Dynamic Modeling and Analysis of Hub-Tapered Beam System

Abstract

The dynamic modeling of a hub-tapered flexible beam system which is rotating in a plane is investigated. In the modeling, the high-order terms related with the nonlinear coupling term are retained, which are ignored in the first-order approximation coupling modeling. So, this model can also be called a high-order approximation coupling model, which cannot only be used in the case of small deformation but also in the case of large deformation. Solving for this mathematical model, we can obtain the dynamics of the system. Examples are given to indicate the differences between the high-order approximation coupling model and the first-order approximation coupling model. Also, the natural frequencies are studied through the transversal bending vibration analysis. The width ratio and the height ratio of the tapered beam are shown to have great influences on the transversal bending natural frequencies.

1. Introduction

Dynamics of the flexible structures can be affected by many factors, such as the shape and the motion. It is found that the dynamics of the flexible structure with large overall motion has an essential difference from the dynamics of that on an immobile base. In the past three decades, the dynamic modeling of a rotating flexible beam has received extensive research efforts [1 –18]. Kane et al. [1] investigated a rotating flexible cantilever beam by using the traditional coupling model, which showed that this model fails to describe the dynamic behavior of the beam when it is at high rotation speed; and “Dynamic Stiffening” [1] was first pointed out. Then, many methodologies were developed to capture the dynamic stiffening term. Yoo and Shin [6] derived the motion equations of a rotating cantilever beam based on a new dynamic modeling method. The derived equations (governing stretching and bending motions) were all linear, so they could be directly used for the vibration analysis including the coupling effect, which could not be considered in the conventional modeling method. References [7 –10] presented the first-order approximation coupling (FOAC) model, which is based on the theory of continuum medium mechanics and the theory of analysis dynamics. The FOAC model considered the second-order coupling term of longitudinal displacement caused by transversal deformation, while the traditional zeroth-order approximation coupling (ZOAC) model assumed small deformation in structural dynamics, where the longitudinal and transversal deformations are uncoupled. However, the FOAC model can only be used in the case of small deformation. Liu and Hong [11] presented a high-order approximation coupling (HOAC) model and showed that the HOAC model can be approximated to the FOAC model in the case of small deformation. However, the distinctions between HOAC model and FOAC model were not indicated.

Great progress has been made in the research on rotating flexible beam, but all the work mentioned above focused on the uniform beam. In some realistic engineering examples, such as turbine blades, the huge flagelliform antennas on the spacecraft, the cross sections of these structures are always variational along the length. We will get a more accurate dynamic analysis if we use the tapered beam model instead of the uniform beam model. Reference [14] investigated the free bending vibration of rotating tapered beams by using the dynamic stiffness method. The range of problems considered included beams for which the height and/or width of the cross section vary linearly along the length. A parametric study was carried out to demonstrate the effects of rotational speed, taper ratio, and hub radius on the natural frequencies and mode shapes of the tapered beams. Reference [15] investigated the intrinsic characteristics and the flexible motion of the tapered Euler-Bernoulli beam with tip mass and a rotating hub. Dynamic equations were derived from the extended Hamilton principle and the FOAC model. By comparing the uniform and the tapered system in the same condition, the obtained results indicated that a little difference in section can cause a significant difference in natural frequencies and dynamics response.

Since the FOAC model neglected some high-order terms in the dynamic equations, it could not be used in the large deformation problems [16]. So an exact dynamic model is needed to solve the large deformation problems. Shabana [17, 18] presented an absolute nodal coordinate method for the large rotation and deformation problems, and some examples were given to demonstrate the use of the absolute nodal coordinate formulation in the large rotation and deformation analysis of flexible bodies. The absolute nodal coordinate method can solve the large deformation problem of the flexible body very well. However, it cannot distinguishthe rigid body motion from the elastic deformation of the flexible body. Small deformation problems also should be treated as large deformation problems.

In this paper, we will study the dynamics of a hub-tapered beam system by using a high-order approximation coupling (HOAC) model, which cannot only be used in the small deformation problems but also large deformation problems. Both the transversal deformation and the longitudinal deformation of the flexible beam are considered. And in the total longitudinal deformation, the nonlinear coupling term, also known as the longitudinal shortening caused by transversal deformation, is considered. The assumed mode method (AMM) is used to describe the deformation of the flexible beam. The rigid-flexible coupling dynamic equations are established via employing the second kind of Lagrange's equation, retaining the high-order terms synchronously. Solving for this mathematical model, we can obtain the dynamics of the system.

2. Dynamic Equations

The system considered here is a flexible tapered beam driven by a hub rotor, as shown in Figure 1, which we call hub-beam system. The hub is assumed to be rigid with radius a and rotary inertia J_oh. Applied at the hub, there is a torque τ. The flexible beam is a tapered Euler-Bernoulli beam (Figure 2) with length L and mass density ρ. The section area of the beam is S(x), and the section moment of inertia about k' axis is I(x). Then, the bending stiffness of the beam is EI(x) and the compressive stiffness is ES(x), where S(x) and I(x) are formed as

\begin{matrix} S (x) = (\frac{x}{L} b_{1} + \frac{L - x}{L} b_{2}) (\frac{x}{L} h_{1} + \frac{L - x}{L} h_{2}), \\ I (x) = \frac{1}{12} {(\frac{x}{L} b_{1} + \frac{L - x}{L} b_{2})}^{3} (\frac{x}{L} h_{1} + \frac{L - x}{L} h_{2}) . \end{matrix}

(1)

Figure 1:

Schematic diagram of the system.

Figure 2:

Schematic diagram of the tapered beam.

The coordinate systems used in developing the model are shown in Figure 3. Wherein o – ij is the inertial reference frame, o′ – i^′j^′ is a rotating frame attached to the hub such that its i^′ axis is directed along the undeformed configuration of the beam. The i^′ axis makes an angle of θ with the i axis. The beam deformation is expressed with respect to the rotating frame. There is an assumption that the beam is constrained in the horizontal plane of ij.

Figure 3:

The deformation of the beam.

2.1. Kinetic Energy

To a generic point P(x) on the beam, the deformed position vector to point o can be expressed as

r = (a + x + w_{1} (x, t) + w_{c} (x, t)) i^{'} + w_{2} (x, t) j^{'},

(2)

where w₁(x, t) is the axial extension quantity, w₂(x, t) is the transversal deformation, and w_c(x, t) is the axial shorten caused by transversal deformation. w_c(x, t) is also called the coupling deformation. In the ZOAC model, we usually used the small deformation assumption in structure dynamics but ignored the coupling deformation w_c(x, t). When the large overall motion is at high speed, the coupling deformation w_c(x, t) will have a significant influence on the dynamic characteristic of the system. w_c(x, t) can be expressed as

w_{c} (x, t) = - \frac{1}{2} \int_{0}^{x} {(\frac{\partial w_{2} (ξ, t)}{\partial ξ})}^{2} d ξ .

(3)

The velocity vector of the point P(x) can be obtained by differentiating Equation (2) as follows:

\dot{r} = ({\dot{w}}_{1} (x, t) + {\dot{w}}_{c} (x, t) - w_{2} (x, t) \dot{θ}) i^{'} + [(a + x + w_{1} (x, t) + w_{c} (x, t)) \dot{θ} + {\dot{w}}_{2} (x, t)] j^{'};

(4)

then, the kinetic energy of the system can be expressed as

K = \frac{1}{2} J_{oh} {\dot{θ}}^{2} + \frac{1}{2} \int_{0}^{L} ρ S (x) {\dot{r}}^{T} \dot{r} d x,

(5)

where the first part is the kinetic energy of the hub and the second part is the kinetic energy of the beam.

2.2. Potential Energy

As the beam is constrained in the horizontal plane, the gravitational potential energy can be ignored here. So, the potential energy of the system can be expressed as

V = \frac{1}{2} {\int_{0}^{L} E S (x) (\frac{\partial w_{1} (x, t)}{\partial x})}^{2} d x + \frac{1}{2} {\int_{0}^{L} E I (x) (\frac{\partial^{2} w_{2} (x, t)}{\partial x^{2}})}^{2} d x,

(6)

where the first part is the stretching energy of the beam and the second part is the bending energy of the beam.

2.3. Dynamic Equations

The assumed mode method (AMM) is used to describe the deformation of the flexible beam. The axial deformation w₁(x, t) and the transversal deformation w₂(x, t) can be expressed as

w_{1} (x, t) = Φ_{x} (x) A (t), w_{2} (x, t) = Φ_{y} (x) B (t),

(7)

where Φ_x(x)∊R^{1 × N} and Φ_y(x)∊R^{1 × N} are the mode functions of axial and transverse vibrations of the beam, respectively. Any compact set of functions, which satisfy the boundary conditions of the cantilever beam, can be used, while A(t)∊R^{N × 1} and B(t)∊R^{N × 1} are the mode coordinates of axial and transverse vibrations of the beam, respectively. So, there is

w_{c} (x, t) = - \frac{1}{2} B^{T} H (x) B,

(8)

where H(x)∊R^{N × N} is the coupling shape function, as follows:

H (x) = \int_{0}^{x} {Φ_{y}^{'}}^{T} (ξ) Φ_{y}^{'} (ξ) d ξ .

(9)

Take the generalized coordinates $q = {(θ, A^{T}, B^{T})}^{T}$ and substitute the kinetic and potential energy expressions (Equations (5) and (6)) into the second kind of Lagrange's equation

\frac{d}{d t} (\frac{\partial K}{\partial \dot{q}}) - \frac{\partial K}{\partial q} = - \frac{\partial V}{\partial q} + F_{q} .

(10)

Then, we will get the dynamic equation of the system written in compact form as

M \overset{‥}{q} = Q,

(11)

where M is called generalized mass matrix and Q is called generalized force matrix. M and Q can be formed as

M = [\begin{bmatrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & M_{23} \\ M_{31} & M_{32} & M_{33} \end{bmatrix}], Q = [\begin{bmatrix} Q_{θ} \\ Q_{A} \\ Q_{B} \end{bmatrix}] + [\begin{bmatrix} τ \\ 0 \\ 0 \end{bmatrix}],

(12)

where

\begin{matrix} M_{11} = J_{oh} + J_{ob} + 2 S_{x} A + A^{T} M_{x} A + B^{T} M_{y} B - \underset{}{B^{T} C B} + \underset{}{\underset{}{\frac{1}{4} \int_{0}^{L} ρ S (x) (B^{T} H B B^{T} H B) d x}} - \underset{}{\underset{}{\int_{0}^{L} ρ S (x) (A^{T} {Φ_{x}}^{T} B^{T} H B) d x}} \\ M_{22} = M_{x} \\ M_{33} = M_{y} + \underset{}{\underset{}{\int_{o}^{L} ρ S (x) (H B B^{T} H) d x}} \\ M_{21} = {M_{12}}^{T} = - M_{x y} B \\ M_{31} = {M_{13}}^{T} = {S_{y}}^{T} + {M_{x y}}^{T} A \underset{}{\underset{}{+ \int_{0}^{L} ρ S (x) (H B Φ_{y} B - \frac{1}{2} {Φ_{y}}^{T} B^{T} H B) d x}} \\ M_{32} = {M_{23}}^{T} = - \underset{}{\underset{}{\int_{0}^{L} ρ S (x) (H B Φ_{x}) d x}} \\ Q_{θ} = - 2 \dot{θ} [S_{x} \dot{A} + A^{T} M_{x} \dot{A} + B^{T} M_{y} \dot{B} - \underset{}{B^{T} C \dot{B}}] - \underset{}{\underset{}{\int_{0}^{L} ρ S (x) (B^{T} {Φ_{y}}^{T} {\dot{B}}^{T} H \dot{B}) d x}} + \underset{}{\underset{}{\dot{θ} \int_{0}^{L} ρ S (x) ({\dot{A}}^{T} {Φ_{x}}^{T} B^{T} H B + 2 A^{T} {Φ_{x}}^{T} B^{T} H \dot{B}}} \underset{}{\underset{}{- B^{T} H B B^{T} H \dot{B}) d x}} \\ Q_{A} = {\dot{θ}}^{2} (S_{x}^{T} + M_{x} A) + 2 \dot{θ} M_{x y} \dot{B} - K_{1} A - \underset{}{\underset{}{\frac{1}{2} {\dot{θ}}^{2} \int_{0}^{L} ρ S (x) ({Φ_{x}}^{T} B^{T} H B) d x}} + \underset{}{\underset{}{\int_{0}^{L} ρ S (x) ({Φ_{x}}^{T} {\dot{B}}^{T} H \dot{B}) d x}} \\ Q_{B} = {\dot{θ}}^{2} (M_{y} \underset{}{- C}) B - 2 \dot{θ} {M_{x y}}^{T} \dot{A} - K_{2} B + \underset{}{\underset{}{{\dot{θ}}^{2} \int_{0}^{L} ρ S (x) (\frac{1}{2} H B B^{T} H B - H B Φ_{x} A) d x}} - \underset{}{\underset{}{2 \dot{θ} \int_{0}^{L} ρ S (x) (H B Φ_{y} \dot{B} - {Φ_{y}}^{T} B^{T} H \dot{B}) d x}} - \underset{}{\underset{}{\int_{0}^{L} ρ S (x) (H B {\dot{B}}^{T} H \dot{B}) d x}} . \end{matrix}

(13)

Equation (11) is the rigid-flexible coupling dynamic equation of the system. The parts with single underline in (13) are called the one-order coupling terms. The parts with double underlines in (13) are called the high-order coupling terms. If without the large overall motion, (11) can be degenerated as the structure dynamic equation. Actually, (11) contains four different models: the high-order approximation coupling (HOAC) model with large overall motion, the first-order approximation coupling (FOAC) model with large overall motion, the zero-order approximation coupling (ZOAC) model with large overall motion, and the structure dynamic model without large overall motion. The constant matrices in (13) can be expressed as

\begin{matrix} S_{x} = \int_{0}^{L} ρ S (x) (a + x) Φ_{x} (x) d x \\ S_{y} = \int_{0}^{L} ρ S (x) (a + x) Φ_{y} (x) d x \\ M_{x} = \int_{0}^{L} ρ S (x) {Φ_{x}}^{T} (x) Φ_{x} (x) d x \\ M_{y} = \int_{0}^{L} ρ S (x) {Φ_{y}}^{T} (x) Φ_{y} (x) d x \\ M_{x y} = \int_{0}^{L} ρ S (x) Φ_{x}^{T} (x) Φ_{y} (x) d x \\ C = \int_{0}^{L} ρ S (x) (a + x) H (x) d x \\ K_{1} = \int_{0}^{L} E S (x) Φ_{x}^{' T} (x) Φ_{x}^{'} (x) d x \\ K_{2} = \int_{0}^{L} E I (x) Φ_{y}^{'' T} (x) Φ_{y}^{''} (x) d x \\ J_{ob} = \int_{0}^{L} ρ S (x) {(a + x)}^{2} d x . \end{matrix}

(14)

3. First-Order Approximation Coupling (FOAC) Model

When evolving the kinetic energy expression, since the coupling deformation w_c(x, t) is the second-order amount of the transversal deformation w₂(x, t), some high-order amount respected to w_c(x, t) can be ignored. Thus, the parts with double underlines in (13) can be ignored. So, (11) can be transferred as another form

[\begin{bmatrix} M_{11} & M_{12} & M_{13} \\ M_{21} & M_{22} & 0 \\ M_{31} & 0 & M_{33} \end{bmatrix}] [\begin{bmatrix} \overset{‥}{θ} \\ \overset{‥}{A} \\ \overset{‥}{B} \end{bmatrix}] + [\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & G_{23} \\ 0 & G_{32} & 0 \end{bmatrix}] [\begin{bmatrix} \dot{θ} \\ \dot{A} \\ \dot{B} \end{bmatrix}] + [\begin{bmatrix} 0 & 0 & 0 \\ 0 & K_{22} & 0 \\ 0 & 0 & K_{33} \end{bmatrix}] [\begin{bmatrix} θ \\ A \\ B \end{bmatrix}] = [\begin{bmatrix} Q_{θ} \\ Q_{1} \\ 0 \end{bmatrix}] + [\begin{bmatrix} τ \\ 0 \\ 0 \end{bmatrix}],

(15)

where

M_{11} = J_{oh} + J_{ob} + 2 S_{x} A + A^{T} M_{x} A + B^{T} M_{y} B - \underset{}{B^{T} C B}

(16)

M_{22} = M_{x}

(17)

M_{33} = M_{y}

(18)

M_{21} = {M_{12}}^{T} = - M_{x y} B

(19)

M_{31} = {M_{13}}^{T} = {S_{y}}^{T} + {M_{x y}}^{T} A

(20)

G_{23} = - G_{32}^{T} = - 2 \dot{θ} M_{x y}

(21)

K_{22} = K_{1} - {\dot{θ}}^{2} M_{x}

(22)

K_{33} = K_{2} - {\dot{θ}}^{2} (M_{y} \underset{}{- C})

(23)

Q_{θ} = - 2 \dot{θ} [S_{x} \dot{A} + A^{T} M_{x} \dot{A} + B^{T} M_{y} \dot{B} - \underset{}{B^{T} C \dot{B}}]

(24)

Q_{1} = {\dot{θ}}^{2} S_{x}^{T} .

(25)

Equation (15) is calledthe FOAC dynamic equations. The parts with single underline in (16), (23), and (24) are called one-order coupling terms. In the traditional ZOAC model, these parts are always ignored.

3.1. Transversal Bending Vibration Analysis Ignoring Longitudinal Deformation Effect

The transversal bending vibration analysis will be studied in this section. For simplifying analysis, the angular speed of the large overall motion is assumed to be constant as $\overset{‥}{θ} = 0$ . Compared to the transversal deformation, the longitudinal deformation has little influence on the transversal bending vibration. So, the transversal bending vibration equation of the system with constant angular speed can be simplified as

M_{33} \overset{‥}{B} + K_{33} B = 0 .

(26)

For convenience of discussion, dimensionless parameters are used in the present analysis and they are defined as

ς = \frac{t}{T}, ξ = \frac{x}{L}, κ_{2} = \frac{B}{L}, δ = \frac{a}{L}, γ = T \dot{θ}, α = \frac{b_{1}}{b_{2}}, β = \frac{h_{1}}{h_{2}},

(27)

where T = (ρS₀L⁴/EI₀)^1/2 and S₀ = b₂h₂, I₀ = (1/12)b₂³h₂. And δ, γ, α, and β will denote the hub radius ratio, the angular speed ratio, the width ratio, and the height ratio, respectively.

The dimensionless form of the free vibration equation is finally obtained by using the dimensionless variables and parameters defined in (27)

\overset{—}{M_{33}} {\overset{‥}{κ}}_{2} + \overset{—}{K_{33}} κ_{2} = 0,

(28)

where

\begin{matrix} \overset{—}{M_{33}} = \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] {Φ_{y}}^{T} (ξ) Φ_{y} (ξ) d ξ \\ \overset{—}{K_{33}} = \int_{0}^{1} {[(1 - ξ) + ξ α]}^{3} [(1 - ξ) + ξ β] Φ_{y}^{'' T} (ξ) {Φ_{y}}^{''} (ξ) d ξ - γ^{2} \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] {Φ_{y}}^{T} (ξ) Φ_{y} (ξ) d ξ + \underset{}{γ^{2} \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] (δ + ξ) \overset{—}{H} (ξ) d ξ} \\ \overset{—}{H} (ξ) = {\int_{0}^{ξ} (\frac{\partial Φ_{y} (η)}{\partial η})}^{T} (\frac{\partial Φ_{y} (η)}{\partial η}) d η . \end{matrix}

(29)

Let κ₂ = e^λωςΘ; we can get the eigenvalue equation

ω^{2} \overset{—}{M_{33}} Θ = \overset{—}{K_{33}} Θ,

(30)

where λ is imaginary, ω is the dimensionless natural frequency, and Θ is a constant column matrix.

Table 1 shows the dimensionless natural frequencies of the uniform beam without hub in the transversal bending vibration under different γ, where δ = 0, and α = β = 1, truncate the first four modes when calculating. From Table 1, we can see that the dimensionless natural frequencies here are the same as the results of paper [13], using another modeling method. That proves the correctness of this model. So we will truncate the first four modes when calculating in the following analysis. Additionally, we can see that the dimensionless natural frequencies increase as the angular speed ratio γ increases.

Table 1:

The dimensionless natural frequencies of the uniform beam in transversal bending vibration with different γ (δ = 0, α = β = 1).

γ	First natural frequency		Second natural frequency		Third natural frequency
γ	Present	Paper [13]	Present	Paper [13]	Present	Paper [13]

0	3.51601	3.51602	22.0345	22.0345	61.6973	61.6972
1	3.54323	3.54324	22.1585	22.1585	61.8338	61.8337
2	3.62181	3.62182	22.5264	22.5264	62.2413	62.2412
3	3.74361	3.74361	23.1268	23.1268	62.9144	62.9144
4	3.89804	3.89804	23.9423	23.9423	63.8443	63.8443
5	4.07471	4.07471	24.9520	24.9520	65.0196	65.0195
6	4.26498	4.26498	26.1328	26.1328	66.4266	66.4265
7	4.46252	4.46251	27.4623	27.4623	68.0502	68.0501
8	4.66305	4.66305	28.9191	28.9191	69.8748	69.8746
9	4.86398	4.86398	30.4841	30.4841	71.8842	71.8840
10	5.06384	5.06384	32.1408	32.1408	74.0628	74.0626

In Table 2, two models are compared, one is the FOAC model in this paper, and the other is the traditional ZOAC model (without the single underline parts). Also, the beam is considered to be a uniform one for simplifying analysis, where δ = α = β = 1. From (28), we can find that the two models are same when γ = 0. So the natural frequencies are equal when γ = 0. When γ increasing, the natural frequencies of the two models become discriminative. When the angular speed of the large overall motion is low (γ = 2), the natural frequencies of the two models have a small distinction. But when the large overall motion is at high speed, the natural frequencies of the ZOAC model will be imaginary, which is impractical, while the natural frequencies of the FOAC model are still real. So we can say that the traditional ZOAC model without the coupling deformation is inapplicable when the large overall motion of the beam is at high speed.

Table 2:

The dimensionless natural frequencies of the uniform beam in the transversal bending vibration with different models (δ = α = β = 1).

γ	First natural frequency		Second natural frequency		Third natural frequency
γ	FOAC model	ZOAC model	FOACmodel	ZOAC model	FOAC model	ZOAC model

0	3.51601	3.51601	22.0345	22.0345	61.6973	61.6973
2	4.40063	2.89177	23.2807	21.9435	63.0368	61.6649
10	13.2934	9.36149i	43.3038	19.6346	88.9661	60.8815
50	62.8131	49.8762i	183.989	44.8829i	321.035	36.1463

Table 3 shows the dimensionless natural frequencies of the tapered beam in the transversal bending vibration under different δ, where α = β = 0.5. It is obvious that the dimensionless natural frequency is proportional to the hub radius ratio δ, especially when the large overall motion is at high speed.

Table 3:

The dimensionless natural frequencies of the tapered beam in the transversal bending vibration with different δ (α = β = 0.5).

δ	First natural frequency		Second natural frequency		Third natural frequency
δ	γ = 2	γ = 10	γ = 2	γ = 10	γ = 2	γ = 10

0	4.75279	6.52741	19.9938	28.3386	49.3721	59.7309
0.5	5.10793	11.2473	20.3400	33.8886	49.7335	66.8171
1	5.43934	14.4910	20.6802	38.6215	50.0924	73.2230
5	7.57406	29.5438	23.2193	64.8417	52.8764	111.989
10	9.57836	41.2091	26.0391	87.0114	56.1639	146.690

Table 4 shows the first dimensionless natural frequencies of the tapered beam with different α when δ = β = 1. As we can see, the first natural frequency is in inverse proportion to the width ratio α. When α changes from 0 to 0.3, the first natural frequency will have a great change. Compared to the change of the first natural frequency when γ = 2, the change of the first natural frequency is much smaller when γ = 10.

Table 4:

The first dimensionless natural frequencies in the transversal bending vibration with different α (δ = β = 1).

α	γ = 2		γ = 10
α	First natural frequency	Change (%)	First natural frequency	Change (%)

0 0.3 0.5 0.7 1	6.20400 4.99493 4.72715 4.56147 4.40063	19 5.4 3.5 3.5	15.8566 14.2455 13.8222 13.5539 13.2934	10 3.0 1.9 1.9

Table 5 shows the first dimensionless natural frequencies of the tapered beam with different β when δ = α = 1. Also, the first natural frequency is in inverse proportion to the height ratio β. As we can see, the changes of the first natural frequency under different β have the same variation as the changes of the first natural frequency under different α.

Table 5:

The first dimensionless natural frequencies in the transversal bending vibration with different β (δ = α = 1).

β	γ = 2		γ = 10
β	First natural frequency	Change (%)	First natural frequency	Change (%)

0	7.74540	27	16.2564	12
0.3	5.65629	9.8	14.3832	3.4
0.5	5.09972	6.9	13.8975	2.2
0.7	4.74692	7.3	13.5916	2.2
1	4.40063		13.2934

3.2. Transversal Bending Vibration Analysis Including Longitudinal Deformation Effect

The transversal bending vibration analysis including longitudinal deformation effect will be studied in this section. Also, the angular speed of the large overall motion is assumed to be constant as $\overset{‥}{θ} = 0$ , and Q₁ is neglected here for the free vibration analysis. So, the free vibration equation of the system with constant angular speed can be formed as

(\begin{pmatrix} M_{22} & 0 \\ 0 & M_{33} \end{pmatrix}) (\begin{pmatrix} \overset{‥}{A} \\ \overset{‥}{B} \end{pmatrix}) + (\begin{pmatrix} 0 & G_{23} \\ G_{32} & 0 \end{pmatrix}) (\begin{pmatrix} \dot{A} \\ \dot{B} \end{pmatrix}) + (\begin{pmatrix} K_{22} & 0 \\ 0 & K_{33} \end{pmatrix}) (\begin{pmatrix} A \\ B \end{pmatrix}) = 0 .

(31)

Other dimensionless parameters are defined as

κ_{1} = \frac{A}{L}, ∊ = {(\frac{S_{0} L^{2}}{I_{0}})}^{1 / 2},

(32)

where ∊ is proportional to the length to thickness ratio of a beam, called the slenderness ratio. Now, the dimensionless form of the free vibration equation can be obtained as

(\begin{pmatrix} \overset{—}{M_{22}} & 0 \\ 0 & \overset{—}{M_{33}} \end{pmatrix}) (\begin{pmatrix} {\overset{‥}{κ}}_{1} \\ {\overset{‥}{κ}}_{2} \end{pmatrix}) + (\begin{pmatrix} 0 & \overset{—}{G_{23}} \\ \overset{—}{G_{32}} & 0 \end{pmatrix}) (\begin{pmatrix} {\dot{κ}}_{1} \\ {\dot{κ}}_{2} \end{pmatrix}) + (\begin{pmatrix} \overset{—}{K_{22}} & 0 \\ 0 & \overset{—}{K_{33}} \end{pmatrix}) (\begin{pmatrix} κ_{1} \\ κ_{2} \end{pmatrix}) = 0,

(33)

where

\overset{—}{M_{22}} = \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] {Φ_{x}}^{T} (ξ) Φ_{x} (ξ) d ξ

(34)

\overset{—}{G_{23}} = - {\overset{—}{G_{32}}}^{T} = - 2 γ \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] {Φ_{x}}^{T} (ξ) Φ_{y} (ξ) d ξ

(35)

\overset{—}{K_{22}} = ∊^{2} \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] Φ_{x}^{' T} (ξ) Φ_{x}^{'} (ξ) d ξ - γ^{2} \int_{0}^{1} [(1 - ξ) + ξ α] [(1 - ξ) + ξ β] \times {Φ_{x}}^{T} (ξ) Φ_{x} (ξ) d ξ .

(36)

Equation (31) can also be transformed into the following form:

M \dot{z} + K z = 0,

(37)

where

M = (\begin{pmatrix} \overset{—}{M_{22}} & 0 & 0 & 0 \\ 0 & \overset{—}{M_{33}} & 0 & 0 \\ 0 & 0 & I & 0 \\ 0 & 0 & 0 & I \end{pmatrix}), K = (\begin{pmatrix} 0 & \overset{—}{G_{23}} & \overset{—}{K_{22}} & 0 \\ \overset{—}{G_{32}} & 0 & 0 & \overset{—}{K_{33}} \\ - I & 0 & 0 & 0 \\ 0 & - I & 0 & 0 \end{pmatrix}), z = (\begin{pmatrix} {\dot{κ}}_{1} \\ {\dot{κ}}_{2} \\ κ_{1} \\ κ_{2} \end{pmatrix}) .

(38)

Let z = Ze^σς; we can get

(σ M + K) Z = 0,

(39)

where σ is the complex eigenvalue.

Table 6 shows the comparison of the first transversal natural frequencies with and without longitudinal deformation effect when α = β = 0.5 and ∊ = 70. The value of ∊ = 70 guarantees the assumptions of Euler-Bernoulli beam. From (35), we can see that matrix $\overset{—}{G_{32}}$ is proportional to the angular speed ratio γ. So, the larger γ is, the greater the influence of the longitudinal deformation on the transversal bending vibration is. When the beam is immobile as γ = 0, (33) can be degenerated to (28). But when the beam has large overall motion, the first natural frequency including longitudinal deformation will be smaller than that ignoring longitudinal deformation. Especially when the large overall motion is at very high speed (γ = 50), the error will be significant. So, we can get a conclusion that the longitudinal deformation will also have an obvious influence on the transversal bending vibration when the large overall motion is at very high speed.

Table 6:

Comparison of the first transversal natural frequencies with and without longitudinal effect (α = β = 0.5, ∊ = 70).

δ	γ	Longitudinal deformation ignored	Longitudinal deformation included	Error (%)

0	0	4.62520	4.62520	0
	2	4.75279	4.75107	0.036
	10	6.52741	6.46607	0.94
	50	14.8390	11.8649	20

1	0	4.62520	4.62520	0
	2	5.43934	5.43737	0.036
	10	14.4910	14.3536	0.95
	50	66.4377	51.7201	22

4. High-Order Approximation Coupling (HOAC) Model

Equation (11) is the rigid-flexible coupling dynamic equation of the system, which is also called the high-order approximation coupling (HOAC) dynamic equation. The parts with double underlines in (13) are called the high-order coupling terms. These terms are always ignored in the FOAC dynamic equations for convenience. But they will have a great influence on the dynamics of the system with large deformation. Here, two examples will be given to approve it.

4.1. Beam Undergoing Prescribed Translation and Rotation

A tapered beam made of aluminum with the dimensions and material properties given in Table 7 is used. The boundary conditions of a cantilever beam are used in the simulations. And the torque applied at the hub τ can be expressed as

τ (t) = {\begin{cases} τ_{0} sin (\frac{2 π}{T} t) & 0 \leq t \leq T \\ 0 & t > T, \end{cases}

(40)

where T = 2s.

Table 7:

Data of the hub-tapered beam system.

Property	Symbol	Value

Effective length of the beam	L	8 m
Tip width of the tapered beam	b ₁	1.5 × 10⁻² m
Bottom width of the tapered beam	b ₂	2.5 × 10⁻² m
Tip height of the tapered beam	h ₁	1.5 × 10⁻² m
Bottom height of the tapered beam	h ₂	2.5 × 10⁻² m
Mass density of the beam	ρ	2.7667 × 10³ kg·m⁻³
Elastic modulus of the beam	E	6.8952 × 10¹⁰ N·m⁻²
Radius of the hub	a	1 m
Rotary inertia of the hub	J _oh	5 kg·m²

Figure 4(a) gives the tip deformations of the tapered beam with HOAC model and FOAC model when τ₀ = 36 N·m. When the torque is applied on the hub, the max tip deformation is 0.56 m, which is only a small deformation to the effective length of the beam. In this case, the tip deformations of the HOAC model and the FOAC model are exactly coincidental, while Figure 4(b) gives the tip deformations of the two models when τ₀ = 136 N·m. When the torque applying on the hub, the max tip deformation of the HOAC model reached 2.32 m, this is obviously a large deformation to the effective length of the beam. So, in this case, the result of the FOAC model becomes divergent as shown in Figure 4(b). On the other hand, the result of the HOAC model is still convergent. Then, we can say that the FOAC model is inapplicable in the situation of large deformation, while the HOAC model is applicable. That is because the high-order terms with double underlines respected to w_c(x, t) were ignored in the FOAC modeling.

Figure 4:

The tip deformation of the tapered beam.

The deformation of the tapered beam in the rotating frame o′ – i^′j^′, with HOAC model when τ₀ = 136 N·m, is shown in Figure 5. And the motion in the inertial reference frame o – ij is shown in Figure 6.

Figure 5:

The deformation in rotating frame.

Figure 6:

The motion in inertial reference frame.

In order to research the dynamic response of the tapered beam, two ratios are defined here, the width ratio R_b = b₁/b₂ and the height ratio R_h = h₁/h₂. And two situations are assumed here, on the assumption that the weight of the tapered beam stays invariable.

Suppose the height of the beam keeps uniform, as h₁ = h₂ = 0.02 m. Then, we will get

(1 + R_{b}) b_{2} = 0.04 .

(41)

Suppose the width of the beam keeps uniform, as b₁ = b₂ = 0.02 m. Then, we will get

(1 + R_{h}) h_{2} = 0.04 .

(42)

Figure 7 gives the max tip deformations and the response frequency of the tapered beam with different R_b and R_h. It is apparent that the max tip deformation increases as R_b and R_h increases, while the response frequency decreases as R_b and R_h increases. Also, we will find that the change of R_b will have a greater influence on the dynamics of the tapered beam on our assumption (the weight of the tapered beam keeps invariable). But in any case, all R_b and R_h will have a great influence on the dynamics of the tapered beam.

Figure 7:

The dynamic impact of R_b, R_h.

4.2. Single Pendulum with Gravity

To illustrate the effect of large deformation, a flexible tapered pendulum with free falling is investigated as shown in Figure 8. The pendulum is assumed to fall under the effect of gravity.

Figure 8:

The flexible tapered pendulum.

To a flexible tapered pendulum with free falling, the gravitational potential energy should be included. So, the gravitational potential energy will be added to the system potential energy

V_{g} = - g^{T} W \int_{0}^{L} ρ S (x) r d x = - g^{T} W R,

(43)

where g is the gravitational acceleration array and W is the homogeneous transfer matrix. g, W, and R can be expressed as

\begin{matrix} g^{T} = [g_{x}, g_{y}] W = [\begin{bmatrix} cos θ & - sin θ \\ sin θ & cos θ \end{bmatrix}] \\ R = \int_{0}^{L} ρ S r d x . \end{matrix}

(44)

Since the FOAC model could not be used in the large deformation problems, paper [16] investigated the application of an absolute nodal coordinate formulation in the coupling dynamics of flexible beams with large deformation. The simulation of a flexible pendulum indicated that the absolute nodal coordinate formulation is suited for the beams with large deformation, while the FOAC model cannot be used in that case. Here, we will simulate the example of Figure 4 in paper [16] by using the HOAC model. In reference to paper [16], the flexible pendulum is a uniform one, ρ = 2.76667 × 10³ kg·m⁻³, L = 1.8 m, S(x) = 2.5 × 10⁻⁴ m², I(x) = 1.3 × 10⁻⁹ m⁴, and E = 6.8952 × 10⁹ N·m⁻².

Figure 9 shows the tip deformation of the flexible pendulum with large deformation. As we can see, the result of the FOAC model is divergent, while the result of the HOAC model is nearly the same as the result of paper [16]. This proves the validity of the HOAC model and the applicability in the large deformation situation. As we know, the absolute nodal coordinate method cannot distinguishthe rigid body motion from the elastic deformation of the flexible body. Small deformation problems also should be treated as large deformation problems, while the FOAC model can only deal with the small deformation problems. However, the HOAC model proposed in this paper can make up for the deficiency of those two methods.

Figure 9:

The tip deformation of the uniform pendulum.

To further prove the applicability of the HOAC model in the large deformation situation, a quiet flexible tapered pendulum will be studied here. The dimensions and material properties are still as shown in Table 7, except E = 6.8952 × 10⁵ N·m⁻², a = 0 m.

Figure 10 shows the deformed shape of the tapered pendulum with HOAC model at different time steps when E = 6.8952 × 10⁵ N·m⁻². As we can see, the deformation is very large at any time step. If we use the FOAC or ZOAC models, the simulation will be divergent.

Figure 10:

The deformed shape at different times when E = 6.8952 × 10⁵ N·m⁻².

Figure 11 shows the energy change of the tapered pendulum. Curve 1 is the kinetic energy, curve 2 is the elastic potential energy, curve 3 is the gravitational potential energy, and curve 4 is the total energy. Obviously, as the pendulum falls, the kinetic energy increases at the beginning stage. But when the pendulum passes someplace, the kinetic energy will decrease. Also, the elastic potential energy changes the same law as the kinetic energy, while the gravitational potential energy decreases before the pendulum falls to the lowest point. When the pendulum passes the lowest point, the gravitational potential energy will increase. Anyway, the total energy keeps invariant at any time.

Figure 11:

The energy change of the tapered pendulum when E = 6.8952 × 10⁵ N·m⁻².

Figure 12 shows the deformed shape of the tapered pendulum with HOAC model at different time steps when E = 6.8952 × 10¹⁰ N·m⁻². As we can see, the deformation is very small at any time step. So, the elastic potential energy is almost 0 (Figure 13). However, the elastic potential energy is very significant when E = 6.8952 × 10⁵ N·m⁻². Also, there are some differences between the kinetic energies when E = 6.8952 × 10⁵ N·m⁻² and E = 6.8952 × 10¹⁰ N·m⁻². The kinetic energy change is more complex when large deformation occurs.

Figure 12:

The deformed shape at different times when E = 6.8952 × 10¹⁰ N·m⁻².

Figure 13:

The energy change when E = 6.8952 × 10⁵ N·m⁻² and E = 6.8952 × 10¹⁰ N·m⁻².

5. Conclusion

A new rigid-flexible coupling model of a hub-tapered beam system, which is also called the high-order approximation coupling (HOAC) model, was established in this paper. The HOAC model could not only be used in the case of small deformation, but also in the case of large deformation, while the previous FOAC model could only be used in the case of small deformation. Based on the HOAC model, the motion and the dynamic response of the tapered beam were studied.

Also, the natural frequencies of the FOAC model were studied through the transversal bending vibration analysis. The dimensionless natural frequencies were shown to increase as the angular speed ratio γ and the hub radius ratio δ increased. However, they decreased as the width ratio α and the height ratio β increased. When the large overall motion was at very high speed, the longitudinal deformation would also have a significant influence on the transversal bending vibration.

Footnotes

Acknowledgments

This work was financially supported by the National Natural Science Foundation of China (nos. 11302192, 10772085, 11132007, and 11272155) and the 333 Project of Jiangsu Province (BRA2011172), for which the authors are grateful.

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