force \(S_a\).

This conclusion is well confirmed by the experimental results shown in Fig. 2. The tests were carried out on the ST-3000 vibration test stand. The impact damper, made of steel, had point contact between the colliding surfaces. The gap was varied from 0 to 10 mm. The ratio of the colliding masses was \(\mu=10\). The decrease in the dynamic coefficient (Fig. 2) can be explained by the fact that, with increasing \(S_a\), the acceleration amplitude of the forced vibrations of the system increases somewhat; accordingly, the impact energy increases and the damping capacity of the impact vibration absorber rises to a certain value.

It should be noted that formula (6) does not take into account the influence of the shape and material of the colliding surfaces. To take them into account, we introduce coefficients \(K_1\) and \(K_2\), which reflect the dependence of the coefficient

the dynamics accordingly on the shape and material. Then expression (6) will take the form

\[ A^0=\frac{k_1\cdot k_2}{\Omega\cdot t_{\mathrm{ud}}\cdot S_a} \left[ \delta+\frac{\pi\cdot\mu\cdot S_a}{\operatorname{tg}\frac{\Omega\cdot t_{\mathrm{ud}}}{2}\left(1-\cos\Omega\cdot t_{\mathrm{ud}}\right)} \right]. \tag{7} \]

Fig. 2. Effect of the gap and excitation amplitude on vibrations

The coefficients \(K_1\) and \(K_2\) can be determined experimentally, and their values for some materials are given in Table 1.

Table 1

Expression (7) makes it possible, to some extent, to analyze the efficiency of vibration damping as a function of the parameters of the impact damper and the magnitude of the disturbance, and also to establish the possibility of using an impact damper in a given object.

REFERENCES

1. S. P. Timoshenko. Vibrations in Engineering. Publishing House “Nauka,” Main Editorial Office of Physico-Mathematical Literature. Moscow, 1967.

2. A. E. Kobrinskii. The principle of action and a brief theory of D. I. Ryzhkov’s vibration damper design. Vestnik mashinostroeniya, No. 9, Moscow, Mashinostroenie Publishing House, 1954.