Modal characteristics of a rotating flexible beam with a concentrated mass based on the absolute nodal coordinate formulation

Abstract

A new dynamic model of a rotating flexible beam with a concentrated mass located in arbitrary position is derived based on the absolute nodal coordinate formulation, and its modal characteristics are investigated in this paper. To consider the concentrated mass at an arbitrary location of the beam, a Dirac’s delta function is used to express the mass per unit length of the beam. Based on the proposed dynamic model, the frequency analysis is performed. The nonlinear equation is transformed into the linear one via employing the linear perturbation analysis method. The stiffness matrix of static equilibrium of the system under the deformed condition is obtained, in which the effect of coupling between the longitudinal deformation and transversal deformation is included. This means even if only the chordwise bending equation is solved, the longitudinal vibration effect can be still considered. As we know, once the longitudinal deformation is large, it will significantly affect the chordwise bending vibration. So the proposed model in this paper is more accurate than the traditional dynamic models which are usually lack of the coupling terms between the longitudinal deformation and transversal deformation. In fact, the traditional dynamic models for the chordwise vibration analysis in the existing literature are usually linear due to neglecting the coupling terms, and consequently, they are only suitable for the modal characteristic analysis of a beam under small deformations. In order to get some general conclusions of the natural frequencies and mode shapes, the equation which governs the chordwise bending vibration of the rotating beam is transformed into a dimensionless form. The dynamic model presented in this paper is nonlinear and can be conveniently used to analyze the modal characteristics of a rotating flexible beam with large deformations. To demonstrate the power of the new dynamic model presented in this paper, the dynamic simulations involving the comparisons between the different frequencies obtained using the model proposed in this paper and the models in the existing literature and the investigating in frequency veering and mode shift phenomena are given. The simulation results show that the angular velocity of the flexible beam will give rise to the phenomena of the natural frequency loci veering and the associated mode shift which is verified in the previous studies. In addition, the phenomena of the natural frequency loci veering rather than crossing can be observed due to the changing of the magnitude of the concentrated mass or of the location of the concentrated mass which are found for the first time. Furthermore, there is an interesting phenomenon that the natural frequency loci will veer more than once due to different types of mode coupling between the bending and stretching vibrations of the rotating beam. At the same time, the mode shift phenomenon will occur correspondingly. Additionally, the characteristics of the vibration nodes are also investigated in this paper.

Appendix

The dimensionless form of the Eq. (40) can be obtained by using the dimensionless variables and parameters defined in Eq. (45)

$$\begin{aligned} \bar{{\varvec{M}}}\frac{\partial ^{2}\bar{{\varvec{q}}}_a }{\partial \tau ^{2}}+\bar{\varvec{{\varvec{Q}}}}_{\mathrm{T}} =\lambda ^{{2}}\delta \bar{\varvec{{\varvec{S}}}}_x \end{aligned}$$

(49)

where

$$\begin{aligned}&\bar{{\varvec{M}}}=\bar{{\varvec{M}}}_a +\bar{\varvec{{\varvec{M}}}}^{{*}} \end{aligned}$$

(50)

$$\begin{aligned}&\bar{\varvec{{\varvec{M}}}}_a =\sum _{e=1}^n {{\varvec{B}}_e ^\mathrm{T}\bar{\varvec{{\varvec{M}}}}_e {\varvec{B}}_e } \end{aligned}$$

(51)

$$\begin{aligned}&\bar{\varvec{{M}}}_e =\frac{1}{n}\int _{ 0}^{ 1} { \bar{\varvec{{S}}}(\eta )^\mathrm{T}\bar{\varvec{{S}}}(\eta )d\eta } \end{aligned}$$

(52)

$$\begin{aligned}&\bar{\varvec{{M}}}^{{*}}=\alpha {\varvec{B}}_k ^{\mathrm{T}}\bar{\varvec{{S}}}^{\mathrm{T}}(\frac{\Delta l}{L})\bar{\varvec{{S}}}(\frac{\Delta l}{L}){\varvec{B}}_k \end{aligned}$$

(53)

$$\begin{aligned}&\frac{\Delta l}{L}=\beta -\frac{k-1}{n} \end{aligned}$$

(54)

$$\begin{aligned}&\bar{\varvec{{S}}}_x =\bar{\varvec{{S}}}_{x1} +\bar{\varvec{{S}}}_{x1}^\mathrm{*} \end{aligned}$$

(55)

$$\begin{aligned}&\bar{\varvec{{S}}}_{x1} =\frac{1}{n}\sum _{e=1}^n {{\varvec{B}}_e^\mathrm{T} } \int _0^1 {\bar{\varvec{{S}}}_1^\mathrm{T} \mathrm{d}\eta } \end{aligned}$$

(56)

$$\begin{aligned}&\bar{\varvec{{S}}}_{x1}^{*} =\alpha {\varvec{B}}_k^\mathrm{T} \bar{\varvec{{S}}}_1^\mathrm{T} (\Delta l) \end{aligned}$$

(57)

$$\begin{aligned}&\bar{\varvec{{Q}}}_{\mathrm{T}} =-\lambda ^{{2}}\bar{\varvec{{M}}\bar{{\varvec{q}}}}_a -\bar{\varvec{{Q}}}_{fa} \end{aligned}$$

(58)

$$\begin{aligned}&\bar{\varvec{{Q}}}_{f a} =\sum _{e=1}^n {{\varvec{B}}_e ^\mathrm{T}\bar{\varvec{{Q}}}_{f e} } \end{aligned}$$

(59)

$$\begin{aligned} \bar{\varvec{{Q}}}_{f e}= & {} -n\mu ^{2}\int _0^1 {\bar{{\varepsilon }}_{x0} \bar{\varvec{{{S}}}}^{{\prime \mathrm{T}}}\bar{\varvec{{{S}'}}} \bar{\varvec{{q}}}_e \mathrm{d}\eta } \nonumber \\&-\int _0^1 {\bar{{\kappa }}\cdot \left( {\frac{n^{2}\bar{\varvec{{\varGamma }}} }{\bar{{f}}^{3}}-3n\frac{\bar{{\kappa }}}{\bar{{f}}^{2}}\bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} } \right) \bar{\varvec{{q}}}_e \mathrm{d}\eta } \end{aligned}$$

(60)

$$\begin{aligned} \bar{\varvec{{\varGamma }}}= & {} \bar{\varvec{{{S}'}}}^{\mathrm{T}}\varvec{\tilde{I}}^\mathrm{T}\bar{\varvec{{{S}''}}}+\bar{\varvec{{{S}''}}}^{\mathrm{T}}\varvec{\tilde{I}} \bar{\varvec{{{S}'}}} \end{aligned}$$

(61)

$$\begin{aligned} \bar{{\varepsilon }}_{x0}= & {} \frac{1}{2}\left[ {n^{2}\bar{\varvec{{q}}}_e^\mathrm{T} \bar{{{\varvec{S}}'}}^{\mathrm{T}}\bar{{{\varvec{S}}'}}\bar{\varvec{{q}}}_e -1} \right] \end{aligned}$$

(62)

$$\begin{aligned} \bar{{f}}= & {} n\sqrt{\bar{\varvec{{q}}}_e^\mathrm{T} \bar{{{\varvec{S}}'}}^{\mathrm{T}}\bar{{{\varvec{S}}'}}\bar{\varvec{{q}}}_e } \end{aligned}$$

(63)

$$\begin{aligned} \bar{{\kappa }}= & {} n^{3}\bar{{{\varvec{S}}'}}\bar{\varvec{{q}}}_e \times \bar{{{\varvec{S}}''}}\bar{\varvec{{q}}}_e /\bar{{f}}^{3} \end{aligned}$$

(64)

$$\begin{aligned} \bar{\varvec{{q}}}_a= & {} {\varvec{q}}_a /L \end{aligned}$$

(65)

$$\begin{aligned} \bar{{\varvec{q}}}_e= & {} \frac{1}{L}\left[ {{\begin{array}{cccccccc} {\bar{{e}}_1 }&{} {\bar{{e}}_2 }&{} {\bar{{e}}_3 }&{} {\bar{{e}}_4 }&{} {\bar{{e}}_5 }&{} {\bar{{e}}_6 }&{} {\bar{{e}}_7 }&{} {\bar{{e}}_8 } \\ \end{array} }} \right] ^{\mathrm{T}} \end{aligned}$$

(66)

In which, the vector of absolute nodal coordinates includes the global position

$$\begin{aligned} \bar{{e}}_1= & {} \left. {r_1 } \right| _{x=0} ,\;\;\bar{{e}}_2 =\left. {r_2 } \right| _{x=0} \nonumber \\ \bar{{e}}_5= & {} \left. {r_1 } \right| _{x=l} ,\;\;\bar{{e}}_6 =\left. {r_2 } \right| _{x=l} \end{aligned}$$

(67)

and the gradients of the global position vector at element nodes multiplied by l,the length of the elements, are defined as

$$\begin{aligned} \bar{{e}}_3= & {} \left. {l\frac{\partial r_1 }{\partial x}} \right| _{x=0} ,\bar{{e}}_4 =l\left. {\frac{\partial r_2 }{\partial x}} \right| _{x=0} \nonumber \\ \bar{{e}}_7= & {} \left. {l\frac{\partial r_1 }{\partial x}} \right| _{x=l} ,\bar{{e}}_8 =l\left. {\frac{\partial r_2 }{\partial x}} \right| _{x=l} \end{aligned}$$

(68)

In which, the global shape function matrix $\bar{{\varvec{S}}}$ can be rewritten as

$$\begin{aligned}&\bar{{\varvec{S}}}=\left[ {{\begin{array}{l} {\bar{\varvec{{s}}}_1 } \\ {\bar{\varvec{{s}}}_2 } \\ \end{array} }} \right] =\left[ {{\begin{array}{cccccccc} {\bar{{s}}_1 }&{} 0&{} {\bar{{s}}_2 }&{} 0&{} {\bar{{s}}_3 }&{} 0&{} {\bar{{s}}_4 }&{} 0 \\ 0&{} {\bar{{s}}_1 }&{} 0&{} {\bar{{s}}_2 }&{} 0&{} {\bar{{s}}_3 }&{} 0&{} {\bar{{s}}_4 } \\ \end{array} }} \right] \end{aligned}$$

(69)

$$\begin{aligned}&\left\{ {{\begin{array}{l} {\bar{{s}}_1 =1-3\eta ^{2}+2\eta ^{3}} \\ {\bar{{s}}_2 =\eta -2\eta ^{2}+\eta ^{3}\;} \\ {\bar{{s}}_3 =3\eta ^{2}-2\eta ^{3}\;\;\;\;} \\ {\bar{{s}}_4 =-\eta ^{2}+\eta ^{3}\;\;\;\;\;} \\ \end{array} }} \right. \end{aligned}$$

(70)

$$\begin{aligned}&\eta =\frac{n}{L}x\;\;,\;\;\eta \in [{\begin{array}{cc} 0&{} 1 \\ \end{array} }] \end{aligned}$$

(71)

$$\begin{aligned}&\bar{{{\varvec{S}}'}}=\frac{\partial \bar{{\varvec{S}}}}{\partial \eta } \end{aligned}$$

(72)

$$\begin{aligned}&\bar{{{\varvec{S}}''}}=\frac{\partial ^{2}\bar{{\varvec{S}}}}{\partial \eta ^{2}} \end{aligned}$$

(73)

The dimensionless form of the Eq. (41) of the system is obtained as Eq. (46).

In Eq. (46)

$$\begin{aligned}&\bar{\varvec{{K}}}_{\mathrm{T}} =\bar{\varvec{{K}}}_a -\lambda ^{2}\bar{\varvec{{M}}} \end{aligned}$$

(74)

$$\begin{aligned}&\bar{\varvec{{K}}}_a =\sum _{e=1}^n {{\varvec{B}}_e^\mathrm{T} \bar{\varvec{{K}}}_e {\varvec{B}}_e } \end{aligned}$$

(75)

$$\begin{aligned}&\bar{\varvec{{\chi }}}(\tau )=\frac{1}{L}\varvec{\chi }(t) \end{aligned}$$

(76)

$$\begin{aligned}&\bar{\varvec{{K}}}_e =\mu ^{2}(n^{3}\int _0^1 { \bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \bar{\varvec{{q}}}_e \bar{\varvec{{q}}}_e^\mathrm{T} \bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \mathrm{d}\eta } +n\int _0^1 { \bar{{\varepsilon }}_{x0} \bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \mathrm{d}\eta )} \nonumber \\&\quad +\frac{1}{n}\int _0^1 \left( {\frac{n^{3}\bar{\varvec{{\varGamma }}} }{\bar{{f}}^{3}}-3n^{2}\frac{\bar{{\kappa }}}{\bar{{f}}^{2}}\bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} } \right) \bar{\varvec{{q}}}_e \bar{\varvec{{q}}}_e^\mathrm{T} \nonumber \\&\quad \times \left( \frac{n^{3}\bar{\varvec{{\varGamma }}}^{\mathrm{T}} }{\bar{{f}}^{3}}-3n^{2}\frac{\bar{{\kappa }}}{\bar{{f}}^{2}}\bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \right) \mathrm{d}\eta \nonumber \\&+\frac{1}{n}\int _0^1 \bar{{\kappa }}\left( \frac{n^{3}\bar{\varvec{{\varGamma }}}^{\mathrm{T}} }{\bar{{f}}^{3}}-3n^{2}\frac{\bar{{\kappa }}}{\bar{{f}}^{2}}\bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \right) \mathrm{d}\eta \nonumber \\&+\frac{1}{n}\int _0^1 \bar{{\kappa }}\left( -6\frac{n^{5}\bar{\varvec{{\varGamma }}} }{\bar{{f}}^{5}}\bar{\varvec{{q}}}_e \bar{\varvec{{q}}}_e^\mathrm{T} \bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}}\right. \nonumber \\&\left. +15\bar{{\kappa }}\frac{1}{\bar{{f}}^{4}}n^{4}\bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}\bar{{q}}}_e \bar{\varvec{{q}}}_e^\mathrm{T} \bar{\varvec{{{S}'}}}^{\mathrm{T}}\bar{\varvec{{{S}'}}} \right) \mathrm{d}\eta \end{aligned}$$

(77)