Study of Oscillations of a Load Striking a Stop

P. M. Alabuzhev, V. I. Kopytov

A number of technical questions connected with forced oscillations of systems (with one or several degrees of freedom) reduce, under certain assumptions, to the problem of the oscillation of a load on a spring with impact against a rigid stop. As an example one may cite the oscillating “frame—striker” system of drilling, rebound, and other types of hammers.

In works [3, 5, 7] problems are considered concerning oscillations of a mass between two stops—elastic and rigid. The problem of the oscillation of a mass on a spring with one rigid stop was considered in work [6]; however, it seems to us that the solution given below is more transparent and simple. We have investigated the indicated question by means of the variational method of Academician B. G. Galerkin [1, 2], the application of which made it possible to determine the character of the nonlinearity of the oscillatory system under consideration. The theoretical solution was checked experimentally on an installation manufactured at the Tomsk Polytechnic Institute (Fig. 1, a).

Fig. 1. Schematic diagram of the installation:
1—rocker mechanism; 2—spring; 3—drum pulling the paper; 4—electric motor rotating the drum; 5–6—pencils for recording curve I of the displacement of the end of the spring and curve II of the displacement of the load 7; 7—load of mass \(m\); 8—oscillation stop.

The results of the calculation and experiment are presented in the form of formulas and graphs.

Theoretical solution of the problem

Consider the oscillatory system shown in Fig. 1. It is required to determine the law of motion of a load of mass \(m\), if the end \(A\) of the spring executes vertical harmonic oscillations according to the law

\[ AA_0 = h \sin \omega t . \]

We shall consider the problem under the following assumptions: there is no damping of the oscillations, and the impact of the load against the stop is inelastic. We shall seek such periodic motions of the mass \(m\) whose frequency and phase coincide with the frequency and phase of the excitation.

The differential equation of motion of the system under consideration may be written in the form

\[ m(\ddot{x} - h \omega^2 \sin \omega t) = mg - F(\lambda_{\text{st}} + x), \]

or

\[ m\ddot{x} + F(\lambda_{\text{st}} + x) = mg + mh\omega^2 \sin \omega t, \tag{1} \]

where
\(x\) is the displacement of the load relative to the position of static equilibrium of the system \(OO\);
\(mg\) is the force of gravity of the load of mass \(m\);
\(F(\lambda_{\text{st}} + x)\) is the elastic force of the spring, which is a linear quantity in the interval of variation of \(x\) from \(-\infty\) to \((a_0 - h)\), i.e., up to the stop;
\(a_0\) is the distance from the position of static equilibrium of the load on the spring to the stop for the mean position of point \(A\);
\(\lambda_{\text{st}}\) is the static deformation of the spring caused by the load of mass \(m\);
\(h\) is the semi-amplitude of the free end of the spring.

In accordance with Academician Galerkin’s method, we shall seek a particular solution of equation (1) in the form of the series

\[ x = x_0 + \sum_{i=0}^{n} a_{2i+1} \sin (2i+1)\omega t, \tag{2} \]

where \(x_0\) is a quantity independent of time \(t\).

Let us find the solution of equation (1), restricting ourselves only to the first term of series (2) (the first approximation),

\[ x = x_0 + a \sin \omega t . \tag{3} \]

Substitute the value of \(x\) into equation (1). We multiply the result by \(\delta x\) and integrate with respect to \(t\) from \(0\) to \(\dfrac{2\pi}{\omega}\); then we obtain

\[ \int_{0}^{\frac{2\pi}{\omega}} \left[ -m\omega^2(a+h)\sin \omega t + F(\lambda_{\text{st}} + x_0 + a\sin \omega t) - mg \right] (\delta x_0 + \delta a \sin \omega t)\,dt = 0, \]

whence, by virtue of the independence of the variations \(\delta x_0\) and \(\delta a_0\), we obtain a system of two equations for determining the quantities \(x_0\) and \(a_0\)

\[ \left. \begin{aligned} &\int_{0}^{\frac{2\pi}{\omega}} \left[F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)-mg\right]\,dt=0,\\[6pt] &\int_{0}^{\frac{2\pi}{\omega}} \left[-m\omega^2(a+h)\sin\omega t +F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\right]\sin\omega t\,dt=0. \end{aligned} \right\} \tag{4} \]

Hence:

\[ \left. \begin{aligned} mg&=\frac{\omega}{2\pi} \int_{0}^{\frac{2\pi}{\omega}} F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\,dt,\\[6pt] m\omega^2(a+h)&=\frac{\omega}{\pi} \int_{0}^{\frac{2\pi}{\omega}} F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\sin\omega t\,dt. \end{aligned} \right\} \tag{4a} \]

To compute the integrals appearing on the right-hand sides of equations (4a), it is necessary to know the concrete expression for the elastic force

\[ F(t)=F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t). \]

It is obvious that if, during oscillation, the mass \(m\) does not reach the stop \((x<a_0-h)\), then

\[ F(\lambda_{\mathrm{st}}+x)=c(\lambda_{\mathrm{st}}+x), \]

or

\[ F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t) =c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t), \tag{5} \]

where \(c\) is the constant stiffness of the spring.

Substituting the value of the elastic force (5) into expressions (4a), after calculations we obtain:

from the first equation

\[ mg=c(\lambda_{\mathrm{st}}+x_0), \tag{6} \]

whence

\[ x_0=0, \]

from the second equation

\[ m\omega^2(a+h)=ca, \]

dividing by \(c\), after transformations we shall have

\[ \left(\frac{\omega}{\omega_0}\right)^2\cdot(a+h)=a, \tag{7} \]

whence

\[ a=\frac{h\left(\dfrac{\omega}{\omega_0}\right)^2}{1-\left(\dfrac{\omega}{\omega_0}\right)^2}, \tag{7a} \]

where \(\omega=\sqrt{\dfrac{c}{m}}\) is the natural frequency of the linear “load–spring” system of stiffness \(c\).

Equation (7a) expresses the dependence of the amplitude \(a\) of the forced oscillations on the oscillation frequency of the free end of the spring, for the motion of the mass \(m\) relative to the position of static equilibrium. It is obvious that for \(\omega_0>\omega\) (oscillations in the subresonance region) the amplitude \(a\) is equal in magnitude to the dynamic deformation of the spring. The maximum displacement of the mass \(m\) relative to any fixed point will henceforth be called the amplitude of the forced oscillations in absolute coordinates and denoted by \(A\). Obviously,

\[ A=a+h=\frac{h\left(\dfrac{\omega}{\omega_0}\right)^2}{1-\left(\dfrac{\omega}{\omega_0}\right)^2}+h. \tag{8} \]

For \(\omega_0<\omega\) (oscillations beyond the linear resonance), the motions of the free end of the spring and of the mass \(m\) are in opposite phase. The amplitude of the forced oscillations of the mass \(m\) in relative coordinates is expressed by the formula

\[ a=\frac{h\left(\dfrac{\omega}{\omega_0}\right)^2}{\left(\dfrac{\omega}{\omega_0}\right)^2-1}. \tag{9} \]

The amplitude of the forced oscillations of the mass \(m\) in absolute coordinates in this case is equal to

\[ A=a-h=\frac{h\left(\dfrac{\omega}{\omega_0}\right)^2}{\left(\dfrac{\omega}{\omega_0}\right)^2-1}-h. \tag{10} \]

If, during the oscillation process, the mass \(m\) strikes a stop, then the integrals in expressions (4a) give a different value after evaluation. In this case the expression for the elastic force is written in the form

\[ F(\lambda_{\mathrm{st}}+x)= \begin{cases} c(\lambda_{\mathrm{st}}+x), & \text{for } x<a_0-h,\\ c(\lambda_{\mathrm{st}}+a_0-h), & \text{for } x=a_0-h. \end{cases} \]

From the condition of contact of the mass \(m\) with the stop, we obtain the dependence

\[ a_0-h=x_0+a, \tag{11} \]

whence it follows that contact occurs at \(t=\dfrac{\pi}{2\omega}\).

Thus, the interval of integration with respect to \(t\) is divided into 4 parts (motion of the mass from the equilibrium position to the stop, contact, departure from the stop, motion in the direction of spring compression and back)

\[ F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)= \begin{cases} c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t), & \text{for } 0<t<\dfrac{\pi}{2\omega}-\sigma,\\[1.2em] c(\lambda_{\mathrm{st}}+a_0-h) =\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{\frac{\pi}{2\omega}-\sigma}^{\frac{\pi}{2\omega}} \varphi\!\left(t-\frac{\pi}{2\omega}+\sigma\right)\,dt =\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{0}^{\sigma}\varphi(\tau)\,d\tau, & \text{for } \dfrac{\pi}{2\omega}-\sigma\le t\le \dfrac{\pi}{2\omega},\\[1.2em] c(\lambda_{\mathrm{st}}+a_0-h) =\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{\frac{\pi}{2\omega}}^{\frac{\pi}{2\omega}+\sigma} \varphi\!\left(\frac{\pi}{2\omega}+\sigma-t\right)\,dt =\displaystyle \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{0}^{\sigma}\varphi(\tau)\,d\tau, & \text{for } \dfrac{\pi}{2\omega}< t\le \dfrac{\pi}{2\omega}+\sigma,\\[1.2em] c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t), & \text{for } \dfrac{\pi}{2\omega}+\sigma\le t\le \dfrac{2\pi}{\omega}. \end{cases} \tag{12} \]

In order to take into account the influence of the stop on the magnitudes of the amplitudes of the forced oscillations of the mass \(m\), in expression (12) the values of the function \(F(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\) at \(t=\dfrac{\pi}{2\omega}\) are represented in integral form, as A. N. Krylov proposed doing in such cases [4, p. 400]. Therefore it appears possible to write the elastic force at \(t=\dfrac{\pi}{2\omega}\) in the form

\[ F(\lambda_{\mathrm{st}}+a_0-h) = \lim_{\sigma\to 0}\frac{\omega}{\pi} \int\limits_{0}^{\sigma}\varphi(\tau)\,d\tau, \]

which means that the force under consideration is characterized by a certain function \(\varphi(\tau)\) over a small interval of time adjacent to the instant of contact (impact).

Substituting expression (12) into the integral appearing on the right in the first equation (4a), and splitting the interval of integration into four parts, we obtain

\[ \begin{aligned} mg={}& \left. \frac{\omega}{2\pi}\int_{0}^{\frac{\pi}{2\omega}-\sigma} c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\,dt \right|_{\sigma=0} +\frac12\lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}-\sigma}^{\frac{\pi}{2\omega}} \varphi\!\left(t-\frac{\pi}{2\omega}+\sigma\right)dt \\ &+\frac12\lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}}^{\frac{\pi}{2\omega}+\sigma} \varphi\!\left(\frac{\pi}{2\omega}+\sigma-t\right)dt +\left. \frac{\omega}{2\pi}\int_{\frac{\pi}{2\omega}+\sigma}^{\frac{2\pi}{\omega}} c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\,dt \right|_{\sigma=0}, \end{aligned} \]

whence, carrying out the corresponding calculations, we have

\[ x_0=-(\lambda_{\mathrm{st}}+a_0-h). \tag{13} \]

From the second equation (4a) we shall have

\[ \begin{aligned} m\omega^2(a+h)={}& \left. \frac{\omega}{\pi}\int_{0}^{\frac{\pi}{2\omega}-\sigma} c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\sin\omega t\,dt \right|_{\sigma=0} \\ &+\lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}-\sigma}^{\frac{\pi}{2\omega}} \varphi\!\left(t-\frac{\pi}{2\omega}+\sigma\right)\sin\omega t\,dt +\lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}}^{\frac{\pi}{2\omega}+\sigma} \varphi\!\left(\frac{\pi}{2\omega}+\sigma-t\right)\sin\omega t\,dt \\ &+\left. \frac{\omega}{\pi}\int_{\frac{\pi}{2\omega}+\sigma}^{\frac{2\pi}{\omega}} c(\lambda_{\mathrm{st}}+x_0+a\sin\omega t)\sin\omega t\,dt \right|_{\sigma=0}, \end{aligned} \]

whence

\[ m\omega^2(a+h)=ca+2c(\lambda_{\mathrm{st}}+a_0-h) \]

or, dividing by \(c\), we obtain

\[ \left(\frac{\omega}{\omega_0}\right)^2(a+h) = a+2(\lambda_{\mathrm{st}}+a_0-h). \tag{14} \]

A certain peculiarity is presented by the calculation of the integrals

\[ \lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}-\sigma}^{\frac{\pi}{2\omega}} \varphi\!\left(t-\frac{\pi}{2\omega}+\sigma\right)\sin\omega t\,dt \tag{a} \]

and

\[ \lim_{\sigma\to 0}\frac{\omega}{\pi} \int_{\frac{\pi}{2\omega}}^{\frac{\pi}{2\omega}+\sigma} \varphi\!\left(\frac{\pi}{2\omega}+\sigma-t\right)\sin\omega t\,dt. \tag{b} \]

To compute integral (a), we make the change of variable, putting

\[ t=\tau+\frac{\pi}{2\omega}-\sigma, \]

then expression (a) is written in a somewhat different form, namely:

\[ \lim_{\sigma\to 0}\frac{\omega}{\pi}\int_0^\sigma \varphi(\tau)\sin\omega t\left(\tau+\frac{\pi}{2\omega}-\sigma\right)\,d\tau . \]

Since the sine does not change sign in the small interval \((0,\sigma)\), using the mean-value theorem we obtain

\[ \lim_{\sigma\to 0}\sin\omega\left(\frac{\pi}{2\omega}-\theta\sigma\right) \frac{\omega}{\pi}\int_0^\sigma \varphi(\tau)\,d\tau = c(\lambda_{\mathrm{st}}+a_0-h). \quad (0\leq \theta \leq 1). \]

The second integral (b) is computed analogously; in it one must make the change of variable according to the formula

\[ t=\frac{\pi}{2\omega}+\sigma-\tau, \]

as a result of which we obtain

\[ \lim_{\sigma\to 0}\sin\omega\left(\frac{\pi}{2\omega}+\theta\sigma\right) \frac{\omega}{\pi}\int_0^\sigma \varphi(\tau)\,d\tau = c(\lambda_{\mathrm{st}}+a_0-h). \]

Equation (14), taking into account (13) and (11), is written in the form

\[ \left(\frac{\omega}{\omega_0}\right)^2(a+h) = a-2x_0 = a+2[a-(a_0-h)] = 3a-2(a_0-h), \]

whence

\[ a= \frac{2(a_0-h)+h\left(\frac{\omega}{\omega_0}\right)^2} {3-\left(\frac{\omega}{\omega_0}\right)^2}. \tag{15} \]

Expression (15) gives, in the first approximation, the dependence of the relative amplitude of the forced oscillations of the mass \(m\), striking the stop, on the frequency of oscillations of the free end of the spring. The amplitude of the forced oscillations in absolute coordinates will be

\[ A=a+h= \frac{2(a_0-h)+h\left(\frac{\omega}{\omega_0}\right)^2} {3-\left(\frac{\omega}{\omega_0}\right)^2} +h. \]

Assigning values to the system parameters \(h\), \(a_0\), and \(\omega_0\) (i.e., \(c\) and \(m\)), we construct the graph (Fig. 2) of the dependence of the amplitude of the forced-

...of these oscillations in absolute coordinates on the excitation frequency \(\omega\) (along the abscissa axis the values of \(\left(\dfrac{\omega}{\omega_0}\right)^2\) are plotted, which, obviously, changes the scale of the curves but does not change their character). The graph shown in Fig. 2 clearly illustrates the picture of the development of the oscillations of the mass \(m\). As \(\omega\), or \(\left(\dfrac{\omega}{\omega_0}\right)^2\), increases, the absolute amplitudes of the forced oscillations of the mass \(m\), determined

Fig. 2. Calculated amplitude curves: the solid line corresponds to the first approximation; the dashed line, to the second approximation. The amplitude values found from the experiment are plotted as separate points. Absolute values of the constant quantities: \(\omega_0 = 16.7\ \mathrm{sec}^{-1}\), \(a_0 = 2.0\ \mathrm{cm}\), \(h = 1.6\ \mathrm{cm}\).

by formula (16), increase up to a certain value. After this, a disruption of the impact oscillations occurs, and the mass \(m\) oscillates without reaching the stop. As a result, we have the linear case of oscillations. The absolute amplitudes of the forced oscillations of the mass \(m\) are then determined by formula (10). At the moment of disruption, a phase shift of the forced oscillations occurs. Conversely, when the frequency of the disturbing force is decreased from the value \(\left(\dfrac{\omega}{\omega_0}\right)^2 > 3\), the absolute amplitudes of the forced oscillations of the mass \(m\), determined by formula (10), increase. When the amplitude reaches such a value that the mass \(m\) touches the stop, a sharp upward jump in the amplitude occurs, and the point falls on the curve determined by equation (16). From the condition of contact with the stop

we determine the value of the quantity $\left(\dfrac{\omega}{\omega_0}\right)^2$ at which a jump-like increase in the amplitude of the forced oscillations of the mass $m$ occurs,

\[ \frac{h\left(\dfrac{\omega}{\omega_0}\right)^2} {\left(\dfrac{\omega}{\omega_0}\right)^2 - 1} - h = a_0, \]

whence

\[ \left(\frac{\omega}{\omega_0}\right)^2 = 1 + \frac{h}{a_0}. \tag{17} \]

Discarding in formula (16) the term characterizing the disturbance, we obtain, in the first approximation, an equation that relates the frequency of free oscillations of the mass $m$ with impact against the stop to the amplitude of the oscillations (the resonance curve),

\[ A = \frac{2(a_0 - h)} {3 - \left(\dfrac{\omega}{\omega_0}\right)^2} + h. \tag{18} \]

Thus, we have obtained a picture typical of a nonlinear problem, in which one value of the disturbance parameter corresponds to two values of the amplitude of the forced oscillations.

Let us note that in carrying out the experiments the following phenomenon was observed. For values of $h$ close in magnitude to $a_0$, as the disturbance frequency increased, after the oscillations broke away, stable impact oscillations of the mass $m$ were observed with a frequency equal to $\dfrac{1}{2}\omega$ and $\dfrac{1}{3}\omega$. In other words, subharmonic oscillations occurred. In the present work the question of determining the subharmonics was not considered.

Figure 3 gives the amplitude curves for various values of $a_0$ and $h$, from which it is seen that if only the magnitude of the clearance $a_0$ is varied, while the magnitude of the half-swing $h$ of the end of the spring is left constant, then the amplitude curves will shift upward or downward ($a_0$ correspondingly increases or decreases). In this case the point $A$ of the break in the amplitude curve, corresponding to the instant of contact of the mass $m$ with the stop, will shift along the curve representing the linear case for the given value of $h$. In this case the resonance curves will also shift, and the point of break $C$ will move along the straight line $\left(\dfrac{\omega}{\omega_0}\right)^2 = 1$, corresponding to linear resonance.

At \(a_0=\mathrm{const}\), when the value \(h\) is changed, the amplitude curves shift either to the right (\(h\) decreases) or to the left (\(h\) increases).

Let us estimate the sufficiency of the first approximation in solving problems of this kind. To do this, we shall find the solution of equation (1), taking two terms in the series (2),

\[ x=x_0+a_1\sin\omega t+a_3\sin3\omega t . \tag{19} \]

Fig. 3. Calculated amplitude curves for various values of the system parameters \(a_0\) and \(h\).

Solid lines correspond to the value \(h=1.6\ \text{cm}\); \(a_0=2.5\) and \(3\ \text{cm}\); dash-dotted lines correspond to the value \(h=1\ \text{cm}\), \(a_0=2.0\) and \(2.5\ \text{cm}\). Heavy lines are resonance curves for the corresponding values of the system parameters \(a_0\) and \(h\).

Having carried out calculations analogous to those performed in the first approximation, we obtain, for determining \(a_1\) and \(a_3\), the following system of algebraic equations:

\[ \left\{ \begin{aligned} \left[\left(\frac{\omega}{\omega_0}\right)^2-1\right]a_1 -2(a_1-a_3)+2(a_0-h)+h\left(\frac{\omega}{\omega_0}\right)^2&=0,\\ -\left[9\left(\frac{\omega}{\omega_0}\right)^2-1\right]a_3 -2(a_1-a_3)+2(a_0-h)&=0. \end{aligned} \right. \tag{20} \]

From expression (19) it follows that the quantity \(x\) has a maximum at \(t=\dfrac{\pi}{2\omega}\),

\[ x_{\max}=x_0+a_1-a_3 . \tag{21} \]

Solving the system of algebraic equations (20) with respect to \(a_1\) and \(a_3\), we obtain:

\[ \left. \begin{aligned} a_1 &= \frac{(21h-18a_0)\left(\dfrac{\omega}{\omega_0}\right)^2+2(a_0-h)-9h\left(\dfrac{\omega}{\omega_0}\right)^4} {9\left(\dfrac{\omega}{\omega_0}\right)^4-\left(30\dfrac{\omega}{\omega_0}\right)^2+5}, \\[1em] a_3 &= \frac{\left(\dfrac{\omega}{\omega_0}\right)^2-1} {9\left(\dfrac{\omega}{\omega_0}\right)^2-1}\,a_1 - \frac{h\left(\dfrac{\omega}{\omega_0}\right)^2} {9\left(\dfrac{\omega}{\omega_0}\right)^2-1}. \end{aligned} \right\} \tag{22} \]

The quantities \(a_1\) and \(a_3\) tend to infinity when the denominator is equal to zero; consequently,

\[ 9\left(\frac{\omega}{\omega_0}\right)^4 - 30\left(\frac{\omega}{\omega_0}\right)^2 + 5 = 0, \]

whence we determine two values of the quantity \(\left(\dfrac{\omega}{\omega_0}\right)^2\):

\[ \left(\frac{\omega}{\omega_0}\right)^2 = 3.16 \quad \text{and} \quad \left(\frac{\omega}{\omega_0}\right)^2 = 0.18. \]

Obviously, the second root should be discarded, since it does not satisfy the requirements of the problem posed.

Thus, taking into account the second term of series (2), the amplitude of the forced oscillations in absolute coordinates will differ only slightly from the corresponding amplitude determined in the first approximation (see Fig. 2). The correction from the second term of series (2) to the total value \(x_{\max}\) is no more than 15% for values \(\left(\dfrac{\omega}{\omega_0}\right)^2 < 2.8\). As experiments show, in real systems, owing to the presence of resistance forces, whose action on the oscillatory system increases with the amplitude of oscillation, the breakdown of stable oscillations with impact against the stop occurs at values

\[ \left(\frac{\omega}{\omega_0}\right)^2 < 2.3. \]

Consequently, in practical calculations one may restrict oneself to the first approximation only. This considerably reduces the amount of computational work.

Experimental studies

To verify the correctness of the solution set forth above, experiments were carried out on an apparatus made at the Tomsk Polytechnic Institute (Fig. 1). A steel load of mass \(m\), suspended on a spring (of constant stiffness \(c\)), stri—

striking the stop (in order to provide conditions close to an inelastic impact, the stop was made of wood—beech). The free end \(A\) of the spring was connected with the frame of the slider-crank mechanism and, together with it, performed a vertical oscillatory motion according to the law \(AA_0 = h \sin \omega t\).

The experimental procedure adopted was as follows:

1) the distance \(a_0\) of the mass \(m\) from the stop was fixed for the mean position of point \(A\);

2) the angular velocity \(\omega\) of rotation of the crank of the slider was varied, and consequently so was the frequency of oscillations of the free end \(A\) of the spring and of the load of mass \(m\);

3) during a certain interval of time (approximately 10 sec), pencils on paper fastened to a rotating drum simultaneously recorded the displacement curves of the mass \(m\) and of the free end \(A\) of the spring.

Fig. 4. Vibrogram of subharmonic oscillations of the mass \(m\) with impact against the stop:
I—the curve of the path of the mass \(m\) (in time); II—the curve of the path of the free end of the spring (in time). The period of oscillation \(T_1\) of the mass \(m\) is equal to twice the period of oscillation \(T_2\) of the free end of the spring \(A\).

The angular velocity of the crank was increased until the oscillations of the mass \(m\) broke away, with impact against the stop (impact oscillations). After the breakaway of the impact oscillations, the frequency of the forced oscillations can be increased further. In this case the mass \(m\), for \(h < a_0\), will “float” in the gap without touching the stop. For \(h > a_0\), beginning with a certain frequency \(\omega\), stable subharmonic oscillations are obtained with frequency equal to

\[ \frac{1}{2}\omega,\quad \frac{1}{3}\omega \]

and so on (Fig. 4).

Conversely, in the case of decreasing the frequency of the forced oscillations, the jump in the amplitude of the forced oscillations of the mass \(m\) occurs at a frequency \(\omega\) smaller than its value at the breakaway of the oscillations, i.e., the phenomenon of “dragging” is observed. The depth of the “dragging” is determined by formula (17), which is confirmed experimentally.

Knowing the number of oscillations of the end \(A\) of the spring and the time of the experiment, we determine the frequency \(\omega\) of the forced oscillations. The value of the amplitude of the forced oscillations of the mass \(m\) is read directly from the record on the paper.

The experimental points are plotted on the theoretical curve (Fig. 2), from which it follows that the solution of the problem in the first approximation gives, within the limits of the experiment, a quite satisfactory result.

Conclusions

This paper gives a somewhat different, new solution to the problem of oscillations of a mass on a spring with one rigid stop [6], using the more general methods proposed by A. N. Krylov and B. G. Galerkin.

The results obtained theoretically and confirmed experimentally show that the introduction of an oscillation stop into the “mass—spring” system makes the latter nonlinear; moreover, the “mass—spring—stop” system becomes “stiffer” (the resonance curves shift to the right).

The formulas presented in the paper for the amplitude of the forced oscillations of the mass \(m\), both in relative and in absolute motion, can be used in calculating systems of this kind, especially if the systems operate in a resonant regime. The most advantageous frequency in this case is the one determined by formula (17). Although the amplitude of forced oscillations increases with increasing frequency of the forced oscillations, a great danger arises of “breakdown” of the impact oscillations.

The paper considers only the case of an inelastic impact against the stop. However, the case of an elastic impact was also subjected to experimental study (a stop made of steel, aluminum, lead); in the latter case the experimental curves deviate from the theoretical one, while remaining almost equidistant from it. From the mathematical standpoint, the problem of oscillations with elastic impact presents considerable difficulties and requires additional conditions and assumptions. In this case it is necessary to know the contact time and the magnitude of the elastic deformation of the stop. The oscillatory process during a cycle must be described by two equations corresponding to the time of contact of the mass with the stop and to the time of motion of the mass between impacts. In such a formulation, the problem should be considered only for the given concrete conditions.

Investigation of the solution of the differential equation of the oscillatory motion of the mass of the system “elastic connection—mass—rigid stop” shows that, when applying B. G. Galerkin’s method to problems of this type, it is sufficient to confine oneself only to the first approximation.

References

1. V. V. Bolotin. Dynamic Stability of Elastic Systems, Gostekhizdat, 1956.

2. A. N. Dinnik. On Galerkin’s method for determining critical forces and frequencies of oscillations. Selected Works, vol. III, Academy of Sciences of the Ukrainian SSR, Kiev, 1956.

3. Yu. I. Iorish. Protection of Aircraft Equipment from Vibrations. Oborongiz, 1949.

1. A. N. Krylov, On Certain Differential Equations of Mathematical Physics Having Applications in Technical Problems. Academy of Sciences of the USSR, 1932.

2. A. I. Lur’e, A. I. Chekmarev, Forced oscillations in a nonlinear system with a characteristic composed of two rectilinear segments. Applied Mathematics and Mechanics, new series, vol. 1, no. 3, 1938.

3. I. G. Rusakov and A. A. Kharkevich, Forced oscillations of a system striking against a stop, ZhTF, vol. XII, nos. 11–12, 1942.

4. A. I. Chekmarev, The influence of a constant force on oscillations in nonlinear systems. Engineering Collection, vol. IV, no. 2, 1948.

NOTED MISPRINTS

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