# Steady oscillations of systems with non-linear and unsymmetrical elasticity

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## Abstract

The paper discusses the oscillations of systems with arbitrary elastic characteristics under arbitrary excitation. It presents a simple method for deriving forced motions from free undamped motions. A detailed treatment of free motions is given first. Special methods are offered for the evaluation of the time integral $t=\int (dt/dx) dx$ over the interval in which $dx/dt$ is small. The substitution $x = A sin \theta$ (with $A$ the amplitude of the motion) is shown to make the integrand finite at all points, and hence to permit a graphical evaluation of the integral. Undamped oscillations under harmonic excitation are derived from free oscillations by changing the excitation from a time function $F(t)$ into a space function $F(x)$ through the assumption that $x$ and $t$ are related as in the free motion. By combining $F(x)$ with the elastic restoring force $E(x)$, a new effective $E(x)$ is obtained --- to which there corresponds a new "free" motion, which in turn furnishes a second approximation for the relation between $x$ and $t$, and hence a new function $F(x)$. A new effective $E(x)$ and a new "free" motion are then found; and the cycle is repeated until the relation between $x$ and $t$ ceases to change. In general, the process converges rapidly. The accuracy can be checked at any stage of the work. Special methods are suggested to facilitate the drawing of response curves for various intensities of the excitation. A discussion is given of the general nature of the curves connecting the excitation intensity $P$ in $F(t) = P \cos \omega t$ with the oscillation frequency $\omega$. For a particularly convenient expression of the relation between $P$ and $\omega$, the use of two parameters $\mu=(P/E(A))/(2 - P/E(A))$ is suggested --- $E(A)$ being the elastic force for $x = A$, and $\omega_0$ being the natural frequency corresponding to $A$. By generally similar processes, damped motions under harmonic excitation are derived from harmonically excited undamped motions or from free motions. A convenient method is offered for the construction of response curves from the curves for the undamped forced motion, by applying a correction based on the minimum force $F$ required to maintain the damped motion at $\omega=\omega_0$. Although a direct superimposition of the effects of different harmonics in $F(t)$ is not possible because of the non-linearity of $E(x)$, the method can be used no matter what form the excitation has --- so long as it is periodic. The method is shown to apply also when the frequencies of the excitation and of the motion are multiples or sub-multiples of each other. Furthermore, the method remains valid when there are nonlinearities in the inertia and damping terms of the equation of motion. At thorough study is made of the stability of the motions discussed. A general criterion of stability is offered --- based on the assumption that a motion is definitely unstable if a small disturbance tends to grow initially, and that stability exists if a disturbance has an initial tendency to annul itself. This criterion is shown to be supported by empirical knowledge as far as such knowledge exists. In an appendix, there are given charts for the harmonic analysis of oscillation curves by a method especially adapted to curves of this type. The method is based on a polygonal approximation to the curve to be analyzed, and works on a subdivision of the area under the curve into triangles. For each of these triangles, the Fourier coefficients can be read from the charts; and the final series is obtained by addition of the series for the individual triangles. See more