PERFIL Y COMPETENCIA PROFESIONAL DEL CICLO Y MÓDULOS

The aim of this section is to illustrate the deficiency of CEA, and the necessity of TDA in the study of the unstable nonlinear friction-induced vibration problem. The comparisons between vibration frequencies solved through both methods (CEA and TDA) are carried out. Results show that stability analysis at the equilibrium point gives a clear indicator for the stability of the system, while the unstable frequencies from CEA are not the actual vibration frequencies of the nonlinear friction-induced vibration system. Therefore, TDA in the frequency domain needs to be performed as well to give the accurate unstable vibration frequencies.

A few examples are run and they all gives a similar phenomenon. So only some typical results with F=40 and knl=180 are illustrated in the following paragraphs. The critical

friction coefficient μc for the bifurcation of the system is 0.29.

4.6.1 Frequencies of the stable vibration

Firstly, the dynamic transient responses with μ=0.2 are calculated. Fig. 4.23 illustrates the vertical vibration of the mass. The vibration is stable which also matches what is expected from CEA, as μ<μc. In this case, no separation happens during the vibration.

Then vibration frequencies which are calculated through the fast Fourier transform of the transient vibration, and natural frequencies by CEA are given by Table 4.3.

Table 4.3 Frequencies of the stable system (rad/s)

Friction coefficient

CEA FFT of TDA

ω1 ω2 ω1 ω2

μ=0.2 5.03 5.72 5.03 5.71 From Table 4.3, it can be seen that frequencies calculated through two different approaches only have a slight difference. This means that when the system is stable, even if CEA are based on the linearised system, its results are nearly the same as the FFT results of TDA.

4.6.2 Frequencies of the unstable vibration

Furthermore, the comparisons between frequencies of the unstable vibration calculated by CAE (case 1), FFT of TDA ignoring separation (case 2) and FFT of TDA considering separation (case 3), are carried out. The time responses of case 2 and case 3 are shown in Figs. 4.24 and 4.25 respectively.

FFT of the time responses of the vertical vibration during two time durations Δt1 (the

displacement within Δt1 is smaller than 1) and Δt2, which represent the motion in

transient state and steady state respectively, are carried out. Results are shown in Table 4.4.

(b)

Fig. 4.24 Time responses of the ignoring separation case (0.8). (a) Horizontal and vertical vibration; (b) Contact force.

(a)

(b)

Fig. 4.25 Time responses of the considering separation case (0.8). (a) Horizontal and vertical vibration; (b) Contact force.

Table 4.4 Frequencies of the unstable system (rad/s)

Friction coefficient

CEA (case 1) FFT of TDA ignoring

separation (case2) FFT of TDA considering separation (case3) Unstable frequency Transient frequency Steady state frequency (dominant) Transient frequency Steady state frequency (dominant) μ=0.8 5.36 5.37 6.33 5.37 4.7

Because the system is unstable with two modes coupling together, the frequencies of the two modes of system are the same. This is also illustrated clearly in the frequency spectrums of case 2 and case 3 (Fig. 4.26 (a)-(d)). So Table 4.4 shows only one of the frequencies. For the steady state, when separation is ignored, although the contribution is tiny, the second harmonic component can be observed in Fig. 4.26 (b). On the other hand, when separation is considered, the contribution of the second harmonic components becomes larger and high-order harmonic component appears, as shown in Fig. 4.26 (d).

Additionally, from Table 4.4, it can be seen that if separation is ignored, the vibration frequency during transient vibration (Δt1) is very close to the unstable frequency

calculated by eigenvalue analysis at the equilibrium point. However, when the vibration settle down, its vibration frequency is 6.33 rad/s which is larger than unstable frequency 5.36 rad/s of CEA. On the other hand, when separation is considered, the frequency during Δt1 (transient vibration) is also close to the unstable frequency of

CEA; however, during steady state (Δt2), the dominant vibration frequency (4.7 rad/s)

is actually smaller than unstable frequency of CEA (5.36 rad/s). Therefore, ignoring separation in the nonlinear friction-induced vibration overestimates the vibration frequency of the system.

The phenomenon reflected from Table 4.4 can be explained. Because complex eigenvalue analysis only relies on the linear term without the higher order terms in the Taylor expansion of the nonlinear term, two consequences can happen: (1) when the vibration is near the equilibrium point, eigenvalue analysis of the linearised system

gives approximate eigenvalues, for example the frequency of case 2 and 3 during Δt1

is close to the result of CEA; (2) when the vibration is far off the equilibrium point, especially, when the amplitude of the vibration is larger than 1, effects of the higher order terms of the Taylor expansion of the original nonlinear term become significant, so the eigenvalues of the linearised system which loses the contributions of the higher term, unlike the FFT of the TDA, are not accurate. Hence, the CEA of the linearised system are trustful when the vibration is small, otherwise FFT of the DTA needs to be carried out.

(a) (b)

(c) (d) Fig. 4.26 FFT spectrums. (a) spectrum during t1 of case 2; (b) spectrum during

2

t

 of case 2; (c) spectrum during t1 of case 3; (d) spectrum during t2 of case 3

For the vibration considering separation (case 3) shown in Fig. 4.25, initially, when the system vibrates around its equilibrium point (vibration amplitude is smaller than

1), according to previous explanations, it makes sense that the frequency given by CEA equals to the frequency through TDA. However, in the steady state, not only vibration is larger than 1, shown in Fig. 4.25 (a), but also the effects of separation are involved. Separation makes the system switch between two states (separate and in-contact), which have their own vibration frequencies. The effects of these two aspects are excluded in the stability analysis at equilibrium points (CEA), which results in its frequency results being not as accurate as the FFT of TDA with considering separation. Additionally, the results of case 2, which ignores the effects of separation, are not accurate either.

To sum up, the analysis of this part illustrates the importance of considering separation from the frequency point of view, and the necessity of stability analysis of the linearised system, as it is efficient and can give a clear indication for bifurcation; The most important finding is that, in nonlinear friction-induced vibration, FFT of DTA are necessary for the determination of unstable frequencies.