First update on vibration fault analysis of centrifugal compressor units
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The rotating part of a rotating machine with a rotor imbalance fault is called the rotor. Due to machining errors, assembly errors, uneven material properties, as well as corrosion, wear, scaling, and the detachment of components during operation, the rotor possesses eccentric masses. Fig. 2-1 Force model of rotating. Let the mass of the system be M, the eccentric mass of the rotor be m, and the eccentricity be e (as shown in Fig. 2-1 [14]). When the rotor rotates at a constant angular velocity ω, the centrifugal force F0 generated by the eccentric mass m is given by F0 = meω². The equations of motion are: Mx” + Cx’ + Kx = meω²cosωt, My” + Cy’ + Ky = meω²sinωt (2-1). The amplitudes X and Y are given by X = Y = meω²/(K – Mω²) + (Cω)². The solutions for x and y are x = X cos(ωt – Φ) and y = Y sin(ωt – Φ) (2-2). Let p = …, λ = ω/p, and ξ = C/(2Mp); then X = Y = … (2-3). The phase difference Φ satisfies tanΦ = 2ξλ/(1 – λ²) (2-4). Equations (2-3) and (2-4) can be represented using amplitude-frequency plots and phase-frequency plots (Figure 2-2 [14]). Fig. 2-2 Amplitude-frequency and Phase-frequency curves. When the rotor rotates at a constant angular velocity ω with a certain deflection a, as the rotor rotates around its axis, the shaft also rotates around the line connecting the centers of the bearings (that is, the shaft moves within the plane formed by the deflection curve of the shaft and the line connecting the bearing centers). This phenomenon is known as precession of the shaft. The equations of motion for the centroid of the rotor are given by [14]: Mx” + Cx’ + Kx = Meω²cosωt, My” + Cy’ + Ky = Meω²sinωt (2-5). The deflection at the center point s of the rotating shaft is given by [14] (2-6). From these equations, it can be seen that when λ << 1, the deflection a is very small, and the phase angle Φ is also low, approaching zero ； When λ=1, if the damping is very small, the deflection a can be large; at this point the phase angle Φ=90° ； When λ>>1, the deflection value a approaches e, the phase angle Φ approaches 180°, and the center of gravity G tends to coincide with the fixed point O, as shown in Figure 2-3 [14]. Fig. 2-3 Phase variation of rotator centroid. The dashed circle in the figure represents the precession trajectory of the rotor’s axis. The precession direction of the rotor can be the same as or opposite to the rotor’s rotation direction, and the precession angular velocity can be equal to or different from the rotational speed of the shaft. As can be seen from the diagram, the trajectory of the rotor’s axis is circular. However, in a real rotor, due to differences in the axial bending stiffness of the shaft, and especially due to variations in the stiffness of the supports, the rotor’s response to unbalanced masses results in different amplitudes in the x and y directions, and the phase difference is not 90°; as a result, the trajectory of the rotor’s axis is elliptical rather than circular. As shown in Fig. 2-4 [14]: Fig. 2-4 The A* orbit when the rotor is unbalanced. The vibration identification characteristics of a unbalanced rotor are a prominent peak at the power frequency in the spectrum; sometimes high-frequency harmonics such as the second harmonic are present as well. The phase remains stable at a constant rotational speed, while the amplitude is proportional to the degree of imbalance and increases or decreases with changes in operating speed (with the exception when exceeding the critical speed). The axis trajectory is that of an ellipse undergoing synchronous precession.
I’m not sure how to upload images within the content; does it necessarily require creating separate image files before uploading them? I hope experienced users can give some guidance.
Reply to 2# 1357911cw: When plain text is pasted, it has no effect on images; therefore, the images need to be uploaded separately, which requires some basic formatting in advanced mode. Alternatively, the information can be turned directly into a screenshot, so that both images and text can appear together; however, the files in this format are not very clear.
That book? Could you just tell me directly?
2.2 Rotor misalignment fault: The rotors of a compression unit are connected to each other by couplings in order to transmit motion and torque. Due to installation errors, as well as deformation during rotor manufacturing and under load, as well as changes in ambient temperature, poor alignment can occur between rotors. The forms of rotor misalignment include radial displacement, angular displacement, and combined displacement, as shown in Figure 2-5 [16]. Fig. 2-5 Types of rotor misalignment: (a) Radial displacement, (b) Declination displacement, (c) Comprehensive displacement. In shafting systems with poor rotor alignment, the forces exerted by the coupling change the actual working positions of the rotor journals and bearings. This not only alters the operating conditions of the bearings but also reduces the natural frequency of the rotor shafting. Therefore, rotor misalignment is a major cause of abnormal vibrations in the rotor as well as premature damage to the bearings. There are many types of couplings; in chemical enterprises, toothed couplings as well as rigid and flexible couplings are commonly used. Due to their different structural characteristics, their vibration mechanisms also vary. 1. Vibration mechanism of misalignment in toothed couplings The forms of misalignment in toothed couplings are shown in Figure 2-6 [18]. Fig. 2-6 Types of gear coupling misalignment: (a) Radial displacement, (b) Declination displacement, (c) Comprehensive displacement. When the alignment between the rotors of a machine set is out of spec, the contact between the inner and outer tooth surfaces of the gear coupling changes, as shown in Fig. 2-7 [18]. Fig. 2-7 Force condition of gear coupling. The normal force on the tooth surfaces is given by P = Mk/d cosα (2-7), where d is the diameter of the coupling’s pitch circle, in mm; α is the pressure angle of the coupling teeth, in degrees; and Mk is the torque transmitted by the coupling, in kgf·cm. The frictional torque generated by the friction force due to tooth surface engagement is given by: MF = μ·P·d = μ·Mk/cosα (2-8), where μ is the friction coefficient. The tilting torque of the intermediate gear sleeve is MT = P·b’ = b·Mk·cosφ/d·cosα (2-9); here, φ is the inclination angle of the intermediate gear sleeve, and b is the outer tooth width. If the effects of other factors are ignored, assuming that MF and MT lie in the same plane and are perpendicular to each other, the radial components resulting from these two torques are FF = MF/t and FT = MT/t (2-10), where t is the center-to-center distance between the teeth at both ends of the intermediate gear sleeve, in mm. The additional radial force acting on the bearing is FX = (2-11). Similarly, due to the effect of friction, the maximum additional axial force is FYmax = μ·P = μ·MKmax/d·cosα (2-12). When there is a radial displacement between the rotor axes, the intermediate gear sleeve of the coupling and the half-coupling form a sliding pair that cannot rotate relative to each other; however, they can slide and move in a circular path in a plane. The center of mass of the intermediate gear sleeve then moves in a circle with a diameter equal to the radial displacement △y of the axes, as shown in Figure 2-8 [18]. The rotation frequency of the intermediate gear sleeve of the coupling is twice that of the rotor’s rotation frequency; that is, as the rotor completes one rotation, the intermediate gear sleeve performs two circular motions in a plane, with the rotor’s radial displacement serving as the diameter of these circles. Fig. 2-8 Radial displacement vibration of rotor. When there is an angular displacement between the axes of the two rotors (as shown in Fig. 2-9 [18]), the rotational angular velocity of the rotor is given by [18]: ω2=ω1cosα/(1-sin2α.cos2φ1) (2-13). The transmission ratio is: i21= cosα/(1-sin2α.cos2φ1) (2-14). Here, ω1 represents the rotational angular velocity of the driving rotor; α represents the angle of deviation of the driven rotor; and φ1 represents the rotation angle of the driving rotor. Fig. 2-9 Declination displacement vibration of rotor. When the rotational angular velocity ω1 of the driving rotor remains constant, the rotational angular velocity ω2 of the driven rotor is not constant, but rather a function of the angle of deviation α and the rotation angle φ1 of the driving rotor. When φ1=0° or 180°, i21= 1/cosα is at its maximum value; whereas when φ1=90° or 270°, i21= cosα is at its minimum value, as shown in Figure 2-10【18】. Fig. 2-10 Change curve of transmission ratio i21. It can be seen that when there is an angular displacement of the rotor axis of the machine, the transmission ratio changes twice for each full rotation of the rotor; moreover, the magnitude of these changes increases as the angle of deviation α increases. As a result, the driven rotor experiences vibration due to the angular acceleration induced by these changes in the transmission ratio, and its radial vibration frequency is also twice the rotor’s rotational frequency. In actual production, the alignment between the rotors of a unit is often a combined displacement that includes both radial and angular misalignments; therefore, the mechanism behind radial vibration of the rotors is the result of these two factors combined. Furthermore, due to the additional axial force generated by the toothed coupling and the effect of the rotor’s angular displacement, the driven rotor undergoes one axial reciprocating motion per revolution, so the frequency of the rotor’s axial vibration is the same as its rotational frequency. 2. Vibration mechanism of misalignment in rigid and flexible couplings: When misaligned rotors are connected using a rigid coupling, as shown in Figure 2-11, the rotor bends due to the forced connection, and the coupling surfaces move relative to each other by two full rotations for every one rotation of the rotor. Therefore, its vibration frequency is twice the operating speed frequency. And the axial vibration frequency is synchronized with the operating frequency. The vibration mechanism of a flexible coupling is similar to that of a rigid coupling; the only difference is that, with each rotation, each bolt in the coupling is stretched and compressed once. Therefore, when the number of bolts N is even, the vibration frequency is N times the fundamental frequency, while when N is odd, the vibration frequency is 2N times the fundamental frequency [1]. In actual machine operation, since most of the bending deformation is absorbed by the flexible connecting elements (such as elastic diaphragms), the excitation forces generated are relatively small, and the machine is less sensitive to vibrations caused by misalignment; as a result, the intensity of the vibrations produced is relatively low. Fig. 2-11 Connection of rigid coupling when rotor misalignment【14】. The vibration identification characteristics in the case of rotor misalignment are as follows【19】: on the spectrum, the radial vibration spectrum consists of the fundamental frequency, its second harmonic, and various modulation harmonics; the amplitude of the second harmonic is relatively large, and it serves as the characteristic frequency. The axial vibration spectrum consists of the fundamental frequency and its harmonics, with the fundamental frequency having a peak. When the rotational speed is constant, the phase remains stable. The bearing vibration near the coupling is high; the axis trajectory consists of a larger circle surrounding a smaller one, and the precession is synchronous and in the positive direction. It is sensitive to changes in load, with vibration increasing as the load increases.
I’m sorry; I wrote it myself. I didn’t want to waste everyone’s money, so I decided to post it directly in the thread, but it turned out to be difficult to share the image.
The original poster still hasn’t uploaded the corresponding spectrum or schematic diagram; as a result, it’s really unclear what exactly you’re trying to say