On the Use of the Multiple Scale Harmonic Balance Method for Nonlinear Energy Sinks Controlled Systems

Abstract

The Multiple Scale Harmonic Balance Method (MSHBM) is discussed here for several paradigmatic systems (primary structures) equipped with a Nonlinear Energy Sink (NES). This is a small-mass oscillator with essentially nonlinear stiffness, used for passive control purpose. The method permits to overcome the difficulties inherent to standard perturbation methods, which occur as a consequence of the nonlinearizable nature of the NES equation. It combines the Multiple Scale Method and the Harmonic Balance Method to furnish Amplitude Modulation Equations ruling the slow asymptotic dynamics of the augmented system. The MSHBM is illustrated here for a general, internally non-resonant, multi d.o.f. structure equipped with a NES and under multiple concurrent actions, namely steady wind inducing Hopf bifurcation , and 1:1 and 1:3 resonant harmonic forces. The relevant Amplitude Modulation Equations are specialized for simpler cases, where the single contributions of each external action is considered separately. The effect of the NES on the dynamics of the system is discussed for each case and numerical results are displayed.

Appendix A: Coefficients of the Equations

The index H indicates the Hermitian (transpose and complex conjugate). The expression of the coefficients of (19) are:

$$\begin{aligned} c_1&= 2i\omega \mathbf {v}^H\mathbf {M}\mathbf {u}+\mathbf {v}^H\mathbf {C}\mathbf {u},\quad c_2=-i\omega \mathbf {v}^H\mathbf {C}_1\mathbf {u}-\mathbf {v}^H\mathbf {K}_\mu \mathbf {u},\nonumber \\ c_3&= -\mathbf {v}^H\mathbf {K}_\sigma \mathbf {u},\quad c_4 = -i\omega \mathbf {v}^H\mathbf {r},\quad c_5=-3\mathbf {v}^H\mathbf {r},\nonumber \\ c_6&= -3\mathbf {v}^H\mathbf {n}(\mathbf {u},\mathbf {u},\bar{\mathbf {u}}),\quad c_7=-3\mathbf {v}^H\mathbf {n}(\mathbf {w}_0,\bar{\mathbf {u}},\bar{\mathbf {u}}),\nonumber \\ c_8&= -6\mathbf {v}^H\mathbf {n}(\mathbf {u},\mathbf {w}_0,\bar{\mathbf {w}}_0),\quad c_9=\frac{1}{2}\mathbf {v}^H\mathbf {f}_1 \end{aligned}$$

(62)

In (20) the column matrices $\mathbf {w}_j$ ($j=1,\ldots ,8$) are the solutions of the following singular algebraic problems:

$$\begin{aligned} \mathbf {w}_1:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_1=-i\omega \Bigl (\mathbf {C}_1\mathbf {u}\nonumber \\&-\frac{1}{c_1}(\mathbf {v}^H\mathbf {C}_1\mathbf {u})(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\nonumber \\&-\Bigl (\mathbf {K}_\mu \mathbf {u}-\frac{1}{c_1}(\mathbf {v}^H\mathbf {K}_\mu \mathbf {u})(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(63)

$$\begin{aligned} \mathbf {w}_2:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_2=-\Bigl (\mathbf {K}_\sigma \mathbf {u}\nonumber \\&-\frac{1}{c_1}(\mathbf {v}^H\mathbf {K}_\sigma \mathbf {u})(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(64)

$$\begin{aligned} \mathbf {w}_3:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_3=-6\Bigl (\mathbf {n}(\mathbf {u},\mathbf {w}_0,\bar{\mathbf {w}}_0)\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {n}(\mathbf {u},\mathbf {w}_0,\bar{\mathbf {w}}_0)(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(65)

$$\begin{aligned} \mathbf {w}_4:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_4=-i\omega \Bigl (\mathbf {r}\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {r}(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(66)

$$\begin{aligned} \mathbf {w}_5:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_5=-3 \Bigl (\mathbf {r}\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {r}(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(67)

$$\begin{aligned} \mathbf {w}_6:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_6=-6\Bigl (\mathbf {n}(\mathbf {u},\mathbf {u},\bar{\mathbf {u}})\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {n}(\mathbf {u},\mathbf {u},\bar{\mathbf {u}})(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(68)

$$\begin{aligned} \mathbf {w}_7:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_7=-6\Bigl (\mathbf {n}(\mathbf {w}_0,\bar{\mathbf {u}},\bar{\mathbf {u}})\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {n}(\mathbf {w}_0,\bar{\mathbf {u}},\bar{\mathbf {u}})(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr )\end{aligned}$$

(69)

$$\begin{aligned} \mathbf {w}_8:\qquad (\mathbf {K}_0&+i\omega \mathbf {C}_0-\omega ^2\mathbf {M})\mathbf {w}_8=-\frac{1}{2} \Bigl (\mathbf {f}_1\nonumber \\&-\frac{1}{c_1}\mathbf {v}^H\mathbf {f}_1(2i\omega \mathbf {M}\mathbf {u}+\mathbf {C}_0\mathbf {u})\Bigr ) \end{aligned}$$

(70)

The solution is made unique by the normalization condition $\mathbf {w}_j^T\mathbf {u}=0$.

Moreover $\mathbf {w}_j$ ($j=9,\ldots ,15$) are the solutions of the following non-singular algebraic problems, in which, however, compatibility is satisfied.

$$\begin{aligned} \mathbf {w}_9:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_9=-3i\mathbf {C}_1\mathbf {w}_0-\mathbf {K}_\mu \mathbf {w}_0\end{aligned}$$

(71)

$$\begin{aligned} \mathbf {w}_{10}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{10}=-\mathbf {K}_\sigma \mathbf {w}_0\end{aligned}$$

(72)

$$\begin{aligned} \mathbf {w}_{11}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{11}=-3i\omega \mathbf {r}\end{aligned}$$

(73)

$$\begin{aligned} \mathbf {w}_{12}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{12}=-\mathbf {r}\end{aligned}$$

(74)

$$\begin{aligned} \mathbf {w}_{13}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{13}=-\mathbf {n}(\mathbf {u},\mathbf {u},\mathbf {u})\end{aligned}$$

(75)

$$\begin{aligned} \mathbf {w}_{14}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{14}=-6\mathbf {n}(\mathbf {w}_0,\mathbf {u},\bar{\mathbf {u}})\end{aligned}$$

(76)

$$\begin{aligned} \mathbf {w}_{15}:\qquad (\mathbf {K}_0&+3i\omega \mathbf {C}_0-9\omega ^2\mathbf {M})\mathbf {w}_{15}=-3\mathbf {n}(\mathbf {w}_0,\mathbf {w}_0,\bar{\mathbf {w}}_0) \end{aligned}$$

(77)

In (45)–(50), the expressions of the right hand side terms are:

$$\begin{aligned} \mathscr {F}_1&=-\zeta \omega a-\frac{3}{32\omega ^3}\eta _3\kappa _c a^2\sin (3\alpha )-\frac{1}{2}\xi b_1\cos (\alpha -\beta _1)\nonumber \\&+\frac{3}{8\omega }\kappa b_1^3\sin (\alpha -\beta _1)+\frac{3}{8\omega }\kappa b_1^2b_3\sin (\alpha +2\beta _1-\beta _3)\nonumber \\&+\frac{3}{4\omega }\kappa b_1b_3^2\sin (\alpha -\beta _1)\end{aligned}$$

(78)

$$\begin{aligned} \mathscr {F}_2&=\frac{3}{64\omega ^5}\eta _3^2\kappa _c a-\frac{\sigma \omega }{2}a-\frac{3}{32\omega ^3}\eta _3\kappa _c a^2\cos (3\alpha )+\frac{3}{8\omega }\kappa _c a^3\nonumber \\&+\frac{1}{2}\xi b_1\sin (\alpha -\beta _1)+\frac{3}{8\omega }\kappa b_1^3\cos (\alpha -\beta _1)\nonumber \\&+\frac{3}{8\omega }\kappa b_1^2b_3\cos (\alpha +2\beta _1-\beta _3)+\frac{3}{4\omega }\kappa b_1b_3^2\cos (\alpha -\beta _1)\end{aligned}$$

(79)

$$\begin{aligned} \mathscr {F}_3&=-\frac{1}{2}m\omega ^2 a\cos (\alpha -\beta _1)+\frac{1}{2}m\omega ^2 b_1\nonumber \\&-\frac{3}{8}\kappa b_1^2b_3\cos (3\beta _1-\beta _3)-\frac{3}{4}\kappa b_1b_3^2\end{aligned}$$

(80)

$$\begin{aligned} \mathscr {F}_4&=\frac{1}{2}m\omega ^2 a\sin (\alpha -\beta _1)+\frac{1}{2}\xi \omega b_1-\frac{3}{8}\kappa b_1^2b_3\sin (3\beta _1-\beta _3)\end{aligned}$$

(81)

$$\begin{aligned} \mathscr {F}_5&=\frac{9}{8}\eta _3 m\cos (\beta _3)-\frac{9}{64}\eta _3 m\sigma \cos (\beta _3)+\frac{27}{32}\eta _3m\zeta \sin (\beta _3)\nonumber \\&+\frac{27}{4096\omega ^6}\eta _3^3m\kappa _c\cos (\beta _3)+\frac{27}{128\omega ^2}\eta _3m\kappa _c a^2\cos (\beta _3)\nonumber \\&-\frac{9}{64}\kappa _c ma^3\cos (3\alpha -\beta _3)-\frac{1}{8}\kappa b_1^3\cos (3\beta _1-\beta _3)\nonumber \\&-\frac{9}{64}\kappa m b_1^3\cos (3\beta _1-\beta _3)+\frac{9}{2}m\omega ^2b_3\nonumber \\&-\frac{3}{4}\Bigl (1+\frac{9}{8}\mu _1\Bigr )\kappa b_1^2b_3-\frac{3}{8}\Bigl (1+\frac{9}{8}\mu _1\Bigr )\kappa b_3^3\end{aligned}$$

(82)

$$\begin{aligned} \mathscr {F}_6&=\frac{27}{32}\eta _3 m \zeta \cos (\beta _3)-\frac{9}{8}\Bigl (1-\frac{1}{8}\sigma \Bigr )\eta _3 m\sin (\beta _3)\nonumber \\&-\frac{27}{4096\omega ^6}\eta _3^3\kappa _c m\sin (\beta _3)-\frac{27}{128\omega ^2}\eta _3\kappa _c ma^2\sin (\beta _3)\nonumber \\&-\frac{9}{64}\kappa _c m a^3\sin (3\alpha -\beta _3)-\frac{1}{8}\Bigl (1+\frac{9}{8}m\Bigr )\kappa b_1^3\sin (3\beta _1-\beta _3)\nonumber \\&-\frac{3}{2}\Bigl (1+\frac{9}{8}m\Bigr )\xi \omega b_3 \end{aligned}$$

(83)