Finding an optimal shape of a curved mechanical beam for enhanced internal resonance

Abstract

One of the most interesting nonlinear phenomena of mode coupling is internal resonances, which can promote directed energy transfer from one eigenmode to another, even at small amplitudes in free or forced oscillations. Internal resonances can be highly beneficial for many engineering applications. However, in most cases, internal resonances are encountered either accidentally or by proper tuning of different control parameters during experiments without prior planning. Therefore, the ability to a priori design a mechanical resonator with intentional internal resonance at given amplitudes holds great promise. Here, we show a simple methodological way to manipulate the eigenfrequencies and the coupling between the eigenmodes of a doubly clamped mechanical beam using a genetic algorithm for shape optimization of the initial curvature of the beam. We demonstrate that our methodology can be applied to both 1-to-2 and 1-to-3 internal resonances of micro-beams. Our results pave the way to a new class of design techniques for internal resonance enhancement based on shape optimization.

Appendices

Derivation of the nonlinearly coupled ordinary differential equations

Assuming that the motion of the beam is dominated by its first two modes, $w(x,t)=\phi _1(x)q_1(t)+\phi _2(x)q_2(t)$, where $\phi _{1,2}$ are calculated from Eq. (8) and satisfy the orthonormality condition $\langle \phi _k,\phi _l\rangle =\int _0^1\phi _k\phi _ldx=\delta _{kl}$, we substitute w(x, t) into Eq. (3) and perform Galerkin projection onto $\phi _{1,2}$. This procedure results in the following pair of ordinary differential equations

$$\begin{aligned}{} & {} \ddot{q}_{1}+\omega _1^2q_{1}-3\chi \left( \int _0^1{\phi _{1}'^2dx}\int _0^1{\phi _{1}'w_0'dx}\right) q_1^2-\nonumber \\{} & {} \quad 2\chi \left( \int _0^1{\phi _{1}'^2dx}\int _0^1{\phi _{2}'w_0'dx}\right. \nonumber \\{} & {} \quad \left. +2\int _0^1{\phi _{1}'\phi _{2}'dx}\int _0^1{\phi _{1}'w_0'dx} \right) q_1q_2\nonumber \\{} & {} \quad -\chi \left( \int _0^1{\phi _{2}'^2dx}\int _0^1{\phi _{1}'w_0'dx} \right. \nonumber \\{} & {} \quad \left. +2\int _0^1{\phi _{1}'\phi _{2}'dx}\int _0^1{\phi _{2}'w_0'dx} \right) q_2^2\nonumber \\{} & {} \quad +\chi \left( \int _0^1{\phi _{1}'^2dx}\right) ^2 q_{1}^{3}\nonumber \\ {}{} & {} \quad +3\chi \left( \int _0^1{\phi _{1}'^2dx}\int _0^1{\phi _{1}'\phi _{2}'dx}\right) q_1^2q_2\nonumber \\{} & {} \quad +\chi \left[ \!\int _0^1{\phi _{1}'^2dx}\! \int _0^1{\phi _{2}'^2dx}\!\!+\!2\left( \!\int _0^1{\phi _1'\phi _{2}'dx} \right) ^2\!\right] q_1q_2^2\nonumber \\{} & {} \quad +\chi \left( \int _0^1{\phi _{2}'^2dx}\int _0^1{\phi _{1}'\phi _{2}'dx}\right) q_2^3=0, \end{aligned}$$

(15)

$$\begin{aligned}{} & {} \ddot{q}_{2}+\omega _2^2q_{2}-3\chi \left( \int _0^1{\phi _{2}'^2dx}\int _0^1{\phi _{2}'w_0'dx}\right) q_2^2\nonumber \\{} & {} \quad -2\chi \left( \int _0^1{\phi _{2}'^2dx}\int _0^1{\phi _{1}'w_0'dx} \right. \nonumber \\{} & {} \quad \left. +2\int _0^1{\phi _{1}'\phi _{2}'dx}\int _0^1{\phi _{2}'w_0'dx} \right) q_1q_2\nonumber \\{} & {} \quad -\chi \left( \int _0^1{\phi _{1}'^2dx}\int _0^1{\phi _{2}'w_0'dx}\right. \nonumber \\{} & {} \quad \left. +2\int _0^1{\phi _{1}'\phi _{2}'dx}\int _0^1{\phi _{1}'w_0'dx} \right) q_1^2\nonumber \\{} & {} \quad +\chi \left( \int _0^1{\phi _{2}'^2dx}\right) ^2 q_{2}^{3}\nonumber \\ {}{} & {} \quad +3\chi \left( \int _0^1{\phi _{2}'^2dx}\int _0^1{\phi _{1}'\phi _{2}'dx}\right) q_2^2q_1\nonumber \\{} & {} \quad +\chi \!\left[ \int _0^1{\phi _{2}'^2dx}\! \int _0^1{\phi _{1}'^2dx}\!\!+\!2\left( \!\int _0^1\!{\phi _1'\phi _{2}'dx} \right) ^2\!\right] q_2q_1^2\nonumber \\{} & {} \quad +\chi \left( \int _0^1{\phi _{1}'^2dx}\int _0^1{\phi _{1}'\phi _{2}'dx}\right) q_1^3=0, \end{aligned}$$

(16)

which can readily be identified with Eqs. (4)–(5).

Without going into technical details, we outline the derivation of the normal form of the 1-to-2 and 1-to-3 IRs. Since we consider a weakly nonlinear system, Eqs. (4)–(5) can be written as

$$\begin{aligned}{} & {} \ddot{q}_1+\omega _1^2q_1=\epsilon F_1(q_1,q_2),\nonumber \\{} & {} \ddot{q}_2+\omega _2^2q_2=\epsilon F_2(q_1,q_2). \end{aligned}$$

(17)

As the pair of equations in Eq. (17) are close to those of a pair of linear uncoupled oscillators, we can expect that their solutions have a nearly sinusoidal (harmonic) form with unknown, and generally time-varying amplitude and phase. Hence, we seek solutions in the following complex form

$$\begin{aligned} q_1(t)\!=\!A_1(t)e^{i\omega _{1} t}\!+\!c.c.,~q_2(t)=A_2(t)e^{i\omega _{2} t}\!+\!c.c., \nonumber \\ \end{aligned}$$

(18)

where c.c. denotes the complex conjugate of the preceding term. Note that we make no restriction on $q_{1,2}(t)$ here, as the observed frequencies may well deviate from $\omega _{1,2}$ if the complex–amplitudes $A_{1,2}$ rotate in the complex plane. Introducing a constraint on the time derivative of the two modes, $\dot{q}_{n}=i\omega _{n}A_{n}e^{i\omega _{n}t}+c.c.,~n=1,2$ (which is dictated by the form of this coordinate change), we find from Eq. (17) the following equations for the complex amplitudes

$$\begin{aligned} \dot{A}_1= & {} \epsilon \frac{e^{-i\omega _1t}}{2i\omega _1} F_1(A_1e^{i\omega _{1} t},A_2e^{i\omega _{2} t}),\nonumber \\ \dot{A}_2= & {} \epsilon \frac{e^{-i\omega _2t}}{2i\omega _2} F_2(A_1e^{i\omega _{1} t},A_2e^{i\omega _{2} t}). \end{aligned}$$

(19)

Note that $F_{1,2}$ contain components that are $2\pi /\omega _1$ and $2\pi /\omega _2$ periodic in t. Hence, representing these functions by their corresponding double Fourier series yield

$$\begin{aligned} \dot{A}_1= & {} \frac{\epsilon }{2i\omega _1}\sum _{k,l}f_{1_{k,l}}A_1^k e^{i(k-1)\omega _{1} t}A_2^{l}e^{il\omega _{2} t},\nonumber \\ \dot{A}_2= & {} \frac{\epsilon }{2i\omega _2} \sum _{q,r}f_{2_{q,r}}A_1^q e^{iq\omega _{1} t}A_2^{r}e^{i(r-1)\omega _{2} t}, \end{aligned}$$

(20)

where $A^{-n}\triangleq \bar{A}^n$ and the over-bar denotes complex-conjugation. Up to this point, the transformations are exact and no approximations have been made.³ We now use the small parameter $\epsilon$ to obtain approximate equations for the evolution of $A_{1,2}$. As the right-hand sides (RHS) of the pair of equations in Eq. (20) are small (of order $\epsilon$ in magnitude), the variations of $A_1$ and $A_2$ can be either slow (if they are large) or small (if they are fast, e.g., with the frequencies $\omega _1,~\omega _2$). We restrict ourselves to large and slow variations, i.e., we neglect all the fast and small terms on the RHS of Eq. (20). Neglecting the terms containing the fast oscillations ($e^{i n \omega _1t}, ~e^{im\omega _2t},~n,m=\pm 1,\pm 2,...$) can also be considered as an averaging over the period of the oscillations $T_{1,2} = 2\pi /\omega _{1,2}$; thus this method is often called the method of averaging.

From Eq. (20), we see that for a non-zero slowly varying RHS, which implies a resonant interaction between the modes, the fast oscillatory terms, $\omega _1$ and $\omega _2$ must be rationally related, e.g., $\omega _2/\omega _1=|(k-1)/l|,~k,l\in \mathbb {Z}$. Note that since k and l can be arbitrary integers, any rationally related frequencies will yield IR. However, due to dissipation, only the lower-order IRs will be observed in practice. For specific values of k and l, we can determine from Eq. (20) the resonant nonlinear coupling terms. The pair $k=-2,~l=1$ gives 1-to-3 IR with the lowest-order resonant terms $f_{1_{-2,1}}{\bar{A}}_1^2A_2$ and $f_{2_{3,0}}A_1^3$ on the RHS of the first and second equations in Eq. (20), respectively. The pair $k=-1,~l=1$ gives 1-to-2 IR with the lowest-order resonant terms $f_{1_{-1,1}}{\bar{A}}_1 A_2$ and $f_{2_{2,0}}A_1^2$ on the RHS of the first and second equations in Eq. (20), respectively. These terms can be readily associated with the single-term coupling potential $U_{\textrm{cpl}}=\alpha q_1^nq_2$ that yields Eqs. (6)–(7).

Fig. 12

Shape functions of the optimal 1-to-2 IR beams that obtained for probabilities combination of $\{20\%,30\%,50\%\}$ (left) and combination of $\{30\%,40\%,30\%\}$ (right)

The frequency of oscillation for the isolated modes

Without modal coupling, the equation of motion for the isolated mode can be written as

$$\begin{aligned} \ddot{q}+U'(q)=0. \end{aligned}$$

(21)

where $U=\omega ^2q^2/2+\beta q^3/3+\gamma q^4/4$ is the potential energy (per unit mass) of the isolated mode. We multiply Eq. (21) by $\dot{q}$ and integrate with respect to time to obtain the conserved quantity $\dot{q}^2/2+U(q)=E$, where E is the total energy (per unit mass) of the isolated mode. From the conservation of energy, we find that

$$\begin{aligned} dt=\pm \frac{dq}{\sqrt{2(E-U(q))}}, \end{aligned}$$

(22)

where the ± reflects the change of sign in the velocity $\dot{q}$ when crossing the turning points $q^{(1)},q^{(2)}$. In half cycle, the resonator starts its motion from the lower turning point $q^{(2)}$ with a positive velocity and arrives to the higher turning point $q^{(1)}$. Therefore, we can write

$$\begin{aligned} \int _{0}^{\frac{T}{2}}{dt}=\int _{q^{(2)}}^{q^{(1)}}{\frac{dq}{\sqrt{2[E-U(q)]}}}. \end{aligned}$$

(23)

Furthermore, we can rewrite the quadratic polynomial under the square root in the following way

$$\begin{aligned} 2[E-U(q)]&=\frac{\gamma }{4}\left( -q^{4}-\frac{4\beta }{3\gamma }q^{3}-\frac{2\omega _{0}^{2}}{\gamma }q^{2}+\frac{4}{\gamma }E\right) \nonumber \\&= \frac{\gamma }{4}(q^{(1)}-q)(q-q^{(2)})\nonumber \\ {}&\quad (q-q^{(3)})(q-q^{(4)}). \end{aligned}$$

(24)

Thus, from Eq. (23), we find that

$$\begin{aligned} \frac{T}{2}&=\sqrt{\frac{2}{\gamma }}\int _{q^{(2)}}^{q^{(1)}}\nonumber \\&\quad \frac{dq}{\sqrt{(q^{(1)}-q)(q-q^{(2)})(q-q^{(3)})(q-q^{(4)})}}. \end{aligned}$$

(25)

We normalize the modal coordinate q by the largest real root $q^{(1)}$, and hence,

$$\begin{aligned}&\frac{T}{2}=\sqrt{\frac{2}{ q^{(1)}\gamma }}\nonumber \\ {}&\quad \int _{q'^{(2)}}^{1}{}\frac{dq'}{\sqrt{(1-q')(q'-q'^{(2)})(q'-q'^{(3)})(q'-q'^{(4)})}} \end{aligned}$$

(26)

where $q'=q/q^{(1)}$, $q'^{(1)}=q^{(1)}/q^{(1)}=1$, $q'^{(2)}=q^{(2)}/q^{(1)}$, $q'^{(3)}=q^{(3)}/q^{(1)}$ and $q'^{(4)}=q^{(4)}/q^{(1)}$. The solution of the elliptic integral in Eq. (26) is given by (cf. Ref. [73], Eq. (259.00))

$$\begin{aligned} T=4g'\sqrt{\frac{2}{\gamma }}K(k'), \end{aligned}$$

(27)

where

$$\begin{aligned} k'^{2}= & {} \frac{1}{4}\frac{(q'^{(1)}-q'^{(2)})^2-(z'^{(1)}-z'^{(2)})^2}{z'^{(1)}z'^{(2)}},\nonumber \\ z'^{(1)^2}= & {} (q'^{(1)}-\mathfrak {R}\{q'^{(3)}\})^2+(\mathfrak {I}\{q'^{(3)}\})^2,\nonumber \\ z'^{(2)^2}= & {} (q'^{(2)}-\mathfrak {R}\{q'^{(3)}\})^2+(\mathfrak {I}\{q'^{(3)}\})^2,\nonumber \\ g'= & {} \frac{1}{\sqrt{(z'^{(1)}z'^{(2)})}}, \end{aligned}$$

(28)

$K(k')$ is an elliptic integral of the first kind, and $\mathfrak {R}\{\cdot \}$ and $\mathfrak {I}\{\cdot \}$ are the real and imaginary parts of $\{\cdot \}$, respectively. Thus, by switching back to the non-normalized variables $q, q^{(1)}, q^{(2)}, q^{(3)}$, and $q^{(4)}$, we find that the fundamental frequency is given by

$$\begin{aligned} \varOmega (E)=\frac{2\pi }{T}=\frac{\pi }{2K(k)}\sqrt{\frac{\gamma }{2}z^{(1)}z^{(2)}}. \end{aligned}$$

(29)

Supplementary figures

See Figs. 12, 13 and 14

Fig. 13

Comparison between the 1-to-2 IR beam of generation 50 (left) and 1-to-2 IR optimal beam of generation 89 (right). The upper panels are the energy-frequency backbone curves of the first and second modes with the frequency crossing conditions, and the lower panels are the projections of the numerical simulations of the displacement field w(x, t) onto the first two modes of the beam

Fig. 14

Projections of the numerical simulations of the displacement field w(x, t) onto the first two modes of the beam for 1-to-2 IR with non-trivial initial conditions in a single mode. Left: the non-trivial initial condition is only on the first mode while the second mode is set to zero initial amplitude. Right: the non-trivial initial condition is only on the second mode while the first mode is set to zero amplitude. The first (blue) and the second (red) modes clearly show the energy exchange due to the resonant interaction for both cases. (Color figure online)

Analytical calculation of resonant envelope modulations

We present here a detailed analysis of the amplitude modulations for the 1-to-2 IR (a similar analysis can be conducted for the 1-to-3, see [58]). By making the ansatz $q_1(t)=A_1(t)e^{i\omega _1t}+cc$ and $q_2(t)=A_2(t)e^{2i\omega _1t}+cc$, where cc denotes the complex-conjugate of the preceding term, and applying the method of averaging [55], we obtain from Eqs. (12)–(13) the following pair of complex-amplitude equations

$$\begin{aligned} \dot{A}_1&=\frac{3i\gamma _1}{2\omega _1}|A_1|^2A_1+\frac{i\alpha }{\omega _1}A_1^*A_2 +\frac{2i\eta }{\omega _1}A_1|A_2|^2, \end{aligned}$$

(30)

$$\begin{aligned} \dot{A}_2&=-i\varDelta \omega _2A_2+\frac{3i\gamma _2}{4\omega _1}|A_2|^2A_2+\frac{i\alpha }{4\omega _1}A_1^2\nonumber \\&\quad +\frac{i\eta }{\omega _1}|A_1|^2A_2, \end{aligned}$$

(31)

where $\varDelta \omega _2=2\omega _1-\omega _2$ is the frequency mismatch between the modes. We normalize the complex amplitude equations by $\ell _{1,2}A_{1,2n}=A_{1,2}$, where $\ell _1=3^{1/4}\omega _1\sqrt{2/\gamma _1}$ and $\ell _2=3^{1/4}\omega _1\sqrt{1/\gamma _1}$, the time by $\tau =\omega _1t$, and obtain the following rescaled equations

$$\begin{aligned} A_{1n}'&=3\sqrt{3}i|A_{1n}|^2A_{1n}+i\frac{\alpha }{\omega _1}\left( \frac{\sqrt{3}}{\gamma _1}\right) ^{\frac{1}{2}}A_{1n}^*A_{2n}\nonumber \\&\quad +\frac{2\sqrt{3}i\eta }{\gamma _1}A_{1n}|A_{2n}|^2=-i\frac{\partial \mathcal {H}}{\partial A_{1n}^*}, \end{aligned}$$

(32)

$$\begin{aligned} A_{2n}'&=-i\varDelta \omega _{2n}A_{2n}+\frac{3i\sqrt{3}}{4}\frac{\gamma _2}{\gamma _1}|A_{2n}|^2A_{2n}\nonumber \\&\quad +i\frac{\alpha }{2\omega _1}\left( \frac{\sqrt{3}}{\gamma _1}\right) ^{\frac{1}{2}}A_{1n}^2+\frac{2\sqrt{3}i\eta }{\gamma _1}|A_{1n}|^2A_{2n}|\nonumber \\&=-i\frac{\partial \mathcal {H}}{\partial A_{2n}^*}, \end{aligned}$$

(33)

where $\varDelta \omega _{2n}=\varDelta \omega _2/\omega _1$, and

$$\begin{aligned} \mathcal {H}&=\varDelta \omega _{2n}|A_{2n}|^2-\frac{3\sqrt{3}}{2}|A_{1n}|^4-\frac{3\sqrt{3}}{8}\frac{\gamma _2}{\gamma _1}|A_{2n}|^4\nonumber \\&\quad -\frac{\alpha }{2\omega _1}\left( \frac{\sqrt{3}}{\gamma _1} \right) ^{1/2}\left( {A_{1n}^*}^2A_{2n}+A_{1n}^2{A_{2n}^*} \right) \nonumber \\&\quad -\frac{2\sqrt{3}\eta }{\gamma _1}|A_{1n}|^2|A_{2n}|^2 \end{aligned}$$

(34)

is the averaged Hamiltonian of the system which, by construction, is a conserved quantity. The second conserved quantity is an analog of the Manly-Rowe invariant that has the form of $\mathcal {M}=I+2|A_{2n}|^2$, where $I=|A_{1n}|^2$ is the action variable. We define the phase difference $\psi =2 \arg (A_{1n})-\arg (A_{2n})$ as the angle variable, and rewrite Eqs. (32)–(33) using action-angle variables

$$\begin{aligned} I'&=\frac{\partial \mathcal {H}}{\partial \psi }=\frac{\alpha }{\sqrt{2}\omega _1}\left( \frac{\sqrt{3}}{\gamma _1} \right) ^{1/2}(\mathcal {M}-I)^{1/2}I\sin \psi , \end{aligned}$$

(35)

$$\begin{aligned} \psi '&=-\frac{\partial \mathcal {H}}{\partial I}=\frac{\varDelta \omega _{2n}}{2}+3\sqrt{3}I-\frac{3\sqrt{3}}{16}\frac{\gamma _2}{\gamma _1}(\mathcal {M}-I)\nonumber \\&\quad \quad +\frac{\alpha }{\sqrt{2}\omega _1}\left( \frac{\sqrt{3}}{\gamma _1} \right) ^{1/2}\frac{2\mathcal {M}-3I}{2(\mathcal {M}-I)^{1/2}}\cos \psi \nonumber \\&\quad \quad + \frac{\sqrt{3}\eta }{\gamma _1}(\mathcal {M}-2I). \end{aligned}$$

(36)

The dynamics of Eqs. (35)–(36) can be mapped onto the motion of a particle trapped in a potential well [58].

$$\begin{aligned}&\frac{1}{2}I'^2+U_{\textrm{eff}}(I)=0,\nonumber \\&U_{\textrm{eff}}(I)=-\frac{\sqrt{3}\alpha ^2}{4\gamma _1\omega _1^2}(\mathcal {M}-I)I^2\nonumber \\&\quad \quad \quad \quad \quad +\frac{1}{2}\left[ \mathcal {H}-\frac{\varDelta \omega _{2n}}{2}(\mathcal {M}-I)+\frac{3\sqrt{3}}{2}I^2\right. \nonumber \\&\quad \quad \quad \quad \quad \left. +\frac{3\sqrt{3}}{32}\frac{\gamma _2}{\gamma _1}(\mathcal {M}-I)^2 + \frac{\sqrt{3}\eta }{\gamma _1}I(\mathcal {M}-I)\right] ^2. \end{aligned}$$

(37)

The potential $U_{\textrm{eff}}(I)$ is a quartic polynomial in I parameterized by $\mathcal {M}$ and $\mathcal {H}$. It can have a single well or two wells separated by a local maximum [58]. In the case where the local maximum of the double potential well lies above zero, the equation $U_{\textrm{eff}}(I)=0$ has four real solutions $I_1>I_2>I_3>I_4$, where depending on the initial condition I(0), the “particle trapped in the potential well” $I(\tau )$ oscillates either between $I_1$ and $I_2$, or between $I_3$ and $I_4$. In the first case, the oscillations $I(\tau )$ between $I_1$ and $I_2$ can be described in terms of Jacobi elliptic functions [73]

$$\begin{aligned}{} & {} I(u)=\frac{(I_1-I_2)I_3{\text {sn}}^2(u|m)-(I_1-I_3)I_2}{(I_1-I_2){\text {sn}}^2(u|m)-(I_1-I_3)},\nonumber \\{} & {} u=\tau \sqrt{\frac{3(48\gamma _1\!+\!3\gamma _2\!-\!32\eta )^2}{1024\gamma _1^2}}\frac{2}{\sqrt{(I_1\!-\!I_3)(I_2\!-\!I_4)},}\nonumber \\{} & {} m=\frac{(I_1-I_2)(I_3-I_4)}{(I_1-I_3)(I_2-I_4)}. \end{aligned}$$

(38)

From the above analytical solution, we find that $q_1=2\ell _1I^{1/2}\cos (\omega _1t+\arg (A_{1n}))$ and $q_2=2^{1/2}\ell _2(\mathcal {M}-I)^{1/2}\cos (2\omega _1t+\arg (A_{2n}))$.