ON THE POSSIBILITY OF APPLYING A DYNAMIC VIBRATION ABSORBER TO IMPACT MACHINES

P. M. Alabuzhev, V. I. Kopytov

The problem of vibrations of the body of a hammer (pneumatic, drilling, and other types) with an attached dynamic absorber, under certain assumptions, reduces to the problem of vibrations of a system “elastic connection—mass—stop” with two degrees of freedom.

Fig. 1. Principal diagram of a mechanical system with a dynamic vibration absorber.

In work [1], vibrations of a similar mechanical system were investigated without taking resistance forces into account. In the present article a solution of the above problem is given with allowance for the forces of resistance to the motion of the attached mass. In doing so, we limited ourselves to considering the case in which the resistance forces are proportional to the first power of the velocity [2, 3, 5].

Let us consider a mechanical oscillatory system consisting of two masses, \(m_1\) and \(m_2\), connected to each other by spring II (an elastic connection) (Fig. 1). Mass \(m_1\), by means of a spring, is ...

is fastened to the fixed point \(A\). In addition, a cylinder with a certain viscous liquid is attached to the mass \(m_1\). In the cylinder with the liquid there moves a piston connected to the mass \(m_2\). The resistance forces of the liquid that develop during the motion of the piston, and consequently of the mass \(m_2\), will be assumed proportional to the first power of the velocity of motion of the piston relative to the cylinder.

Let us investigate the forced oscillations of the masses \(m_1\) and \(m_2\), if the mass \(m_1\) has a fixed rigid limiter of oscillations. The disturbing force acting on the mass \(m_1\) varies according to the law \(H\sin\omega t\).

We shall seek the solution of the problem under the following assumptions: the impact of the mass \(m_1\) against the limiter is inelastic; the elastic forces in springs I and II vary according to a linear law.

The differential equations of motion of the system are written in the form:

\[ \left. \begin{aligned} m_1 \ddot{x}_1-\alpha(\dot{x}_2-\dot{x}_1)+F(x_1)-c_2(x_2-x_1)-H\sin\omega t&=0;\\ m_2 \ddot{x}_2+\alpha(\dot{x}_2-\dot{x}_1)+c_2(x_2-x_1)&=0, \end{aligned} \right\} \tag{1} \]

where

\[ \begin{aligned} x_1&\text{ is the displacement of the mass }m_1\text{ relative to the position of the unstretched spring I, point }O_1;\\ x_2&\text{ is the displacement of the mass }m_2\text{ relative to the position of the unstretched spring II, point }O_2;\\ F(x_1)&\text{ is the elastic force of spring I;}\\ \alpha(\dot{x}_2-\dot{x}_1)&\text{ is the resistance force developing during the motion of the mass }m_2;\\ \alpha&\text{ is the coefficient of proportionality (resistance);}\\ c_2&\text{ is the stiffness coefficient of spring II.} \end{aligned} \]

We shall seek the forced oscillations in the first approximation of the method of Academician B. G. Galerkin [1] in the form of sinusoidal functions having the same frequency as the disturbing force, but shifted in phase relative to the force:

\[ \left. \begin{aligned} x_1&=A+a_1\sin(\omega t-\gamma_1);\\ x_2&=B+a_2\sin(\omega t-\gamma_2). \end{aligned} \right\} \tag{2} \]

Here \(A\) and \(B\) are certain quantities independent of the time \(t\).

Substitute the values of \(x_1, x_2\) and their derivatives into equations (1). Multiply the resulting expressions by the first variations \(\delta x_1\) and \(\delta x_2\), and integrate the sum of these expressions with respect to \(t\) from \(0\) to \(\dfrac{2\pi}{\omega}\). Owing to the independence of the variations \(\delta A,\ \delta B,\ \delta a_1,\ \delta a_2,\ \delta\gamma_1\) and \(\delta\gamma_2\), we obtain

a system of six equations for determining the six quantities \(A;\ B;\ a_1;\ a_2;\ \gamma_1;\ \gamma_2\):

\[ \left. \begin{aligned} c_2(B-A) &= \frac{\omega}{2\pi} \int_{0}^{\frac{2\pi}{\omega}} F\left[A+a_1\sin(\omega t-\gamma_1)\right]\,dt;\\[4pt] m_1\omega^2 a_1+\alpha\omega a_2\sin(\gamma_2-\gamma_1) +c_2a_2\cos(\gamma_2-\gamma_1)-c_1a_1+H\cos\gamma_1 &= \frac{\omega}{\pi} \int_{0}^{\frac{2\pi}{\omega}} F\left[A+a_1\sin(\omega t-\gamma_1)\right]\sin(\omega t-\gamma_1)\,dt;\\[4pt] -\alpha\omega a_1+\alpha\omega a_2\cos(\gamma_2-\gamma_1) -c_2a_2\sin(\gamma_2-\gamma_1)+H\sin\gamma_1 &= \frac{\omega}{\pi} \int_{0}^{\frac{2\pi}{\omega}} F\left[A+a_1\sin(\omega t-\gamma_1)\right]\cos(\omega t-\gamma_1)\,dt;\\[4pt] c_2(B-A)&=0;\\ -m_2\omega^2a_2+\alpha\omega a_1\sin(\gamma_2-\gamma_1)+c_2a_2-c_2a_1\cos(\gamma_2-\gamma_1)&=0;\\ \alpha\omega a_2-\alpha\omega a_1\cos(\gamma_2-\gamma_1)-c_2a_1\sin(\gamma_2-\gamma_1)&=0. \end{aligned} \right\} \tag{3} \]

To compute the integrals appearing in the first three equations of system (3), it is necessary to specify a concrete expression for the elastic force

\[ F(x_1)=F\left[A+a_1\sin(\omega t-\gamma_1)\right]. \]

Introducing into consideration a new parameter \(a_0\), as the distance from the position of the unstretched first spring to the stop, the elastic force \(F(x_1)\) can be written in the form

\[ F(x_1)= \begin{cases} c_1x_1, & \text{for } x_1<a_0,\\ c_1a_0, & \text{» } x_1=a_0. \end{cases} \tag{4} \]

where \(c_1\) is the stiffness coefficient of the first spring.

From the condition that the mass \(m_1\) comes into contact with the stop, we have

\[ A+a_1=a_0, \tag{5} \]

i.e., contact occurs at the instant of time \(t_1\), determined from the equation

\[ \omega t_1-\gamma_1=\frac{\pi}{2}, \]

whence

\[ t_1=\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}. \tag{6} \]

Consequently, the interval of integration with respect to \(t\) is divided into four parts, analogously to the problem of the oscillations of a two-mass system without resistance forces [4]:

\[ F[A+a_1\sin(\omega t-\gamma_1)]= \begin{cases} c_1[A+a_1\sin(\omega t-\gamma_1)], \\[0.4em] \qquad \text{for } 0<t<\dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}-\sigma; \\[1.0em] c_1a_0=\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}-\sigma}^{\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}} \varphi\!\left(t-\frac{\pi}{2\omega}-\frac{\gamma_1}{\omega}+\sigma\right)\,dt = \\[1.0em] \qquad\displaystyle =\lim_{\sigma\to 0}\frac{\omega}{\pi}\int\limits_0^\sigma \varphi(\tau)\,d\tau, \\[0.8em] \qquad \text{for } \dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}-\sigma<t<\dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}, \\[1.0em] c_1a_0=\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}}^{\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}+\sigma} \varphi\!\left(\frac{\pi}{2\omega}+\frac{\gamma_1}{\omega}+\sigma-t\right)\,dt \\[1.0em] \qquad\displaystyle =\lim_{\sigma\to 0}\frac{\omega}{\pi}\int\limits_0^\sigma \varphi(\tau)\,d\tau, \\[0.8em] \qquad \text{for } \dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}<t<\dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}+\sigma, \\[1.0em] c_1[A+a_1\sin(\omega t-\gamma_1)], \\[0.4em] \qquad \text{for } \dfrac{\pi}{2\omega}+\dfrac{\gamma_1}{\omega}+\sigma<t<\dfrac{2\pi}{\omega}. \end{cases} \tag{7} \]

Substituting this expression for the elastic force into the integrals of expressions (3), after calculations analogous to those set forth in works [1; 4], we obtain

\[ \left. \begin{aligned} &c_2(B-A)=c_1A+c_1a_0;\\ &m_1\omega^2a_1-c_2a_1+\alpha\omega a_2\sin(\gamma_2-\gamma_1) +c_2a_2\cos(\gamma_2-\gamma_1)+H\cos\gamma_1 =c_1a_1+2c_1a_0;\\ &\alpha\omega a_1-\alpha\omega a_2\cos(\gamma_2-\gamma_1) +c_2a_2\sin(\gamma_2-\gamma_1)-H\sin\gamma_1=0;\\ &c_2(B-A)=0;\\ &-m_2\omega^2a_2+\alpha\omega a_1\sin(\gamma_2-\gamma_1)+c_2a_2 -c_2a_1\cos(\gamma_2-\gamma_1)=0;\\ &\alpha\omega a_2-\alpha\omega a_1\cos(\gamma_2-\gamma_1) -c_2a_1\sin(\gamma_2-\gamma_1)=0. \end{aligned} \right\} \tag{8} \]

From the first and fourth equations of the given system and condition (5) we have the relation

\[ c_1 a_0 = c_1(a_1-a_0). \tag{9} \]

Substitute expression (9) into the second equation of system (8) and write the given equations in dimensionless quantities, which, obviously, reduces the number of parameters of the mechanical system and makes it possible to present the system of equations in a form more convenient for solution:

\[ \left\{ \begin{aligned} &\left(3-p_1^2+\frac{c_2}{c_1}\right)\frac{a_1}{h} -\left[2\beta p_2\sin(\gamma_2-\gamma_1)+\cos(\gamma_2-\gamma_1)\right]\frac{c_2}{c_1} \frac{a_2}{h} -\cos\gamma_1 =2\frac{a_0}{h}; \\[4pt] &2\zeta p_1\,\frac{c_2}{c_1}\frac{a_1}{h} -\left[2\beta p_2\cos(\gamma_2-\gamma_1)-\sin(\gamma_2-\gamma_1)\right]\frac{c_2}{c_1}\frac{a_2}{h} -\sin\gamma_1=0; \\[4pt] &2\beta p_2\,\frac{a_1}{h}\sin(\gamma_2-\gamma_1) -\frac{a_1}{h}\cos(\gamma_2-\gamma_1) =(p_2^2-1)\frac{a_2}{h}; \\[4pt] &\frac{a_1}{h}\sin(\gamma_2-\gamma_1) +2\beta p_2\,\frac{a_1}{h}\cos(\gamma_2-\gamma_1) =2\beta p_2\,\frac{a_2}{h}. \end{aligned} \right. \tag{10} \]

where the following designations have been introduced:

\[ p_1=\frac{\omega}{\omega_{01}};\qquad p_2=\frac{\omega}{\omega_{02}};\qquad \omega_{01}=\sqrt{\frac{c_1}{m_1}};\qquad \omega_{02}=\sqrt{\frac{c_2}{m_2}}; \]

\[ 2\zeta=\frac{z}{m_2\omega_{02}};\qquad h=\frac{H}{c_1}. \tag{11} \]

From the last two equations of system 10 we obtain the relation

\[ (1+4\zeta^2p_2^2)\frac{a_1}{h} = \sqrt{\,4\beta^2p_2^6+\left[1-4\zeta^2\right]^2p_2^2\,}\left(\frac{a_2}{h}\right). \tag{12} \]

The first two equations of system (10), after certain transformations, are reduced to a quadratic equation with respect to the quantity \(\frac{a_1}{h}\), from which we find

\[ \frac{a_1}{h} = \frac{ 2\chi\frac{a_0}{h} \pm \sqrt{ \chi^2+ \left[1-4\left(\frac{a_0}{h}\right)^2\right]\varepsilon^2 } }{ \chi^2+\varepsilon^2 }, \tag{13} \]

where the quantities \(\chi\) and \(\varepsilon\) have the expressions

\[ \chi= \left(3-p_1^2+\frac{c_2}{c_1}\right) -\frac{c_2}{c_1}\cdot \frac{ (1+4\beta^2p_2^2)\left[2\zeta^2(2+p_2^2)p_2^2+(1-p_2^2)\right] }{ 4\beta^2p_2^6+\left[1-(1-4\beta^2)p_2^2\right]^2 }; \tag{14} \]

\[ \varepsilon = 2\zeta p_1^2\frac{c_2}{c_1} \left\{ 1- \frac{ (1+4\beta^2p_2^2)\left[1-2(1-2\zeta^2)p_2^2\right] }{ 4\beta^2p_2^6+\left[1-(1-4\beta^2)p_2^2\right]^2 } \right\}. \]

From expression (12) we have

\[ \frac{a_2}{h} = \frac{1+4\beta^2p_2^2} {\sqrt{\,4\beta^2p_2^6+\left[1-\left(1-4\beta^2\right)p_2^2\right]^2\,}} \cdot \frac{a_1}{h}, \tag{12a} \]

where \(\dfrac{a_1}{h}\) is determined by relation (13). The minus sign before the radical should be discarded, since it does not satisfy the problem under consideration.

Thus, the amplitudes \(\dfrac{a_1}{h}\) and \(\dfrac{a_2}{h}\) of the forced vibrations of the masses \(m_1\) and \(m_2\), as functions of the frequency of the exciting force and of the parameters of the system \(\left(\dfrac{a_0}{h}; \dfrac{c_2}{c_1}; \beta\right)\), are expressed by equations (13) and (12, a).

The resonance curves for the masses \(m_1\) and \(m_2\) are obtained from the system of equations (10), in which the term characterizing the action of the external force must be set equal to zero. Denoting the amplitudes of the vibrations of the masses \(m_1\) and \(m_2\) by the same letters, but enclosed in parentheses, we obtain

\[ \left(\frac{a_1}{h}\right) = \frac{2\,\dfrac{a_0}{h}}{\chi}; \tag{15} \]

\[ \left(\frac{a_2}{h}\right) = \frac{1+4\beta^2p_2^2} {\sqrt{\,4\beta^2p_2^6+\left[1-\left(1-4\beta^2\right)p_2^2\right]^2\,}} \left(\frac{a_1}{h}\right). \tag{16} \]

If, in the course of the vibrations, the mass \(m_1\) does not reach the stop, then we have an ordinary linear mechanical system with two degrees of freedom, taking into account the forces resisting the motion of the mass \(m_2\). In this case the elastic force is

\[ F(x)=c_1x_1. \]

Therefore the system of equations (3), in dimensionless quantities, is written in the form:

\[ \left. \begin{aligned} &\left(1-p_1^2+\frac{c_2}{c_1}\right)\frac{a_1}{h} - \left[2\beta p_2\sin(\gamma_2-\gamma_1)+\cos(\gamma_2-\gamma_1)\right]\frac{c_2}{c_1} \times \frac{a_2}{h} = \cos\gamma_1; \\[6pt] &2\beta p_2\frac{c_2}{c_1}\cdot\frac{a_1}{h} - \left[2\beta p_2\cos(\gamma_2-\gamma_1)-\sin(\gamma_2-\gamma_1)\right]\frac{c_2}{c_1} \times \frac{a_2}{h} = \sin\gamma_1; \\[6pt] &2\beta p_2\frac{a_1}{h}\sin(\gamma_2-\gamma_1) - \frac{a_1}{h}\cos(\gamma_2-\gamma_1) = (p_2^2-1)\frac{a_2}{h}; \\[6pt] &\frac{a_1}{h}\sin(\gamma_2-\gamma_1) + 2\beta p_2\frac{a_1}{h}\cos(\gamma_2-\gamma_1) = 2\beta p_2\frac{a_2}{h}. \end{aligned} \right\} \tag{17} \]

Carrying out calculations analogous to those set forth above, we obtain the expressions for \(\dfrac{a_1}{h}\) and \(\dfrac{a_2}{h}\) in the form:

\[ \frac{a_1}{h}=\frac{1}{\sqrt{\chi^2+\varepsilon^2}} ; \tag{18} \]

\[ \frac{a_2}{h} = \frac{1+4\beta^2 p_2^2} {\sqrt{4\beta^2 p_2^6+\left[1-\left(1-4\beta^2\right)p_2^2\right]^2}} \cdot \frac{a_1}{h}, \tag{19} \]

where

\[ \chi= \left(1-p_1^2+\frac{c_2}{c_1}\right) - \frac{c_2}{c_1}\cdot \frac{ \left(1+4\beta^2 p_2^2\right) \left[2\beta^2\left(2+p_2^2\right)p_2^2+\left(1-p_2^2\right)\right] }{ 4\beta^2 p_2^6+\left[1-\left(1-4\beta^2\right)p_2^2\right]^2 }; \]

\[ \varepsilon = 2\beta p_2 \frac{c_2}{c_1} \left\{ 1- \frac{ \left(1+4\beta^2 p_2^2\right) \left[1-2\left(1-2\beta^2\right)p_2^2\right] }{ 4\beta^2 p_2^6+\left[1-\left(1-4\beta^2\right)p_2^2\right]^2 } \right\}. \tag{20} \]

Thus, the values of the amplitudes \(\dfrac{a_1}{h}\) and \(\dfrac{a_2}{h}\) of the forced oscillations of the masses \(m_1\) and \(m_2\), as functions of the frequency of the disturbing force and of the parameters \(\left(\dfrac{c_2}{c_1}; \beta\right)\) for a linear system, are determined by equations (18) and (19).

Determination of the frequency regions in which stable oscillations of the mass \(m_1\) occur with impact against the stop is, in the general case, very cumbersome; therefore this question is not considered in the present work. However, in order to estimate the boundaries of the stability region for oscillations with impact of the mass \(m_1\) against the stop, one may, in a first approximation, use the results obtained in the investigation of a system with two degrees of freedom without taking into account the resistance forces [4].

Assigning specific values to the system parameters \(\left(\dfrac{a_0}{h}; \dfrac{c_2}{c_1}; \dfrac{m_2}{m_1}; \beta\right)\), we can construct graphs that clearly illustrate the process of development of the system oscillations as the frequency of the disturbing force changes.

Let \(p_1=p_2=p\); then \(\dfrac{c_2}{c_1}=\dfrac{m_2}{m_1}=\mu\). In addition, let us assign the following numerical values of the system parameters:

\[ \frac{a_0}{h}=2;\qquad \mu=0.2;\qquad \beta=0.1. \tag{21} \]

In Fig. 2 the dependences of the amplitudes of oscillation of the masses \(m_1\) and \(m_2\) as the frequency \(p\) changes are presented graphically for the selected values of the system parameters (21). Let us trace how the oscillations of the system develop. As the frequency \(p\) is increased from zero, the system begins to oscillate. If, in this process, the mass \(m_1\) does not reach

impacts the stop, then we have the linear case. We construct the amplitude curves according to formulas (18). When the mass \(m_1\) strikes the stop, we construct the curves according to formulas (13). The resonant curves (15), which originate at

Fig. 2. Amplitude curves of the oscillations of the mass \(m_1\) and of the mass \(m_2\) of the dynamic absorber.

first as straight lines corresponding to the resonant straight lines of the linear system with allowance for resistance forces, are marked by bold lines. With a further increase in the frequency \(p\), the system enters a state of resonance (the first resonance), and at a certain value of \(p\) a “breakdown” of the oscillations of the mass \(m_1\) with impact against the stop occurs.

... limiter, i.e., the mass \(m_1\) oscillates without reaching the limiter. We have the usual linear case. The amplitude curves are constructed from formulas (18). Increasing the frequency \(p\) further, we find that the mass \(m_1\) again begins to strike the limiter. The subsequent construction of the curves is carried out according to formulas (13). When a second resonance is reached at a certain value of \(p\), a second “breakaway” of the oscillations of the mass \(m_1\) occurs, with impact against the limiter. As a result, we again have the linear case. The construction of the amplitude curves is again carried out according to formulas (18). With a further increase in the frequency \(p\), the amplitudes of oscillation of the masses \(m_1\) and \(m_2\) tend to zero.

Conversely, decreasing the frequency \(p\), at a certain value of it, when \(\dfrac{a_1}{h} = \dfrac{a_0}{h}\), there will occur a sharp jump in the amplitudes of oscillation of the masses \(m_1\) and \(m_2\). Moreover, the frequency \(p\) corresponding to the jump in the amplitude of oscillation of the masses \(m_1\) and \(m_2\) will be less than the frequency at which the “breakaway” of the oscillations occurs; that is, we have the phenomenon of “dragging” of the oscillation amplitude, characteristic of a nonlinear system. Decreasing the value of \(p\) still further, we find that the mass \(m_1\) again enters such a state in which it will not touch the limiter. At a certain value of the frequency \(p\), the mass \(m_1\) will again begin to strike the limiter, and a jump in the magnitude of the amplitudes of oscillation of the masses \(m_1\) and \(m_2\) will occur.

It is evident from the graphs that the amplitude curves do not go to infinity, but have a definite maximum.

CONCLUSIONS

From the calculations and graphs presented, it is evident that the introduction into the system “elastic connection—mass—limiter” of a dynamic vibration absorber, with allowance for resistance forces, is entirely possible. By attaching such a dynamic absorber, it is possible to reduce the amplitude of oscillation of the mass \(m_1\) of the system with a limiter when the latter is in a state of resonance. In this case the frequency of the external force may vary within certain limits, which is not observed in a dynamic absorber without resistance forces [4].

The presence of resistance forces leads to dissipation of the energy of the oscillating masses of the system, as a result of which the amplitudes of oscillation of the latter in the state of resonance have finite values. In this case the work (dissipation energy) performed by the resistance force is equal to this force multiplied by the displacement over which it acts [2].

The question of choosing the “best damping coefficient” \(\beta\) [2], depending on the parameters of the system \(\left(\dfrac{c_2}{c_1}; \dfrac{m_2}{m_1}; \dfrac{a_0}{h}\right)\), for which the amplitudes of oscillation of the mass \(m_1\) will have minimum values, is not considered in the present work. This problem is the subject of further investigations.

References

1. P. M. Alabuzhev, V. I. Kopytov. Investigation of the vibrations of a load striking a stop. Izvestiya TPI, vol. 106, Metallurgizdat, 1958.

2. Den Hartog. Theory of Vibrations. OGIZ, 1942.

3. T. Karman, M. Biot. Mathematical Methods in Engineering. Gostekhizdat, 1948.

4. V. I. Kopytov. Forced vibrations of a system with two degrees of freedom under impact of one of the masses against a stop. Izvestiya vuzov, Mining Journal, No. 8, 1958.

5. S. P. Timoshenko. Theory of Vibrations in Engineering. GNTI, 1931.