Dynamic modeling and nonlinear free vibration analysis of a rotating 3D beam induced by adjacent two revolute joints

In this article, a 3D dynamic model and nonlinear free vibration analysis of a uniform beam attached to a rigid hub rotating at two adjacent rotary joints are presented. The nonlinear model is performed by considering the effects of steady-state axial deformation, Coriolis terms, and geometric nonlinearity by means of von Karman’s strain–displacement relations. The coupled axial, chordwise, and flapwise equations of motion are derived from Hamilton’s principle. Then, Galerkin’s procedure is utilized to discretize the governed partial differential equations. The effects of hub radius and rotational speeds are incorporated into the governing equations of motion. The relative effects of the Coriolis term on the flapwise and chordwise deformations of a considered 3D rotating beam are investigated. The validity of the present numerical results is verified by comparing them with the available results of the cantilever beam with and without the hub radius at various rotational speeds. The quantitative data obtained within the limit of speed of rotation reveals that the Coriolis factor has a significant impact on the flapwise bending frequencies but not really on the chordwise bending frequencies.


Introduction
The rotating beam has received considerable attention in many diverse engineering disciplines due to its ability to represent the major components of structures such as flexible arms, propellers, helicopter blades, and other various mechanical systems.As a result, to develop a proper mechanical design, the dynamic characteristics, i.e., the natural frequencies, of the structures should be thoroughly investigated.There have been a lot of motion models made for analyzing the vibrations of the rotating beams as a whole.
Many previous studies on rotating beams focused on out-of-plane (flapwise) vibration.Hoa [1] investigated the vibration frequency of a rotating beam with a tip mass using the finite element method, including the effects of the hub radius, the setting angle, and the tip mass.Fung and Shi [3] analyzed the natural vibration frequencies of a constrained clampedfree Euler-Bernoulli beam with a tip mass.Ozgumus and Kaya [7] studied the flapwise vibration analysis of a double-tapered Euler-Bernoulli beam that is mounted on a rotating rigid hub using the differential transform method.Bulut [10] examined the dynamic stability of a parametrically activated rotating tapered beam with periodically changing velocity using the monodromy matrix technique.
In the existing literature, mainly linear flapwise free vibration has been studied.However, a number of researchers have also looked into nonlinear free vibrations that deal with the effects of coupling.Yokoyama [2] investigated the in-plane and out-of-plane free vibrations of a rotating Timoshenko beam by using a finite element technique that took into account the effects of hub radius, setting angle, shear deformation, and rotary inertia.Yoo and Shin [4] derived a new dynamic modeling method for a rotating cantilever beam based on stretching and bending motions.So, three sets of linear equations of motion were obtained and, hence, flapwise bending motion, which was uncoupled from the other two sets.The effect of a concentrated mass located in an arbitrary position on the modal characteristics of a rotating cantilever beam was investigated by Yoo et al. [5].Yoon and Son [8] studied the vibration system of a rotating cantilever pipe conveying fluid and a tip mass.In this work, the equation of motion is derived using the Lagrange equation, employing hybrid deformation variables.Berzeri and Shabana [6] studied the centrifugal stiffening effect on rotating twodimensional beams using the finite element absolute nodal coordinate formulation.Kim et al. [9] derived a new modeling method using the nonlinear von-Karman strain and the corresponding linear stress for the axial, chordwise, and flapwise equations of motion of rotating beams.Zhang et al. [11] analyzed the modal characteristics of a rotating flexible beam with a concentrated mass located in an arbitrary position by using the absolute nodal coordinate formulation.In this work, the effect of coupling between the longitudinal deformation and the transversal deformation is also included.Zhao and Wu [12] established the coupling equations of motion of a rotating three-dimensional cantilever beam using fully nonlinear Green strain-displacement relationships.Zhao and Wu [12] concluded that the steady-state axial deformation has a remarkable effect on the chordwise bending frequency but not on the flapwise bending frequency.Lee et al. [13] presented a transfer matrix expression to analyze the bending vibration of a rotating tapered beam.Zhou et al. [14] analyzed the nonlinear forced vibration analysis of a rotating three-dimensional tapered Euler-Bernoulli cantilever beam subjected to a uniformly distributed load via Galerkin's procedure to study the effects of the taper ratio, rotating velocity, radius of hub, and external excitation.Yun-dong et al. [15] developed a new nonlinear model of a rotating uniform cantilever beam with tip mass in which the effects of axial geometric nonlinearity and large curvature are capture via nonlinear Green strain-displacement relations.Efafi et al. [16] obtained 3D dynamic modeling and robust control of a special flexible robotic arm.In this work, the proposed model is similar to that in our present paper according to the kinematic structure (joint location).In the strain energy, they carry out nonlinear curvature in the transverse directions, neglecting the coupling effects and longitudinal deformations.García-Vallejo et al. [18] investigated the critical velocities of rotating 3D beam dynamics according to both the linear theory of elasticity and the nonlinear theory of elasticity through a nonlinear finite element model based on the absolute nodal coordinate formulation.Warminski et al. [19] analyzed the dynamics of a rotor consisting of a flexible beam connected to a rotating rigid hub on the basis of the extended Bernoulli-Euler theory, which incorporates a nonlinear curvature, associated transverse and longitudinal oscillations, and a nonconstant angular velocity of the hub.
González-Carbajal et al. [20] presented simplified models of the rotating beam that capture the centrifugal stiffening effect caused by the inclusion of the axial and transverse kinematic coupling of the beam centerline.The results of the simplified models are compared with the results of the finite element method.Arvin [21] analyzed the nonlinear free vibration and fundamental parametric resonance of rotating beams with lag-axial coupling motion arising from the Coriolis force employing the multiple scales method.
Previous research has thoroughly examined beams rotating about one axis, particularly a vertical axis with a constant angular speed.In practice, such derived models associated with rotating beams can be regarded essentially as 2D dynamic models of a rotating beam in contrast to a 3D model.Some work considers the case of in-plane vibration while accounting for the setting angle effect.Existing works that have been developed, such as a helicopter rotor blade or a turbine blade, can be accurately represented.On the other hand, extending the previous models to a flexible arm case without any changes is inappropriate.The flexible arm can perform sophisticated motions because it has multiple rotating joints that are situated on orthogonal axes.Therefore, actuated joints bring about varied angular velocities along these axes.Hence, employing directly derived, existing models for the vibration analysis in both in-plane and out-of-plane directions does not provide satisfactory results due to the change in the dynamic nature of the system.
The aim of this study is to model and analyze both in-plane and out-of-plane free vibrations of a rotating Euler-Bernoulli beam with two adjacent rotary joints.Hamilton's principle is employed to derive the nonlinear partial differential equations of motion based on von Karman's strain-displacement relations.Then, Galerkin's method is used to discretize the governing motion equations.The effects of the Coriolis terms on the dimensionless linear natural frequencies of flapwise and chordwise vibrations of a 3D rotating beam are investigated.The frequency values obtained in this paper were calculated numerically, and five modes were taken on each axis to improve the accuracy of the numerical study.If these values were obtained from a single generic matrix, the actual frequency results would not be correctly determined because the frequencies were too close to each other.Unlike a related study (e.g., [14]), the overall matrix was divided into two different matrices in the chordwise-longitudinal and flapwise-longitudinal directions.It is sufficient to investigate at a linear natural frequency when dealing with small oscillations as opposed to those other large oscillation studies.The numerical results of rotating beams with and without the hub radius are given for various values of rotational speed and compared to previous works.It is shown that the proposed method is a good way to solve problems with a 3D rotating beam's free vibration.

Kinematic relationship
In this part, we figure out the equations of motion for the rotating beam we are looking at with the following assumptions: As illustrated in Fig. 1, the considered beam undergoes 3D motion through the composing of a rotating hub arising from the external torques τ 1 and τ 2 , oscillating in two planes (flapwise and chordwise directions).A novel mathematical model needs to be developed here because the dynamics of the studied 3D beam differs from its cantilever beam counterpart.Thus, the dynamic equations of the considered rotating beam are derived here.The motion of the considered system is described by a global inertial frame O − X i Y i Z i and a floating reference frame O − X f Y f Z f located at the hub center.The rigid The position vector of any point in the beam with respect to the X f Y f Z f frame is expressed as follows: where R is the radius of the rigid hub, û, v, and ŵ are the longitudinal, lateral, and flapwise arbitrary displacements along the x-, y-, and z-directions, respectively.According to the Euler-Bernoulli beam theorem, the displacements of an arbitrary point in the beam can be described as follows: where u, v, and w are the longitudinal, lateral, and flapwise displacements, respectively.The relationship between the inertial and floating reference frames can be represented by two rotation matrices.Let 0 R 1 (θ 1 ) and 1 R 2 (θ 2 ) to be the rigid joint rotation matrices for rotating degrees of θ 1 and θ 2 as follows: where s( * ) and c( * ) represent sin and cos trigonometric functions, respectively.A generic point's position vector in the inertial coordinate frame can be calculated as where θ 1 and θ 2 are rotation angles about the Y i and Z i axes.In addition, the corresponding angular velocity of the floating reference frame X f Y f Z f with respect to the global inertial frame is expressed as (5)

Dynamic modeling
Assuming that such a beam is composed of an isotropic material with linear material characteristics and is slender, the Euler-Bernoulli beam assumption may be implemented into this model without considering the effects of shear deformation and rotating inertia.The straindisplacement relations can be written using von Kármán [17] geometric nonlinear theory as follows: And, substituting Eq. ( 1) into Eq.( 6), the nonlinear strain-displacement relations can be written as Therefore, the following expression for the potential energy of the system can be obtained: where E denotes the Young modulus, A is the cross-sectional area, and I y and I z are the cross-sectional moments of inertia about the y and z axes, respectively.The system's kinetic energy, comprising the hub and beam, may be expressed as follows: where ρ and I h are the mass per unit length and the inertia of the joints, respectively.A dot means differentiation with respect to time t .Presuming the symmetry of the beam, the inertia of the joints may be expressed as follows: The examined rotating beam's motion equations are derived from the Hamilton principle By substituting Eq. ( 8) and Eq. ( 9) into Eq.( 11), the governing equations of axial, chordwise, and flapwise motions can be obtained as The corresponding boundary conditions are given by Eqs. ( 12)-( 14) Apparently, formula (12) comprises a time-independent term: The time-independent portion in Eq. ( 12) results in a time-independent axial deformation; hence, the overall axial displacement u(x, t) is constituted of the axial static displacement and axial dynamic displacement, which might be written as where u s (x) and u d (x) represent the static and dynamic axial displacements, respectively.For the sake of simplicity in the numerical calculations, we will assume that I y = I z .To simplify motion equations, the accompanying dimensionless variables are given: Note that is the axial stiffness.Substituting Eqs. ( 16)-( 17) into Eqs.( 12)-( 14) and ignoring the asterisk symbol of eloquence, we can obtain the corresponding dimensionless equations: The variables * i v and * i ẇ in Eqs. ( 19)-( 21) characterize the Coriolis impact of the system.The centrifugal stiffening impact of the rotating beam is represented by the variables * i k v, * i k ẇ, and * i k us,d .The variables * ˙ i u s,d , * ˙ i v, and * ˙ i w characterize the tangential impact.The variables i and k represent 1 or 2 and * the constant variables.
Partially similar dynamic equations are employed in references [9] and [12].The associated dimensionless boundary conditions are stated by Eqs. ( 19)-( 21) Zhao and Wu [12] reveal that the rotational speed has a substantial influence on the steady-state axial deformation.As an essential component of axial displacement, steadystate axial deformation has received considerable study.In the subsequent section, a more comprehensive examination of the effect of rotational motions on axial motion is presented.

Steady-state axial deformation and rotational speed limit
The corresponding axial middle point equation is a second-order nonlinear differential equation as follows: The following is an accurate representation of this equilibrium equation solved using the software package Mathematica, and the whole numerical calculations in this paper were obtained via this package. where The axial strain corresponding to u s can be represented as follows: It can be seen that Eq. ( 24) represents a highly complicated connection between the steadystate axial deformation and the rotation speed.With reasonable simplification, the following axial static equilibrium equation model can be accomplished as u s can be represented as follows: The axial strain corresponding to u s can be represented as follows: Equation (28) provides a cubic relationship.In a related work, Zhao and Wu [12] highlighted that, contrary to a trigonometric nonlinear model, a nonlinear cubic model can offer a straightforward representation of the link between steady-state axial deformation and rotating speed.In the meantime, a comparison of the numerical precision presented in Table 1 reveals that it is compatible with the outcomes produced from the two models.In other words, Eq. ( 27) is a more appropriate formulation for steady-state axial deformation due to its simplicity and precision.Therefore, the solution of the model, Eq. ( 27), is employed as the expression for the steady-state axial deformation in the ensuing analysis.Table 1 compares the steady-state axial deformation of the rotating beam according to the current models and Zhang et al. [11].Zhang et al. [11] derived the tip axial displacement by employing the ANCF finite element method.ANCF is designed exclusively for large deformation analysis in engineering structures.Calculated discrepancies in tip axial deformation reveal a considerable difference in high angular velocity but a seemingly negligible difference in lower speed increase.As Table 1 shows, these model results are the same when the speed of rotation is low, but they are different when the speed of rotation is high.Equation ( 24) is closer to those mentioned by Zhang et al. [11].
Here, we also analyze the maximum bound of the dimensionless speeds of rotation, which is essential to understanding the relationship between steady-state axial strain and rotational speed indicated in Eq. ( 29).As for a linearly elastic material, the constitutive stress Ref. [11] Eq. ( 28 increases with axial displacement.As the stress approaches a particular level, the structure will lose its capability to withstand displacement and may even be destroyed or damaged.Equation (29) demonstrates that the highest strain occurs at the fixed end (x = 0) for any given rotational speed.Considering (x = 0) in Eq. (29), we can get the following formula for the maximum rotational speed, which relies on generalized Hooke's law:

Solution procedure
The dynamic axial, chordwise, and flapwise deformations are approximated by linear combinations of the comparison functions to reveal approximate solutions in finite-dimensional function spaces through the Galerkin technique.Typically known as trial functions, these approximated functions can be represented as where N represents the total number of modal coordinates; p i (τ ), q i (τ ), and r i (τ ) represent the ith generalized coordinate to be evaluated.The comparison functions are φ i (x), ψ i (x), and ϕ i (x).In this inquiry, the comparison functions for the axial deformation are chosen as the mode functions for the longitudinal vibration of a cantilever beam, whereas the comparison functions for the chordwise and flapwise motions are chosen as the mode functions for the transverse vibration of a cantilever beam.These mode characteristic functions are provided by Wright et al. [22].Yo and Shin [4] demonstrated that if the second field moments of the section I y are equal to those of the section I z , the frequency relationship in the chordwise and flapwise directions can be obtained as follows: Equation ( 34) applies only in the case of a z direction rotation.In the present study, there are rotations on two different axes.Hence, it cannot be said that the frequency in the flapwise direction is greater than the frequency in the chordwise direction in all cases.The frequency values also found in the present paper were calculated numerically, and five modes were taken on each axis to increase the accuracy of the numerical study.If these values were obtained from a general single matrix, the right frequency results would not be extracted properly since the frequencies are quite close to each other, as mentioned.Therefore, the overall matrix structure was divided into two matrices to see approximate results and their effect on each other and to do preliminary research.The results indicated that this process was not entirely inaccurate; in fact, the vibrations in these two directions are minimally discrete in real life.
The discretized Eqs. ( 19)-( 21) are given by the matrix-vector equation shown below. where The discretized equations that correspond to the linearized equations of Eqs. ( 19)-( 21) can be expressed in matrix-vector form as follows: The only differences between Eqs. ( 35)-( 36) and ( 39)-( 40) are on the right-hand side.In Appendix, each parameter of Eq. ( 38) may be attached.To calculate the bending frequency in the chordwise direction, real modal analysis has been used [Eq.( 39)].Considering that the coordinate vector T 1 is a harmonic function of τ , it may be represented as where ω c is the chordwise bending circular frequency, T c is the mode shape associated with ω c , and i is an imaginary number.This definition yields The general characteristic equation for the chordwise vibration could be generated by converting the equation of chordwise motion [Eq.(39)]: The equation of the flapwise motion is associated with the equation of the axial motion [Eq.( 40)].The real modal analysis approach cannot be utilized to evolve the bending frequency because the gyroscopic matrix is really not symmetric.To apply the complex modal analysis approach, Eq. ( 40) is converted into the state-space form shown below: where where I stands for the 3 × 3 unit matrix.By resolving the complex eigenvalue and eigenvector of the state vector, the flapwise bending frequency can be computed.It examined the influence of the Coriolis term on the frequency of flapwise and chordwise bending.

Comparative studies and discussion
As proof of the present model's accuracy, natural frequencies derived from the present research are examined against those found in the literature.Since the chordwise/flapwise bending natural frequencies of a rotating beam attached to a rigid hub rotating by two neighboring rotary joints are absent in the literature, only the −z (vertical) values of rotating beams are compared.Using Mathematica, symbolic and numeric calculations were successfully executed.Every one of the numerical findings described in the subsequent parts is derived from 10 modes, with N = 5 for each direction.
Table 2 shows comparison of dimensionless natural frequencies for chordwise motions with varying rotational speeds to the reference [4], demonstrating that the natural frequencies for chordwise motions in the current research are consistent with those in the literature.The dimensionless natural frequencies for the flapwise motion with various parameters derived from the current investigation are compared with the proper referencing [4] and [13] in Tables 3-4.The results show that the current research's natural frequencies for various hub radii and rotational velocities are in excellent agreement with those reported in the literature.
Zhao and Wu [12] demonstrated a spinning beam with a high rotational angular speed.When the angular speed reaches its maximum, the beam begins to lose susceptibility to displacement and may break.Yun-dong et al. [15] showed that when the parameters of a rotating cantilever beam are set to = 4900 and R = 0, the limit of the dimensionless angular velocity occurs at = 110.Yun-Dong et al. [15] arrived at the conclusions using numerical values rather than analytic methods.For the comparison, if the same parameter values are used, then = 99 produces dimensionless critical values for the angular velocity of rotation.10% is the difference between the computed critical velocities of this study's hypothesis and those of Yun-Dong et al. [15].There exists a resemblance between the corresponding steadystate expression model and the model of Zhao and Wu [12].In their research, neither hub radius nor rotational velocity are accounted for in both directions.In our study, the rotational critical speed is represented by max = 2 1 + 2 2 .The difference of 10% is apparent in the results.This may be the result of the assumptions made at the beginning of the calculations.This difference needs further analysis and may be investigated in future research studies.
The external excitation is typically primarily applied in the horizontal plane, and thus the flapwise motion is unrelated to the other two paths for the linear components of the mathematical model.Therefore, many researchers disregarded the impacts of the Coriolis term and steady-state axial deformations here on flapwise natural frequency.With varying the hub radius and rotational speeds, the influence of the Coriolis effect was evaluated just on the fundamental natural frequencies of the bending vibrations (in the flapwise and chordwise directions) of a rotating beam.
To explore the impact of rotational velocity on the chordwise direction, the computed numerical frequency of the beam at various rotational speeds is studied in Table 5.Table 5 displays the varied rotational speeds in a chordwise direction even though the Coriolis force has no apparent influence in this direction.
The dimensionless hub radius varied from 0 to 0.5, whereas the dimensionless rotational speeds varied between 20 and 100.Using Eq. (30), the rotational speeds are chosen within the critical angular velocity.At a larger hub radius, the critical angular velocity decreases, i.e., (for R = 0; max = 98; for R = 0.1; max = 90; for R = 0.5; max = 70).Tables 5,  6, 7, 8 include the calculated numerical outcomes.The relative inaccuracy is obtained by determining the difference between the two frequencies with and without Coriolis effects and dividing the result by the value with Coriolis.Table 6 shows that when the parameters 1 = 80 and 2 = 20 are used, the maximum relative inaccuracy of the fundamental natural frequency in the flapwise direction varies by 34% between the two cases.The discrepancy grows as the angular velocity increases.Furthermore, Coriolis terms have an extra impact on the natural frequency of the first order.Tables 7-8 also include numerical examples discussing the influence of hub radius on the fundamental natural frequencies.For this reason, the rotational speeds that provide the greatest difference between the Coriolis effects with and without these vibration frequencies are selected.So, the natural frequencies that were calculated show that the difference gets smaller as the hub radius gets bigger.

Conclusion
To investigate the impacts of the Coriolis term on coupling vibration, the coupling equations of motion of a rotating three-dimensional cantilever beam are derived.In contrast to a previously disclosed approach, the current method involves both horizontal and vertical rotating motion.The numerical findings obtained within the rotational speed restriction indicate that the Coriolis term has a significant influence on the frequency of bending in the flapwise direction but not in the chordwise direction.The reliability of the calculated outputs was confirmed by comparing the results with those of prior research, which accounted for differences in the hub radius and the speed of rotation.

Fig. 1
Fig. 1 Schematic diagram of a rotating 3D flexible beam

Table 1
Comparison of the dimensionless tip axial deformation of the flexible beam ( 1

Table 2
Comparison of the dimensionless natural frequencies in chordwise motion of the flexible beam ( 1 = 0, = 4900)

Table 3
Comparison of first and second natural frequencies in the flapwise bending vibration ( 1 = 0, R = 0,

Table 5
Effect of the Coriolis term on the fundamental chordwise bending frequency ( = 4900, R = 0,

Table 6
Effect of the Coriolis term on the fundamental flapwise bending frequency ( = 4900, R = 0,

Table 7
Effect of the Coriolis term on the first flapwise bending frequency ( = 4900, R = 0.1, max =

Table 8
Effect of the Coriolis term on the first flapwise bending frequency ( = 4900, R = 0.5, max =