On the Transfer of Energy by Impact Through an Elastic Link

A. P. TROPOTOV

(Submitted by the Department of Mining Machines and Mine Transport)

A number of practical problems associated with the calculation and design of impact-action machines reduce to the study of the process of collision of two masses through an elastic element.

To clarify the quantitative and qualitative aspects of collision under the usual assumptions that simplify the problem, let us consider the vibrations of the system shown in Fig. 1.

Fig. 1.

In the process of collision, the motion of the masses occurs according to the equations

\[ \begin{aligned} m_1 \ddot{x}_1 &= -k(x_1 - x_2),\\ m_2 \ddot{x}_2 &= k(x_1 - x_2), \end{aligned} \tag{1} \]

where

\[ k \]

is the stiffness coefficient of the spring;

\[ x_1 \text{ and } x_2 \]

are the coordinates of the masses, measured from the equilibrium position, which is characterized by the presence of contact of both masses with the spring in the absence of its deformation.

We shall seek particular solutions of equations (1) in the form

\[ \begin{aligned} x_1 &= B_1 \sin pt,\\ x_2 &= B_2 \sin pt. \end{aligned} \tag{2} \]

Substituting (2) into (1) and carrying out the usual mathematical operations necessary for finding the general solution of equations (1), we obtain the frequency equation of system (1) in the form

\[ m_1 m_2 p^4 - (m_1 + m_2) k p^2 = 0 . \tag{3} \]

Solving equation (3) with respect to \(p^2\), we find two values of the root

\[ p_1^2 = 0; \qquad p_2^2 = \frac{k}{m_1} + \frac{k}{m_2}. \tag{4} \]

The root of the frequency equation (3), \(p_1^2 = 0\), signifies the possibility of translational motion of the elastic system along the \(x\)-axis [1] and corresponds to the solution of equations (1) in the form

\[ x_{1p_1} = x_{2p_1} = at + b, \tag{5} \]

where \(a, b\) are constants.

The second particular solution, corresponding to the frequency value

\[ p_2 = \pm \sqrt{\frac{k}{m_1} + \frac{k}{m_2}}, \]

with the amplitude ratio

\[ \frac{B_{1p_2}}{B_{2p_2}} = -\frac{m_2}{m_1} \tag{6} \]

is found as

\[ \begin{aligned} x_{1p_2} &= B_{1p_2}\sin p_2 t,\\ x_{2p_2} &= -\frac{m_1}{m_2} B_{1p_2}\sin p_2 t . \end{aligned} \tag{7} \]

The general solution of equations (1) is determined by the sum of the particular solutions (5) and (7):

\[ \begin{aligned} x_1 &= at + b + B_{1p_2}\sin p_2 t,\\ x_2 &= at + b - \frac{m_1}{m_2} B_{1p_2}\sin p_2 t . \end{aligned} \tag{8} \]

The arbitrary constants \(a, b, B_{1p_2}\) must satisfy the initial conditions

\[ \begin{aligned} x_{1,0} &= 0, \qquad &x_{2,0} &= 0,\\ \dot{x}_{1,0} &= \pm U_1, \qquad &\dot{x}_{2,0} &= \mp V_2, \end{aligned} \tag{9} \]

taking which into account, the general solutions (8) take the form

\[ \begin{aligned} x_1 &= \frac{\pm V_1 m_1 \mp V_2 m_2}{m_1 + m_2}\,t + \frac{(\pm V_1 \pm V_2)m_2}{(m_1 + m_2)p_2}\sin p_2 t, \\[6pt] x_2 &= \frac{\pm V_1 m_1 \mp V_2 m_2}{m_1 + m_2}\,t - \frac{(\pm V_1 \pm V_2)m_1}{(m_1 + m_2)p_2}\sin p_2 t . \end{aligned} \tag{10} \]

Relations (10) are the equations of motion of the masses \(m_1\) and \(m_2\) in the pro-

during impact, describing all possible cases of collision. The first derivatives of equations (10) determine the laws governing the change in the velocities of the masses

\[ \dot{x}_1=\frac{\pm V_1 m_1+V_2 m_2}{m_1+m_2} +\frac{(\pm V_1\pm V_2)m_2}{m_1+m_2}\cos p_2 t, \]

\[ \dot{x}_2=\frac{\pm V_1 m_1\mp V_2 m_2}{m_1+m_2} -\frac{(\pm V_1\pm V_2)m_1}{m_1+m_2}\cos p_2 t . \tag{11} \]

As follows from equations (11), the absolute velocities of the masses during impact are composed of the constant velocity of the center of oscillation

\[ \frac{\pm V_1 m_1+V_2 m_2}{m_1+m_2} \]

(the translational component) and of the velocities of motion relative to the center of oscillation

\[ \frac{(\pm V_1\pm V_2)m_2}{m_1+m_2}\cos p_2 t,\qquad \frac{(\pm V_1\pm V_2)m_1}{m_1+m_2}\cos p_2 t \]

(the relative components).

The expressions obtained make it possible to determine the magnitude of the deformation of the spring, the force developed during impact, the velocities of the masses during impact, the displacement of the center of oscillation over the time of impact, and the amount of energy transmitted for given values of the velocities and of the masses and stiffness of the spring.

In Fig. 2 graphs are given of the variation of the components of the velocities of the masses during impact for the cases:

\[ \text{a) } m_2=2m_1;\qquad \text{b) } m_2=m_1;\qquad \text{c) } m_2=\frac{m_1}{2}, \]

where \(m_1\) is the mass of the striker delivering the blow to the stationary mass \(m_2\) through a spring of stiffness \(k\). Lines \(1_1, 1_2\) determine the dependence of the translational component of the velocities of the masses on time; curves \(2_1, 2_2\) characterize the change in the relative velocities of the striker \(m_1\) and of the mass \(m_2\). Curves \(3_1\) and \(3_2\) show how the absolute velocities of the colliding masses change during impact.

The ordinates of the points \(V_{m1}, V_{m2}\) correspond to the values of the velocities of the masses \(m_1\) and \(m_2\) at the instant they separate from the spring.

It follows from the graphs that for \(m_2=2m_1\), after impact the masses will move in opposite directions with velocities

\[ V_{m1}=-\frac{V_1}{3}, \]

\[ V_{m2}=\frac{2}{3}V_1 . \]

For \(m_2=m_1\), during the time \(t=\frac{T}{2}\) there will occur a complete exchange of the velocities of the colliding bodies: the mass \(m_2\), stationary before impact, will acquire the velocity

\[ \frac{V_1}{2}+\frac{V_1}{2}=V_1, \]

while the striker \(m_1\) will stop. Finally, for

\[ m_2=\frac{m_1}{2}, \]

after impact both masses will move in the direction of impact with velocities

\[ V_{m1}=\frac{V_1}{3};\qquad V_{m2}=\frac{4}{3}V_1 . \]

Curves 1 (Fig. 2, g) express the dependence between the ratios of the masses \(m_2\), stationary before impact, to the masses of the strikers \(m_1\), and the corresponding ratios of their velocities at the moment of time \(t=\dfrac{T}{2}\).

Fig. 2. Graphs of the change in the velocities of the masses during impact (a, b, v) and a diagram for determining the amount of energy transmitted during impact.

Thus, for

\[ \frac{m_2}{m_1}=3 \]

at the moment when the loads separate from the spring, the latter will have velocities equal in magnitude but opposite in sign,

\[ \frac{V_{m2}}{V_{m1}}=-1. \]

Curves 2 (Fig. 2, g) serve to determine the amount of energy transmitted during impact. For example, when

\[ \frac{m_2}{m_1}=2, \]

the ratio of the kinetic energy \(A_{m2}\) of mass \(m_2\) to the kinetic energy \(A_{m1}\) of mass \(m_1\) after impact will be

\[ \frac{A_{m2}}{A_{m1}}=8. \]

It is evident that

\[ A_{m2}+A_{m1}=A, \]

where \(A\) is the energy of the striker before impact. Taking into account that after impact \(A_{m2}=8A_{m1}\), we find

\[ 8A_{m1}+A_{m1}=A. \]

Hence

\[ A_{m1}=\frac{A}{9}. \]

Consequently, \(1/9\) of the energy remains with the striker, while to the mass \(m_2\) there is transmitted

\[ \frac{8}{9} \]

of the striker’s energy.

The same amount of energy can be transferred to the mass \(m_2\) when

\[ \frac{m_2}{m_1}=\frac{1}{2}. \]

However, the qualitative aspects are entirely different: in the first case the masses after impact move in opposite directions; in the second case, after impact the masses move in one and the same direction. Similar reasoning is valid for all other ratios

\[ \frac{m_2}{m_1}>1 \quad \text{and} \quad \frac{m_2}{m_1}<1. \]

An analysis of relations (10, 11) and of the graphs shown in Fig. 2 makes it possible to draw the following conclusions:

1. The spring may be regarded as a link possessing two characteristic properties. The first of these is due to its ability to bring the velocities of the colliding bodies practically instantaneously to the velocity of the center of mass in an inelastic impact. The second presupposes the participation of the masses in free oscillations directed in different directions, with amplitudes inversely proportional to the masses.

2. The velocity of the center of oscillation, the relative and absolute velocities of the masses, and the amount of energy transmitted in the impact do not depend on the stiffness of the elastic link and are determined only by the ratios of the colliding masses.

3. The striker can transfer one and the same amount of energy both to a larger and to a smaller mass. When \(\frac{m_2}{m_1}>1\), after impact the masses will move in opposite directions, whereas when \(\frac{m_2}{m_1}<1\) the masses will move in one and the same direction.

4.

\[ \lim_{\frac{m_2}{m_1}\to 0}\frac{V_{m_2}}{V_{m_1}}=2, \]

i.e., however small the mass \(m_2\) may be, its velocity \(V_{m_2}\) after impact cannot become equal to or greater than twice the initial velocity of the striker.

1. The time

\[ t=\frac{T}{2}=\frac{\pi}{p_2}, \]

during which the masses are in contact with the spring, as well as the maximum value of its deformation,

\[ \frac{\pm V_1 \pm V_2}{p_2}, \]

depend on the stiffness coefficient of the spring,

\[ t=f_1(k), \quad l_{\max}=f_2(k). \tag{12} \]

1. From (12) it follows that by selecting the stiffness coefficient \(k\) of the elastic link in an impact, one can develop any forces \(F\), irrespective of the kinetic energy stored by the striker, since

\[ F=kB_{\max}=f_3(k). \]

1. The relative velocity of the masses \(m_1\) and \(m_2\) after collision is equal to their relative velocity before collision.

LITERATURE

1. Ponomarev S. D., Biderman V. L., Likharev K. K., Makushin V. I., Fedos’ev V. N., Malinin N. N., Fundamentals of Modern Methods of Strength Calculation in Mechanical Engineering. Mashgiz, 1952.