Functionally graded beam
Consider a doubly symmetric cross-section functionally graded beam with length, L, width, w
0, and height, h
0, at the root of the beam and width, w, and height, h, at the tip of the beam as shown in Fig. 1. The graded material properties vary symmetrically along thickness direction from the core towards the surface according to a power law:
$$ {P}_{(z)}={P}_{(m)}+\left({P}_{(c)}-{P}_{(m)}\right){\left|\frac{2\times z}{h}\right|}^n $$
(1)
where P
(z) represents an effective material property (i.e., density, P
(z), or Young’s modulus, E
(z)) and P
(m) and P
(c) are metallic and ceramic properties, respectively.
The volume fraction exponent or power law index, n, is a variable whose value is greater than or equal to zero and the variation in properties of the beam depends on its magnitude. Structure is constructed with functionally graded material with ceramic rich at top and bottom surfaces (at z = h/2 and −h/2) with protecting metallic core (at z = 0).
Equations of motion
For the beam considered in this study, the equations of motion are obtained with the following assumptions:
-
1.
There is a smooth and continuous variation in properties along thickness direction.
-
2.
The neutral and centroidal axes in the cross section of the rotating beam coincide so that effects due to eccentricity and torsion can be negligible.
-
3.
Cross section of the beam varies uniformly along its length with the uniform taper.
-
4.
Shear and rotary inertia effects of the beam are neglected due to slender shape of the beam.
Figure 1 shows the deformation of the neutral axis of a beam fixed to a rigid hub rotating about the axis z. The tapered beam is attached to a rigid hub which rotates with constant angular speed and no external force acts on the beam. The beam rotates at an angular speed of Ω (t) about the z-axis. In the present work, a hybrid set of Cartesian variable w and a non-Cartesian variable s is approximated by spatial functions and corresponding coordinates are employed to derive the equations of motion. The position of a generic point on the neutral axis of the double-tapered beam before deformation located at P
0 changes to P after deformation and its elastic deformation is denoted as \( \widehat{d} \) that has two components in two-dimensional space.
Approximation of deformation variables
Using the Rayleigh-Ritz assumed mode method, the deformation variables are approximated as
$$ s\left(x,t\right)={\displaystyle \sum_{j=1}^{\mu_1}{\phi}_{1j}(x){q}_{1j}(t)} $$
(2)
and
$$ w\left(x,t\right)={\displaystyle \sum_{j=1}^{\mu_3}{\phi}_{3j}(x){q}_{3j}(t)} $$
(3)
In the above equations, ϕ
1j
and ϕ
3j
are the assumed modal functions (test functions) for s and w, respectively. Any compact set of functions which satisfies the essential boundary conditions of the cantilever beam can be used as the test functions. The q
ij
s are the generalized coordinates and μ
1 and μ
3 are the number of assumed modes used for s and w, respectively. The total number of modes, μ, is equal to the sum of individual modes, i.e., μ = μ
1
+ μ
3
.
The geometric relation between the arc length stretch s and Cartesian variables u and w presented by Yoo and Shin (1998) as
$$ s=u+\frac{1}{2}{\displaystyle \underset{0}{\overset{x}{\int }}\left[{\left({w}^{\hbox{'}}\right)}^2\right]}\kern0.5em d\sigma $$
(4)
where a symbol with a prime (′) represents the partial derivative of the symbol with respect to the integral domain variable.
Strain energy of the system
The total elastic strain energy of the beam considering the assumptions given in “Functionally graded beam” section can be written as
$$ {E}_s=\frac{1}{2}E(z)A(x){\displaystyle \underset{L}{\int }{\left(\frac{ds}{dx}\right)}^2dx+\frac{1}{2}E(z){I}_y(x){\displaystyle \underset{L}{\int }{\left(\frac{d^2w}{d{x}^2}\right)}^2dx\left(\mathrm{i}\kern0.5em =\kern0.5em 1,\kern0.5em 2,\upmu \right)}} $$
(5)
where I
y(x) is the second moment of area of the tapered beam.
For the beam tapers in two planes, the general parameters of the beam are given by
$$ h(x)={h}_o\left(1-{\tau}_h\frac{x}{L}\right) $$
(6)
$$ w(x)={w}_o\left(1-{\tau}_w\frac{x}{L}\right) $$
(7)
$$ A(x)={A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right) $$
(8)
$$ {I}_y(x)={I}_{0y}\left(1-{\tau}_w\frac{x}{L}\right){\left(1-{\tau}_h\frac{x}{L}\right)}^3 $$
(9)
where τ
w
and τ
h
are the width taper ratio and height taper ratio respectively and are defined as
$$ {\tau}_w=\left(1-\frac{w}{w_0}\right) $$
(10)
$$ {\tau}_h=\left(1-\frac{h}{h_0}\right) $$
(11)
where A
0 and I
0y
are area of cross section and area moment of inertia of the beam at the root of the tapered beam respectively and are defined as
$$ {A}_0={w}_0{h}_0 $$
(12)
$$ {I}_{0y}=\frac{b_0{w}_0^3}{12} $$
(13)
Kinetic energy of the system
The velocity of a generic point P can be obtained as
$$ {\overrightarrow{v}}^P={\overrightarrow{v}}^O+\frac{{}^Ad\overrightarrow{p}}{dt}+{\overrightarrow{\omega}}^A\times \overrightarrow{p} $$
(14)
where \( {\overrightarrow{v}}^O \) is the velocity of point O that is a reference point identifying a point fixed in the rigid frame A; \( {\overrightarrow{\omega}}^o \) vector \( \overrightarrow{P} \) in the reference frame A and the terms \( \overrightarrow{P} \), \( {\overrightarrow{v}}^o \), and \( {\overrightarrow{\omega}}^A \) can be expressed as follows
$$ \overrightarrow{p}=\left(x+u\right)\widehat{i}+v\widehat{j} $$
(15)
$$ {\overrightarrow{v}}^O=r\varOmega \widehat{j}; $$
(16)
$$ {\overrightarrow{\omega}}^A=\varOmega \widehat{k} $$
(17)
$$ {\overrightarrow{v}}^p=\left(\dot{u}-\varOmega v\right)\widehat{i}+\left[\dot{v}+\varOmega \left(r+x+u\right)\right]\widehat{j} $$
(18)
where \( \widehat{\iota} \), ĵ, and \( \widehat{k} \) are orthogonal unit vectors fixed in A, r is the distance from the axis of rotation to point O (i.e., radius of the rigid frame), and Ω is the angular speed of the rigid frame. Using Eq. 14, the kinetic energy of the rotating beam is derived as
$$ {E}_k=\frac{1}{2}{\displaystyle \underset{0}{\overset{L}{\int }}\rho (z)}A(x){\overrightarrow{v}}^p.{\overrightarrow{v}}^pdx $$
(19)
In which, A
(x) is the area of cross section of the tapered beam, P
(z) is the mass density, and L is the length of the beam.
Equation of motion
Substituting Eqs. 2 and 3 into Eqs. 5 and 19 and taking partial derivatives of E
k
and E
s
with respect to q
ij
and q
ij
, neglecting the higher order non-linear terms, the equation of motion for the tapered beam is formulated using the Lagrange’s method. The Lagrange’s equation of motion for free vibration of distributed parameter system can be written as
$$ \frac{d}{dt}\left(\frac{\partial {E}_k}{\partial \dot{q}i}\right)-\frac{\partial {E}_k}{\partial qi}+\frac{\partial {E}_s}{\partial qi}=0\kern0.5em \mathrm{i}\kern0.5em =\kern0.5em 1,2,3\kern0.5em \mu $$
(20)
The linearized equations of motion can be obtained as follows
$$ \begin{array}{l}{\displaystyle \sum_{j=1}^{\mu_1}\left[\left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{1i}{\phi}_{1j}dx}\right){\ddot{q}}_{1j}-2\varOmega \left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{1i}{\phi}_{2j}dx}\right){\dot{q}}_{ij}\right.}\\ {}-{\varOmega}^2\left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{1i}{\phi}_{1j}dx}\right){q}_{1j}+\left({E}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{I}_{0z}{\left(1-{\tau}_b\frac{x}{L}\right)}^3\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{1i}^{\hbox{'}}{\phi}_{1j}^{\hbox{'}}dx}\right){q}_{1j}\\ {}={\varOmega}^2{\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right)x{\phi}_{1i}dx}+{\varOmega}^2\rho (z){\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{1i}dx}\end{array} $$
(21)
$$ \begin{array}{l}{\displaystyle \sum_{j=1}^{\mu_3}\left[\left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right){\phi}_{3i}{\phi}_{3j}dx}\right){\ddot{q}}_{3j}+\left({E}_{(z)}{I}_{0z}{\displaystyle \underset{0}{\overset{L}{\int }}\left(1-{\tau}_b\frac{x}{L}\right){\left(1-{\tau}_h\frac{x}{L}\right)}^3{\phi}_{3i}^{\hbox{'}\hbox{'}}{\phi}_{3j}^{\hbox{'}\hbox{'}}dx}\right)\right.}{q}_{3j}\\ {}+{\varOmega}^2\left\{r\left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right)\left(L-x\right){\phi}_{3i}^{\hbox{'}}{\phi}_{3j}^{\hbox{'}}dx}\right)\right.{q}_{3j}\\ {}\left.+\left.\left({\rho}_{(z)}{\displaystyle \underset{0}{\overset{L}{\int }}{A}_0\left(1-{\tau}_w\frac{x}{L}\right)\left(1-{\tau}_h\frac{x}{L}\right)\frac{1}{2}\left({L}^2-{x}^2\right){\phi}_{3i}^{\hbox{'}}{\phi}_{3j}^{\hbox{'}}dx}\right){q}_{3j}\right\}\right]=0\end{array} $$
(22)
where a symbol with double prime (″) represents the second derivative of the symbol with respect to the integral domain variable.
Dimensionless transformation
For the analysis, Eqs. 21 and 22 may be obtained in dimensionless form by introducing following dimensionless variables in to the equations:
$$ \tau =\frac{t}{T} $$
(23)
$$ \xi =\frac{x}{L} $$
(24)
$$ {\theta}_j=\frac{q_j}{L} $$
(25)
$$ \delta =\frac{r}{L} $$
(26)
$$ \gamma =T\varOmega $$
(27)
where τ, δ, and γ refer to dimensionless time, hub radius ratio, and dimensionless angular speed, respectively.
