Abstract
Problems related to the transverse vibration of a rotating tapered cantilever beam with hollow circular cross-section are addressed, in which the inner radius of cross-section is constant and the outer radius changes linearly along the beam axis. First, considering the geometry parameters of the varying cross-sectional beam, rotary inertia, and the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. Next, dimensionless parameters and variables are introduced for the differential equation and boundary conditions, and the differential quadrature method (DQM) is employed to solve this differential equation with variable coefficients. Combining with discretization equations for the differential equation and boundary conditions, an eigen-equation of the system including some dimensionless parameters is formulated in implicit algebraic form, so it is easy to simulate the dynamical behaviors of rotating tapered beams. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.
1. Introduction
The dynamical problem of rotating uniform and nonuniform solid beam is widely used in many practical engineering, such as helicopter rotor blades and wind turbine blades. Also, the dynamics of rotating tapered hollow beams is of practical significant, for example, rotating tank gun barrel (hollow circular cross-section). As pointed out in [1], in dynamical analysis, a rotating beam differs from a nonrotating beam because it also possesses centrifugal stiffness and Coriolis effects that influence its dynamical characteristics. Besides the above effects, there are some complicated factors, including the secondary coupling deformation term, coupling effect, and the variable coefficient differential equation. Therefore, the methodologies and solutions for rotating nonuniform beam turn out to be cumbersome.
The dynamic analysis of rotating uniform beams has been the subject of many articles and received much attention. Yoo and Shin [2] investigated the effect of centrifugal force for rotating uniform cantilever beams and used a modal formulation to obtain the natural frequencies and mode shape. Tsai et al. [3] proposed the corotational finite element method combined with floating frame method to derive differential equation of motion for the rotating inclined Euler uniform beams at constant angular speed and investigated the steady-state deformation and the natural frequencies of infinitesimal free vibration. Vinod Kumar and Ganguli [4] used the static part of the homogeneous differential equation of violin strings to obtain new shape functions for the finite element analysis of rotating Timoshenko beams. Aksencer and Aydogdu [5] studied flapwise vibration of rotating composite beams, which are used in different beam theories, including Euler-Bernoulli, Timoshenko, and Reddy beam theories, and obtained some results for different orthotropy ratios, rotation speed, hub ratio, length to thickness ratio of the rotating composite beam, and different boundary conditions. Li et al. [6] developed a new dynamic model of a planar rotating hub-beam system, where the beam is of an Euler-Bernoulli type and the deformation of the beam is described by the slope angle and stretch strain of the centroid line of the beam. They obtained four corresponding spatially discretized models, that is, ESA, FOSA, SOSA, and SSOSA model, and calculated natural frequencies and mode shapes of the system with the chordwise bending and stretching coupling effect. J. W. Lee and J. Y. Lee [7] investigated the effects of cracks on the natural frequencies of a rotating Bernoulli-Euler beam using a new numerical method in which these effects can be computed simply using the transfer matrix method.
In recent years, more studies related to transverse vibration of rotating nonuniform beams can be found in the following papers. Gunda and Ganguli [8] developed new interpolating functions which satisfy the static part of the homogenous governing differential equation for rotating uniform and tapered beams and imposed as a constraint equation in the derivation of the shape functions. Cheng et al. [9] investigated vibration characteristics of cracked rotating tapered beam by p-version finite element method and analyzed the effects of crack location, crack size, rotating speed, and hub radius on vibration characteristics of the beam. Bulut [10] considered out-of-rotation plane bending vibrations of rotating composite beam with periodically varying speed and further examined the effect of taper ratio on dynamic stability of this parametrically system. Banerjee and Jackson [11] addressed the free vibration problem of a rotating tapered Rayleigh beam by developing its dynamic stiffness matrix. In their analysis, the effects of centrifugal stiffening, an outboard force, an arbitrary hub radius, and importantly, the rotary inertia (Rayleigh beam) are included. Sarkar and Ganguli [12] proposed an inverse problem approach for dynamics of the rotating nonuniform Euler-Bernoulli beam and showed that there exists a certain class of rotating Euler-Bernoulli beam, having cantilever and pinned-free boundary conditions, which has a closed-form polynomial solution to its governing differential equation. At the same year, they also studied the free vibration of a nonhomogeneous rotating Timoshenko beam, having uniform cross-section, using an inverse problem approach, for both cantilever and pinned-free boundary conditions [13]. Tang et al. [14] studied free vibration of rotating tapered cantilever beams with rotary inertia using the integral equation method and analyzed the effects of the rotary inertia, angular speed, taper ratio, and hub radius. Li and Zhang [15] developed a new rigid-flexible coupled dynamic model to study dynamics of rotating axially functionally graded (FG) tapered beams by using the B-spline method (BSM) and observed some new interesting phenomena of frequency veering and mode shift in a rotating axially FG tapered beam when the B-S coupling effect is included. Huo and Wang [16] derived the nonlinear dynamic equations of a rotating, double-tapered, cantilever Timoshenko beam and analyzed the effect of angular speed, hub radius, slenderness ratio, and the height and width taper ratios on the natural frequencies of the rotating Timoshenko beam when the rotation beam is in a steady state, in which the extensional deformation of the beam is considered. Panchore et al. [17, 18] investigated free vibration problem of a rotating Euler-Bernoulli beam and a rotating Timoshenko beam using meshless local Petrov-Galerkin method and introduced a locking-free shape function formulation with an improved radial basis function interpolation. Ghafarian and Ariaei [19] presented a new procedure for determining natural frequencies and mode shapes of a system of elastically connected multiple rotating tapered beams through a differential transform method, which obey the Timoshenko beam theory, and discussed the effects of the rotational speed, hub radius, taper ratios, rotary inertia, shear deformation, slenderness ratio, and elastic layer stiffness coefficients on the natural frequencies. Ghafari and Rezaeepazhand [20] presented free vibration analysis of rotating composite beams with arbitrary cross-section using dimensional reduction method. Adair and Jaeger [1] used the computational approach of AMDM to analyze the free vibration of nonuniform Euler-Bernoulli beams under various boundary conditions, rotation speeds, and hub radii and simultaneously obtained the natural frequencies and corresponding closed-form series solution of the mode shape. Panchore and Ganguli [21] studied the free vibration problem of a rotating Rayleigh beam using the quadratic B-spline finite element method. Other researchers also investigated the relevant second-order coupling term that represents longitudinal shrinking of the rotating beam caused by the transverse displacement. Li et al. [22] introduced a dynamic model of a rotating hub-functionally graded material beam system with the dynamic stiffening effect. In their work, the dynamic stiffening effect of the rotating hub-FGM beam system is captured by a second-order coupling term. Zhao and Wu [23] established the coupling equations of motion of a rotating three-dimensional cantilever beam to study the effects of Coriolis term and steady-state axial deformation on coupling vibration, which considered the longitudinal shrinkage caused by flapwise and chordwise bending displacement. At present, a large amount of articles relating to free vibration of rotating functionally graded plates or disk can be found (see, for instance, [24–26]).
In the above referenced articles, the model of rotating uniform beam and nonuniform beam have been considered, especially for rotating tapered beam, which has rectangular cross-section with linearly varying width and constant height, with linearly varying height and constant width, and with linearly varying width and height. However, to the best of the authors’ knowledge, no research work related to the dynamics of a rotating beam with varying hollow circular cross-section (or rotating tapered hollow beam) has been yet presented. The dynamical of the system is of practical significant because rotating tapered hollow beams are widely used as structural components in the engineering field.
In this paper, the investigation proceeds as follows. First the geometry parameters of a rotating tapered cantilever beam with hollow circular cross-section are described, and the governing differential equation of motion for transverse free vibration of a rotating tapered Rayleigh beam is derived using Hamilton variational principle. Next, for harmonic oscillation, the differential equation with variable coefficients is solved using the differential quadrature method, and an eigen-equation of the system for dimensionless parameters is formulated in explicit algebraic form. Finally, for rotating solid tapered beams, comparisons with previously reported results demonstrate that the results obtained by the present method are in close agreement; for rotating tapered hollow beams, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are more further depicted.
2. Parameters of Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section
Figure 1 shows the schematic diagram of a rotating tapered cantilever beam with hollow circular cross-section, which has length L, elastic modulus , and density ρ and is fixed at point of a rigid hub with radius . The hub is rotating in the horizontal plane around point with a rotating angular speed . A fixed (inertial) planar coordinate system OXY through the fixed point and a floating coordinate system oxy that is tangent to the attachment point of the beam to the hub are prescribed, respectively. The latter relative to the former rotates with a rotation angle of large range motion.
The rotating beam with varying hollow circular cross-section is considered, whose outer diameter varies linearly and the inner diameter keeps unchanged along its longitudinal -axis, as shown in Figure 2. The beam has the root radius (at x = 0) and the end radius (at x = L), the wall thickness versus the coordinate , and the radius at the middle line of the wall thickness for any cross-section. A local coordinate system with a normal direction and tangential direction at the central line of hollow circular cross-section is adopted.
(a)
(b)
The average radius and the wall thickness can be expressed, respectively, as follows:where is called the taper ratio of cross-section. It is also stipulated that the section size of the beam decreases and increases linearly from the root to the end, that is, , in which is denoted by (called the ratio of inner radius to the root radius). There are two particular cases: one is a uniform beam when , that is, , and the other is a particular varying cross-section beam when , that is, .
The area of any cross-section and its moment of inertia with respect to axis can be expressed, respectively, as where and are the area and the moment of inertia with respect to axis of the root cross-section of the beam, respectively, , and are given by
Any hollow circular cross-section of the beam is shown as Figure 3. The vertical coordinate of any point M can be expressed, respectively, as
(a)
(b)
3. Differential Equation of Motion
3.1. The Description of the Deformation Field
Figure 4 shows that is a radius vector of the original point o of floating coordinate system oxy with respect to the point O of the inertial coordinate system OXY, is a radius vector of any point , which is on the axis of the beam before deformation, the point P is the positions of the point after deformation, and is a displacement vector of the point .
The vector of point relative to original point O of inertial coordinate system OXY can be expressed aswherein which and are the axial displacement and the transverse bending deflection, respectively; is axial displacement of any point caused by transverse bending, in which the prime denotes spatial derivatives with respect to ; is a second-order coupling term that represents longitudinal shrinking of the rotating tapered beam with hollow circular cross-section beam caused by the transverse displacement . It includes the coupling effect between the axial displacement and transverse displacement of rotating tapered hollow beam.
Taking the derivative with respect to time for (4), the velocity vector of the point at the inertial coordinate system can be obtainedwhere is a antisymmetric matrix relating to angular speed , in which the over dot denotes derivative with respect to time .
3.2. Differential Equation of Motion
3.2.1. Kinetic Energy of System
Kinetic energy of system consists of two parts: one is the kinetic energy of the hub and the other is the kinetic energy of the beam with hollow circular cross-section; namely,
The kinetic energy of the hub is given bywhere is rotary inertia of the hub with respect to central axis.
Neglecting the axis displacement of the beam with hollow circular cross-section, the kinetic energy of the per unit length of the beam can be expressed aswhere is the effective mass of the per unit length of the beam; it can be expressed as
The kinetic energy of the beam with hollow circular cross-section can be rewritten asThus, substituting (9) and (12) into (8), the total kinetic energy of the system can be expressed as
3.2.2. Strain Energy of System
Neglecting the deformation energy caused by shear deformation, the strain energy of rotating beam with hollow circular cross-section is written aswhere is the elastic modulus of material; and represent the normal stress and normal strain in direction, respectively.
In (6), ignoring the axial displacement and nonlinear term, the normal strain can be got by the relationship between strain and displacement: namely,
According to Figure 3, a geometrical relationship is given by
Thus, the strain energy of the beam with hollow circular cross-section can be rewritten as
3.2.3. Derivation of Differential Equation of Motion
In this paper, Hamilton variational principle for elastic system is used to derive the differential equation of motion. The basis form of Hamilton variational principle can be showed as
Substituting (13) and (17) into (18) and implementing a lot of variational operation and integration by parts, a variational expression is given by where .
Because second-order coupling deformation term is a second-order small quantity, we can neglect some nonlinear terms and time-varying coupling terms in (19) to simplify the equation appropriately. Thus, the differential equation of motion of the rotating beam with hollow circular cross-section can be derived
Taking uniform rotation into consideration, that is, , , a differential equation of motion of the beam can be expressed asin which , /.
The boundary conditions of the cantilever beam are as follows:
3.3. Dimensionless Method of the Equation
For simplicity, the following dimensionless quantities are introduced: , , , (called dimensionless angular speed of the hub), (called ratio of hub radius to beam length), and (called slenderness ratio).
Dimensionless expression of (21) can be expressed aswhere , +
Let the solution of (23) be ; a differential equation of mode shape can be written aswhere is dimensionless natural frequency.
The dimensionless forms of the boundary conditions (22) are rewritten as
4. Differential Quadrature Method
In order to solve the differential equation with variable coefficients (24) and deal with the boundary conditions (25), the differential quadrature method (DQM) and the method are used, respectively. Selecting nonuniform nodes, the node coordinates are as follows [27, 28]:where is the numbers of nodes and is small parameter.
According to the DQM procedures, (24) can be discretized as follows:where , , , .
The boundary conditions (25) can also be discretized as follows:
Equations (27), (28a), (28b), (28c), and (28d) can be rewritten as a simpler matrix formwhere , , and are all square matrix; is column matrix.
Equation (29) denotes a generalized eigenvalue problem. Based on the linear algebra theory, the sufficient and necessary conditions of homogeneous linear algebraic equations which create the nonzero solution are that the determinant of coefficients equals zero; thus, one can arrive to the following generalized eigen-equation:where the square matrices , , and involve some parameters, such as ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section, each dimensionless natural frequencies, and dimensionless angular speed of the hub.
5. Numerical Results and Analyses
5.1. Rotating Tapered Cantilever Beam with Solid Circular Cross-Section
In this paragraph, setting in (24), we can obtain the differential equation of motion for rotating tapered beam with solid circular cross-section, where taper ratio is defined. It becomes evident that if the taper ratio and , they are varying solid circular cross-section beam with zero radius at free end and entirely uniform solid circular cross-section, respectively. Prior to the presentation of our numerical results, let us first consider three particular cases to confirm the effectiveness of the present approach: a simple uniform nonrotating cantilever beam, a rotating tapered Euler-Bernoulli cantilever beam, and a rotating tapered Rayleigh cantilever beam, which is given by setting parameters , , , and , parameters , , and parameters , , respectively. In three cases, we calculate the first three-order dimensionless natural frequencies by selecting for different taper ratio of cross-section and dimensionless angular speed, and some numerical results are tabulated in Tables 1 and 2 . From the two tables, we can see that the numerical results in the present coincide well with the existing ones [14]. These verify that the method presented in this paper is efficient and accurate. The first three dimensionless mode shapes are shown as Figure 5 for , , , , and .
Furthermore, in the case of , for example, , that is, the radius of beam cross-section at free end is more than one at the cantilevered end, the first three-order dimensionless natural frequencies of a rotating tapered Rayleigh cantilever solid beam for three dimensionless angular speeds are tabulated in Table 3. It can be seen in Table 3 that as a whole, the first three-order dimensionless natural frequencies increase with the increase of dimensionless angular speeds.
Figure 6 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different ratios of hub radius to beam length at , . It can be found from Figure 6 that, with the increase of the ratios of hub radius to beam length, the first three-order dimensionless natural frequencies increase. Figure 7 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different slenderness ratios at , . It can be seen from Figure 7 that, for two different , the increase of slenderness ratio has scarce influence on the first-order dimensionless natural frequencies; however, it has significant influence on the second- and the third-order dimensionless natural frequency. This shows that the increase of rotary inertia makes the system natural frequency decrease, and this conclusion is consistent with those given by Timoshenko beam. Figure 8 shows the variation of the first three-order dimensionless natural frequencies of rotating tapered solid beams with dimensionless angular speed of the hub for three different taper ratios of cross-section at , . It can be seen from Figure 8 that, with the increase of taper ratio, the first three-order dimensionless natural frequencies of the system decrease. It is noted that the increase of taper ratio has scarce influence on the first-order dimensionless natural frequency, it has slight influence on the second-order dimensionless natural frequency, and it has obviously an effect on the third-order dimensionless natural frequency. It should be pointed out that, as shown in Figures 6–8, in the case of the given ratios of hub radius to beam length, the slenderness ratio, and taper ratio of cross-section, the first three dimensionless natural frequencies of rotating tapered solid beam monotonically increase with dimensionless angular speed of the hub.
5.2. Rotating Tapered Cantilever Beam with Hollow Circular Cross-Section
For a rotating tapered cantilever beam with hollow circular cross-section, its inner diameter , that is, , and the taper ratio is defined. When , the outer radius of beam at free end is the same as its inner radius. When , the beam is entirely uniform hollow circular cross-section. This section will mainly discuss the effect of ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three dimensionless natural frequencies of rotating tapered hollow beams.
As a particular case of hollow circular cross-section, Table 4 gives the variation of the first three dimensionless natural frequencies of rotating uniform thin-wall cross-section beams with dimensionless angular speed of the hub for three different slenderness ratios at , , and . It should be pointed out that the first three dimensionless natural frequencies of nonrotating beam in Table 4 equal the first three dimensionless natural frequencies 3.5156, 22.0336, and 61.7010 [29] of cantilever beam multiplied by . We can see that the numerical results in the present coincide well with the existing ones [29]. The first three dimensionless natural frequencies of the rotating uniform thin-wall cross-section beam decreased as the slenderness ratio was raised.
5.2.1. Effect of the Hub Dimensionless Angular Speed and Taper Ratio of Cross-Section
Table 5 gives the variation of the first three dimensionless natural frequencies of rotating tapered hollow beams with dimensionless angular speed of the hub for two different slenderness ratios and three ratios of inner radius to the root radius at . With this table, it is obvious that the first three dimensionless natural frequencies increase with the ratio of inner radius to the root radius, except the second-order dimensionless natural frequencies at , .
Figure 9 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for and at , , . With this figure, it is also understood that the first three-order dimensionless natural frequencies of rotating tapered hollow beam monotonically increase with dimensionless angular speed of the hub. Meanwhile, it is further noted in Figure 9(a) that, with the increase of the taper ratio of cross-section, the first-order dimensionless natural frequency of the system is reduced slightly. However, in Figures 9(b) and 9(c), with the increase of the taper ratio of cross-section, the values of the second- and third-order dimensionless natural frequencies of the system are increased.
(a)
(b)
(c)
In addition, it is also observed in Figure 9 that the increase of slenderness ratio makes three dimensionless natural frequencies of the system decrease, contrasting Figures 9(a), 9(b), and 9(c), obviously, and the effect of the slenderness ratio on the second- and third-order dimensionless natural frequencies is relatively greater than the first-order dimensionless natural frequency.
5.2.2. Effect of Ratios of Hub Radius to Beam Length and Slenderness Ratio
As it was expected in Table 6, the change of ratios of hub radius to beam length has no effect on the natural frequencies of nonrotating tapered beam . It is also observed in Table 6 that, for rotating tapered beam, the dimensionless natural frequencies of the system increase with ratios of hub radius to beam length.
Figure 10 plots the curves between the first three-order dimensionless natural frequencies of rotating tapered hollow beam and dimensionless angular speed of the hub for at , . With this figure, the first three-order dimensionless natural frequencies of the system increase with dimensionless angular speed of the hub and ratios of hub radius to beam length.
Figure 11 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different taper ratios of cross-section , at , , . Figure 12 plots the curves of the first three-order dimensionless natural frequencies with ratios of hub radius to beam length for two different slenderness ratios at , , . It can be seen in Figures 11 and 12 that, for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of the system are almost linearly increased. Meanwhile, it is noted that the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.
6. Conclusion
In this paper, a new type of transverse vibration of a rotating tapered cantilever beam with linearly varying solid and hollow circular cross-section, that is, rotating tapered beam, was presented. The rotating beam, which is considered a tapered cantilever beam, is modeled by the Rayleigh beam theory. Considering the secondary coupling deformation term, the differential equation of motion for the transverse vibration of rotating tapered beam with solid and hollow circular cross-section is derived by Hamilton variational principle, which includes some complex variable coefficient terms. A differential quadrature method to solve the above-mentioned differential equation with variable coefficients was employed to simulate the dynamical behaviors of tapered rotating beams. Also, for two types of rotating tapered beams with solid and hollow circular cross-section, the effects of the hub dimensionless angular speed, ratios of hub radius to beam length, the slenderness ratio, the ratio of inner radius to the root radius, and taper ratio of cross-section on the first three-order dimensionless natural frequencies are depicted. The main results of this study are summarized as follows.
When the rotating angular speed is constant, in the case of the given ratios of the hub radius to beam length, the slenderness ratio, and the taper ratio of cross-section, the first three-order dimensionless natural frequencies of rotating tapered solid and hollow beams monotonically ascend as the hub dimensionless angular speed increases. For a rotating tapered hollow beam at a constant angular speed, the first-order dimensionless natural frequency of the system is reduced slightly with the increase of the taper ratio of cross-section, the values of the second- and the third-order dimensionless natural frequencies of the system are increased; for different taper ratio of the cross-section and slenderness ratios, with the increase of ratios of hub radius to beam length, the first three-order dimensionless natural frequencies of system almost linearly increase, and the influence of the slenderness ratio on the third-order natural frequency of the system is more obvious than that of the first order and the second order.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The authors are grateful for the support from the National Natural Science Foundation of China (Grant no. 11472211).