4.4 Linear, non-classical boundary conditions
4.5.1 Example: nonlinear spring-supported beam
where Rn|s(•) is the tensor resultant of a system, •, of n polynomials of order s, and νaug is
the notation used for the augmented matrix defined in Eq. (4.53). It can now be recalled that the coefficient tensor νBCis defined in terms of c1 and κ. Given that this model assumes that
the mode shapes of the system will be scaled as the system vibrates, it is possible to obtain values for κ by setting c1 = 1. In doing so, Eq. (4.53) provides a solvability condition that
can be numerically solved, as has been seen in the linear BC models.
Although a numerical solution to Eq. (4.53) is possible, as Nmaxincreases, it can become
difficult to ensure the accuracy of the solution to a sufficient number of decimal places. As such, it can be useful to express this large determinant in terms of elementary, 2 × 2 determinants, called Pl¨ucker relations, as outlined in [220].
4.5.1
Example: nonlinear spring-supported beam
To further assess the ability of the proposed method for treating nonlinear BCs, it is useful to consider a model in which both BCs are nonlinear. To that end, the following beam configuration is introduced, in which both ends of the beam are supported by springs with linear, quadratic, and cubic stiffnesses. Although the consideration of this beam may seem somewhat theoretical, in reality, it can simply be thought of as an investigation into the
applicability of the method, giving insight into its validity for more complicated systems. In particular, there are multiple microscale BCs that can easily arise, and this methodology introduces an efficient way of investigating the associated dynamics. The BCs for this beam are given by
φ00(0) = φ00(`) = 0, φ000(0) = ˆK1(0)φ(0) + ˆK2(0)φ(0)2+ ˆK3(0)φ(0)3, φ000(`) = ˆK1(`)φ(`) + ˆK2(`)φ(`)2+ ˆK3(`)φ(`)3,
(4.54) where ˆKn(x) defines the nth-order spring stiffness at x.
By first addressing the second derivative BCs, it is possible to rewrite the mode shape equation in Eq. (4.10) as
φ(y) = c1
cos(κy) + cosh(κy) + cos(κ)
sinh(κ) − coth(κ) sinh(κy) +c2 sin(κy) + sin(κ) sinh(κ)sinh(κy) . (4.55)
The BCs defined in Eq. (4.54) take the exact form given in Eq. (4.41), so the derivation is qualitatively identical to the general form given in Section 4.5 and is not repeated here. Instead, it is possible to immediately consider the system in the form given in Eq. (4.52). Once more, the coefficients in ν0 and ν`are complicated expressions in terms of ˆK
(x) n and κ,
but can be easily treated using symbolic mathematical solvers, such as Wolfram Mathematica or Maple. The explicit terms for these are as follows:
ν00,3 = 2 ˆK1(0)+ κ3 coth(κ) − cos(κ) sinh(κ) X + 4 ˆK2(0)X2+ 8 ˆK3(0)X3, ν01,2 = κ3 1 − sin(κ) sinh(κ) , ν02,1 = 0, ν03,0 = 0,
ν`0,3 = (2 ˆK1(`)cos(κ) + κ3[(cosh(κ) − cos(κ)) coth(κ) − sin(κ) − sinh(κ)])X + 4 ˆK2(`)cos2(κ)X2+ 8 ˆK3(`)cos3(κ)X3,
ν`1,2 = κ3(cos(κ) − sin(κ) coth(κ)) + 2 ˆK1(`)sin(κ) + 8 ˆK2(`)sin(κ) cos(κ)X + 24 ˆK3(`)sin(κ) cos2(κ)X2,
ν`2,1 = 4 ˆK2(`)sin2(κ) + 24 ˆK3(`)sin2(κ) cos(κ)X, ν`3,0 = 8 ˆK3(`)sin3(κ).
The expressions in Eq. (4.56) are more algebraically complicated than in previous exam- ples, leading to a resultant that can be particularly intensive to solve numerically. As an initial investigation into the validity of the proposed methodology, it is useful to consider the symmetric case of this beam, so that it may be compared to the classical free-free and pinned-pinned models. To that end, the beam stiffnesses will be set to ˆKn(0) = ˆKn(`) = ˆKn,
with the value of these constants being varied so that the classical cases can be approached. To aid this investigation, a single parameter, γ, will be used to express the spring stiffnesses as
ˆ
Figure 4.8: Normalised mode shapes of the first mode of the nonlinear spring-supported beam for γ ∈ [1e-9, 1e-6, 1e-3, 1].
Figs. 4.8–4.10 display the first, second, and third mode shapes, respectively, for a number of values of γ. Namely, these are γ ∈ [1e-9, 1e-6, 1e-3, 1]. As γ → ∞, it is expected that the behaviour should become increasingly similar to that of the pinned-pinned beam. This trend is visible in Figs. 4.8–4.10, in which, for the case γ = 1, the mode shapes are very close to those expected from the fully pinned beam. Interestingly, although the beam tip supports are symmetric, the positions of the tips in the mode shapes are not. This suggests that the variable position of the beam tips allows the beam stiffness to influence the symmetry of the modes. This holds true even when the spring stiffnesses are extremely high. However, it
Figure 4.9: Normalised mode shapes of the second mode of the nonlinear spring-supported beam for γ ∈ [1e-9, 1e-6, 1e-3, 1].
should be noted that, as γ is increased, they become closer to one another, suggesting that the pinned-pinned beam represents an upper limit case for this configuration. At the other end of the scale, one might expect the behaviour to tend to that of a free-free beam. However, it can be seen that the spring stiffnesses still have a significant influence. In fact, these shapes are qualitatively similar to those found if the spring-supported beam is assumed to be rigid, as there is negligible bending of the beam when γ = 1e-6 or 1e-9.
The influence that the value of γ has on the free response of the system is investigated in Fig. 4.11 through the consideration of the backbone curves for γ ∈ [1e-9, 1e-6, 1e-3]. As
Figure 4.10: Normalised mode shapes of the third mode of the nonlinear spring-supported beam for γ ∈ [1e-9, 1e-6, 1e-3, 1].
expected, the extent to which the system behaves nonlinearly increases with γ. Not only does the variation in frequency at higher amplitudes become more pronounced, but interactions between the modes also arise. In the previous section, it was noted that the introduction of a rotational spring at one end of a pinned-pinned beam leads to substantial changes in the α and β matrices, causing internal resonances to arise. In the current arrangement, the beam itself is symmetric. Therefore, it might understandably be assumed that it is unlikely that such a resonance would occur. However, the asymmetric nature of the mode shapes has a similar effect on the γ and β matrices. As such, it should be concluded that, in the initial discussion
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