From equations (4) and (5) and using F(x)=-kl5y/5x, the distribution expression of the spring force and vibration velocity can be obtained: the analog equivalent network of uniform spring longitudinal vibration F(x)=F1coskx- jZv1sinkxv(x)=v1coskx-jF1Zsinkx(7) The above equation is similar to the solution of the uniform lossless transmission line equation in electricity, where the particle vibration velocity is analogous to the current at a certain point on the line, and the force is analogous to the line-to-line voltage at a certain point on the line. It can be seen that the propagation process of the longitudinal vibration in the spring is completely similar to the propagation of the electric wave in the transmission line.
The discussion of the spring eigenfrequency now considers the general condition of the spring mass m, the two-end force and the vibration velocity are F1, F2 and v1, v2, respectively, and one end is attached with a uniform spring longitudinal vibration with a load mass of M. Assuming that the spring length l, the cross-sectional area S, the equivalent body density Q is constant in a small longitudinal vibration, the corresponding analog circuit is as shown.
The eigenfrequency at the time of fixing both ends is a fixed condition at both ends, which corresponds to the case where the input impedance Zi => when the load Z1 =>. It is obvious from equation (8) that the resonance condition, that is, the eigenfrequency equation is: sinkl=0>knl=nP, n=1, 2,3, and the eigenfrequency Mn=ncl2l, n=1, 2, 3 will be obtained. Spring longitudinal wave velocity cl=(k/m)l Substituting into equation (4) has Mn=n2km, n=1,2,3, (9) The above formula is the eigenfrequency formula when the two ends of the spring are fixed. Where m is the mass of the spring itself and k is the stiffness factor of the spring. n=1, the corresponding M1=(k/m)/2 is the spring inherent fundamental frequency. It can be seen from equation (9) that the spring has an infinite number of eigenfrequencies, that is, an infinite number of normal vibration modes. The longitudinal wave in the spring can be regarded as a superposition of an infinite number of normal vibrations of different frequencies, and the spring is in a standing vibration state at the time of resonance. It is not difficult to push the spring eigenfrequency for the free ends (Zl=0, Zi=0) exactly the same as the above equation, except that the displacement distribution is different. When fixed, the two ends are nodes; when free, the ends are amplitudes.
The eigenfrequency when one end is fixed at one end is free. The condition at this time is Zl=>, Zi=0, and it is not difficult to obtain the resonance condition from equation (8): coskl=0>knl=(2n-1)P2, n=1, 2,3,, the eigenfrequency is Mn=(2n-1)cl4l, n=1,2,3, and the wave velocity cl is substituted into the above formula to get Mn=(2n-1)4km, n=1,2,3, (10) When n=1, the corresponding M1=(k/m)/4 is the inherent fundamental frequency, which is half of the fixed or free ends, and there are an infinite number of normal ways. For the spring to be fixed at one end, the eigenfrequency when the one end is subjected to the harmonic force is also the same as the above formula. This situation is equivalent to a force source in the load end, the condition is still Zl =>, Zi = 0. As for the eigenfrequency is the same, it can be considered that, when the mechanical damping loss is not considered, the harmonic force When the frequency is equal to an eigenfrequency of the spring, the coupling between the harmonic force and the end face of the spring is zero, and the harmonic end is equivalent to the free end.
In conclusion, the eigenfrequency equation of the uniform mass spring under several specific boundary conditions is derived by using the electromechanical analogy, and the corresponding eigenfrequency formula is given. The process is simple and clear. It can be seen from the discussion that for different boundary conditions, only the input impedance or loop impedance of the corresponding analog equivalent circuit should be discussed. It can thus be seen that it is convenient to use an equivalent network method to analyze the natural frequencies of the elastic elements or mechanical vibration systems connected by some of the elastic elements.
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