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Figure 5.2 shows that the shaft centers itself at high rotating speeds. The diskrotates about its axis through the center of gravity at speeds above the critical(Ωω1). This fact is utilized in centrifuges and other machines in which bal-ancing is not an option due to the eccentric stock (Fig. 5.9). The system is tunedto run above the critical speed using soft springs in neck bearings. The disk rotatesabout its axis through the center of gravity, i. e. the springs in the neck bearings aredeflected at the rotational frequency by the magnitude of eccentricity and apply acorresponding force to the frame. It is considerably smaller for a soft neck bearingthan the respective unbalancing force that would occur with a rigid bearing.5.2.2 Passing through the Resonance PointMany drives are operated at speeds above the critical, i.e. the rotors have to passthrough one or several resonance points during start-up until they reach their oper-ating speed, and the same is true for coasting down and braking. This applies to thebending vibrations of rotors (such as turbomachines, textile mandrils, spin-driers,centrifuges), but also to machine foundations (see, for example, Fig. 3.14), torsionaloscillators (such as vehicle drive trains, fans), and coupled-mass oscillators (suchas screens, belt drives). Extreme loads are reached during this passing through res-onance. Of primary interest is the behavior of the rotor near resonance since thelargest dynamic deflections occur there.If one can assume that the rotor starts with aconstant angular accelerationα,the resulting amplitudes are as shown in Fig. 5.3.Fig. 5.3Amplitude when passingthrough resonance as a functionof acceleration
3165 Bending OscillatorsThis is to show the result of the calculation only, see Fig. 5.3. The maximum am-plitude is not reached when excitation frequency and natural frequency coincide,but slightly later. The maxima shift towards higher speeds during acceleration andtowards lower speeds during deceleration processes. The faster the passing throughresonance the smaller are the amplitudes. The dimensionless characteristic param-eterα/ω21serves as a measure. If one simulates the start-up process consideringthe motor characteristic, it turns out that the speed increases at a slower rate nearresonance because the drive has to generate energy for moving the resonating foun-dation. The rotor puts up a larger resistance than the one that would match its ownrotational inertia when passing through the resonance point.The reverse effect occurs when decelerating, i. e. at declining speed: a momentis exerted by the foundation onto the rotor so that the latter accelerates in excess ofwhat the input torque would cause. The excess energy of the vibrating foundation isthen transferred onto the rotor. The effect that the foundation acts as an additionaldrive can be utilized for the efficient operation of vibrating machines as oscillators,e. g. in a vibrating compactor, see Sect. 5.4.5 “Passing through resonance” in .