Simulation results

q_{i j}+ ω_{i j}²q_{i j}+

(M,N,G,H,T,U

∑

^{µi j,ghtumnqghqtuqmn}

)

= f_{i j}, (6.16)

where

µ_{i j,ghtumn}=

R,S

∑

^Γghturs_ϒ^{Θmnrsi j}

rs

. (6.17)

Eq. (6.16) describes the motion of the undamped {i, j}^th modes of the plate. These EoMs can be sorted by the magnitude of the linear natural frequencies and written in our familiar matrix form, see Eq. (3.1), with the imposed modal damping terms, as

¨q + Cq + ΛΛΛq + N_q(q) = F_m, (6.18) where C is a diagonal matrix of damping coefficients, ΛΛΛ is a diagonal matrix of the squares of modal natural frequencies, N_qis the column vector containing the nonlinear terms whose n^th element may be written

N_q,n=

N r=1

∑

N s=r

∑

N t=s

∑

α_ℓ^[n]q_rq_sq_t, (6.19)

and F_mis a vector of modal forcing terms.

Substituting Eqs. (6.4) and (6.6) with the example plate parameters values in Table 6.1 into Eqs. (6.11), (6.13), (6.15) and (6.19) may result in the values of linear natural frequen-cies and coefficients of nonlinear stiffness terms. Table 6.2 lists the values of the linear natural frequencies and the non-zero coefficients of nonlinear terms for the first four modes of the plate. The configurations of the linear modeshapes of the first four modes, plotted using Eqs. (6.4), are shown in Fig. 6.5. It can be seen that these results are almost identical to those of the FE simulation, see Fig. 6.2.

6.3.2 Simulation results

In the EoM of the n^th mode, the nonlinear terms can be mainly classified into two types, i.e. single-mode terms, q³_n, and mixed-mode terms, q_iq_jq_k where i ̸= n, j ̸= n or k ̸= n, see Table 6.2. It is known that the single-mode terms can only affect the resonant frequency of its corresponding mode; while the mixed-mode ones may affect the response of other modes via the modal interaction for the situation of the single-mode-single-frequency excitation.

Mode No. ω_n[rad/s] Coefficients (×10⁹) NL term Mode I

i= 1 j= 1

58.9

α₁^I = 5.45 q³₁

α₂^I = 23.6 q₁q²₂

α₃^I = 22.7 q₁q²₃

α₄^I = 24.4 q₁q²₄

α₅^I = 74.3 q₂q₃q₄ Mode II

i= 1 j= 2

143.9

α₁^II= 23.6 q²₁q₂

α₂^II= 31.4 q³₂

α₃^II= 65.1 q₂q²₃ α₄^II = 124.3 q₂q²₄ α₅^II= 74.3 q₁q₃q₄ Mode III

i= 2 j= 1

150.8

α₁^III = 22.7 q²₁q₃ α₂^III = 65.1 q²₂q₃

α₃^III = 31.4 q³₃

α₄^III = 132.4 q₃q²₄ α₅^III = 74.3 q₁q₂q₄ Mode IV

i= 2 j= 2

235.8

α₁^IV= 24.4 q²₁q₄ α₂^IV = 124.3 q²₂q₄ α₃^IV = 132.4 q²₃q₄

α₄^IV= 55.8 q³₄

α₅^IV= 74.3 q₁q₂q₃ Table 6.2 Linear natural frequencies and nonlinear parameters for the first four modes of the example plate.

The result in Fig. 6.4 has already demonstrated that there exists a modal interaction between Mode I and other three modes of the plate structure. Hence, to investigate the effects of the single- and mixed-mode nonlinear terms on the resonant frequency shift for the multi-mode forced situation, two kinds of nonlinear four-mode truncation models for the example plate are employed to qualitatively compare with the FE model, i.e. a coupled model including nonlinear mixed-mode terms and an uncoupled model excluding mixed-mode term. Their

(a) (b)

(c) (d)

Fig. 6.5 The linear modeshape configurations of the first four modes of the plate depicted in Fig. 6.1 with the properties described in Table 6.1, used in the Galerkin decomposition. (a) Mode I, (b) Mode II, (c) Mode III and (d) Mode IV.

respective EoMs are stated as,

Coupled NROM : ¨q + C ˙q + Λq + Nq(q) = F_m(t), (6.20a) Uncoupled NROM : ¨˜q + C ˙˜q + Λ ˜q + ˜Nq˜( ˜q) = F_m(t), (6.20b) where the modal force vector Fm(t) may be written

F_m= P_rr(t) + P_hcos(Ωt), (6.21)

where r(t) is the scaled random input signal (i.e. the random amplitude input in FE sim-ulation), P_r is the vector magnitude of the modal random force component and P_h is the vector amplitude of the harmonic component. Nqand ˜N_q˜ are the vectors of nonlinear terms,

written as

It is noteworthy that due to unstable responses, another NROM candidate which only con-sists of the mixed-mode terms is unable to be employed for comparison.

In both NROMs, viscous damping is added, and the damping ratio is assumed to be ζ = 0.1% for all modes. These two equations are integrated over the same force time history defined at discrete data points for each specific excitation case using the fourth-order Runge-Kutta integration operator. The random data is generated identically for that used in the previous FE simulation. The discrete-time period between consecutive time history points is 10⁻⁴s and the integration was performed over a period of 50 s. The displacement response of the same point considered in the FE simulation is again used here, and similarly, the spectrum results are employed to demonstrate the frequency-shift phenomenon.

Firstly, the excitations considered in Fig. 6.3 are simulated using two NROMs, and the result is shown in Fig. 6.6. For this excitation case, the magnitudes of the random com-ponents are P_r = ⁴

π²ρ h[1, 1, 1, 1]^⊺× 10⁻² and P_r = ⁴

π²ρ h[1, 1, 1, 1]^⊺ for the simulation (i) and (ii) respectively. The magnitudes of harmonic component for both cases are P_h= [0]^⊺. From the results of the simulation (i), it can be seen that the responses of the uncoupled and coupled models are nearly identical and their resonant frequencies are close to the linear natural frequencies. As expected, this observation confirms that the effect of the coupled-mode terms is insignificant for the low-response-amplitude situation. For the high-level excitation situation, the result (ii), the resonant frequencies for both models tend to shift to higher values. However, the shift level of the coupled model is more significant than that of the uncoupled model. Note that the frequency shift for the results of the uncoupled model is due to the backbone curve distortion. This result demonstrates that both single- and mixed-mode terms can cause frequency shifting in the nonlinear region. Based on the qualitative comparison between Fig. 6.3 and Fig. 6.6, it is unable to distinguish which NROM is more

0 5 10 15 20 25 30 35 40 45 50 Frequency (Hz)

-160 -150 -140 -130 -120 -110 -100 -90 -80 -70

Power/frequency (dB/Hz)

(ii)

(i)

Fig. 6.6 The power spectral density of the response of the plate depicted in Fig. 6.1 to random excitations in the first four modes, computed using NROM simulation data. The two force amplitudes used are: (i) P_r= ^4×10−2

π²ρ h (1, 1, 1, 1)^⊺and (ii) P_r= ⁴

π²ρ h(1, 1, 1, 1)^⊺. The displacement response x = ¹₂q₁−

√ 2 2 q₂−

√ 2

2 q₃+ q₄is considered, which is equivalent to that at the point with the coordinates (x, y) = ³₄(a, b). The blue and red lines represent the results of the uncoupled and coupled models respectively, and the dashed-black lines denote the linear natural frequencies.

accurate for describing the nonlinear behaviour of the example plate.

Then, the NROMs are used to simulate the plate under the hybrid excitation considered in Fig. 6.4. The corresponding force amplitudes used are P_r = ⁴

π²ρ h(1, 1, 1, 1)^⊺× 10⁻² and P_h= _{ρ hab}⁴ (5, 0, 0, 0)^⊺× 10⁻³. Fig. 6.7 shows the simulation results for this case. We can see that for Mode II, III and IV, compared with linear natural frequencies, their resonant frequencies of the coupled model result have increased, while those in the uncoupled model result have not changed. For this case, it is the coupled model that can more accurately rep-resent the nonlinear behaviour of the full-order FE model, see from the comparison between Fig. 6.4 and Fig. 6.6.

The NROM simulation results in Fig. 6.6 and Fig. 6.7 have further demonstrated that the mixed-mode terms would cause modal interactions, especially for the multi-mode exci-tation situation. However, it is not clear whether this modal interaction is the resonant one investigated in Chapter 5. Note that the one-to-one modal interaction is excluded as it may only occur between Mode II and III. By examining the linear frequencies ratio and nonlin-ear terms, it can be predicted that the multi-mode auto-parametric interaction involving the

0 5 10 15 20 25 30 35 40 45 50 Frequency (Hz)

-160 -150 -140 -130 -120 -110 -100 -90 -80 -70

Power/frequency (dB/Hz)

Fig. 6.7 The power spectral density of the response of the plate depicted in Fig. 6.1 to a hybrid excitation in the first four modes, computed using NROM simulation data. The hy-brid excitation scenario is that all four modes are randomly excited and Mode I is sinusoidal forced simultaneously: P_r = ^4×10−1

π²ρ h (1, 1, 1, 1)^⊺, P_h= ^4×10_{ρ hab}⁻³(5, 0, 0, 0)^⊺ and Ω = ω_n1. The displacement response x = ¹₂q₁−

√2 2 q₂−

√2

2 q₃+ q₄is considered. The denotations are the same as those of Fig. 6.6.

four modes of interest may occur. For example, the nonlinear terms q₂q₃q₄ in the EoM of Mode I may give birth to a complex conjugate u_2pu_3pu_4mand u_2mu_3mu_4pduring the nonlin-ear nnonlin-ear-identity transform of the direct normal form technique application. These derived terms respond at the fundamental frequency of Mode I, thus causing a resonant modal in-teraction. Therefore, a group of extra NROM simulations is performed in which only Mode I and Mode III are directly excited:

• In Fig. 6.8(a), Mode I and III are randomly forced at two levels: (i) P_r= ⁴

π²ρ h(1, 0, 1, 0)^⊺× 10⁻²and P_h= (0)^⊺, and (ii) Pr= ⁴

π²ρ h(1, 0, 1, 0)^⊺and P_h= (0)^⊺.

• In Fig. 6.8(b), Mode I and III are randomly forced at a relatively low level and an extra sine-wave force is applied to Mode I: P_r = ⁴

π²ρ h(1, 0, 1, 0)^⊺× 10⁻², P_h =

4

ρ hab(5, 0, 0, 0)^⊺× 10⁻³and Ω = ω_n1.

• In Fig. 6.8(c), Mode I and III are randomly forced at a relatively low level and an extra sine-wave force is applied to Mode III: P_r = ⁴

π²ρ h(1, 0, 1, 0)^⊺× 10⁻², P_h =

4

ρ hab(0, 0, 5, 0)^⊺× 10⁻³and Ω = ω_n3.

0 5 10 15 20 25 30 35 40 45 50 Frequency (Hz)

-160 -140 -120 -100 -80

Power/frequency (dB/Hz)

(i) (ii)

(a)

0 10 20 30 40 50

Frequency (Hz) -160

-140 -120 -100 -80

Power/frequency (dB/Hz)

(b)

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz) -160

-140 -120 -100 -80

Power/frequency (dB/Hz)

(c)

Fig. 6.8 The power spectral density of the response of the plate depicted in Fig. 6.1, com-puted using the NROM simulation data, when (a) Mode I and III are randomly forced, (b) only Mode I is additionally forced harmonically and (c) only Mode III is additionally forced harmonically.

As with Fig. 6.6, Fig. 6.8(a) shows a similar result that for the low-level forcing case, the results of coupled and uncoupled models are close and for the high-level case, they both present frequency-shift phenomena and the shift level of the coupled model results is higher.

Also, the observation from Fig. 6.8(b) and Fig. 6.8(c) is similar to that of Fig. 6.7 that the power increment of one mode due to the harmonic forcing results in a resonant frequency increasing of the other mode. Note that in Fig. 6.8(b), the resonant peak at around 30 Hz is due to the three-times harmonic response of Mode I. From the results illustrated in Fig. 6.8, we may conclude that:

• the frequency shift is not caused by the double-mode one-to-one or multi-mode auto-parametric modal interaction because Mode II and IV are always still for this case.

• the occurrence of the frequency shift is not affected by the number of modes involved but the power level of the nonlinear dynamic system.

• the effect of modal interactions in the example plate due to mixed-mode terms is bilateral, see from Fig. 6.8(b) and Fig. 6.8(c).

6.4 Effect of the nonlinear coupled-mode terms

In document Backbone Curve Analysis of Nonlinear Mechanical Systems (Page 157-164)