Investigation of the Effective Forces of Resonating Cylindrical Helical Springs

G. S. Zhartovsky

(Presented by the scientific seminar of the Department of Mining Machines, Mine Transport, and Mining Mechanics)

This paper gives a quantitative estimate of the dynamic forces of resonating cylindrical helical springs. Many instruments and machines in which springs serve as pressure devices are subjected to external vibrations. The vibrations are transmitted to the points at which the springs are fastened and, when the frequencies of the forced oscillations coincide with the natural frequencies of the elastic links, they can cause resonances. There are a number of mechanisms and machines—for example, cam mechanisms, electrical commutator machines, etc.—which themselves serve as sources of spring oscillations. Springs that have entered into resonance (whether pre-stretched or compressed) produce additional alternating-sign forces, the magnitude of which, as will be shown below, is so considerable that the effective force of the spring may vary from zero to twice the static value. Thus, in studying the dynamics of cam mechanisms, it has been noted that the effective force of the springs may decrease so much at resonance that the tappets will separate from the cams, despite the presence of an excess force of the springs when operating away from resonance [1].

Consider a compressed helical cylindrical spring in Fig. 1, a, taking into account that practically all compression springs may be regarded as springs with fixed ends [2]. Let

\[ P_{\mathrm{st}} = \kappa \lambda \]

be the static force developed by the compressed spring;

\[ \lambda \]

be the magnitude of the preliminary compression;

\[ \kappa = \frac{G d^{4}}{8 i D^{3}} \]

be the stiffness of the spring;

\[ G \]

be the modulus of elasticity of the second kind for the spring material;

\[ d \]

be the diameter of the wire cross section;

\[ D \]

be the mean diameter of the spring;

\[ H \]

be the height of the compressed spring;

\[ M \]

be the mass of the spring.

Replace the spring by an equivalent straight bar of the same height \(H\) [3], as shown in Fig. 1, b, and consider the longitudinal oscillations of a mass

\[ dm = m\,dx \]

with coordinate \(x\), where \(m\) is the mass per unit length of the equivalent bar. If the forces of internal friction in the spring material are neglected, an inertial force \(dF_{\mathrm{in}}\) and a restoring spring force \(dF_{\mathrm{r}}\) will act on the mass \(dm\). Since, at the moment of resonance, the inertial and restoring forces are in antiphase [4], they must be taken with opposite signs.

Solving the wave equation of longitudinal oscillations

\[ \frac{\partial^{2}u}{\partial t^{2}}=a^{2}\frac{\partial^{2}u}{\partial x^{2}}, \]

where \(u=V(x,t)\) is the amplitude of the longitudinal oscillations of the spring coils

Fig. 1. Calculation scheme of a cylindrical helical spring

is sought in the form of a product of two functions—a function only of the coordinate \(V(x)\) and a function only of time \(V(t)\).

The coordinate function is determined from the natural modes of oscillation. Taking into account only the fundamental mode for the case of boundary conditions under consideration, as shown in Fig. 1, c, we obtain

\[ u(x,t)=V(x)V(t)=x_{0}\sin\frac{\pi}{H}x\cdot \sin\omega_{p}t, \]

where \(x_{0}\) is the amplitude of oscillations of the mass;

\(\omega_{p}\) is the angular frequency of oscillations of the equivalent bar (i.e., the spring) corresponding to the fundamental mode of oscillation.

Then the acceleration of the mass \(dm\) at resonance is

\[ \ddot{u}(x,t)=-\omega_{p}^{2}x_{0}\sin\frac{\pi}{H}x\cdot \sin\omega_{p}t. \]

Considering the amplitude values of the force \(dF_{\text{in}}\) (for \(\sin\omega_{p}t=1\)), we obtain

\[ dF_{\text{in}}=m\,dx\cdot x_{0}\omega_{p}^{2}\sin\frac{\pi}{H}x; \]

\[ F_{\text{in}}=\int_{0}^{H}dF_{\text{in}}=\frac{2Hm\omega_{p}^{2}x_{0}}{\pi}=\frac{2Mx_{0}\omega_{p}^{2}}{\pi}. \]

The amplitude value of the restoring force (for \(\sin\omega_{p}t=1\)):

\[ dF_{\text{r}}=\kappa\frac{du(x,t)}{dx}=\kappa\frac{\pi}{H}x_{0}\cos\frac{\pi}{H}x\,dx; \]

\[ F_{\text{r}}=2\kappa x_{0}\frac{\pi}{H}\int_{0}^{H/2}\cos\frac{\pi}{H}x\,dx=2\kappa x_{0}. \]

The amplitude value of the effective force of the spring at the moment of resonance, for both ends fixed,

\[ P_{\text{eff}}=\pm\left(F_{\text{in}}-F_{\text{v}}\right)+P_{\text{st}} =\pm\left(\frac{2Mx_0\omega_p^2}{\pi}-2kx_0\right)+P_{\text{st}}. \tag{1} \]

Accordingly,

\[ P_{\text{eff}}(t)=\left(\frac{2M\omega_p^2x_0}{\pi}-2kx_0\right)\sin\omega_p t+P_{\text{st}}. \tag{2} \]

For springs preliminarily stretched by the force \(P_{\text{st}}\), the fundamental form of longitudinal oscillations will be analogous to that considered above; therefore formulas (1) and (2) may also be applied to this case of spring deformation.

The experimental investigation of dynamic forces was carried out with the aid of setups, two of which are shown in Figs. 2 and 3. The springs were loaded (in compression and tension) by strain gauges, which were attached at one end to a rigid base, and at the other, by means of tungsten loops, to the free ends of the springs. The vibrations were excited on an ST-1000 vibration stand in the frequency range from 20 to 1000 cps. The acceleration was monitored by a PIU-2M piezoelectric vibration meter with a TsDU-1 type pickup. In the static state, the strain gauges were preliminarily stressed by a force \(P_{\text{st}}\), the magnitude of which depended on the stiffness and the preliminary compression of the spring being tested. With two strain gauges it is possible to set a force of up to 5 kg. The measurements were made on a UTS-VT-12 strain-gauge station using an N-102 loop oscillograph. The setups make it easy to calibrate the measuring channel by the static method (removing the load from the strain gauges). The limitation of this measurement method is quite obvious: the permissible value for recording a half-swing of the dynamic force, i.e., the quantity \(P_{\text{dyn}}=|F_{\text{in}}-F_{\text{v}}| \leq P_{\text{st}}\), otherwise \(0<P_{\text{eff}}<2P_{\text{st}}\), since strain gauges with such fastening do not operate in compression. The swing of the resonating coils was recorded optically. Time marks were made every 0.002 sec.

The oscillogram in Fig. 4 shows how \(P_{\text{dyn}}\) changes as the forcing frequency approaches the natural frequency of the spring. In the post-resonance zone, with a rapid change in the excitation frequency, beats are characteristic. Verification of the correctness of the calculation formulas (1) and (2) was carried out on preliminarily compressed and stretched springs. The experimental and calculated data are brought together in a table, and the oscillograms of the dynamic forces for cases of spring compression are shown in Fig. 5; here the case “compression” corresponds to the oscillogram in Fig. 5, a, and the case “compression” to the oscillogram in Fig. 5, b. The distortion of the harmonic form of \(P_{\text{dyn}}\) can be explained by the presence of a small transverse rocking of the suspended strain gauges. It is sufficient to reduce the acceleration of the vibration stand to \(5g\), which is associated with an approximately twofold decrease in the quantity \(x_0\), for the form of variation of \(P_{\text{dyn}}\) to approach a harmonic one. This is shown in the oscillogram of Fig. 5, v, corresponding to the case “compression” at \(x_0=0.25\) mm.

As is evident from the table, at the moment of resonance the force developed by the springs does not remain constant, but varies over wide limits from \(P_{\text{eff}}^{\min}\) to \(P_{\text{eff}}^{\max}\). It should be noted that these limits depend on the acting vibrational acceleration. Friction in the material of the spring coils limits the value of \(x_0\), and the relationship between \(x_0\) and the external action is nonlinear. In some cases \(P_{\text{eff}}^{\min}\) may approach zero, signifying complete unloading of the preliminarily defor-

Fig. 2. Experimental setup for investigating springs

Fig. 3. Experimental setup for investigating springs

of the deformed spring. The calculated data agree well with the experiment, which proves the possibility of such an approach to estimating the effective forces of resonating springs. This method can also be applied to cases of springs of another type, for example, flat springs of contact

Fig. 4. Oscillogram of the change in the spring pressing force when passing through resonance

pairs of relays operating under vibration conditions; in this case dependencies (1) and (2) will take on a different form.

Table 1

REFERENCES

1. G. A. Rotbart. Cam Mechanisms (trans. from English, ed. Prof. Kolchin I. I.). Sudpromgiz, 1960.
2. V. L. Bileman. Transverse vibrations of springs. In: Strength Calculations, issue 8. Mashgiz, 1962.
3. Strength Calculations in Mechanical Engineering, vol. III, ed. S. D. Ponomarev. Mashgiz, 1959.
4. S. P. Strelkov. Introduction to the Theory of Vibrations. Nauka, 1964.

Fig. 5, a, b, c. Oscillograms of the dynamic forces of resonating springs