For a usual damping of a few percents, conservative and non-conservative NNMs give almost the same results. However, for higher values of equivalent damping, the obtained NNMs are different and the calculation method has to be chosen appropriately. Moreover, when the damping is nonlinear, a high damping can be obtained at high amplitudes of oscillations. Therefore, efficient methods to compute non-conservative NNMs are needed. In the literature, some researches concern the influence of damping on the NNM. In [KUE 15], an energy balance is applied to conservative NNMs to explore the con- nection between NNMs and damped frequency responses. Others researchers take non- conservative NNMs as periodical solutions at iso-energy level. To obtain periodical solutions without losses of energy, an input of energy must be added to the equation. Krack proposed to insure energy balance by introducing a fictive negative damping [KRA 15]. The corresponding energy added to the NNM equation must not modify the non-conservative energy.
In order to obtain the exact energy to be injected in the system, the following approach is used. The conservative part of the nonlinear forces is first discriminated from the non- conservative one. Secondly, Phase and Energy resonances are computed directly from equations properties. The resulting equations characterizing the non-conservative NNMs are conservative. Therefore, standard methods for NNM computation can be used. To link the non-conservative resonances with frequency-responses, a fictive force Ff ic is
introduced. This fictive force can be seen as the input of energy needed to insure the energy balance of the non-conservative equations of motion. Moreover, this fictive force can be used to target a chosen NNM with a forced response.
3.2.1
Characterization of conservative and non-conservative forces
An energy balance is used to characterize the conservative and non-conservative parts of the nonlinear forces. To obtain non-conservative NNMs, the nonlinear force fnl has
to be separated into conservative fnl c and non-conservative fnl nc nonlinear forces. To discriminate such forces, an energy balance is made on the equation of motion. We use the fat that viscous damping is a pure dissipating force and the terms associated with the mass and the stiffness are pure conservative forces. A conservative force is characterized by the following property:
RT 0 x˙∗fnl(x, ˙x)dt = 0 ⇔ RT(1−a) 0 x˙∗fnl(x, ˙x)dt = RTa 0 x˙∗fnl(x, − ˙x)dt = 0 (3.19)
with a ∈ [0, 1]. For periodic solutions, only a =12 and forces symmetric with respect to the speed vector ˙x are solutions to Eq. (3.19). Therefore, the nonlinear force fnl can be separated into non-conservative fnl ncand conservative fnl c nonlinear forces represented
Generalization of Nonlinear Normal Modes to non-conservative systems
respectively by the following symmetric and anti-symmetric functions:
fnl= fnl c+ fnl nc with "
fnl c= 12(fnl(x, ˙x) + fnl(x, − ˙x))
fnl nc=12(fnl(x, ˙x) − fnl(x, − ˙x)) (3.20) The conservative and non-conservative parts of the equation of motion are then introduced as:
Now that conservative and non-conservative terms have been separated, the non- conservative NNMs can be characterized.
3.2.2
Phase resonance
In this subsection, the non-conservative NNM corresponding to the phase resonance is presented. The phase φ between the damped NNM X and the forcing vector F of Eq. (1.5) is calculated: φ = arctan ((∇bis⊗ In)X) T(R nc) XT(R c) with ∇bis= 0 01×2L 02L×1 IH⊗ 0 −1 −1 0 (3.21) with (Rc, Rnc) representing the conservative and non-conservative part of Eq. (1.5) cal-
culated in Eq. (3.20). In order to have a phase resonance, XT(Rc) has to be equal to zero
with XT(Rnc) 6= 0. Since X 6= 0, the equation corresponding to the Phase resonance is
obtained (3.22).
Rφ = ω2∇2⊗ M + I ⊗ K X + Fnl c= 0L (3.22)
One can note that Eq. (3.22) corresponds to the classical equation used to calculate un- damped unforced NNMs of an equation with viscous damping. Since Eq. (3.22) is au- tonomous, methods for conservative NNMs computation, presented in Section 3.1, can be used to compute the non-conservative NNMs.
3.2.3
Energy resonance
In this subsection, the non-conservative NNM corresponding to the energy resonance is presented.
Characterization of the energy resonance Let’s make the hypothesis that at the energy resonance, all the energy related to the displacement X is provided by the wanted non- conservative NNM XE, resulting in X = XE. The energy resonance occurs when:
∂ XTX
∂ ω = 0 (3.23)
First, the Eq. (1.5) is conventionally rewritten as:
with A a matrix such that AX − F = R and MatFnl= MatFnlc+ MatFnlnc. The ma-
trices (MatFnlc, MatFnlnc) do not need to be calculated since they will not be used in the
characterizing equation obtained at the end of the development. After some calculation explain in Annexes 4.5, the equation characterizing the energy resonance is obtained:
RE = ω2∇2⊗ M + I ⊗ K X + (Fnl c) − 1 2 ∇21⊗ M + 1 2ω ∂ (MatFnlTc)1 ∂ ω −1 ∇ ⊗ C +∂ MatFnl T nc ∂ ω (∇ ⊗ C) X +Fnl nc ω = 0L (3.25)
with ∇1 and (MatFnlTc)1 defined in Appendix 4.5. The calculation of the partial deriv-
atives in Eq. (3.25) are presented in Appendix 4.5. Since Eq. (3.25) is autonomous, methods for conservative NNMs computation, presented in Section 3.1, can be used to compute the non-conservative NNMs.
Example 1: case of a viscous damping and a pure conservative forcing. In this case, the non-conservative part of the nonlinear forces is null and Eq. (3.25) is reduced to:
ZX + Fnl− 1 2 I2H+1⊗ M −1C2 X = 0 L (3.26)
3.2.4
Fictive Force
In this work, non-conservative NNMs are defined as non-necessary synchronized period- ical solutions at iso-energy level. To obtain periodical solutions without loss of energy, an input of energy must be added to the equation of motion. However, the corresponding energy added to the NNM equation must not modify the invariant manifold. To do so, the equations corresponding to the phase and energy resonances are compared with the forced equation of motion. Then, the input of energy, called fictive forces Ff ic, is identified:
R = ZX + Fnl− Ff ic= 0L
Rφ = 0 ou RE = 0L
⇒ Ff ic= ZX + Fnl− (Rφ ou RE) (3.27)
with (R, Rφ, RE) corresponding respectively to the equations used to compute the forced
responses, the phase and the energy resonances. The input of energy, represented by the identified forcing vector Ff ic, corresponds to the required forcing vector that has to be
applied to the system in order to pass by the corresponding resonance point. The fictive forces do not modify the non-conservative NNMs since they are identified from equations Eqs. (3.22) and (3.25) characterizing them.
The fictive forces corresponding to the non-conservative NNMs are presented: • Phase resonance
By performing the calculation with Eqs. (1.5) and (3.22), the force vector that has to be applied to the system in order to pass through the phase resonance point is given by:
Shifted quadratic eigenvalue problem for stability analysis
• Energy resonance
By performing the calculation with the Eqs. (1.5) and (3.25), the force vector that has to be applied to the system in order to pass through the energy resonance point is given by: Ff icE = ω (∇ ⊗ C) X + Fnl nc+ 1 2 ∇2⊗ M + 1 2ω ∂ (MatFnlc) ∂ ω −1 ∇ ⊗ C +∂ MatFnlnc ∂ ω (∇ ⊗ C) X + 1 ωFnl nc (3.29)
In previous work, only the phase resonances were computed. To observe the influ- ence of a specific NNM onto the responses associated with a particular forcing vector, an energy balance was made, see [KUE 15]. The energy balance permits to calculated the value of a multiplier d that need to be applied to whatever forcing vector F in order for the forced responses to pass by a specific point of NNM X. The energy balance is interesting for experimental purpose since it can provide forcing vector dF simpler that the fictive force Ff ic which can be complex to realize experimentally. Since the non-conservative
NNMs have been extended to take in account energy resonance, the method is adapted as follow: RT 0 x˙∗ff icdt = RT 0 x˙∗df dt ⇔ XT(∇ ⊗ I) F f ic= XT(∇ ⊗ I) dF ⇔ d =XT(∇⊗I)Ff ic XT(∇⊗I)F