7. Anexos
7.6. Anexo VI. Contexto interno y externo
4.3.1
Example: pinned-pinned beam with rotational spring
The clamped-clamped beam in the previous section provides an insight into the hysteretic nature of nonlinear systems, though further interesting, and potentially damaging, behaviour can be exhibited if the symmetry of the system is broken. To investigate this, the boundary conditions of the beam are changed. At both ends, the beam will rest upon pinned supports, so that the displacement is fixed, but the rotation is not. To break the symmetry, a rotational spring of stiffness ˆk is added at the right-hand end. The schematic for this system is shown in Fig. 4.5. w(x, t) x = 0 ˆ k x = ` ρ, ˆA, E, I
Figure 4.5: Schematic for a pinned-pinned beam with rotational spring at the right-hand tip. This system has been more thoroughly derived in [159], and the approach taken in that paper
is summarised here. The equations of motion are now defined by ρ ˆA∂ 2w ∂t2 + EI ∂4w ∂x4 − " E ˆA 2L Z ` 0 ∂w ∂x 2 dx # ∂2w ∂x2 + δ(x − `)ˆkψ(`, t) = 0, (4.29)
where ψ(x, t) denotes the rotation of the beam and δ(•) is the Dirac delta function, used here to represent the fact that there is only a spring at the tip. The spring stiffness, and associated asymmetry, can be incorporated into the mode shapes of the system, so that the ith mode is
defined by φi(x) = v u u t 2 1 − 2EIˆ k` sin 2(κ`) −sin(κ`) sinh(κ`) 2 sin(κx) − sin(κ`) sinh(κ`)sinh(κx) . (4.30) This result is taken directly from [159], which also defines the solvability condition for κ, given by
cot(κ) − coth(κ) +2EIκ ˆ
k = 0. (4.31) By projecting the system onto these mode shapes and applying the Galerkin approximation, the equations of motion in Eq. (4.30) can be written, in modal coordinates, as
¨ qn+ EIα4,n ρ ˆA` qn+ E 2ρ`2 N X i=1 N X j=1 N X k=1 βi,jα2,k,nqiqjqk = 0, (4.32)
where α4,n has been updated to include the linear stiffness provided by the spring. As in
the previous case, the terms denoted α and β are nonlinear coefficients that can be found by integrating products of mode shapes and their respective derivatives. Depending on the number of equations included in the model, N , these equations can now easily be solved either by the analytical methods discussed in Chapter 3 or via numerical continuation. The latter has been used to create the backbone curves in Fig. 4.6.
In [159], a two-mode model is used and a 1:3 internal resonance is observed, in which energy is transferred from the first to the second mode. In Fig. 4.6, an expanded modal basis is used and it can be seen that, as ωr approaches 155 Hz, energy is actually transferred to the third
and fourth mode. However, it should be noted that energy transfer to the former is relatively minimal.
This interaction between the modes is a behaviour that is not exhibited by the clamped- clamped beam, as seen in Fig. 4.4. Although not shown here, this absence of internal res-
125 130 135 140 145 150 155 160 0 0.5 1 1.5 2 2.5 3 3.5 10 -4 Mode 1 Mode 2 Mode 3 Mode 4
Figure 4.6: Modal contributions of the first four modes in the first backbone curve. onance is also the case for the pinned-pinned beam without the rotational spring (see [9], for example). It is, therefore, possible to conclude that such an interaction is caused by the introduction of the symmetry-breaking, rotational spring. Further confirmation of this has been found by considering a pinned-pinned beam with identical rotational springs at both ends, which also shows no sign of such an interaction; due to this response and that of the clamped-clamped beam, it is not shown here.
Given that the only difference in these models is the BCs, it is the terms of Eqs. (4.21) and (4.32) defined by these expressions that must be investigated. Here, it should be noted that
EIα4,n
ρ ˆAL = ω
2
n. (4.33)
The only remaining terms to be considered are the nonlinear coupling terms and, in partic- ular, the α2 and β parameters by which these are defined. The values of α2,i,j and βi,j are
given in Tables 4.1 and 4.2 for the coupling between the first five modes in the pinned-pinned model, both with and without the rotational spring.
Since the only difference between these systems is the addition of the spring, the nature of these matrices offers some enlightening insight. This is summarised as follows:
Table 4.1: Parameter values for α2,i,j for i, j ∈ {1, 2, 3, 4, 5}.
Pinned-pinned
8.8877E+09 -3.3059E+09 -2.9327E+09 -2.5354E+09 -2.2060E+09 -3.3059E+09 3.3116E+10 -6.0361E+09 -5.9017E+09 -5.5322E+09 -2.9325E+09 -6.0365E+09 7.2597E+10 -8.6215E+09 -8.7148E+09 -2.5354E+09 -5.9017E+09 -8.6209E+09 1.2732E+11 -1.1143E+10 -2.2060E+09 -5.5322E+09 -8.7155E+09 -1.1143E+10 1.9728E+11
Pinned-pinned with spring
1.5183E+09 2.6873E+09 3.1945E+09 2.7973E+09 1.3608E+09 2.9764E+10 4.0117E+10 2.3824E+10 1.0268E+10 4.4290E+10 2.2281E+10 1.7504E+10 8.0044E+09 2.3590E+10 1.4446E+10 3.0198E+10 2.6464E+09 3.0099E+10 5.4013E+09 2.9842E+10 3.6546E+10 2.7304E+10 3.2184E+10 3.9700E+10 2.7516E+10
Table 4.2: Parameter values for βi,j for i, j ∈ {1, 2, 3, 4, 5}.
Pinned-pinned
4.6050E+01 -1.7129E+01 -1.5195E+01 -1.3137E+01 -1.1430E+01 -1.7129E+01 1.7159E+02 -3.1276E+01 -3.0578E+01 -2.8664E+01 -1.5195E+01 -3.1276E+01 3.7615E+02 -4.4668E+01 -4.5158E+01 -1.3137E+01 -3.0578E+01 -4.4668E+01 6.5967E+02 -5.7735E+01 -1.1430E+01 -2.8664E+01 -4.5158E+01 -5.7735E+01 1.0222E+03
Pinned-pinned with spring
2.6216E+04 4.5241E+04 3.4668E+04 -1.8453E+04 -1.6901E+05 4.5241E+04 7.8698E+04 6.0091E+04 -3.2040E+04 -2.9316E+05 3.4668E+04 6.0091E+04 4.6125E+04 -2.4564E+04 -2.2437E+05 -1.8453E+04 -3.2040E+04 -2.4564E+04 1.3155E+04 1.1940E+05 -1.6901E+05 -2.9316E+05 -2.2437E+05 1.1940E+05 1.0937E+06
are differences in their signs. In particular, in the no-spring system, only the lead diagonal terms are positive and the others are negative. In the spring case, all of the terms are positive.
• Furthermore, for each i, it is true that max
j {|α2,i,j|} = α2,i,i in the classical model, but
this is not necessarily the case once the spring is added.
• The pinned-pinned α2 matrix is symmetric, but the spring removes this symmetry.
• For the β matrix, there is a noticeable difference in the order of the terms, of the magnitude 102 − 104.
• Once more, the only positive terms are on the lead diagonal for the pinned-pinned beam, but this is not true of the spring case.
The equivalent terms for the clamped-clamped beam are not given, though all of the prop- erties of the pinned-pinned model listed here remain true for that case. As previously men-
Table 4.3: Linear natural frequencies for the clamped-clamped, pinned-pinned, and pinned- pinned with rotational spring beam configurations.
Mode C-C ωn,k/ωn,1 P-P ωn,k/ωn,1 P-P w/ spring ωn,k/ωn,1 1 1015.1 1 699.6 1 977.0 1 2 2798.3 2.757 2267.0 3.241 3246.8 3.323 3 5485.7 5.404 4730.0 6.761 6470.8 6.623 4 9068.1 8.933 8088.5 11.653 10557.9 10.806 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 10 -4 Free Forced 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 10 -4 120 140 160 0 0.5 1 1.5 2 2.5 3 3.5 10 -4
Figure 4.7: Forced responses for three forcing cases.
tioned, it is a 1:3 internal resonance in the spring beam, so the response frequency of the second mode is approximately three times that of the first. In fact, as shown in Table 4.3, ωn,2is slightly greater than 3ωn,1for both pinned configurations, and there is minimal differ-
ence between the extent to which this is the case. As such, this highlights the importance of the nonlinear coupling coefficients, as discussed above.
The influence of the modal interaction on the forced response is presented in Fig. 4.7. The forcing levels, P, are varied and displayed in the figure, and the damping level across all three cases is ζ = 1e-3. The forcing in panel (b) is four times that in panel (a), but it can be seen that the maximum value of q1 is only fractionally greater. This suggests that, for this beam,
the internal resonance introduces some limiting behaviour in the modal displacement, so that the amount of energy required to reach the higher amplitudes of the fundamental tongue is greater than in the symmetric case with no rotational spring.