Vibration of deploying rectangular cross-sectional beam made of functionally graded materials

Abstract

Transverse vibration and stability of deploying rectangular cross-sectional cantilever beam made of functionally graded material are investigated. The functionally graded material beam is assumed to be constructed with ceramics and metal phases, and the corresponding equivalent parameters of functionally graded material are found to continuously vary across the cross-sectional height with a simple power law. The differential equations of motion of deploying functionally graded material cantilever beam are derived by Hamilton’s principle. Based on the assumed modal method, the beam deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. The eigenfunctions of cantilever beam in which the length of the beam is time-dependent are chosen as admissible functions. Galerkin method is employed to discretize the differential equation to a set of time-coordinate-dependent ordinary differential equations, and then the eigenvalue problem depending on time is obtained. The deployment motion of beam is considered as a constant speed in this study, and some numerical results, which are variations of tip deflection response and complex frequencies, are obtained. Finally, the effects of gradient index of functionally graded material, deploying speed, initial length and protruded length, the cross-sectional height on dynamical characteristics, and divergence instability of the deploying functionally graded material beam are discussed.

Keywords

Deploying functionally graded material beam transverse vibration divergence instability complex frequency

Introduction

Axially moving beams appear in various applications such as telescopic robotic manipulators, deployment of flexible antennas or appendages of spacecraft, as well as the rolling process of plates, and so on. Some axially moving beams with fixed boundary conditions slide along the axial direction but do not stretch, so the mass of the systems can be conserved in the fixed domains. Many methods have been used for the analysis of the dynamic behavior and stability of the axially moving beams. Lee et al.¹ formulated the spectral element model for an axially moving Timoshenko beam under a uniform axial tension. Sze et al.² employed Galerkin method to discretize the governing equations of an axially moving beam and formulated the incremental harmonic balance method for nonlinear vibration. Cepon and Boltezar³ applied an approximate Gelerkin finite element method to solve the initial/boundary-value problem of a viscously damped and axially moving beam with pre-tension. An and Su⁴ employed the generalized integral transform technique (GITT) to obtain a hybrid analytical-numerical solution for dynamical response of clamped axially moving beams, and discussed the influence of axial translation velocity and flexural stiffness on vibration amplitude and frequency of the beam. Öz et al.⁵ used the multiple scales method to investigate the nonlinear vibration and stability of an axially moving beam with time-dependent velocity. Recently, Li et al.^6–9 studied the dynamic characteristics of axially moving nanoscale beams, and obtained many meaningful results. Wu et al.,¹⁰ Wang et al.,¹¹ and Ma et al.¹² studied the vibration and stability of axial motion membrane and plates, respectively.

Some axially moving beams are cantilevered beams where the length of the beams is variable, and so the beams are called deploying beams. In this class of problem, the conservation of mass is not automatically satisfied because the mass either enters or leaves. In order to study the dynamics of the deploying beam, some scholars put forward various modeling methods suitable for this problem, such as Hamilton principle, finite element method, Lagrange and assumed mode methods, etc. The earliest research work in this field was done by Tabarrok et al.¹³ in their work, where a dynamics model for deploying beam was established based on Newton’s second law and Lagrange function. Because of the changing length of the beam in deployment, Stylianou et al.¹⁴ put forward the concept of variable region finite element method, and studied the response and stability of an axially moving beam using the finite element method. Fung et al.¹⁵ derived the vibration equation of axially moving beam with a tip mass by Hamilton’s principle, which laid the important foundation for the dynamical analysis of the deploying beam. Gosselin et al.¹⁶ used Newton method to analyze the stability of a deploying/extruding beam with uniform circular cross section, extending axially in a horizontal plane at a known rate, while immersed in an incompressible fluid. In their equations, an “axial added mass coefficient” was implemented in order to better approximate the mass of fluid, which stayed attached to the beam, and the dynamical behaviors of the system was observed for cases of constant extension rate and for a trapezoidal deployment rate profile. Wang and Ni¹⁷ investigated vibration and stability for an axially moving beam in fluid and constrained by simple supports with torsion springs, and analyzed the effects of axially moving speed, axial added mass coefficient, and several other system parameters on the dynamics and instability of the beam. It was shown that when the moving speed exceeds a certain value, the beam is subject to buckling-type instability. Chang et al.¹⁸ employed variable region finite element method to derive the equations of motion of an axially moving beam based on Rayleigh beam theory, and investigated vibration and dynamical stability of the beam with two kinds of axial motion including constant-speed extension deployment and back-forth periodical motion by Runge–Kutta method and Floquet theory. Wang et al.¹⁹ derived the transverse vibration equation of the axially moving cantilever beam with a tip mass by D’Alembert’s principle, and put forward the $H_{\infty}$ control strategy for the suppression of transverse vibration of the beam by initial deformation excitations. Park et al.²⁰ derived the equations of motion for vibration of a deploying beam from the Eulerian and Lagrangian descriptions. Kim et al.²¹ presented a method for reducing the residual vibration of a flexible beam deployed from a translating hub, and the numerical results showed the effect of the vibration reduction method proposed for both constant-length and variable-length deploying translating beams. Huo et al.²² studied a vertical cantilevered pipe conveying fluid with deploying or retracting motion along the axial line of pipe, and analyzed the effects of the deploying or retracting speed, flow velocity, instantaneous length of pipe, gravity, and mass ratio on dynamical responses and stability. The above research work on the axially moving beam or deploying beam is mainly the uniform beams made of isotropic homogeneous material.

For the past few years, functionally graded material (FGM) has been widely used in mechanical and aerospace structures due to its unique superiority of heat resistance, high strength, and light weight. Sui et al.²³ analyzed the transverse vibration characteristics of axially moving FGM beam with simple supports at both ends based on Timoshenko beam theory. For dynamical analysis of deploying cantilever beam made of FGM, there are very few reports on this new research topic. To the best of the authors’ knowledge, the work by Piovan and Sampaio²⁴ is the first to deal with the deploying beams made of FGM. They studied the vibration of axially moving flexible beams made of FGMs by a finite element scheme, in which a thin-walled beam with annular cross section and a continuously graded variation in the composition of ceramic and metal phases across the wall thickness were considered, and the effects of the beam flexibility, tip mass, and material constituents on the dynamics of the axially moving beams were discussed. However, the deploying FGM cantilever beam with a rectangular cross section was not taken into account in their model.

In the present paper, a study on the vibration and stability of deploying cantilever FGM beam with rectangular cross section are performed. The differential equation of motion of a deploying FGM beam is derived based on Hamilton’s principle, and Galerkin method is employed to discretize the governing differential equation to a set of time coordinate-dependent ordinary differential equations. The effects of gradient index of FGM, deploying speed, initial length, protruded length, and the cross-sectional height on dynamical characteristics and divergence instability of the deploying FGM beam are discussed. The numerical method can be used for the study of this kind of problem, and the research results can be helpful for engineering practical design.

Coupled differential equation of motion

Consider a deploying rectangular cross-sectional cantilever beam made of FGMs. Let the beam be of width $b$ , height h, and variable length $L (t)$ . The Euler coordinate system oxz is depicted in Figure 1, in which the coordinate axis $x$ and $z$ are along the axial line and the thickness direction of the beam, respectively.

Figure 1.

Deploying rectangular cross-sectional cantilever FGM beam and the coordinate system.

In this paper, the FGM is considered to be composed of two materials, such as ceramic and metal, i.e. the upper surface of the beam is metal-rich, and the lower surface is ceramic-rich. The effective material properties of the FGM beam can be expressed as²⁵

X = (X_{c} - X_{m}) {(\frac{z}{h} + \frac{1}{2})}^{p} + X_{m}

(1)

where

X_{c}

and

X_{m}

denote the material properties of ceramic and metallic materials respectively, and

p

is called the gradient index of FGM.

According to the Euler–Bernoulli hypotheses for the FGM beam, the displacement fields take the following forms

{\begin{array}{l} u (x, z, t) = u_{0} (x, t) - z \frac{\partial w_{0} (x, t)}{\partial x} \\ v = 0 \\ w (x, z, t) = w_{0} (x, t) \end{array}

(2)

where u₀ and w₀ denote the axial displacement and the transverse displacement in the axial line of the beam, respectively.

The axial strain–displacement component relation and the stress–strain relation for the FGM beam are given by the following expression respectively

ε_{x} = \frac{\partial u_{0}}{\partial x} - z \frac{\partial^{2} w_{0}}{\partial x^{2}}

(3)

σ_{x} = E (z) ε_{x}

(4)

where

E (z)

is the equivalent Young’s modulus of the FGM beam, which is determined by equation (1).

An axial force $N$ and a bending moment $M$ of the beam due to the axial normal stress are expressed as

(N, M) = \int_{A} σ_{x} [1, z] d A

(5)

where

A

is the area of the rectangular cross section.

Substituting equation (4) into equation (5), the axial force and the bending moment are expressed by displacements as

{\begin{matrix} N = A_{0} \frac{\partial u_{0}}{\partial x} - A_{1} \frac{\partial^{2} w_{0}}{\partial x^{2}} \\ M = A_{1} \frac{\partial u_{0}}{\partial x} - A_{2} \frac{\partial^{2} w_{0}}{\partial x^{2}} \end{matrix}

(6)

where

(A_{0}, A_{1}, A_{2}) = b \int_{- \frac{h}{2}}^{\frac{h}{2}} E (z) [1, z, z^{2}] d z

The strain energy of the system can be described as

U_{d} = \frac{1}{2} \int_{V} σ_{x} ε_{x} d V = \frac{1}{2} \int_{0}^{L (t)} \int_{- \frac{h}{2}}^{\frac{h}{2}} E (z) ε_{x}^{2} b d z d x

(7)

The kinetic energy of the system can be expressed as

T = \frac{1}{2} \int_{0}^{L (t)} \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) [v_{x}^{2} + v_{z}^{2}] b d z d x

(8)

in which

v_{x} = \frac{d x}{d t} + \frac{d u}{d t} = \frac{d x}{d t} + \frac{\partial u_{0}}{\partial t} + \frac{d x}{d t} \frac{\partial u_{0}}{\partial x} - z \frac{\partial^{2} w_{0}}{\partial x \partial t} - z \frac{d x}{d t} \frac{\partial^{2} w_{0}}{\partial x^{2}}

v_{z} = \frac{d w}{d t} = \frac{\partial w_{0}}{\partial t} + \frac{d x}{d t} \frac{\partial w_{0}}{\partial x}

(9)

where

\frac{d x}{d t}

is the deploying speed of the beam.

Substituting equation (9) into equation (8), the kinetic energy of the system is rewritten as

T = \frac{1}{2} \int_{0}^{L (t)} \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) [{(\frac{d x}{d t} + \frac{\partial u_{0}}{\partial t} + \frac{d x}{d t} \frac{\partial u_{0}}{\partial x} - z \frac{\partial^{2} w_{0}}{\partial x \partial t} - z \frac{d x}{d t} \frac{\partial^{2} w_{0}}{\partial x^{2}})}^{2} + {(\frac{\partial w_{0}}{\partial t} + \frac{d x}{d t} \frac{\partial w_{0}}{\partial x})}^{2}] b d z d x

(10)

The potential energy due to the axial inertia force can be expressed as²⁴

U_{a} = - \frac{1}{2} \int_{0}^{L (t)} \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) [L (t) - x] \frac{d^{2} x}{d t^{2}} {(\frac{\partial w_{0}}{\partial x})}^{2} b d z d x

(11)

Substituting equations (7), (10), and (11) into Hamilton’s principle $δ [\int_{t_{1}}^{t_{2}} (T - U_{a} - U_{d}) d t] = 0$ , and performing variational calculus, one can obtain the coupled differential equations of motion

{\begin{cases} A_{0} \frac{\partial^{2} u_{0}}{\partial x^{2}} - A_{1} \frac{\partial^{3} w_{0}}{\partial x^{3}} = I_{0} [\frac{d^{2} x}{d t^{2}} + \frac{\partial^{2} u_{0}}{\partial t^{2}} + \frac{d^{2} x}{d t^{2}} \frac{\partial u_{0}}{\partial x} + 2 \frac{d x}{d t} \frac{\partial^{2} u_{0}}{\partial x \partial t} + {(\frac{d x}{d t})}^{2} \frac{\partial^{2} u_{0}}{\partial x^{2}}] \\ - I_{1} [\frac{\partial^{3} w_{0}}{\partial x \partial t^{2}} + \frac{d^{2} x}{d t^{2}} \frac{\partial^{2} w_{0}}{\partial x^{2}} + 2 \frac{d x}{d t} \frac{\partial^{3} w_{0}}{\partial x^{2} \partial t} + {(\frac{d x}{d t})}^{2} \frac{\partial^{3} w_{0}}{\partial x^{3}}] \\ A_{1} \frac{\partial^{3} u_{0}}{\partial x^{3}} - A_{2} \frac{\partial^{4} w_{0}}{\partial x^{4}} = I_{0} [(L (t) - x) \frac{d^{2} x}{d t^{2}} \frac{\partial^{2} w_{0}}{\partial x^{2}} - \frac{d^{2} x}{d t^{2}} \frac{\partial w_{0}}{\partial x}] \\ - I_{2} [\frac{\partial^{4} w_{0}}{\partial x^{2} \partial t^{2}} + \frac{d^{2} x}{d t^{2}} \frac{\partial^{3} w_{0}}{\partial x^{3}} + 2 \frac{d x}{d t} \frac{\partial^{4} w_{0}}{\partial x^{3} \partial t} + {(\frac{d x}{d t})}^{2} \frac{\partial^{4} w_{0}}{\partial x^{4}}] \\ + I_{1} [\frac{\partial^{3} u_{0}}{\partial t^{2} \partial x} + \frac{d^{2} x}{d t^{2}} \frac{\partial^{2} u_{0}}{\partial x^{2}} + 2 \frac{d x}{d t} \frac{\partial^{3} u_{0}}{\partial t \partial x^{2}} + {(\frac{d x}{d t})}^{2} \frac{\partial^{3} u_{0}}{\partial x^{3}}] \\ + I_{0} [\frac{\partial^{2} w_{0}}{\partial t^{2}} + \frac{d^{2} x}{d t^{2}} \frac{\partial w_{0}}{\partial x} + 2 \frac{d x}{d t} \frac{\partial^{2} w_{0}}{\partial x \partial t} + {(\frac{d x}{d t})}^{2} \frac{\partial^{2} w_{0}}{\partial x^{2}}] \end{cases}

(12)

where

(I_{0}, I_{1}, I_{2}) = b \int_{- \frac{h}{2}}^{\frac{h}{2}} ρ (z) [1, z, z^{2}] d z

, and

ρ (z)

denotes the equivalent mass density of FGM beam, which can be determined by equation (1).

In the following work, the axial displacement $u_{0}$ is neglected, i.e. the deploying FGM beam is assumed to perform rigid motion along its axial direction, and be inextensible along the axial line. Thus in equation (12), $\frac{d x}{d t} = \frac{d L}{d t} = V (t),$ $\frac{d^{2} x}{d t^{2}} = \frac{d^{2} L}{d t^{2}} = \frac{d V}{d t}$ , in which $V (t)$ is called the sliding speed or deploying speed along x-axis.

Implementing a derivation with respect to x on both sides of the first equation of equation (12), and substituting it into the second equation of equation (12), a differential equation of motion for the deploying FGM beam can be rewritten as

\begin{matrix} [I_{2} {(\frac{d L}{d t})}^{2} + \frac{A_{1}^{2}}{A_{0}} - A_{2} - \frac{A_{1} I_{1}}{A_{0}} \frac{d L}{d t}] \frac{\partial^{4} w_{0}}{\partial x^{4}} \\ = I_{0} [L (t) - x] \frac{d^{2} L}{d t^{2}} \frac{\partial^{2} w_{0}}{\partial x^{2}} + I_{0} [\frac{\partial^{2} w_{0}}{\partial t^{2}} + 2 \frac{d L}{d t} \frac{\partial^{2} w_{0}}{\partial x \partial t} + {(\frac{d L}{d t})}^{2} \frac{\partial^{2} w_{0}}{\partial x^{2}}] \\ + (\frac{A_{1} I_{1}}{A_{0}} - I_{2}) (\frac{\partial^{4} w_{0}}{\partial x^{2} \partial t^{2}} + \frac{d^{2} L}{d t^{2}} \frac{\partial^{3} w_{0}}{\partial x^{3}} + 2 \frac{d L}{d t} \frac{\partial^{4} w_{0}}{\partial x^{3} \partial t}) \end{matrix}

(13)

Discretization of the equation

Based on the assuming modal method, the deflection functions $w_{0} (x, t)$ can be expanded as

w_{0} (x, t) = \sum_{i = 1}^{N} η_{i} (t) φ_{i} (x, t)

(14)

where

η_{i} (t)

is the ith generated coordinate, N denotes the number of term,

φ_{i} (x, t)

is the ith corresponding eigenfunction of the cantilever Euler–Bernoulli beam, used here as a suitable set of base functions, i.e.

φ_{i} (x, t) = \cos \frac{β_{i}}{L (t)} x - \cosh \frac{β_{i}}{L (t)} x + r_{i} [\sin \frac{β_{i}}{L (t)} x - \sinh \frac{β_{i}}{L (t)} x]

(15)

where

r_{i} = - \frac{\cos β_{i} + \cosh β_{i}}{\sin β_{i} + \sinh β_{i}}, (i = 1, 2, 3, \dots, N)

, and

β_{i}

are the eigenvalues given by the roots of the transcendental equation

1 + \cosh β_{i} \cos β_{i} = 0

Substituting equation (15) into equation (14), and applying Galerkin’s procedure for equation (13), after some manipulations, one can obtain the second-order ordinary differential equations with time-variable coefficients

\sum_{i = 1}^{N} a_{i j} {\ddot{η}}_{i} (t) + \sum_{i = 1}^{N} b_{i j} {\dot{η}}_{i} (t) + \sum_{i = 1}^{N} c_{i j} η_{i} (t) = 0 (j = 1, 2, \dots, N)

(16)

in which

\begin{matrix} a_{i j} = I_{0} \int_{0}^{L (t)} φ_{i} φ_{j} d x + (\frac{A_{1} I_{1}}{A_{0}} - I_{2}) \int_{0}^{L (t)} \frac{\partial^{2} φ_{i}}{\partial x^{2}} φ_{j} d x \\ b_{i j} = 2 I_{0} \int_{0}^{L (t)} \frac{\partial φ_{i}}{\partial t} φ_{j} d x + 2 I_{0} \frac{d L}{d t} \int_{0}^{L (t)} \frac{\partial φ_{i}}{\partial x} φ_{j} d x + 2 (\frac{A_{1} I_{1}}{A_{0}} - I_{2}) \int_{0}^{L (t)} \frac{\partial^{3} φ_{i}}{\partial x^{2} \partial t} φ_{j} d x \\ + 2 (\frac{A_{1} I_{1}}{A_{0}} - I_{2}) \frac{d L}{d t} \int_{0}^{L (t)} \frac{\partial^{3} φ_{i}}{\partial x^{3}} φ_{j} d x \\ c_{i j} = [A_{2} - \frac{A_{1}^{2}}{A_{0}} + \frac{A_{1} I_{1}}{A_{0}} {(\frac{d L}{d t})}^{2} - I_{2} {(\frac{d L}{d t})}^{2}] \int_{0}^{L (t)} \frac{\partial^{4} φ_{i}}{\partial x^{4}} φ_{j} d x + I_{0} \frac{d^{2} L}{d t^{2}} \int_{0}^{L (t)} [L (t) - x] \frac{\partial^{2} φ_{i}}{\partial x^{2}} φ_{j} d x \\ + I_{0} [\int_{0}^{L (t)} \frac{\partial^{2} φ_{i}}{\partial t^{2}} φ_{j} d x + 2 \frac{d L}{d t} \int_{0}^{L (t)} \frac{\partial^{2} φ_{i}}{\partial x \partial t} φ_{j} d x + {(\frac{d L}{d t})}^{2} \int_{0}^{L (t)} \frac{\partial^{2} φ_{i}}{\partial x^{2}} φ_{j} d x] \\ + (\frac{A_{1} I_{1}}{A_{0}} - I_{2}) [\int_{0}^{L (t)} \frac{\partial^{4} φ_{i}}{\partial x^{2} \partial t^{2}} φ_{j} d x + 2 \frac{d L}{d t} \int_{0}^{L (t)} \frac{\partial^{4} φ_{i}}{\partial x^{3} \partial t} φ_{j} d x + \frac{d^{2} L}{d t^{2}} \int_{0}^{L (t)} \frac{\partial^{3} φ_{i}}{\partial x^{3}} φ_{j} d x] \end{matrix}

(17)

Introducing dimensionless coordinates $ξ = \frac{x}{L}$ , $ς = \frac{z}{h}$ , $(d_{1}, d_{2}, d_{3}) = \int_{- \frac{1}{2}}^{\frac{1}{2}} \bar{E} (ς) [1, ς, ς^{2}] d ς$ , $(s_{1}, s_{2}, s_{3}) = \int_{- \frac{1}{2}}^{\frac{1}{2}} \bar{ρ} (ς) [1, ς, ς^{2}] d ς$ , while $\bar{E} (ς) = (1 - \frac{E_{m}}{E_{c}}) {(ς + \frac{1}{2})}^{p} + \frac{E_{m}}{E_{c}}, \bar{ρ} (ς) = (1 - \frac{ρ_{m}}{ρ_{c}}) {(ς + \frac{1}{2})}^{p} + \frac{ρ_{m}}{ρ_{c}}$ , and multiplying both sides of equation (16) by the factor $L^{3} (t) / (ρ_{c} b h)$ , the above coefficients equation (17) can be expressed as

\begin{matrix} a_{i j} = s_{1} L^{4} (t) δ_{i j} + (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} L^{2} (t) \int_{0}^{1} \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ \\ b_{i j} = 2 s_{1} \dot{L} (t) L^{3} (t) \int_{0}^{1} (1 - ξ) \frac{d {\bar{φ}}_{i}}{d ξ} {\bar{φ}}_{j} d ξ + 4 (s_{3} - \frac{d_{2} s_{2}}{d_{1}}) h^{2} L (t) \dot{L} (t) \int_{0}^{1} \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ \\ - 2 (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} L (t) \dot{L} (t) \int_{0}^{1} ξ \frac{d^{3} {\bar{φ}}_{i}}{d ξ^{3}} {\bar{φ}}_{j} d ξ \\ c_{i j} = [\frac{E_{c} d_{3}}{ρ_{c}} - \frac{E_{c} d_{2}^{2}}{ρ_{c} d_{1}} + \frac{d_{2} s_{2}}{d_{1}} {\dot{L}}^{2} (t) - s_{3} {\dot{L}}^{2} (t)] h^{2} \int_{0}^{1} \frac{d^{4} {\bar{φ}}_{i}}{d ξ^{4}} {\bar{φ}}_{j} d ξ - s_{1} \ddot{L} (t) L^{3} (t) \int_{0}^{1} ξ \frac{d {\bar{φ}}_{i}}{d ξ} {\bar{φ}}_{j} d ξ \\ + s_{1} {\dot{L}}^{2} (t) L^{2} (t) \int_{0}^{1} {(1 - ξ)}^{2} \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ + s_{1} \ddot{L} (t) L^{3} (t) \int_{0}^{1} (1 - ξ) \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ \\ - 2 s_{1} {\dot{L}}^{2} (t) L^{2} (t) \int_{0}^{1} (1 - ξ) \frac{d {\bar{φ}}_{i}}{d ξ} {\bar{φ}}_{j} d ξ + 6 (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} {\dot{L}}^{2} (t) \int_{0}^{1} ξ \frac{d^{3} {\bar{φ}}_{i}}{d ξ^{3}} {\bar{φ}}_{j} d ξ \\ - 2 (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} \ddot{L} (t) L (t) \int_{0}^{1} \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ + 6 (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} {\dot{L}}^{2} (t) \int_{0}^{1} \frac{d^{2} {\bar{φ}}_{i}}{d ξ^{2}} {\bar{φ}}_{j} d ξ \end{matrix}

(18)

\begin{matrix} + (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} \ddot{L} (t) L (t) \int_{0}^{1} (1 - ξ) \frac{d^{3} {\bar{φ}}_{i}}{d ξ^{3}} {\bar{φ}}_{j} d ξ - 6 (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} {\dot{L}}^{2} (t) \int_{0}^{1} \frac{d^{3} {\bar{φ}}_{i}}{d ξ^{3}} {\bar{φ}}_{j} d ξ \\ + (\frac{d_{2} s_{2}}{d_{1}} - s_{3}) h^{2} {\dot{L}}^{2} (t) \int_{0}^{1} (ξ^{2} - 2 ξ) \frac{d^{4} {\bar{φ}}_{i}}{d ξ^{4}} {\bar{φ}}_{j} d ξ \end{matrix}

In the above expressions, $δ_{i j}$ is Kronecker delta symbol; ${\bar{φ}}_{i} = \cos β_{i} ξ - \cosh β_{i} ξ + r_{i} (\sin β_{i} ξ - \sinh β_{i} ξ)$ .

Denoting $η = {[η_{1}, η_{2}, \dots, η_{N}]}^{T}, A^{T} = (a_{i j}), B^{T} = (b_{i j}), C^{T} = (c_{i j})$ , equation (16) can be expressed as the second ordinary differential equations with time-variable coefficients

A \ddot{η} + B \dot{η} + C η = 0

(19)

Denoting $Y = {[η, \dot{η}]}^{T}$ , equation (19) can be reduced to the first-order differential equations with time-variable coefficients

\dot{Y} = D Y

(20)

in which the square matrix

D

is given by

D = [\begin{matrix} O & I \\ - A^{- 1} C & - A^{- 1} B \end{matrix}]

(21)

where

I

is a unitary matrix of order

N \times N

O

is a null matrix of order

N \times N

A solution in equation (20) may be taken as

Y = F \exp (λ t)

(22)

where

λ

is generally a complex eigenvalue or complex frequency, and

F

is a nonzero constant vector.

Substituting equation (22) into equation (20) leads to homogeneous linear algebraic equations

(λ I - D) F = 0

(23)

Based on the linear algebra theory, the sufficient and necessary conditions of homogeneous linear algebraic equations existing in the nonzero solution are that the determinant of coefficients equals to zero, thus one can arrive at the following generalized complex eigenequation

| λ I - D | = 0

(24)

where the square matrix

D

involves some parameters such as time variable, gradient index of FGM, protruded length, sectional dimension, and deploying speed.

By solving the generalized complex eigenequation (24), one can obtain the complex frequencies of deploying FGM beam and the type of stability from the signs of the real part and imaginary part. On the other hand, by directly solving equation (20) using Runge–Kutta method, the tip deflection response of deploying FGM beam can also be obtained.

Numerical results and discussion

In this section, the deploying cantilever FGM beam with rectangular cross section is assumed to be made of aluminum and ZrO₂ with the following properties: $ρ_{m} = 2707 kg / m^{3}$ , $E_{m} = 70 GPa$ , $ρ_{c} = 3000 kg / m^{3}$ , $E_{c} = 151 GPa$ , where the subscripts m and c refer to the aluminum and ZrO₂ materials, respectively. The material properties are assumed to vary along the beam height with a power law relation. The deployment motion of beam is considered as constant speed in this study, and the time-dependent length $L (t)$ of the beam can be expressed as $L (t) = L_{0} + L_{t} t$ , in which $L_{0}$ is the initial length of the beam, and $L_{t}$ is the constant speed, i.e. $V = L_{t}$ .

When the gradient index $p = 0$ and deploying speed $L_{t} = 0$ , the deploying FGM beam degenerates to homogenous ceramic (ZrO₂) cantilever Rayleigh beam. In this case, taking $L_{0} = 1.5 m$ , $h = 10 mm$ , the first three natural frequencies of the cantilever Rayleigh beam can be calculated as $Im (λ_{1}) = 32.00413 rad / s$ , $Im (λ_{2}) = 200.56126 rad / s$ , $Im (λ_{3}) = 561.54342 rad / s$ , respectively. It is noted that the first three natural frequencies $Im (λ_{i}) (i = 1, 2, 3)$ obtained in the present paper have dimensional quantity. For convenience of comparison with the study of Li et al.,²⁶ $Im (λ_{i}) (i = 1, 2, 3)$ are expressed to dimensionless $Ω_{i} = \sqrt{ρ A / (E I)} L^{2} Im (λ_{i}),$ $i = 1, 2, 3$ , as shown in Table 1. It can be seen that the present results coincide well with the results of Li et al.²⁶ It shows that the present method is effective. In this section, we let $N = 3$ .

Table 1.

The first three dimensionless natural frequencies of cantilever Rayleigh beam.

	$Ω_{1}$	$Ω_{2}$	$Ω_{3}$
Present	3.51602	22.03395	61.69197
Li et al.²⁶	3.51602	22.03449	61.69721

Dynamical response analyses

The influence of gradient index on the tip deflection response of deploying FGM beam

In the case of $L_{0} = 1.8 m, L_{t} = 0.5 m/s, h =9 .5 mm$ , Figure 2 gives the tip deflection of deploying FGM beam versus time for gradient index $p = 0, 2, 10, 10^{6}$ , respectively. It is well known that the FGM beam becomes homogenous ceramic or metal beam as gradient index $p$ approaches zero or infinity. When $p = 10^{6}$ , the FGM beam can be thought as the metal beam approximately. It can be seen from Figure 2 that the vibrational frequencies of deploying FGM beam decrease with the increase in the gradient index and time, and gradually change from the homogenous ceramic beam to the metal beam, while the tip deflection amplitudes are almost not affected by the change of gradient index. It shows that the influence of gradient index on the vibration frequency of deploying FGM beam is larger, and more the metal components of the FGM beam, smaller will be the vibration frequency.

Figure 2.

Tip deflection of deploying FGM beam versus time for different gradient indexes ( $L_{0} = 1.8 m, L_{t} = 0.5 m/s, h =9 .5mm$ ).

The influence of deploying speed on the tip deflection response of deploying FGM beam

For the deploying FGM beam with gradient index $p = 1$ , initial length $L_{0} = 1.8 m, h =9 .5 mm$ and four different deploying speeds $L_{t} = 0.5, 0.8, 1.2, 2.0 m / s$ , Figure 3 shows the tip deflection versus time respectively.

Figure 3.

Tip deflection of deploying FGM beam versus time for four different deploying speeds（ $p = 1, L_{0} = 1.8 m, h =9 .5 mm$ ).

It can be seen from Figure 3 that the tip deflection amplitude of deploying FGM beam increases with the increase in the deploying speed, and the beam tends to divergence instability with the increase in the deploying speed. It shows that deploying speed has the most significant influence on the vibration response of the FGM beam.

The influence of initial length on the tip deflection response of deploying FGM beam

For the deploying FGM beam with gradient index $p = 1$ , deploying speed $L_{t} = 0.5 m / s$ , $h = 9.5 mm$ , and three different initial lengths $L_{0} = 0.5, 1.0, 1.5 m$ , the tip deflections versus time are shown in Figure 4, respectively. It can be seen from Figure 4 that the vibrational frequencies and tip deflection amplitudes of the deploying FGM beam decrease with the increase in the initial length. It shows that increase in the initial length can reduce the vibration frequency and amplitude of the beam.

Figure 4.

Tip deflection of deploying FGM beam versus time for three different initial lengths ( $p = 1, L_{t} = 0.5 m / s, h =9 .5 mm$ ).

The influence of the cross-sectional height on the tip deflection response of deploying FGM beam

In the case of $L_{0} = 1.8 m, L_{t} = 0.5 m/s, p =2$ , Figure 5 gives the tip deflection of deploying FGM beam versus time for the cross-sectional height of the FGM beam $h = 10 mm, 15 mm, 20 mm$ , respectively. It can be seen from Figure 5 that the vibrational frequencies of deploying FGM beam increase with the increase in the cross-sectional height of the beam, while the tip deflection amplitudes are almost not affected by the change in the cross-sectional height. So, we can control the vibration frequency of the beam by adjusting the cross-sectional height of the beam.

Figure 5.

Tip deflection of deploying FGM beam versus time for three different cross-sectional heights ( $p = 2, L_{0} = 1.8 m, L_{t} = 0.5 m / s$ ).

Stability analyses

For some parameters $L_{0} = 0.5 m, L_{t} = 10 m / s, h = 10 mm, p = 2$ , the curves of the first three complex frequencies with the protruded length of beam are shown in Figure 6. It can be seen that the imaginary parts of the first three complex frequencies are greater than zero at the beginning stage, and decrease with the increase in the protruded length, eventually tending to zero. Meanwhile, the real parts of the first three complex frequencies increase with the increase in the protruded length, and the real part of the second- and the third-order complex frequencies are less than zero consistently, but the curve of real part of the first-order complex frequency is divided into two curves when $L (t) \geq 2.6651 m$ . With the increase in the protruded length, one of the two curves changes from negative to positive when $L (t) ≍ 3.575 m$ , the other curve remains negative. It is shown that the FGM beam occurs as the first-order divergence instability when $L (t) ≍ 3.575 m$ (we may call the length as the critical length), while the second-order and the third-order mode keeps stable.

Figure 6.

Real part and imaginary part of the first-order three complex frequencies versus the protruded length ( $L_{0} = 0.5 m, L_{t} = 10 m / s, h = 10 mm, p = 2$ ).

The influence of gradient index on the stability of the deploying FGM beam

For $L_{0} = 0.5 m, L_{t} = 10 m / s, h = 9.5 mm$ , Figure 7 shows the curves of the first-order complex frequencies with the protruded length or time of beam for different gradient indexes $p = 0, 1, 5, 10^{6}$ . It can be seen that with the increase in the gradient index, the critical length of the divergence instability and the imaginary part (vibrational frequencies) decrease, and meanwhile, the imaginary part of the first-order complex frequency decrease with the increase in the protruded length. This conclusion agrees well with the one obtained in the section “The influence of gradient index on the tip deflection response of deploying FGM beam”. It shows that more the ceramic components of the FGM beam, the more stabler will be the beam.

Figure 7.

The first-order complex frequencies versus the protruded length for different gradient indexes ( $L_{0} = 0.5 m, L_{t} = 10 m / s, h = 9.5 mm$ ).

The influence of deploying speed on the stability of the deploying FGM beam

For $L_{0} = 1.5 m, h = 10 mm, p = 2$ , Figure 8 shows the curves of the first-order complex frequencies versus the protruded length for different deploying speeds $L_{t} = 10 m/s, 15 m/s, 20 m/s$ . It can be seen that the critical length of divergence instability decreases with the increase in the deploying speed. Therefore, to control the stability of the beam, the deploying speed must be controlled firstly.

Figure 8.

The first-order complex frequencies versus the protruded length for different deploying speeds ( $L_{0} = 1.5 m, h = 10 mm, p = 2$ ).

The influence of initial length on the stability of the deploying FGM beam

For $L_{t} = 10 m/s, h = 10 mm, p = 2$ , Figure 9 shows the curves of the first complex frequencies versus the protruded length for different initial lengths $L_{0} = 0.5 m, 1.0 m, 1.5 m$ . It can be seen that the critical length of divergence instability decreases with the increase in the initial length. In other words, the longer the initial length is, the more unstable will be the beam.

Figure 9.

The first-order complex frequencies versus the protruded length for different initial lengths ( $L_{t} = 10 m / s, p = 2, h = 10 mm$ ).

The influence of the cross-sectional height on the stability of the deploying FGM beam

For $L_{0} = 0.5 m, L_{t} = 10 m / s, p = 2$ , Figure 10 shows the curves of the first-order complex frequencies versus the protruded length for different cross-sectional heights $h = 10 mm, 15 mm, 20 mm$ . It can be seen that the critical length of divergence instability increases with the increase in the cross-sectional height h. Therefore, increasing the cross-sectional height of the beam can improve the stability of the beam.

Figure 10.

The first-order complex frequencies versus the protruded length for different cross-sectional heights ( $L_{0} = 0.5 m, L_{t} = 10 m / s, p = 2$ ).

Conclusion

In Euler coordinate system, a new coupled dynamic model is developed to study the vibration characteristics and stability of the deploying rectangular cross-section cantilever beam made of FGM, and the coupled differential equations of motion of the system, which includes the axial displacement and transverse displacement that are derived based on Hamilton’s principle. In this model, the corresponding equivalent parameters of FGM continuously vary across the cross-sectional height with a simple power law, where the deploying FGM beam becomes homogenous ceramic or metal beam when gradient index approaches zero or infinity. In view of the complicacy of the equations due to variable length and both coupled displacements, only transverse bending displacement is considered in the equations, which involves the variable coefficients (time variable and coordinate variable) and tension and bend coupling effect. Galerkin’s method is employed to discretize the differential equation of motion to a set of time coordinate-dependent first-order ordinary differential equations with time-variable coefficient. By using Runge–Kutta method and solving the generalized complex eigenequation, the tip deflection response, complex frequencies, and stability of deploying FGM beam can obtained. For the deploying FGM beam with constant speed, the effects of gradient index of FGM, deploying speed, initial length, protruded length, and the cross-sectional height on tip deflection response and stability of deploying FGM beam are discussed. The conclusions are as follows: (1) gradient index has greater influence on the vibration frequency and has less influence on the tip deflection amplitude of deploying FGM beam. Reducing the metal components of the FGM beam, the vibration frequency of the beam will increase, but the stability is better. (2) The deploying speed of the FGM beam has obvious influence on its vibration response and stability. Therefore, to control the stability of the beam, the deploying speed must be controlled firstly. (3) The increase in the initial length of the deploying FGM beam will decrease the vibration frequency and amplitude of the beam, but it will cause divergence instability of the beam. (4) The increase in the cross-sectional height of the deploying FGM beam can cause the vibration frequency to increase, but has little effect on the amplitude. Increasing the cross-sectional height will improve the stability of the beam.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (No. 11472211) and Shaanxi Science and Technology Research Projects (No.2015GY004).

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