Asymptotic formulation of the nonlinear bifurcation scenarios in thermomechanically coupled plates

Abstract

The nonlinear dynamics of composite plates with thermomechanical coupling is analytically addressed in order to describe the main bifurcation phenomena triggering the involved pre- and post-buckling response scenario. The static buckling occurrence and two resonance conditions around the unbuckled and buckled equilibria are investigated by means of the asymptotic multiple scale method, together with the double-zero bifurcation marking the occurrence of dynamical buckling. The resulting modulation equations and the steady-state mechanical and thermal responses are determined and compared with the numerical outcomes in order to verify the adequacy and effectiveness of the refined scalings adopted in the multiple scale analyses to describe the various bifurcation scenarios.

Appendices

Appendix A Material properties and equation coefficients

The dynamical behavior of the thermomechanical model is investigated by considering an epoxy/carbon fiber composite plate of dimensions $a=b=1$ m and $h=0.01$ m. The material’s elastic and thermal properties, which are assumed to be independent of the temperature, are taken from [36], and read:

$$\begin{aligned} \begin{aligned}&Y_1=1.72\cdot 10^{11}\frac{N}{m^2}, \, \nu _{12}=0.25, \, \rho =1940\frac{kg}{m^3}, \\&\lambda _{11}=36.42\frac{W}{m\cdot K}, \\&\alpha _1=0.57 \cdot 10^{-6}\frac{1}{K}, \, Y_2=6.91\cdot 10^{9}\frac{N}{m^2}, \\&G_{12}=3.45\cdot 10^{9}\frac{N}{m^2}, \\&\lambda _{22}=0.96\frac{W}{m\cdot K}, \, \alpha _2=35.6 \cdot 10^{-6}\frac{1}{K}, \\&c_v=400\frac{J}{kg\cdot K}, \\&\delta =330\frac{N\cdot s}{m^3}, \, H=100\frac{W}{m^2\cdot K} \end{aligned} \end{aligned}$$

(19)

where $Y_1,Y_2,G_{12}$ are longitudinal modulus of rigidity in x and y direction and shear modulus, respectively; $\nu _{12}$ is the Poisson’s ratio; $\rho$ and $\delta$ are mass density and dam** coefficient; $\lambda _{11}, \lambda _{22}, \lambda _{33}$ are the thermal conductivities along the x, y, and z directions; $\alpha _1,\alpha _2$ are the thermal expansions along x and y directions; $c_v$ is the specific heat at constant strain, and H is the boundary conductance. The subsequent value of the mechanical natural frequency is 286.67Hz. After nondimensionalization, the numerical coefficients of Eq. (1) are:

$$\begin{aligned}{} & {} a_{12}=0.0593, \, a_{14}=0.6859, \, a_{15}=-0.2729,\nonumber \\{} & {} a_{16}=-0.9036, \,a_{22}=7.81 \cdot 10^{-5},\nonumber \\{} & {} a_{23}=-1.2391, a_{24}=1.08\cdot 10^{-4},\nonumber \\{} & {} a_{32}=6.06 \cdot 10^{-4}, \, a_{33}=0.001195 \end{aligned}$$

(20)

Appendix B Multiple Scale analysis of the pre-buckling equilibrium

To develop the asymptotic procedure to the system (4), three time scales are introduced, i.e., $T_{0}=t, \; T_{1}= \epsilon t, \; T_{2}=\epsilon ^2 t$, and, consistently, the time derivatives are expressed as

$$\begin{aligned} \begin{aligned}&d/dt = D_0 + \epsilon D_1 + \epsilon ^2D_2 \\&d^2/dt^2 = D^2_0 +2\epsilon D_0D_1+\epsilon ^2 D^2_1 +2\epsilon ^2D_0D_2 \end{aligned} \end{aligned}$$

(21)

where $D_i = \partial /\partial T_i$. Due to the presence of only cubic nonlinear term in the mechanical equation, and in order to account for the different time evolution of the mechanical variable with respect to the thermal ones, variables are scaled as follows:

$$\begin{aligned}{} & {} \tilde{W}=\epsilon ^{1/2}\; \hat{W}, \qquad \tilde{T}_{R0}=\epsilon ^{3/2} \; \hat{T}_{R0},\nonumber \\{} & {} \tilde{T}_{R1}=\epsilon ^{3/2}\; \hat{T}_{R1} \end{aligned}$$

(22)

so that their expression as perturbation of the reference equilibrium reads:

$$\begin{aligned} \begin{aligned} \tilde{W}(t)&=\epsilon ^{1/2}W_0 (T_0,T_1,T_2)+\epsilon ^{3/2}W_1 (T_0,T_1,T_2)\\&\quad +\epsilon ^{5/2}W_2 (T_0,T_1,T_2) \\ \tilde{T}_{R0}(t)&=\epsilon ^{3/2}\text {T0}_0 (T_0,T_1,T_2)+\epsilon ^{5/2}\text {T0}_1 (T_0,T_1,T_2)\\&\quad +\epsilon ^{7/2}\text {T0}_2 (T_0,T_1,T_2) \\ \tilde{T}_{R1}(t)&=\epsilon ^{3/2}\text {T1}_0 (T_0,T_1,T_2) +\epsilon ^{5/2}\text {T1}_1 (T_0,T_1,T_2)\\&\quad +\epsilon ^{7/2}\text {T1}_2 (T_0,T_1,T_2) \end{aligned}\nonumber \\ \end{aligned}$$

(23)

Parameter scaling is performed by assuming small dam** and small transversal excitation, while coupling terms are scaled to properly account for the different time scale at which thermal variables evolve with respect to the mechanical one:

$$\begin{aligned} a_{12}= & {} \epsilon \; \hat{a}_{12}, \quad f=\epsilon ^{3/2} \; \hat{f},\quad a_{15}=\epsilon \;\hat{a}_{15}, \nonumber \\ a_{16}= & {} \epsilon ^{1/2}\;\hat{a}_{16}, \quad a_{24}= \epsilon ^{1/2}\;\hat{a}_{24}, \quad a_{33}=\epsilon \;\hat{a}_{33} \end{aligned}$$

(24)

To study the response around primary resonance, detuning parameter $\sigma$ is introduced:

$$\begin{aligned} \omega ^2=\varOmega ^2+\epsilon \; \sigma \end{aligned}$$

(25)

Once scaled by $\epsilon ^{-1/2}$, the resulting perturbation equations at each order read:

$$\begin{aligned}&\bullet \;\text {Order}\;\; \epsilon \quad \quad D_0^2W_0+\varOmega ^2 W_0\,\,\, =0 \end{aligned}$$

(26)

$$\begin{aligned}&\bullet \;\text {Order}\;\; \epsilon ^2 \quad D_0^2W_1+\varOmega ^2 W_1 =-a_{12} D_0W_0 -a_{14} W_0^3\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad -2D_0 D_1W_0-\sigma W_0\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad +f \cos (\varOmega T_0) \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~D_0\text {T0}_0+a_{22} \text {T0}_0 =-a_{24} W_0 D_0W_0 \end{aligned}$$

(27)

$$\begin{aligned}&~~~~~~~~~~~~~~~~~~~~~~~~D_0\text {T1}_0+a_{32} \text {T1}_0 =-a_{33} D_0W_0 \nonumber \\&\bullet \;\text {Order}\;\; \epsilon ^3\quad D_0^2W_2+W_2 \varOmega ^2 =-a_{12} D_0W_1-a_{12} D_1W_0\nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-3 a_{14} W_1 W_0^2-a_{15} \text {T1}_0 \nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ -a_{16} \text {T0}_0 W_0-2 D_0 D_1W_1\nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-D_1^2W_0\nonumber \\&\quad ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-2 D_0 D_2W_0-\sigma W_1 \\&~~~~~~~~~~~~~~~~~~~~~~~D_0\text {T0}_1+a_{22} \text {T0}_1=-a_{24} W_0 D_0W_1\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad -a_{24} W_0 D_1W_0\nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad -a_{24} W_1 D_0W_0-D_1\text {T0}_0 \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~D_0\text {T1}_1+a_{32} \text {T1}_1=-a_{33} D_0W_1 \nonumber \\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\quad -a_{33} D_1W_0-D_1\text {T1}_0 \nonumber \end{aligned}$$

(28)

At first order, the solution of the mechanical Eq. (26) reads

$$\begin{aligned} W_0 = A(T_1, T_2) \textrm{e}^{i \varOmega T_0} + c.c. \end{aligned}$$

(29)

with $A(T_1, T_2)$ undetermined function of the slow time scales and c.c. complex conjugate terms (the overbar will denote the complex conjugate and i is the imaginary unit). Substituting $W_0$ in the first equation of (27), and imposing the solvability condition implies

$$\begin{aligned} D_1 A=\frac{i \left( -f+2 (\sigma +i a_{12} \varOmega )A +6 a_{14} A^2 \bar{A}\right) }{4 \varOmega } \end{aligned}$$

(30)

The particular solutions at order $\epsilon ^2$ are

$$\begin{aligned} \begin{aligned}&W_1=\frac{a_{14} A^3 \textrm{e}^{3 i \varOmega T_0 }}{8 \varOmega ^2}+c.c.\\&\text {T0}_0=\frac{a_{24} \varOmega A^2 \textrm{e}^{2 i \varOmega T_0 }}{ i a_{22}-2 \varOmega }+ c.c.\\&\text {T1}_0=\frac{ a_{33} \varOmega A \textrm{e}^{i \varOmega T_0 }}{ i a_{32}-\varOmega }+c.c. \end{aligned} \end{aligned}$$

(31)

In view of Eqs. (29), (31), the solvability condition of the mechanical problem at the third order (28) yields

$$\begin{aligned} D_2 A= & {} c_{1r}+ i c_{1i} + (c_{2r}+ i c_{2i}) A+ i c_{3i} A^2\nonumber \\{} & {} -2 i c_{3i} A \bar{A}+(c_{4r}+ i c_{4i}) A^2 \bar{A}+ i c_{5i} A^3 \bar{A}^2\nonumber \\ \end{aligned}$$

(32)

where

$$\begin{aligned} c_{1r}&= \frac{a_{12} f}{16 \varOmega ^2},\; c_{1i}= \frac{f \sigma }{16 \varOmega ^3}, \; c_{2r}= \frac{a_{15} a_{32} a_{33}}{2 \left( a_{32}^2+\varOmega ^2\right) },\nonumber \\ c_{2i}&= -\frac{a_{12}^2}{8 \varOmega }-\frac{a_{15} a_{33} \varOmega }{2 \left( a_{32}^2+\varOmega ^2\right) }-\frac{\sigma ^2}{8 \varOmega ^3},\\ c_{3i}&= -\frac{3 a_{14} f}{16 \varOmega ^3},\; c_{4r}= \frac{3 a_{12} a_{14}}{4 \varOmega ^2}+\frac{a_{16} a_{22} a_{24}}{2 \left( a_{22}^2+4 \varOmega ^2\right) },\nonumber \\ c_{4i}&= -\frac{3 a_{14} \sigma }{4 \varOmega ^3}-\frac{a_{16} a_{24} \varOmega }{a_{22}^2+4 \varOmega ^2},\; c_{5i}= -\frac{15 a_{14}^2}{16 \varOmega ^3} \nonumber \end{aligned}$$

(33)

According to the usual reconstitution procedure [37], the amplitude derivatives with respect to time t are obtained from (21)

$$\begin{aligned} \dot{A}= \epsilon D_1A + \epsilon ^2 D_2A \end{aligned}$$

(34)

The $\epsilon$ parameter is completely reabsorbed through a backward rescaling, and recalling Eqs. (30) and (32), the complex amplitude modulation equation for the mechanical variable results

$$\begin{aligned} \dot{A}= & {} c_{1r}+ i c_{11i} + (c_{21r}+ i c_{21i}) A+ i c_{3i} A^2\nonumber \\{} & {} -2 i c_{3i} A \bar{A}+(c_{4r}+ i c_{41i}) A^2 \bar{A}+ i c_{5i} A^3 \bar{A}^2 \end{aligned}$$

(35)

where

$$\begin{aligned} c_{11i}= & {} -\frac{f}{4 \varOmega }+c_{1i},\; c_{21r}= c_{2r}-\frac{a_{12}}{2},\nonumber \\ c_{21i}= & {} c_{2i}+\frac{\sigma }{2\varOmega },\; c_{41i}= \frac{3 a_{14}}{2 \varOmega }+c_{4i} \end{aligned}$$

(36)

The complex-valued modulation equation for the mechanical amplitude A can be conveniently expressed in polar form applying the following transformation

$$\begin{aligned} A=\frac{1}{2} a(t) \textrm{e}^{i \theta (t)},\; \bar{A}=\frac{1}{2} a(t) \textrm{e}^{-i \theta (t)} \end{aligned}$$

(37)

Separating real and imaginary parts leads

$$\begin{aligned}&\dot{a}=2 (c_{11i} \sin \theta + c_{1r} \cos \theta )+c_{21r} a\nonumber \\&\quad -\frac{3 c_{3i} a^2 \sin \theta }{2}+\frac{ c_{4r} a^3}{4}\\&a\dot{\theta }=2 (c_{11i} \cos \theta - c_{1r} \sin \theta )+ c_{21i} a\nonumber \\&\quad -\frac{ c_{3i} a^2 \cos \theta }{2} +\frac{ c_{41i}a^3}{4}+\frac{ c_{5i}a^5}{16} \nonumber \end{aligned}$$

(38)

where the time dependence of a and $\theta$ has been omitted for the sake of readability. Finally, the system asymptotic solutions can be reconstructed at second order by recalling Eq. (29) and (31). Moreover, remembering that $W(t)=W_{e_1}+\tilde{W}$, $T_{R0}(t)=T_{R0e_1}+\tilde{T}_{R0}$, $T_{R1}(t)=T_{R1e_1}+\tilde{T}_{R1}$, mechanical and thermal solutions can be expressed in trigonometric form as

$$\begin{aligned}&W(t)=a \cos \psi +c_6 \; a^3 \cos (3 \psi ) \end{aligned}$$

(39)

$$\begin{aligned}&T_{R0}(t)=T_{R0e_1}+c_7 \; a^2 \cos (2 \psi )\nonumber \\&~~~~~~~~~~~~~~+c_8 \; a^2 \sin (2 \psi ) \end{aligned}$$

(40)

$$\begin{aligned}&T_{R1}(t)=c_9 \; a \cos \psi + c_{10} \; \sin \psi \end{aligned}$$

(41)

where $\psi =\varOmega t +\theta$ and

$$\begin{aligned} c_6&=\frac{a_{14}}{32 \varOmega ^2}, \; c_7=-\frac{a_{24} \varOmega ^2}{a_{22}^2+4 \varOmega ^2}, c_8=\frac{a_{22} a_{24} \varOmega }{2 \left( a_{22}^2+4 \varOmega ^2\right) }, \\ c_9&=-\frac{a_{33} \varOmega ^2}{a_{32}^2+\varOmega ^2}, \; c_{10}=\frac{a_{32} a_{33} \varOmega }{a_{32}^2+\varOmega ^2} \end{aligned}$$

Appendix C Multiple Scale analysis of the post-buckling equilibrium

Due to the presence of quadratic and cubic nonlinear terms in the mechanical Eq. (10), five time scales are introduced, i.e., $T_{0}=t, \; T_{1}= \epsilon t, \; T_{2}=\epsilon ^2 t,\; T_{3}=\epsilon ^3 t,\; T_{4}=\epsilon ^4 t$, so that time derivatives are expressed as

$$\begin{aligned} \begin{aligned} d/dt&= D_0 + \epsilon D_1 + \epsilon ^2D_2 +\epsilon ^3D_3+\epsilon ^4D_4\\ d^2/dt^2&= D^2_0 +2\epsilon D_0D_1+\epsilon ^2 (D^2_1 +2D_0D_2)\\&\quad +2\epsilon ^3 (D_1D_2+D_0D_3)\\&\quad +\epsilon ^4 (D_2^2+2D_1D_3+2 D_0D_4) \end{aligned} \end{aligned}$$

(42)

where $D_i = \partial /\partial T_i$. To account for the different time evolution of the mechanical variable with respect to the thermal ones, variables are scaled as follows:

$$\begin{aligned} \tilde{W}=\epsilon \; \hat{W}, \qquad \tilde{T}_{R0}=\epsilon ^2 \; \hat{T}_{R0}, \qquad \tilde{T}_{R1}=\epsilon ^2\; \hat{T}_{R1} \end{aligned}$$

(43)

so that their expression as perturbation of the reference equilibrium reads:

$$\begin{aligned} \begin{aligned} \tilde{W}(t)&=\epsilon W_0 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^2 W_1 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^3 W_2 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^4 W_3 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^5 W_4 (T_0,T_1,T_2,T_3,T_4) \\ \tilde{T}_{R0}(t)&=\epsilon ^2 \text {T0}_0 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^3 \text {T0}_1 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^4 \text {T0}_2 (T_0,T_1,T_2,T_3,T_4)\\&\quad + \epsilon ^5 \text {T0}_3 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^6 \text {T0}_4 (T_0,T_1,T_2,T_3,T_4) \\ \tilde{T}_{R1}(t)&=\epsilon ^2 \text {T1}_0 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^3 \text {T1}_1 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^4 \text {T1}_2 (T_0,T_1,T_2,T_3,T_4)\\&\quad + \epsilon ^5 \text {T1}_3 (T_0,T_1,T_2,T_3,T_4)\\&\quad +\epsilon ^6 \text {T1}_4 (T_0,T_1,T_2,T_3,T_4) \end{aligned} \end{aligned}$$

(44)

As for the pre-buckling resonance analysis, parameter scaling is performed by assuming small dam** and small transversal excitation, while coupling terms are scaled to properly account for the different time scale at which thermal variables evolve with respect to the mechanical one. Due to the increased number of time scales considered and to the different contributions into mechanical and membrane thermal equations, parameter scaling is adjusted as follows:

$$\begin{aligned} a_{12}= & {} \epsilon ^2 \; \hat{a}_{12}, \quad a_{15}=\epsilon ^2 \;\hat{a}_{15}, \quad a_{16}=\epsilon ^2\;\hat{a}_{16}, \nonumber \\ f= & {} \epsilon ^3 \; \hat{f},\quad a_{24}= \epsilon \;\hat{a}_{24}, \quad a_{33}=\epsilon \;\hat{a}_{33} \end{aligned}$$

(45)

To study the response around primary resonance, detuning parameter $\sigma$ is introduced:

$$\begin{aligned} \omega ^2=\varOmega ^2+\epsilon ^2 \; \sigma \end{aligned}$$

(46)

while $\omega$ is derived from the square root of (46) through series expansion to the first order around $\sigma =0$: $\omega = \varOmega +\epsilon ^2 \sigma /(2\varOmega )+\mathcal {O}(\sigma ^2)$.

The resulting perturbation equations at each order read:

$$\begin{aligned}&\bullet \;\text {Order}\;\;\epsilon{} & {} D_0^2W_0+\varOmega ^2 W_0 =0 \end{aligned}$$

(47)

$$\begin{aligned}&\bullet \;\text {Order}\;\;\epsilon ^2{} & {} D_0^2W_1+\varOmega ^2 W_1 =-2 D_0 D_1W_0 -\frac{3 \varOmega W_0^2 }{2 a_{141}} \nonumber \\{} & {} {}&D_0\text {T0}_0+a_{22} \text {T0}_0=-a_{141}a_{24} \varOmega D_0W_0 \end{aligned}$$

(48)

$$\begin{aligned}{} & {} {}&D_0\text {T1}_0+a_{32} \text {T1}_0=-_{33} D_0W_0 \nonumber \\&\bullet \;\text {Order}\;\;\epsilon ^3{} & {} D_0^2W_2+ \varOmega ^2 W_2=-a_{12} D_0W_0\nonumber \\{} & {} {}&-\frac{3 \varOmega W_1 W_0}{a_{141}} -a_{14} W_0^3 \nonumber \\{} & {} {}&-2 D_0 D_1W_1-D_1^2W_0-2 D_0 D_2W_0\nonumber \\{} & {} {}&-\sigma W_0+f \cos (\varOmega T_0) \end{aligned}$$

(49)

$$\begin{aligned}{} & {} {}&D_0\text {T0}_1+a_{22} \text {T0}_1=-a_{141} a_{24} \varOmega D_0W_1\nonumber \\{} & {} {}&-a_{141} a_{24} \varOmega D_1W_0 \nonumber \\{} & {} {}&-a_{24} W_0 D_0W_0-D_1\text {T0}_0 \nonumber \\{} & {} {}&D_0\text {T1}_1+a_{32} \text {T1}_1=-a_{33} D_0W_1\nonumber \\{} & {} {}&-a_{33} D_1W_0-D_1\text {T1}_0 \nonumber \\&\bullet \;\text {Order}\;\;\epsilon ^4{} & {} D_0^2W_3+ \varOmega ^2 W_3=-a_{12} D_0W_1-a_{12} D_1W_0\nonumber \\{} & {} {}&-3 a_{14} W_1 W_0^2-a_{141} a_{16} \text {T0}_0 \varOmega \nonumber \\{} & {} {}&-\frac{3 \sigma W_0^2}{4 a_{141} \varOmega }-\frac{3 W_2 W_0 \varOmega }{a_{141}}-\frac{3 W_1^2 \varOmega }{2 a_{141}}\nonumber \\{} & {} {}&-a_{15} \text {T1}_0-2 D_0 D_1W_2-D_1^2W_1\nonumber \\{} & {} {}&-2 D_0 D_2W_1-2 D_1 D_2W_0-2 D_0 D_3W_0\nonumber \\{} & {} {}&-\sigma W_1 \end{aligned}$$

(50)

$$\begin{aligned}{} & {} {}&D_0\text {T0}_2+a_{22} \text {T0}_2=-\frac{a_{141} a_{24} \sigma D_0W_0}{2 \varOmega }\nonumber \\{} & {} {}&-a_{141} a_{24} \varOmega D_0W_2-a_{141} a_{24} \varOmega D_1W_1\nonumber \\{} & {} {}&-a_{141} a_{24} \varOmega D_2W_0\nonumber \\{} & {} {}&-a_{24} W_0 D_0W_1-a_{24} W_0 D_1W_0-a_{24} W_1D_0W_0\nonumber \\{} & {} {}&-D_1\text {T0}_1-D_2\text {T0}_0 \nonumber \\{} & {} {}&D_0\text {T1}_2+a_{32} \text {T1}_2=-a_{33} D_0W_2-a_{33} D_1W_1\nonumber \\{} & {} {}&-a_{33} D_2W_0-D_1\text {T1}_1-D_2\text {T1}_0 \nonumber \\&\bullet \;\text {Order}\;\;\epsilon ^5{} & {} D_0^2W_4+ \varOmega ^2 W_4=-a_{12} D_0W_2\nonumber \\{} & {} {}&-a_{12} D_1W_1-a_{12} D_2W_0-3 a_{14} W_2 W_0^2\nonumber \\{} & {} {}&-3 a_{14} W_1^2 W_0-a_{141} a_{16} \text {T0}_1 \varOmega \nonumber \\{} & {} {}&-\frac{3 \sigma W_1 W_0}{2 a_{141} \varOmega }-\frac{3 W_3 W_0 \varOmega }{a_{141}}-\frac{3 W_1 W_2 \varOmega }{a_{141}}\nonumber \\{} & {} {}&-a_{15} \text {T1}_1-a_{16} \text {T0}_0 W_0-2 D_0 D_1W_3\nonumber \\{} & {} {}&-D_1^2W_2-2 D_0 D_2W_2-2 D_1 D_2W_1\nonumber \\{} & {} {}&-D_2^2W_0-2 D_0 D_3W_1\nonumber \\{} & {} {}&-2 D_1 D_3W_0-2 D_0 D_4W_0-\sigma W_2 \\{} & {} {}&D_0\text {T0}_3+a_{22} \text {T0}_3=-\frac{a_{141} a_{24} \sigma D_0W_1}{2 \varOmega }\nonumber \\{} & {} {}&-\frac{a_{141} a_{24} \sigma D_1W_0}{2 \varOmega }-a_{141} a_{24} \varOmega D_0W_3 \nonumber \\{} & {} {}&-a_{141} a_{24} \varOmega D_1W_2-a_{141} a_{24} \varOmega D_2W_1\nonumber \\{} & {} {}&-a_{141} a_{24} \varOmega D_3W_0-a_{24} W_0 D_0W_2 \nonumber \\{} & {} {}&-a_{24} W_0 D_1W_1-a_{24} W_0 D_2W_0\nonumber \\{} & {} {}&-a_{24} W_1 D_0W_1-a_{24} W_1 D_1W_0 \nonumber \\{} & {} {}&-a_{24} W_2 D_0W_0-D_1\text {T0}_2-D_2\text {T0}_1-D_3\text {T0}_0 \nonumber \\{} & {} {}&D_0\text {T1}_3+a_{32} \text {T1}_3=-a_{33} D_0W_3\nonumber \\{} & {} {}&-a_{33} D_1W_2-a_{33} D_2W_1-a_{33} D_3W_0 \nonumber \\{} & {} {}&-D_1\text {T1}_2-D_2\text {T1}_1-D_3\text {T1}_0 \nonumber \end{aligned}$$

(51)

At first order, the solution of the mechanical equation (47) reads

$$\begin{aligned} W_0 = A(T_1, T_2, T_3, T_4) \textrm{e}^{i \varOmega T_0} + c.c. \end{aligned}$$

(52)

with $A(T_1, T_2, T_3, T_4)$ undetermined function of the slow time scales. Substituting $W_0$ in the first equation of (48), and imposing the solvability condition implies

$$\begin{aligned} D_1 A=0 \end{aligned}$$

(53)

The particular solutions at order $\epsilon ^2$ are

$$\begin{aligned} \begin{aligned}&W_1=-\frac{3 A \bar{A}}{a_{141} \varOmega }+\frac{A^2 \textrm{e}^{2 i \varOmega T_0} }{2 a_{141} \varOmega }+c.c.\\&\text {T0}_0=\frac{ a_{141} a_{24} \varOmega ^2 A\textrm{e}^{i \varOmega T_0} }{i a_{22}-\varOmega }+ c.c.\\&\text {T1}_0=\frac{a_{33} \varOmega A e^{i \varOmega T_0}}{i a_{32}-\varOmega }+c.c. \end{aligned} \end{aligned}$$

(54)

Moving to the third order, Eqs. (52), (54) are substituted into Eq. (49); removing secular terms into the mechanical equation leads to

$$\begin{aligned} D_2A= \frac{i \left( -f+2 (\sigma +i a_{12} \varOmega )A-24 a_{14} A^2 \bar{A}\right) }{4 \varOmega } \end{aligned}$$

(55)

Mechanical and thermal solutions at third-order result

$$\begin{aligned}&W_2=\left( \frac{a_{14}}{8 \varOmega ^2}+\frac{3}{16 a_{141}^2 \varOmega ^2}\right) A^3 \textrm{e}^{3 i \varOmega T_0 } +c.c. \nonumber \\&\text {T0}_1=\frac{2 a_{24} \varOmega }{+i a_{22}-2 \varOmega }A^2 \textrm{e}^{2 i \varOmega T_0 }+c.c.\\&\text {T1}_1=\frac{ a_{33} }{a_{141}(i a_{32}-2 \varOmega )}A^2\textrm{e}^{2 i \varOmega T_0 }+c.c. \nonumber \end{aligned}$$

(56)

At fourth order, after substitution of solvability conditions and particular solutions at the previous orders, eliminating secular terms implies

$$\begin{aligned} D_3A=\frac{i}{2}\left( \frac{ a_{141}^2 a_{16} a_{24} \varOmega ^2}{i a_{22}-\varOmega }+\frac{a_{15} a_{33}}{i a_{32}-\varOmega }\right) A \end{aligned}$$

(57)

Particular solutions at fourth order read

$$\begin{aligned}&W_3= \frac{3 \sigma A \bar{A}}{4 a_{141} \varOmega ^3}+\frac{3 \left( 20 a_{14} a_{141}^2-19\right) }{8 a_{141}^3 \varOmega ^3}A^2 \bar{A}^2\nonumber \\&+\left( \frac{ f A}{3 a_{141}\varOmega ^3}+\frac{7 \left( 14 a_{14} a_{141}^2-3\right) }{16 a_{141}^3 \varOmega ^3}A^3 \bar{A}\right. \nonumber \\&\quad \left. -\frac{3 \sigma +4 i a_{12} \varOmega }{12 a_{141} \varOmega ^3}A^2 \right) \textrm{e}^{2i \varOmega T_0}\nonumber \\&+\frac{2 a_{14} a_{141}^2+1}{16 a_{141}^3 \varOmega ^3}A^4 \textrm{e}^{4 i \varOmega T_0 }+c.c.\\&\text {T0}_2=\left( \frac{i a_{141} a_{22} a_{24} f}{4 (a_{22}+i \varOmega )^2}\right. \nonumber \\&\left. +\frac{ a_{141} a_{24} (a_{12} a_{22} \varOmega -2 i a_{22} \sigma +\sigma \varOmega )A}{2 (a_{22}+i \varOmega )^2} \right. \nonumber \\&\left. +\frac{i a_{24} \left( a_{22} \left( 12 a_{14} a_{141}^2+5\right) +5 i \varOmega \right) A^2\bar{A} }{2 a_{141} (a_{22}+i \varOmega )^2}\right) \textrm{e}^{i \varOmega T_0 }\nonumber \\&-\frac{3 i a_{24} \left( 2 a_{14} a_{141}^2+11\right) A^3 \textrm{e}^{3 i \varOmega T_0}}{16 a_{141} (a_{22}+3 i \varOmega )}+c.c. \nonumber \\&\text {T1}_2=\left( \frac{i a_{32} a_{33} f}{4 \varOmega (a_{32}+i \varOmega )^2}+\frac{ a_{32} a_{33} (a_{12} \varOmega -i \sigma )A}{2 \varOmega (a_{32}+i \varOmega )^2}\right. \nonumber \\&\left. +\frac{6 i a_{14} a_{32} a_{33} A^2\bar{A}}{\varOmega (a_{32}+i \varOmega )^2}\right) \textrm{e}^{i \varOmega T_0 } \nonumber \\&-\frac{3 i a_{33} \left( 2 a_{14} a_{141}^2+3\right) A^3 \textrm{e}^{3i \varOmega T_0 }}{16 a_{141}^2 \varOmega (a_{32}+3 i \varOmega )}+c.c. \nonumber \end{aligned}$$

(58)

Finally, at fifth order, solvability condition for the mechanical equation of (51) provides

$$\begin{aligned} D_4A&=\frac{f (a_{12} \varOmega +i \sigma )}{16 \varOmega ^3} -\frac{i\left( a_{12}^2 \varOmega ^2+\sigma ^2\right) A}{8 \varOmega ^3}\nonumber \\&\quad +\frac{3 i a_{14} f A^2 }{4 \varOmega ^3} -\frac{i a_{14} f A \bar{A}}{2 \varOmega ^3}\\&\quad +\frac{ a_{14}(-2 a_{12} \varOmega +3 i \sigma ) A^2\bar{A} }{\varOmega ^3} -\frac{27 i a_{14}^2 A^3\bar{A}^2}{\varOmega ^3} \nonumber \end{aligned}$$

(59)

According to the usual reconstitution procedure [37], the amplitude derivatives with respect to time t are obtained from (21)

$$\begin{aligned} \dot{A}= \epsilon D_1A + \epsilon ^2 D_2A+\epsilon ^3 D_3A+\epsilon ^4 D_4 A \end{aligned}$$

(60)

The $\epsilon$ parameter is completely reabsorbed through a backward rescaling, and recalling Eqs. (53), (55), (57), (59), the complex amplitude modulation equation for the mechanical variable results

$$\begin{aligned} \dot{A}= & {} c_{1r}+ i c_{1i} + (c_{2r}+ i c_{2i}) A+ i c_{3i} A \bar{A}\nonumber \\{} & {} -\frac{3}{2} i c_{3i} A^2+(c_{4r}+ i c_{4i}) A^2 \bar{A}+ i c_{5i} A^3 \bar{A}^2 \end{aligned}$$

(61)

where

$$\begin{aligned} c_{1r}&= \frac{a_{12} f}{16 \varOmega ^2}, \; c_{1i}= \frac{f \left( \sigma -4 \varOmega ^2\right) }{16 \varOmega ^3},\; \nonumber \\ c_{2r}&= -\frac{a_{12}}{2}+\frac{a_{16} a_{22} a_{24} \varOmega ^2}{4 a_{14} \left( a_{22}^2+\varOmega ^2\right) }+\frac{a_{15} a_{32} a_{33}}{2 \left( a_{32}^2+\varOmega ^2\right) },\nonumber \\ c_{2i}&= -\frac{a_{12}^2 \varOmega ^2+\sigma ^2-4 \sigma \varOmega ^2}{8 \varOmega ^3}-\frac{a_{16} a_{24} \varOmega ^3}{4 a_{14} \left( a_{22}^2+\varOmega ^2\right) }\nonumber \\&\quad -\frac{a_{15} a_{33} \varOmega }{2 \left( a_{32}^2+\varOmega ^2\right) },\; c_{3i}= -\frac{a_{14} f}{2 \varOmega ^3},\nonumber \\ c_{4r}&= -\frac{2 a_{12} a_{14}}{\varOmega ^2},\; c_{4i}= \frac{3 a_{14} \left( \sigma -2 \varOmega ^2\right) }{\varOmega ^3},\nonumber \\ c_{5i}&= -\frac{27 a_{14}^2}{\varOmega ^3} \end{aligned}$$

(62)

The complex-valued modulation equation for the mechanical amplitude A can be conveniently expressed in polar form applying the following transformation

$$\begin{aligned} A=\frac{1}{2} a(t) \textrm{e}^{i \theta (t)},\; \bar{A}=\frac{1}{2} a(t) \textrm{e}^{-i \theta (t)} \end{aligned}$$

(63)

Separating real and imaginary parts leads

$$\begin{aligned} \dot{a}&=+2(c_{1i} \sin \theta +c_{1r} \cos \theta )+c_{2r} a\nonumber \\&\quad +\frac{5 c_{3i} a^2 \sin \theta }{4} +\frac{c_{4r} a^3}{4} \end{aligned}$$

(64)

$$\begin{aligned} a\dot{\theta }&=2( c_{1i} \cos \theta - c_{1r} \sin \theta )+c_{2i} a\nonumber \\&\quad -\frac{c_{3i} a^2 \cos \theta }{4} +\frac{c_{4i} a^3}{4} +\frac{ c_{5i} a(t)^5}{16} \end{aligned}$$

(65)

where the time dependence of a and $\theta$ has been omitted for the sake of readability. Finally, the system asymptotic solutions can be reconstructed at fourth order by recalling Eqs. (52), (54), (56), (58). Moreover, remembering that $W(t)=W_{e_2}+\tilde{W}$, $T_{R0}(t)=T_{R0e_2}+\tilde{T}_{R0}$, $T_{R1}(t)=T_{R1e_2}+\tilde{T}_{R1}$, mechanical and thermal solutions can be expressed in trigonometric form as

$$\begin{aligned} W(t)&=W_{e_2}-3 c_{6} a^2-27 c_{7} a^4+a \cos \psi \nonumber \\&\quad +c_{8} a \cos (2 \psi -\theta ) +(c_{6} a^2 +14 c_{7} a^4 ) \cos (2 \psi ) \nonumber \\&\quad +c_{9} a^2 \sin (2 \psi )+c_{10} a^3 \cos (3 \psi )\nonumber \\&\quad +c_{7} a^4 \cos (4 \psi ) \end{aligned}$$

(66)

$$\begin{aligned} T_{R0}(t)&=T_{R0e_2}+c_{11} \cos (\psi -\theta )+c_{12} a \cos \psi \nonumber \\&\quad +c_{13} a^3 \cos \psi +c_{14} a^2 \cos (2 \psi )+c_{15} a^3 \cos (3 \psi )\nonumber \\&\quad +c_{16} \sin (\psi -\theta ) +c_{17} a \sin \psi \nonumber \\&\quad +c_{18} a^3 \sin \psi +c_{19} a^3 \sin (3 \psi ) +c_{20} a^2 \sin (2 \psi ) \end{aligned}$$

(67)

$$\begin{aligned} T_{R1}(t)&=c_{21} \cos (\psi -\theta )+c_{22} a \cos \psi \nonumber \\&\quad +c_{23} a^3 \cos \psi +c_{24} a^2 \cos (2 \psi )+c_{25}a^3 \cos (3 \psi )\nonumber \\&\quad +c_{26} \sin (\psi -\theta ) +c_{27} a \sin \psi \nonumber \\&\quad +c_{28} a^3 \sin \psi +c_{29} a^2 \sin (2 \psi )+c_{30} a^3 \sin (3 \psi ) \end{aligned}$$

(68)

where $\psi =\varOmega t +\theta$ and

$$\begin{aligned} c_{6}&=\frac{\sqrt{a_{14}}}{2 \sqrt{2} \varOmega }-\frac{\sqrt{a_{14}} \sigma }{4 \sqrt{2} \varOmega ^3},\qquad c_{7}=\frac{\sqrt{a_{14}^3}}{16 \sqrt{2} \varOmega ^3}, \\ c_{8}&= \frac{\sqrt{2} \sqrt{a_{14}} f}{3 \varOmega ^3},\qquad c_{9}= \frac{a_{12} \sqrt{a_{14}}}{3 \sqrt{2} \varOmega ^2}, \qquad c_{10}= \frac{a_{14}}{8 \varOmega ^2}, \\ c_{11}&= \frac{a_{22}^2 a_{24} \varOmega f }{\sqrt{2} \sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\ c_{12}&=-\frac{a_{24} \varOmega \left( -a_{12} a_{22}^3+a_{12} a_{22} \varOmega ^2+3 a_{22}^2 \sigma +2 a_{22}^2 \varOmega ^2+\sigma \varOmega ^2+2 \varOmega ^4\right) }{2 \sqrt{2}\sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\ c_{13}&= \frac{\sqrt{a_{14}} a_{24} \varOmega \left( 17 a_{22}^2+5 \varOmega ^2\right) }{4 \sqrt{2} \left( a_{22}^2+\varOmega ^2\right) ^2}, \qquad c_{14}= -\frac{2 a_{24} \varOmega ^2}{a_{22}^2+4\varOmega ^2}, \\ c_{15}&= -\frac{27 \sqrt{a_{14}} a_{24} \varOmega }{8 \sqrt{2} \left( a_{22}^2+9 \varOmega ^2\right) }, \\ c_{16}&=-\frac{a_{22} a_{24} \left( a_{22}^2-\varOmega ^2\right) f}{2 \sqrt{2} \sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\ c_{17}&= \frac{a_{22} a_{24} \left( a_{12} a_{22} \varOmega ^2+a_{22}^2 \left( \sigma +\varOmega ^2\right) +\varOmega ^4\right) }{\sqrt{2} \sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\ c_{18}&= -\frac{\sqrt{a_{14}} a_{22} a_{24} \left( 11 a_{22}^2-\varOmega ^2\right) }{4 \sqrt{2} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\ c_{19}&= \frac{9 \sqrt{a_{14}} a_{22} a_{24}}{8\sqrt{2} \left( a_{22}^2+9 \varOmega ^2\right) }, \qquad c_{20}= \frac{a_{22} a_{24} \varOmega }{a_{22}^2+4 \varOmega ^2}, \\ c_{21}&= \frac{a_{32}^2 a_{33} f}{\left( a_{32}^2+\varOmega ^2\right) ^2}, \\ c_{22}&= \frac{a_{33} \left( a_{12} a_{32} (a_{32}-\varOmega ) (a_{32}+\varOmega )-2 \left( a_{32}^2 \left( \sigma +\varOmega ^2\right) +\varOmega ^4\right) \right) }{2\left( a_{32}^2+\varOmega ^2\right) ^2}, \\ c_{23}&= \frac{3 a_{14} a_{32}^2 a_{33}}{\left( a_{32}^2+\varOmega ^2\right) ^2}, \qquad c_{24}= -\frac{\sqrt{2} \sqrt{a_{14}} a_{33} \varOmega }{a_{32}^2+4 \varOmega ^2}, \\ c_{25}&= -\frac{9 a_{14} a_{33}}{8\left( a_{32}^2+9\varOmega ^2\right) }, \\ c_{26}&=- \frac{a_{32} a_{33} (a_{32}-\varOmega ) (a_{32}+\varOmega ) f}{2 \varOmega \left( a_{32}^2+\varOmega ^2\right) ^2}, \\ c_{27}&= \frac{a_{32} a_{33} \left( \varOmega ^2 (2 a_{32}(a_{12}+a_{32})-\sigma )+a_{32}^2 \sigma +2 \varOmega ^4\right) }{2 \varOmega \left( a_{32}^2+\varOmega ^2\right) ^2}, \\ c_{28}&= \frac{3 a_{14} a_{32} a_{33} \left( \varOmega ^2-a_{32}^2\right) }{2 \varOmega \left( a_{32}^2+\varOmega ^2\right) ^2}, \qquad c_{29}= \frac{\sqrt{a_{14}} a_{32} a_{33}}{\sqrt{2} \left( a_{32}^2+4 \varOmega ^2\right) }, \\ c_{30}&= \frac{3 a_{14} a_{32} a_{33}}{8 a_{32}^2 \varOmega +72 \varOmega ^3} \end{aligned}$$

As alternative asymptotic procedure aimed at improving the description of the membrane temperature dynamics, resonance condition is imposed only to the terms related to W and $W^2$ in the mechanical Eq. (10). Using the same asymptotic scheme as the one previously described, the analytical response of the membrane temperature, which is formally identical to (67), shows different expressions of the following coefficients:

$$\begin{aligned} \begin{aligned}&c_{11}= -\frac{a_{17} a_{22}^2 a_{24} \omega }{\sqrt{2} \sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2},\\&c_{12}= \frac{a_{24} \omega \left( a_{12} \left( a_{22}^3-a_{22} \varOmega ^2\right) -2 \left( a_{22}^2 \left( \sigma +\varOmega ^2\right) +\varOmega ^4\right) \right) }{2 \sqrt{2} \sqrt{a_{14}} \left( a_{22}^2+\varOmega ^2\right) ^2},\\&c_{13}= \frac{\sqrt{a_{14}} a_{24} \left( a_{22}^2 (12 \omega +5 \varOmega )+5 \varOmega ^3\right) }{4 \sqrt{2} \left( a_{22}^2+\varOmega ^2\right) ^2}, \\&c_{14}= -\frac{a_{24} \varOmega (\omega +\varOmega )}{a_{22}^2+4 \varOmega ^2},\\&c_{15}= -\frac{9 \sqrt{a_{14}} a_{24} (\omega +2 \varOmega )}{8 \sqrt{2} \left( a_{22}^2+9 \varOmega ^2\right) }, \\&c_{16}= \frac{a_{17} a_{22} a_{24} \omega \left( a_{22}^2-\varOmega ^2\right) }{2 \sqrt{2} \sqrt{a_{14}} \varOmega \left( a_{22}^2+\varOmega ^2\right) ^2},\\&c_{17}= \frac{a_{22} a_{24} \omega \left( 2 a_{12} a_{22} \varOmega ^2+a_{22}^2 \left( \sigma +2 \varOmega ^2\right) -\sigma \varOmega ^2+2 \varOmega ^4\right) }{2 \sqrt{2} \sqrt{a_{14}} \varOmega \left( a_{22}^2+\varOmega ^2\right) ^2},\\&c_{18}= -\frac{\sqrt{a_{14}} a_{22} a_{24} \left( a_{22}^2 (6 \omega +5 \varOmega )+\varOmega ^2 (5 \varOmega -6 \omega )\right) }{4 \sqrt{2} \varOmega \left( a_{22}^2+\varOmega ^2\right) ^2},\\&c_{19}= \frac{3 \sqrt{a_{14}} a_{22} a_{24} (\omega +2 \varOmega )}{8 \sqrt{2} \varOmega \left( a_{22}^2+9 \varOmega ^2\right) }, \; \qquad c_{20}= \frac{a_{22} a_{24} (\omega +\varOmega )}{2 \left( a_{22}^2+4 \varOmega ^2\right) } \end{aligned} \end{aligned}$$

(69)

In turn, the coefficients of mechanical and bending thermal solutions are identical to those of the fully resonant asymptotic procedure, apart from a very minor difference. Thus, they are not reported here, for the sake of brevity.

Appendix D Multiple Scale analysis around double-zero bifurcation

Analysis near the bifurcation point is performed by introducing fractional power series expansions to study Eq. (4). Accordingly, three time scales have been introduced, i.e., $T_{0}=t, \; T_{1}= \epsilon ^{1/2} t, \; T_{2}=\epsilon t$, and consistently, the time derivatives are expressed as

$$\begin{aligned} \begin{aligned}&d/dt = D_0 + \epsilon ^{1/2} D_1 + \epsilon D_2 \\&d^2/dt^2 = D^2_0 +2\epsilon ^{1/2} D_0D_1+\epsilon D^2_1 +2\epsilon D_0D_2 \end{aligned} \end{aligned}$$

(70)

where $D_i = \partial /\partial T_i$. In order to account for the different time evolution of the mechanical variable with respect to the thermal ones, variables are scaled as follows:

$$\begin{aligned} \tilde{W}=\epsilon ^{1/2}\; \hat{W}, \qquad \tilde{T}_{R0}=\epsilon \; \hat{T}_{R0}, \qquad \tilde{T}_{R1}=\epsilon \; \hat{T}_{R1} \end{aligned}$$

(71)

so that their expression as perturbation of the reference equilibrium reads:

$$\begin{aligned} \begin{aligned} \tilde{W}(t)\,\,\,&=\epsilon ^{1/2}W_0 (T_0,T_1,T_2)+\epsilon W_1 (T_0,T_1,T_2)\\&\quad +\epsilon ^{3/2}W_2 (T_0,T_1,T_2) \\ \tilde{T}_{R0}(t)&=\epsilon \text {T0}_0 (T_0,T_1,T_2)+\epsilon ^{3/2}\text {T0}_1 (T_0,T_1,T_2)\\&\quad +\epsilon ^{2}\text {T0}_2 (T_0,T_1,T_2) \\ \tilde{T}_{R1}(t)&=\epsilon \text {T1}_0 (T_0,T_1,T_2)\\&\quad +\epsilon ^{3/2}\text {T1}_1 (T_0,T_1,T_2)+\epsilon ^{2}\text {T1}_2 (T_0,T_1,T_2) \end{aligned} \end{aligned}$$

(72)

Since analysis is developed far from resonance regions (nonresonance condition), the forcing term is scaled to the generating order, i.e. $f=\epsilon ^{1/2} \; \hat{f}$, while the other parameters are scaled as follows:

$$\begin{aligned} \omega ^2= & {} \epsilon \; \hat{\omega }^2, \quad a_{12}=\epsilon \; \hat{a}_{12}, \quad a_{15}=\epsilon ^{1/2} \;\hat{a}_{15}, \nonumber \\ a_{33}= & {} \epsilon ^{1/2} \;\hat{a}_{33} \end{aligned}$$

(73)

The resulting perturbation equations at each order read:

$$\begin{aligned}&\bullet \;\text {Order}\;\;\epsilon{} & {} D_0^2W_0 = f \cos (\varOmega T_0) \end{aligned}$$

(74)

$$\begin{aligned}&\bullet \;\text {Order}\;\;\epsilon ^{3/2}{} & {} D_0^2W_1 = -2 D_0 D_1W_0 \nonumber \\{} & {} {}&D_0\text {T0}_0+a_{22} \text {T0}_0=-a_{24} W_0 D_0W_0 \end{aligned}$$

(75)

$$\begin{aligned}{} & {} {}&D_0\text {T1}_0+a_{32} \text {T1}_0=-a_{33} D_0W_0 \nonumber \\&\bullet \;\text {Order}\;\;\epsilon ^2{} & {} D_0^2W_2=-\omega ^2 W_0-a_{14} W_0^3\nonumber \\{} & {} {}&-a_{12} D_0W_0-2 D_0D_1W_1-D_1^2W_0 \nonumber \\{} & {} {}&-2D_0D_2W_0-a_{15} \text {T1}_0-a_{16} \text {T0}_0 W_0 \\{} & {} {}&D_0\text {T0}_1+a_{22} \text {T0}_1=-a_{24} W_0 D_0W_1\nonumber \\{} & {} {}&-a_{24} W_0 D_1W_0-a_{24} W_1 D_0W_0-D_1\text {T0}_0\nonumber \\{} & {} {}&D_0\text {T1}_1+a_{32} \text {T1}_1=-a_{33} D_0W_1\nonumber \\{} & {} {}&-a_{33} D_1W_0-D_1\text {T1}_0 \nonumber \end{aligned}$$

(76)

At first order, the solution of the mechanical Eq. (74) is combination of complementary and particular solution and reads

$$\begin{aligned} W_0 = A(T_1, T_2)-\frac{f}{2\varOmega ^2} \textrm{e}^{i \varOmega T_0} + c.c. \end{aligned}$$

(77)

with $A(T_1, T_2)$ undetermined real amplitude which is function of the slow time scales, and c.c. complex conjugate terms.

At order $\epsilon ^{3/2}$, substituting $W_0$ in the first equation of (75) implies $D_0^2W_1 = 0$, which furnishes null contribution to the modulation equation and to the mechanical solution, as well. In turn, solving the thermal equations yields the following particular solutions:

$$\begin{aligned} \begin{aligned}&\text {T0}_0=-\frac{a_{24} f A \textrm{e}^{ i \varOmega T_0 }}{ 2 i a_{22}\varOmega -2 \varOmega ^2}-\frac{a_{24} f^2 \textrm{e}^{2 i \varOmega T_0 }}{ 4 i a_{22}\varOmega ^3-8 \varOmega ^4}+ c.c.\\&\text {T1}_0=-\frac{ a_{33}f \textrm{e}^{i \varOmega T_0 }}{2 i a_{32}\varOmega -2 \varOmega ^2}+c.c. \end{aligned} \end{aligned}$$

(78)

In view of Eqs. (77), (78), the solvability condition of the mechanical problem at the order $\epsilon ^2$ (76) yields

$$\begin{aligned} D_1^2A=c_{1}A+c_3A^3 \end{aligned}$$

(79)

where

$$\begin{aligned}&c_{1}= -\omega ^2+\frac{3 a_{14} f^2}{2\varOmega ^4}-\frac{2a_{16} a_{24}f^2}{4a_{22}^2\varOmega ^2+4\varOmega ^4},\; c_{3}= -a_{14} \end{aligned}$$

Moving to the real time scale t, the resulting amplitude modulation equation is

$$\begin{aligned} \ddot{A}=c_{1}A+c_3A^3 \end{aligned}$$

(80)