MECHANICS

V. A. SVEKLO

ON THE THEORY OF THE IMPACT OF CYLINDERS

(Presented by Academician L. M. Sedov on 31 XII 1957)

The problem of the impact of circular cylinders identical in size and material was considered by A. N. Dinnikov \((^{1})\). Below an approximate theory is set forth for the collision of arbitrary cylindrical bodies in the case of initial contact of the bodies of order \(n > 1\).

We start from the simplest scheme: during the impact time \(t_2\) the pressure region \(2l\), the pressure \(p(x,t)\) distributed over it, and the pressure force \(P\) remain unchanged, equal to their value under static compression for the case of initial contact of order \(n\):

\[ p(x_0,t_0)=\frac{P}{\pi l^2}\sqrt{l^2-x_0^2} \left[ \frac{2n}{2n-1} + \frac{2n(2n-2)}{(2n-1)(2n-3)}\frac{x_0^2}{l^2} +\cdots \right. \]
\[ \left. \cdots+ \frac{2n(2n-2)\cdots 2}{(2n-1)\cdots 1}\, \frac{x^{2n-2}}{l^{2n-2}} \right]; \tag{1} \]

\[ p=\frac{(2n-1)!!}{(2n-2)!!}\,L_n l^{2n}; \qquad L_n=\frac{f_1^{(2n)}(0)+f_2^{(2n)}(0)} {(2n)!(\theta_1+\theta_2)}; \]

\(f_j(x)\) determine the shape of the directing curves of the cylinders; \(\theta_i=\dfrac{2}{\pi E_i}(1-\nu_i^2)\); \(E_i\) is Young’s modulus; \(\nu_i\) is Poisson’s constant.

Let us cut from the cylinders disks of thickness \(h=1\). Let \(m_1\) and \(m_2\) be their masses, and let \(\alpha(t)\) be the approach of the axes. Then

\[ M\frac{d^2\alpha}{dt^2}=-hP,\qquad M=\frac{m_1m_2}{m_1+m_2},\qquad 0\le t\le t_2 . \tag{2} \]

Integrating with the condition \(\alpha(0)=0\), we find

\[ \alpha(t)=v_0t-\frac{hP}{2M}t^2,\qquad v_0=\left(\frac{d\alpha}{dt}\right)_{t=0}. \tag{3} \]

If \(t_1=\tfrac12 t_2\) is the moment of completion of the first act of impact, then \((d\alpha/dt)_{t_1}=0\), and we obtain:

\[ t_1=\frac{Mv_0}{hP},\qquad \alpha(t_1)=\alpha_{\max}=\alpha_0=\frac12\,\frac{Mv_0^2}{hP}. \tag{4} \]

To find the unknowns \(t_1, P, l\), let us establish one more relation between them by turning to the vibrations that arise during impact in the contacting sections of the disks. In the calculations these sections are replaced by half-planes. For the first half-plane, in the absence of initial data and body forces, we obtain for the displacement in the direction of the \(Oy\) axis \((^{2})\)

\[ v_1(x_0,y_0,t_0)=\frac{1}{2\pi\rho_1} \left\{ \frac{\partial}{\partial y_0} \left[ \iint_{S_1} v_1^0 p(x,t)\,dx\,dt \right] - \frac{\partial}{\partial x_0} \left[ \iint_{S_1} v_2^0 p(x,t)\,dx\,dt \right] \right\}, \tag{5} \]

where \(v_i^0\) are analogous displacements determined by the fundamental solutions for the half-plane.

It is not difficult to justify the possibility of bringing the derivative sign under the integral sign and to write finally, for both half-planes,

\[ v_j(x_0,0,t_0) = \frac{1}{\pi \rho_j b_j^4} \iint_{S_j} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(\theta)} \frac{p(x,t)}{x_0-x}\,dx\,dt \qquad (j=1,2), \tag{6} \]

where \(\theta=\dfrac{t_0-t}{x-x_0}\); \(S_j\) are determined by the inequalities \(r_j\leqslant a_j(t_0-t)\), \(0\leqslant t\leqslant t_0\); \(a_j\) are the velocities of longitudinal waves; \(F_j(\theta)\) is the Rayleigh function.

On the contact segment we easily obtain, as in the static case, the basic relation:

\[ v_1(x_0,0,t)+v_2(x_0,0,t)=\alpha(t)-f_1(x_0)-f_2(x_0). \tag{7} \]

Substituting \(v_1\) and \(v_2\) from (6), we shall have:

\[ \sum_{j=1}^{2} \frac{1}{\pi \rho_j b_j^4} \iint_{S_j} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(\theta)} p(x,t)\frac{dx\,dt}{x_0-x} = \alpha(t_0)-f_1(x_0)-f_2(x_0). \tag{8} \]

Put \(x_0=0\), and let \(a_jt_0/l<1\); then \(S_j\) will be the triangle shown in Fig. 1. Expand (1) in a series:

\[ p(x_0,t_0) = \frac{P}{\pi l} \left[ \frac{2n}{2n-1} + B_2(n)\frac{x_0^2}{l^2} + B_4(n)\frac{x_0^4}{l^4} +\cdots \right], \tag{9} \]

where \(B_2, B_4,\ldots\) are known coefficients. Substituting in (8), we obtain \((f_1(0)=f_2(0)=0)\)

\[ \frac{P}{\pi^2 l} \sum_{j=1}^{2} \frac{1}{\rho_j b_j^4} \int_{0}^{t_0} \int_{-a_j(t_0-t)}^{a_j(t_0-t)} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(\theta)} \times \]

\[ \times \left[ \frac{2n}{2n-1} + B_2(n)\frac{x^2}{l^2} +\cdots \right] \frac{dx}{-x} = \alpha(t_0). \tag{10} \]

Fig. 1

Putting \(y=\dfrac{x}{t_0-t}\), we shall have

\[ \int_{0}^{a_j(t_0-t)} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(\theta)} \frac{dx}{-x} = \int_{0}^{a_j} \operatorname{Re} \frac{i\sqrt{1/a_j^2-1/y^2}}{F_j(1/y)} \frac{dy}{-y}; \]

\[ \int_{0}^{a_j(t_0-t)} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(\theta)} \,x\,dx = (t_0-t)^2 \int_{0}^{a_j} \operatorname{Re} \frac{i\sqrt{1/a_j^2-\theta^2}}{F_j(1/y)} \,y\,dy,\ldots \]

Condition (10), after carrying out the integrations with respect to \(t\) and reducing by \(t_0\), is rewritten in the form

\[ \frac{2P}{\pi^2 l} \sum_{j=1}^{2} \frac{1}{\rho_j b_j^4} \left[ \Phi_0(k)\frac{2n}{2n-1} + \frac{1}{3}B_2(n)\frac{a_j^2t_0^2}{l^2}\Phi_1(k) +\cdots \right] = v_0-\frac{hP}{2M}t_0. \tag{11} \]

Here we have used equality (3) and introduced the notation

\[ \Phi_0(k) = -\int_{0}^{1} \operatorname{Re} \frac{\sqrt{1/\eta^2-1}}{F_j(1/\eta)} \frac{d\eta}{\eta}; \qquad \Phi_1(k) = \int_{0}^{1} \operatorname{Re} \frac{\sqrt{1/\eta^2-1}}{F_j(1/\eta)} \,\eta\,d\eta; \]

\[ \eta=\frac{y}{a_j}, \qquad k_j=\frac{b_j^2}{a_j^2}. \]

Letting \(t_0\) tend to zero in (11), we obtain the desired additional relation for the case of contact of order \(n\):

\[ \frac{4n}{2n-1}\,\frac{P}{\pi^2 l}\sum_{j=1}^{2}\frac{1}{\rho_j b_j}\Phi_0(k_j)=v_0. \tag{12} \]

For what follows it is convenient to introduce the dimensionless quantities:

\[ \bar{\rho}_j=\frac{\rho_j}{\rho_0};\qquad \bar{b}_j=\frac{b_j}{b_0}; \]

\[ \frac{1}{N_1N_2}=\sum_{j=1}^{2}\frac{\Phi_0(k_j)}{\bar{\rho}_j\bar{b}_j};\qquad \frac{1}{N_2}=\sum_{j=1}^{2}\frac{1}{\bar{\rho}_j\bar{b}_j(1-k_j)};\qquad \psi_n=\frac{(2n-1)!!}{(2n-2)!!}, \tag{13} \]

where \(\rho_0, b_0\) are the characteristic density and velocity of transverse waves of the given system. In the new notation we write

\[ P=\psi_n\frac{A_n}{\theta_1+\theta_2}l^{2n};\qquad A_n=\frac{f_1^{(2n)}(0)+f_2^{(2n)}(0)}{(2n)!};\qquad \frac{1}{\theta_1+\theta_2}=2\pi\rho_0 b_0^2 N_2, \tag{14} \]

and from (3) and (5) we find:

\[ l=\left(\frac{\pi}{2}\right)^{\frac{1}{2n-1}} \left(\frac{2n-1}{2n}\right)^{\frac{1}{2n-1}} \psi_1^{-\frac{1}{2n-1}}N_1^{-\frac{1}{2n-1}}A_n^{-\frac{1}{2n-1}} \left(\frac{v_0}{b_0}\right)^{\frac{1}{2n-1}}; \tag{15} \]

\[ P=\pi\left(\frac{\pi}{2}\right)^{\frac{2n}{2n-1}} \left(\frac{2n-1}{2n}\right)^{\frac{2n}{2n-1}} \psi_n^{-\frac{1}{2n-1}}N_2N_1^{-\frac{2n}{2n-1}} \rho_0 A_n^{-\frac{1}{2n-1}}b_0^{2\frac{n-1}{2n-1}}v_0^{\frac{2n}{2n-1}}; \tag{16} \]

\[ t_1= \frac{\psi_n^{\frac{1}{2n-1}}}{\pi\left(\frac{\pi}{2}\right)^{\frac{1}{2n-1}}} \frac{A_n^{\frac{1}{2n-1}}}{N_2N_1^{\frac{2n}{2n-1}}} \frac{M}{\rho_0h} \frac{\left(\frac{2n}{2n-1}\right)^{\frac{2n}{2n-1}}}{b_0^{2\frac{n-1}{2n-1}}v_0^{\frac{1}{2n-1}}}. \tag{17} \]

\(P\) is expressed in terms of \(\alpha_0\) by the formula

\[ P=\pi^{\frac{2n-1}{n-1}} \left(\frac{\pi}{2}\right)^{\frac{2n}{n-1}} \left(\frac{2n-1}{2n}\right)^{\frac{2n}{n-1}} \psi_n^{-\frac{1}{n-1}} N_1^{\frac{2n}{n-1}}N_2^{\frac{2n-1}{n-1}} \times \]

\[ \times b_0^2\left(\frac{2h}{M}\right)^{\frac{n}{n-1}} \rho_0^{\frac{2n-1}{n-1}}A_n^{-\frac{1}{n-1}}\alpha_0^{\frac{n}{n-1}}. \tag{18} \]

It follows from this that, as \(n\) increases, the relation between \(P\) and \(\alpha_0\) tends to a linear one. In general, in the case of contact of order \(n\), we arrive here at the same conclusions as in the three-dimensional problem based on Hertz’s hypothesis: 1) the dependence of \(t_1\) on the initial velocity \(v_0\) weakens as \(n\) increases; 2) for \(n=\infty\) the pressure force \(P\) proves to be proportional to \(\alpha_0\), and the latter is proportional to the ratio of the initial velocity \(v_0\) to \(b_0\). For \(n=1\) formula (18) loses its meaning.

Example. As an example, consider the impact of a circular disk

\[ y_1=f_1(x)=R\left(1-\sqrt{1-\frac{x^2}{R^2}}\right)\simeq \]

\[ \simeq \frac{1}{2R}x^2+\frac{1}{8R^3}x^4 \]

against the parabola

\[ y_2=f_2(x)=-\frac{1}{2R}x^2 \]

(Fig. 2). We take the materials of the bodies to be identical. We have: \(n=2\);

\[ k=b^2/a^2;\qquad A_2=\frac{1}{8R^3};\qquad \bar{\rho}_j=1;\qquad \bar{b}_j=1;\qquad \rho_0=\rho_j,\quad b_0=b_j, \]

\[ \frac{1}{N_1}=(1-k)\Phi_0;\qquad N_2=\frac{1-k}{2};\qquad m_2=\infty. \]

Formulas (15)—(18) will take the form:

\[ l=\sqrt[3]{2\pi N_1\frac{v_0}{b}R};\qquad P=\frac{3}{8}\pi^2 N_1N_2\sqrt[3]{2\pi N_1b^2v_0^4\rho R}; \]

\[ t_1=\frac{8R}{3\pi N_1N_2\sqrt[3]{2\pi N_1b^2v_0^4}};\qquad \alpha_0=\frac{4R}{3\pi N_1N_2\sqrt[3]{2\pi N_1}}\left(\frac{v_0}{b}\right)^{2/3}; \tag{19} \]

\[ P=\pi^2\left(\frac{3\pi}{4}\right)^3\frac{N_1^4N_2^3}{R}\rho b^2\alpha_0^2. \]

Table 1

To compute \(N_1\), one must know

\[ \Phi_0(k)=-\int_0^1 \operatorname{Re}\frac{\sqrt{1/\eta^2-1}}{F(1/\eta)}\,\frac{d\eta}{\eta}. \]

This integral at the point \(\eta=c/a\) (\(c\) is the velocity of Rayleigh surface waves) exists in the sense of the Cauchy principal value. Table 1 gives some values of \(\Phi_0(k)\), and Table 2 gives the principal quantities for the impact of a disk on a parabola (both bodies are made of steel; in all cases \(R=1\) cm).

Table 2

Steel (\(E=21\cdot 10^{11}\) dyn/cm\(^2\), \(\rho=7.8\) g/cm\(^3\), \(\nu=0.2901\), \(M=24.5\) g)

Let us note that, as is seen from Table 2, the impact time \(t_2\) is greater than the period of the slowest natural vibrations of the disk.

Petrozavodsk
State University

Received
7 VI 1957

References

1. A. N. Dinnik, Impact and Compression of Elastic Bodies, Selected Works, 1, Publishing House of the Academy of Sciences of the Ukrainian SSR, 1952.
2. F. Frank, F. Mises, Differential and Integral Equations of Mathematical Physics, Part II, Ch. XII, 1937.
3. A. Ya. Shtaerman, Contact Problems of the Theory of Elasticity, 1949.