Nonlinear resonances of axially functionally graded beams rotating with varying speed including Coriolis effects

Abstract

The purpose of the current study was to develop an accurate model to investigate the nonlinear resonances in an axially functionally graded beam rotating with time-dependent speed. To this end, two important features including stiffening and Coriolis effects are modeled based on nonlinear strain relations. Equations governing the axial, chordwise, and flapwise deformations about the determined steady-state equilibrium position are obtained, and the rotating speed variation is considered as a periodic disturbance about this equilibrium condition. Multi-mode discretization of the equations is performed via the spectral Chebyshev approach and the method of multiple scales for gyroscopic systems is employed to study the nonlinear behavior. After determining the required polynomial number based on convergence analysis, results obtained are verified by comparing to those found in literature and numerical simulations. Moreover, the model is validated based on simulations carried out by commercial finite element software. Properties of the functionally graded material and the values of average rotating speed leading to 2:1 internal resonance in the system are found. Time and steady-state responses of the system under primary and parametric resonances caused by the time-dependent rotating speed are investigated when the system is tuned to 2:1 internal resonance. A comprehensive study on the time response, frequency response, and stability behavior shows that the rotating axially functionally graded beam exhibits a complicated nonlinear behavior under the effect of the rotating speed fluctuation frequency, damping coefficient, and properties of the functionally graded material.

Appendices

Appendix A: FGM profiles

Some of FGM profiles used in this research are illustrated via numerical examples in Fig. 14. In this figure variations of the ceramic volume fraction are provided for six sets of a, b, and c parameters as a function of the volume fraction index p. It can be seen that the exploited distribution function can construct symmetric/asymmetric volume fraction variations along the beam.

Fig. 14

Variations of the ceramic volume fraction $V_c$ along the beam for different values of the parameters a, b, c and the volume fraction index p: a $\mathrm {FGM}_{(a=1/b=0.8/c=1/p=0.1k)}$, b $\mathrm {FGM}_{(a=1.5/b=0.8/c=2/p=0.1k)}$, c $\mathrm {FGM}_{(a=1.8/b=1.8/c=2/p=0.1k)}$, d $\mathrm {FGM}_{(a=2/b=2/c=3/p=0.1k)}$, e $\mathrm {FGM}_{(a=1.8/b=1.6/c=5/p=0.1k)}$, and f $\mathrm {FGM}_{(a=4/b=3/c=3/p=2k)}$, for $k=0,1,2,...,50$

Appendix B: Matrix elements

Here, the details of the matrices in Eq. (33) are given:

$$\begin{aligned}&{\mathbf {M}}=\begin{bmatrix} \mathbf {M_{uu}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} \mathbf {M_{vv}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} \mathbf {M_{ww}}\end{bmatrix},\ {\mathbf {G}}=\begin{bmatrix} {\mathbf {0}} &{} \mathbf {G_{uv}} &{} {\mathbf {0}}\\ \mathbf {G_{vu}} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\end{bmatrix},\nonumber \\&{\mathbf {K}}_1=\begin{bmatrix} {\mathbf {K}}_{\mathbf {uu}_1} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {K}}_{\mathbf {vv}_1} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {K}}_{\mathbf {ww}_1}\end{bmatrix},\nonumber \\&{\mathbf {K}}_2=\begin{bmatrix} {\mathbf {K}}_{\mathbf {uu}_2} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {K}}_{\mathbf {vv}_2} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {K}}_{\mathbf {ww}_2}\end{bmatrix},\ {\mathbf {K}}_3=\begin{bmatrix} {\mathbf {0}} &{} \mathbf {K_{uv}} &{} {\mathbf {0}}\\ \mathbf {K}_\mathbf{vu} &{} {\mathbf {0}} &{} {\mathbf {0}}\\ {\mathbf {0}} &{} {\mathbf {0}} &{} {\mathbf {0}}\end{bmatrix},\nonumber \\&{\mathbf {C}}\!=\!\lambda ({\mathbf {K}}_1\!+\!\omega _0^2{\mathbf {K}}_2),\ \mathbf {NL}({\mathbf {q}}_s)\!=\!\begin{Bmatrix} {\mathbf {0}}\\ \mathbf {NL_{v}}\\\mathbf {NL_{w}}\end{Bmatrix},\ {\mathbf {f}}\!=\!\begin{Bmatrix} {\mathbf {0}}\\ \mathbf {f_{v}}\\{\mathbf {0}}\end{Bmatrix} \end{aligned}$$

(B.1)

where the elements are calculated by

$$\begin{aligned}&\mathbf {M_{uu}}=-{\mathbf {K}}_{\mathbf {uu}_2}=A{\mathbf {P}}_1^\mathrm {T}{\mathbf {V}}_1{\mathbf {P}}_1 \end{aligned}$$

(B.2)

$$\begin{aligned}&\mathbf {M_{vv}}=-{\mathbf {K}}_{\mathbf {vv}_2}=A{\mathbf {P}}_2^\mathrm {T} {\mathbf {V}}_1{\mathbf {P}}_2-I_z({\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_1 {\mathbf {Q}}_2{\mathbf {P}}_2\nonumber \\&\quad +{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_2{\mathbf {Q}}_1 {\mathbf {P}}_2) \end{aligned}$$

(B.3)

$$\begin{aligned}&\mathbf {M_{ww}}=A{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_1{\mathbf {P}}_3-I_y({\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_1{\mathbf {Q}}_2{\mathbf {P}}_3+{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_2{\mathbf {Q}}_1{\mathbf {P}}_3) \end{aligned}$$

(B.4)

$$\begin{aligned}&\mathbf {G_{uv}}=\mathbf {K_{uv}}=-A{\mathbf {P}}_1^\mathrm {T}{\mathbf {V}}_1{\mathbf {P}}_2 \end{aligned}$$

(B.5)

$$\begin{aligned}&\mathbf {G_{vu}}=\mathbf {K_{vu}}=A{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_1{\mathbf {P}}_1 \end{aligned}$$

(B.6)

$$\begin{aligned}&{\mathbf {K}}_{\mathbf {uu}_1}=-A({\mathbf {P}}_1^\mathrm {T}{\mathbf {V}}_3{\mathbf {Q}}_2{\mathbf {P}}_1+{\mathbf {P}}_1^\mathrm {T}{\mathbf {V}}_4{\mathbf {Q}}_1{\mathbf {P}}_1) \end{aligned}$$

(B.7)

$$\begin{aligned}&{\mathbf {K}}_{\mathbf {vv}_1}=I_z({\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_3{\mathbf {Q}}_4{\mathbf {P}}_2+2{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_4{\mathbf {Q}}_3{\mathbf {P}}_2+{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_5{\mathbf {Q}}_2{\mathbf {P}}_2)\nonumber \\&\qquad \qquad -A({\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_6{\mathbf {Q}}_2{\mathbf {P}}_2+{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_7{\mathbf {Q}}_1{\mathbf {P}}_2+{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_8{\mathbf {Q}}_1{\mathbf {P}}_2) \end{aligned}$$

(B.8)

$$\begin{aligned}&{\mathbf {K}}_{\mathbf {ww}_1}=I_y({\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_3 {\mathbf {Q}}_4{\mathbf {P}}_3+2{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_4 {\mathbf {Q}}_3{\mathbf {P}}_3+{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_5 {\mathbf {Q}}_2{\mathbf {P}}_3)\nonumber \\&\qquad -A({\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_6{\mathbf {Q}}_2{\mathbf {P}}_3 +{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_7{\mathbf {Q}}_1{\mathbf {P}}_3 +{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_8{\mathbf {Q}}_1{\mathbf {P}}_3)\nonumber \\ \end{aligned}$$

(B.9)

$$\begin{aligned}&{\mathbf {K}}_{\mathbf {ww}_2}=I_y({\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_1 {\mathbf {Q}}_2{\mathbf {P}}_3+{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_2{\mathbf {Q}}_1 {\mathbf {P}}_3) \end{aligned}$$

(B.10)

$$\begin{aligned}&\mathbf {NL}_{\mathbf {v}}({\mathbf {q}}_s) =-A{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_3\big [({\mathbf {Q}}_2{\mathbf {P}}_1 {\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_1{\mathbf {P}}_2{\mathbf {q}}_{d_2})\big ]\nonumber \\&-A{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_3\big [({\mathbf {Q}}_1{\mathbf {P}}_1 {\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_2{\mathbf {P}}_2{\mathbf {q}}_{d_2})\big ]\nonumber \\&\qquad \qquad -A{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_4 \big [({\mathbf {Q}}_1{\mathbf {P}}_1{\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_1{\mathbf {P}}_2 {\mathbf {q}}_{d_2})\big ] \end{aligned}$$

(B.11)

$$\begin{aligned}&\mathbf {NL}_{\mathbf {w}}({\mathbf {q}}_s)=-A{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_3 \big [({\mathbf {Q}}_2{\mathbf {P}}_1{\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_1{\mathbf {P}}_3 {\mathbf {q}}_{d_3})\big ]\nonumber \\&\qquad -A{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_3 \big [({\mathbf {Q}}_1{\mathbf {P}}_1{\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_2{\mathbf {P}}_3 {\mathbf {q}}_{d_3})\big ]\nonumber \\&\qquad -A{\mathbf {P}}_3^\mathrm {T}{\mathbf {V}}_4 \big [({\mathbf {Q}}_1{\mathbf {P}}_1{\mathbf {q}}_{d_1})\circ ({\mathbf {Q}}_1{\mathbf {P}}_3 {\mathbf {q}}_{d_3})\big ] \end{aligned}$$

(B.12)

$$\begin{aligned}&\mathbf {f_v}\!=\!(I_z{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_2\!-\!Ar_h{\mathbf {P}}_2^\mathrm {T} {\mathbf {V}}_1-A{\mathbf {P}}_2^\mathrm {T}{\mathbf {V}}_9){\mathbf {f_I}} \end{aligned}$$

(B.13)

In all the above equations, ${\mathbf {Q}}_i$ is the differentiation matrix representing the ith derivative with respect to $\xi$, $\circ$ symbol shows the element-wise multiplication, ${\mathbf {f_I}}$ is a vector of ones, and weighted inner product matrices are defined by

$$\begin{aligned}&\int _{\xi }\big \{\rho (\xi ),\rho ^\prime (\xi ),E(\xi ),E^\prime (\xi ), E^{\prime \prime }(\xi ),\nonumber \\&\qquad E(\xi )u_s^\prime (\xi ),E(\xi )u_s^{\prime \prime }(\xi ), E^{\prime }(\xi )u_s^{\prime }(\xi ),\nonumber \\&\qquad \rho (\xi )(\xi +u_s(\xi ))\big \}f(\xi )g(\xi )\mathrm {d}\xi \nonumber \\&\quad = {\mathbf {f}}^\mathrm {T}\big \{{\mathbf {V}}_1,{\mathbf {V}}_2,{\mathbf {V}}_3, {\mathbf {V}}_4,{\mathbf {V}}_5,{\mathbf {V}}_6,{\mathbf {V}}_7,{\mathbf {V}}_8, {\mathbf {V}}_9\big \}{\mathbf {g}} \end{aligned}$$

(B.14)

Appendix C: Coefficients $\mu$ and $\Delta$

By inserting solutions (43) and (44) into Eq. (41) for $i=1,2,..N_e-1$, and equating the coefficients of $\exp (I\omega _n^cT_0)$ on both sides of the resulting equations for $n=1,2,...,N_e$, $N_e$ number of linear algebraic systems including $N_e-1$ number of equations are derived as

$$\begin{aligned}&2I\omega _0\sum _{j=1,j\ne i}^{N_e}\omega _n^c{\hat{g}}_{i,j}\mu _{j,n}+(\omega _i^2-{\omega _n^c}^2)\mu _{i,n}=0,\nonumber \\&\quad n = 1,2,...,N_e,\ i =1,2,...,N_e-1 \end{aligned}$$

(C.1)

One should note that according to the assumed solutions $\mu _{1,n} = 1$. Based on the same balancing for coefficients of $\exp (I\Omega T_0)$, we have

$$\begin{aligned}&2I\omega _0\Omega \sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\Delta _j+(\omega _i^2-\Omega ^2)\Delta _i=\frac{\gamma \Omega {\hat{f}}_i}{2},\nonumber \\&\quad i = 1,2,...,N_e \end{aligned}$$

(C.2)

By solving these linear algebraic systems of equations, coefficients $\mu _{i,j}$ and $\Delta _i$ can be determined.

Appendix D: Right-hand side terms

• For non-resonant $\Omega$ without internal resonance:

$$\begin{aligned}&R_{i,l}=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l}\mathrm {D}_1A_l+2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l\nonumber \\&\quad +\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\quad i,l = 1,2,...,N_e \end{aligned}$$

(D.1)

• For non-resonant $\Omega$ with internal resonance ($\omega _m^c = 2\omega _n^c+\epsilon \sigma _0$):

$$\begin{aligned} R_{i,l}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l}\mathrm {D}_1A_l\nonumber \\&\quad +2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l\nonumber \\&\quad +\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\nonumber \\ R_{i,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,n}\mathrm {D}_1A_n+2I{\omega _n^c}\mu _{i,n}\mathrm {D}_1A_n\nonumber \\&\quad +\lambda I\omega _i^2\omega _n^c\mu _{i,n}A_n\nonumber \\&\quad +\exp (I\sigma _0T_1)\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}} {\bar{\mu }}_{k,n}{\bar{A}}_n\mu _{j,m}A_m,\nonumber \\ R_{i,m}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,m}\mathrm {D}_1A_m\nonumber \\&\quad +2I{\omega _m^c}\mu _{i,m}\mathrm {D}_1A_m\nonumber \\&\quad +\lambda I\omega _i^2\omega _m^c\mu _{i,m}A_m\nonumber \\&\quad +\exp (-I\sigma _0T_1)\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}\mu _{j,n}\mu _{k,n}A_n^2,\nonumber \\&\quad i,l=1,2,...,N_e,\ l\ne m,n,\ 2\le m\le N_e,\nonumber \\&\quad 1\le n\le N_e-1 \end{aligned}$$

(D.2)

• For case with $\Omega$ near $\omega _n^c$ ($\Omega = \omega _n^c+\epsilon \sigma _1$):

$$\begin{aligned} R_{i,l}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l} \mathrm {D}_1A_l+2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l\nonumber \\&\quad +\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\nonumber \\ R_{n,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{n,j}\mu _{j,n}\mathrm {D}_1A_n+2I{\omega _n^c}\mu _{n,n}\mathrm {D}_1A_n\nonumber \\&\quad +\lambda I\omega _n^2\omega _n^c\mu _{n,n}A_n-\exp (I\sigma _1T_1)\frac{\gamma {\hat{f}}_n\Omega }{2},\nonumber \\&i,l=1,2,...,N_e,\ i\ne n,\ 1\le n\le N_e \end{aligned}$$

(D.3)

• For case with $\Omega$ near $\omega _n^c$ and internal resonance ($\Omega = \omega _n^c+\epsilon \sigma _1$, $\omega _m^c = 2\omega _n^c+\epsilon \sigma _0$):

$$\begin{aligned} R_{i,l}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l} \mathrm {D}_1A_l+2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l\nonumber \\&\quad +\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\nonumber \\ R_{i,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,n}\mathrm {D}_1A_n\nonumber \\&\quad +2I{\omega _n^c}\mu _{i,n}\mathrm {D}_1A_n+\lambda I\omega _i^2\omega _n^c\mu _{i,n}A_n\nonumber \\&+\exp (I\sigma _0T_1)\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}{\bar{\mu }}_{k,n}{\bar{A}}_n\mu _{j,m}A_m,\nonumber \\ R_{n,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{n,j}\mu _{j,n}\mathrm {D}_1A_n+2I{\omega _n^c}\mu _{n,n}\mathrm {D}_1A_n\nonumber \\&\quad +\lambda I\omega _n^2\omega _n^c\mu _{n,n}A_n\nonumber \\&\quad +\exp (I\sigma _0T_1)\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{n,j,k}}{\bar{\mu }}_{k,n}{\bar{A}}_n\mu _{j,m}A_m\nonumber \\&\quad -\exp (I\sigma _1T_1)\frac{\gamma {\hat{f}}_n\Omega }{2},\nonumber \\ R_{i,m}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,m}\mathrm {D}_1A_m\nonumber \\&\quad +2I{\omega _m^c}\mu _{i,m}\mathrm {D}_1A_m+\lambda I\omega _i^2\omega _m^c\mu _{i,m}A_m\nonumber \\&\quad +\exp (-I\sigma _0T_1)\sum _{j=1}^{N_e}\nonumber \\&\quad \sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}\mu _{j,n}\mu _{k,n}A_n^2,\nonumber \\&\quad i,l=1,2,...,N_e,\ i\ne n,\ l\ne m,n,\nonumber \\&\quad 2\le m\le N_e,\ 1\le n\le N_e-1 \end{aligned}$$

(D.4)

• For case with $\Omega$ near $2\omega _n^c$ ($\Omega = 2\omega _n^c+\epsilon \sigma _2$):

$$\begin{aligned} R_{i,l}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l}\mathrm {D}_1A_l\nonumber \\&\quad +2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l+\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\nonumber \\ R_{i,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,n}\mathrm {D}_1A_n\nonumber \\&\quad +2I{\omega _n^c}\mu _{i,n}\mathrm {D}_1A_n+\lambda I\omega _i^2\omega _n^c\mu _{i,n}A_n\nonumber \\&\quad -\exp (I\sigma _2T_1)\bigg (\gamma \omega _n^c\sum _{j=1,j\ne i}^{N_e}\nonumber \\&\quad {\hat{g}}_{i,j}{\bar{\mu }}_{j,n}{\bar{A}}_n+I\gamma \omega _0\sum _{j=1}^{N_e}{\hat{k}}_{2_{i,j}}{\bar{\mu }}_{j,n}{\bar{A}}_n\nonumber \\&-\frac{1}{2}\gamma \Omega \sum _{j=1,j\ne i}^{N_e}{\hat{k}}_{3_{i,j}}{\bar{\mu }}_{j,n}{\bar{A}}_n\nonumber \\&\quad -\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}\Delta _j{\bar{\mu }}_{k,n}{\bar{A}}_n\bigg ),\nonumber \\&\quad i,l=1,2,...,N_e,\ l\ne n,\ 1\le n\le N_e \end{aligned}$$

(D.5)

• For case with $\Omega$ near $2\omega _n^c$ and internal resonance ($\Omega = 2\omega _n^c+\epsilon \sigma _2$, $\omega _m^c = 2\omega _n^c+\epsilon \sigma _0$):

$$\begin{aligned} R_{i,l}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,l}\mathrm {D}_1A_l\nonumber \\&\quad +2I{\omega _l^c}\mu _{i,l}\mathrm {D}_1A_l+\lambda I\omega _i^2\omega _l^c\mu _{i,l}A_l,\nonumber \\ R_{i,n}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,n}\mathrm {D}_1A_n\nonumber \\&\quad +2I{\omega _n^c}\mu _{i,n}\mathrm {D}_1A_n+\lambda I\omega _i^2\omega _n^c\mu _{i,n}A_n\nonumber \\&\quad -\exp (I\sigma _2T_1)\bigg (\gamma \omega _n^c\nonumber \\&\quad \sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}{\bar{\mu }}_{j,n}{\bar{A}}_n\nonumber \\&\quad +I\gamma \omega _0\sum _{j=1}^{N_e}{\hat{k}}_{2_{i,j}}{\bar{\mu }}_{j,n}{\bar{A}}_n-\frac{1}{2}\gamma \Omega \nonumber \\&\quad \sum _{j=1,j\ne i}^{N_e}{\hat{k}}_{3_{i,j}}{\bar{\mu }}_{j,n}{\bar{A}}_n\nonumber \\&\quad -\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}\Delta _j{\bar{\mu }}_{k,n} {\bar{A}}_n\bigg )\nonumber \\&\quad +\exp (I\sigma _0T_1)\sum _{j=1}^{N_e}\sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}{\bar{\mu }}_{k,n}{\bar{A}}_n\mu _{j,m}A_m,\nonumber \\ R_{i,m}&=2\omega _0\sum _{j=1,j\ne i}^{N_e}{\hat{g}}_{i,j}\mu _{j,m}\mathrm {D}_1A_m\nonumber \\&\quad +2I{\omega _m^c}\mu _{i,m}\mathrm {D}_1A_m+\lambda I\omega _i^2\omega _m^c\mu _{i,m}A_m\nonumber \\&\quad +\exp (-I\sigma _0T_1)\sum _{j=1}^{N_e}\nonumber \\&\quad \sum _{k=1}^{N_e}{\hat{f}}_{nl_{i,j,k}}\mu _{j,n}\mu _{k,n}A_n^2,\nonumber \\&\quad i,l=1,2,...,N_e,\ l\ne m,n,\ 2\le m\le N_e,\ \nonumber \\&\quad 1\le n\le N_e-1 \end{aligned}$$

(D.6)

Appendix E: Convergence analysis

To determine the number of polynomials used in the solution technique, a convergence analysis is carried out. In this analysis, the polynomial number is gradually increased, and the predicted natural frequencies are compared with a reference values calculated using a large polynomial number. To quantitatively assess the level of convergence, a relative logarithmic assessment approach given by the following equation is used

$$\begin{aligned} LCV_N = \log \bigg (\frac{1}{K}\sum _{i=1}^K\frac{\big |\omega _{i_N}^c-\omega _{i_{ref}}^c\big |}{\omega _{i_{ref}}^c}\bigg ) \end{aligned}$$

(E.1)

Here, LCV is the logarithmic convergence value, $\omega _{i_N}^c$ refers to the natural frequency of the $i^{th}$ vibration mode calculated using N number of polynomials, $\omega _{i_{ref}}^c$ is the natural frequency of the reference case (obtained using large polynomial numbers), and K shows the number of interested modes. Note that this convergence analysis can be performed for a single mode or several selected modes of the system. To determine the sufficient number of polynomials, $LCV_N<\varepsilon$ should be satisfied (where $\varepsilon$ is the threshold value selected by the user).

Fig. 15

Convergence plots for dimensionless natural frequencies of an axially FG rotating beam based on spectral Chebyshev approach as a function of dimensionless rotating speed for $\delta =0$, ${\hat{\omega }}_0=10$, and $\alpha =70$: a Isotropic material i.e. $p=0$, b $\mathrm {FGM}_{(a=1/b=0.8/c=1/p=0.4)}$, c $\mathrm {FGM}_{(a=1/b=1/c=2/p=1)}$, and d $\mathrm {FGM}_{(a=1.5/b=0.8/c=2/p=1.6)}$

Convergence plots for different volume fraction parameters as a function of dimensionless angular velocity are depicted in Fig. 15. Note that, to present the convergence in terms of contour plots, a linear interpolation is used since LCVs are defined only for integer number of polynomials. In this figure, LC values are obtained as an average of the convergence of the first eight natural frequencies in the system, i.e. $K=8$, and for the reference case, $N=30$ is considered. Each colored contour corresponds to a level of LCV, and the numbers are the corresponding values. According to this figure, the FGM distribution parameters does not affect the convergence behavior; however, as the rotating velocity increases, the number of polynomials necessary to obtain results with the same convergence performance increases. Accordingly, by considering the minimum error threshold to be $0.01\%$ ($LCV = -4$) and maximum value to be $0.4\%$ ($LCV = -2.4$), the number of polynomials is set to 11 shown with red horizontal line.