4. Optimization of FGM drive shafts against torsional buckling and whirling

One of the important design issues in mechanical industries is the buckling and whirling instabilities that may arise from the loads applied to a power transmission shaft. These instabilities result in a reduced control of the vehicle, undesirable performance, and often cause damage, sometimes catastrophic, to the vehicle structure. Therefore, by incorporating such considerations into an early design optimization [29], the design space of a power transmission shaft will be reduced such that undesirable instability effects can be avoided during the range of the vehicle's mission profile. Figure 8 shows an idealized structural model of a long, slender composite shaft having circular thin-walled cross section. The main structure is constructed solely of functionally graded, fibrous composite materials. The laminate coordinates are defined by x parallel to the shaft axis, y points to the tangential direction, and z points to the radial direction. Predictions of both torsional buckling and whirling instabilities are based on simplified analytical solutions of equivalent beam and shell structures. The coupling between bending and torsional deformations, introduced by the composite construction, and its influence on such instabilities is considered.

#### 4.1. Torsional buckling optimization problem

Solutions obtained by applying the power-law model of Eq. (3) have shown that no improvements can be achieved using grading of the fiber volume fraction in the thickness direction. On the other hand, grading in spanwise direction has shown some interesting results. Considering spanwise grading according to Eq. (4), Figure 7 depicts the level curves of the fundamental

observed that the feasible domain is bounded from below and above by the constraint curves corresponding to the upper and lower bounds imposed on the fiber volume fractions at tip and root. The horizontal line Δ<sup>f</sup> = 1.0 (i.e., Vf = 50% at root and tip) split the domain into two zones.

The lower zone encompasses the constrained optimum solution: ( ffiffiffiffiffiffi

<sup>p</sup> combined with the mass constraint in the design space <sup>Δ</sup><sup>f</sup> ; <sup>p</sup> � �. It is

<sup>p</sup> function augmented with <sup>M</sup>^ <sup>s</sup> <sup>¼</sup> 1 in <sup>Δ</sup><sup>f</sup> ; <sup>p</sup> � � design space (Ns = NL = 1) with spanwise

ω<sup>1</sup> � � p

max, gain %

ω^ <sup>1</sup>

<sup>p</sup> )max <sup>=</sup> 2.01875 at the

frequency parameter ffiffiffiffiffiffi

172 Optimum Composite Structures

Figure 7. Level curves of ffiffiffiffiffiffi

Spanwise grading (Eq. 5)

grading "Eq. 4."

ω^ <sup>1</sup>

Table 3. Optimal solutions using different grading patterns (M^ <sup>s</sup> = 1).

Vf—power-law model (Δf, p)opt., ffiffiffiffiffiffi

Thickness grading (Eq. (3)) (1.0, 0.0), 1.8751 0.0% Spanwise grading (Eq. (4)) (0.34, 1.01), 2.01875, 7.66%

n = 1 (0.34, 1.02), 2.01938, 7.70% n = 2 (0.34, 2.425), 2.04813, 9.23% n = 3 (0.34, 5.175), 2.06125, 9.93%

design point (Δf, P)opt. = (0.34, 1.01).

ω^ <sup>1</sup>

Bert and Kim [30] derived the governing differential equations of torsional buckling in the form:

$$\begin{aligned} N\_{x,x} + N\_{yx,y} - 2T u\_{,xy} &= 0\\ N\_{xy,x} + N\_{y,y} + \left( M\_{xy,x}/R \right) + \left( M\_{y,y}/R \right) - 2T \left( v\_{,y} + w\_{,x}/R \right) &= 0\\ M\_{x,xx} + \left( M\_{xy} + M\_{yx} \right)\_{,xy} + M\_{y,yy} - N\_y/R + 2T \left( v\_{,x}/R - w\_{,xy} \right) &= 0 \end{aligned} \tag{18}$$

where Nx and Ny are the normal forces, Nxy and Nyx are shear forces, Mx and My are bending moments, and Mxy and Myx are torsional moments. All are applied to the midsurface and measured per unit wall thickness of the shaft. T is the applied torque, R is the mean radius, and (u, v, w) are the displacements of a generic point on the middle surface of the shaft wall. An iterative process is outlined in Ref. [30] for calculating the buckling torque for specified boundary conditions. There are other simple empirical equations based on experimental studies that can give a reasonable estimate of the buckling torque. The most commonly used formula for the case of simply supported shaft is [31]:

$$T\_{cr} = \left(2\pi R^2 H\right) \left(0.272\right) \left(E\_x\right)^{0.25} \left(E\_y\right)^{0.75} \left(H/R\right)^{1.5} \tag{19}$$

where Tcr is the critical buckling torque and H is the total wall thickness of the shaft. Expressions of the equivalent modulii of elasticity in the axial (Ex) and hoop (Ey) directions for

the shaft are restricted by the available interior space of the vehicle and will be considered as preassigned parameters in the present model formulation. The first case study to be examined herein is a shaft with discrete thickness grading constructed from eight plies (�θ �θ)s with the same properties of carbon/epoxy composites (see Table 2) and same thicknesses. This sequence is applied in filament wound circular shells, as such a process demands adjacent (�θ) layers. Figure 9 depicts the obtained contours in the (Vf1-θ) design space, which are, as seen, monotonic and symmetric about the zero ply angle. A local maximum of T^cr can be observed near the

studies including both discrete and continuous grading with several optimal solutions can be found in Ref. [31]. At the start of the optimization process in each case, the shaft wall was divided into a large number of layers with equal thicknesses, for example, NL = 32. It has been found that the optimization algorithm treats the number of layers as an additional implicit variable. Sometimes the computer discards one or more layers by letting their thicknesses sink to the lower limits and sometimes makes some consecutive layers identical, that is, having the same fiber orientation and volume fraction. Such a situation was repeated for many cases of study. It was found that the appropriate number to be taken for the shaft problem under consideration is NL = 8. This would eliminate much of the numerical effort necessary for performing structural analysis in each optimization cycle and, consequently, reduces the com-

fraction in the eight layers reached its upper value of 70%. The optimal dimensionless ply

Figure 9. <sup>T</sup>^cr—contours in (Vf1-θ) design space under mass constraint <sup>M</sup>^ <sup>¼</sup> 1. (Case of drive shaft with eight symmetric,

buckling torque can be achieved when the fiber orientation angle is close to 90o

The final attained optimal solution was a cross ply layup [900

) with T^cr = 1. This figure illustrates that the maximum critical

Optimization of Functionally Graded Material Structures: Some Case Studies

http://dx.doi.org/10.5772/intechopen.82411

/00

]4 with the fiber volume

. Other case

175

design point (Vf1,θ)=(0.7, 90o

putational time considerably.

balanced, carbon/ epoxy layers)

Figure 8. Shaft model and definition of reference axes.

symmetric and balanced laminates are given in Ref. [31]. The various parameters and variables are normalized with respect to known baseline design, which is constructed from cross-ply laminates [0o /900 ]<sup>N</sup> with equal volume fractions of the fibers and matrix materials, that is, Vf = Vm = 50%. Optimized shaft designs shall have the same transmitted power, length, outer diameter, boundary conditions, and material properties of those known for the baseline design. The different dimensionless quantities are defined in Ref. [31]. The optimal torsional buckling problem is to find the design variables vector X ! <sup>¼</sup> Vf ; <sup>θ</sup>; ^<sup>h</sup> � � k¼1, 2,…NL , which minimizes the objective function:

$$\begin{array}{ll}\text{Minimize} & F = -\hat{T}\_{cr} \\ \text{subject to} & \text{mass limitation}: & \hat{M} - 1 \le 0 \end{array} \tag{20}$$

Torsional strength : <sup>τ</sup>max <sup>τ</sup>allow � � � <sup>1</sup>:<sup>0</sup> <sup>≤</sup> <sup>0</sup> Whirling : <sup>Ω</sup>^ max � <sup>Ω</sup>^ cr <sup>≤</sup> <sup>0</sup> (21)

$$\begin{aligned} \text{Side constraints}: \quad \left(0.30, -\frac{\pi}{2}, 0.015\right) \leq \left(V\_f, \Theta, \hat{h}\right)\_{k=1,2,\dots,N\_\mathbb{L}} \leq \left(0.70, \frac{\pi}{2}, 0.20\right) \\\ 0.75 \leq \sum\_{k=1}^{N\_\mathbb{L}} \hat{h}\_k \leq 1.25 \end{aligned} \tag{22}$$

where <sup>T</sup>^cr <sup>¼</sup> Tcr=Tcro, <sup>Ω</sup>^ <sup>¼</sup> ð Þ <sup>Ω</sup> <sup>∗</sup> <sup>2</sup><sup>π</sup> <sup>=</sup> <sup>60</sup>ω1, <sup>o</sup> � � are the dimensionless critical torque and rotational speed, respectively. The baseline design parameters are denoted by subscript "o." <sup>τ</sup>max <sup>¼</sup> Tmax=2πR<sup>2</sup> H � � is the maximum shear stress, Tmax is the maximum applied torque, and τallow is the allowable shear stress that can be calculated according to the embedded material properties and volume fraction of the fiber [23]. This optimization problem may be thought as a search in an (3NL) dimensional space for a point corresponding to the minimum value of the objective function and such that it lies within the region bounded by subspaces representing the constraint functions. It must be noted that the outside dimensions (outer diameter and length) of the shaft are restricted by the available interior space of the vehicle and will be considered as preassigned parameters in the present model formulation. The first case study to be examined herein is a shaft with discrete thickness grading constructed from eight plies (�θ �θ)s with the same properties of carbon/epoxy composites (see Table 2) and same thicknesses. This sequence is applied in filament wound circular shells, as such a process demands adjacent (�θ) layers. Figure 9 depicts the obtained contours in the (Vf1-θ) design space, which are, as seen, monotonic and symmetric about the zero ply angle. A local maximum of T^cr can be observed near the design point (Vf1,θ)=(0.7, 90o ) with T^cr = 1. This figure illustrates that the maximum critical buckling torque can be achieved when the fiber orientation angle is close to 90o . Other case studies including both discrete and continuous grading with several optimal solutions can be found in Ref. [31]. At the start of the optimization process in each case, the shaft wall was divided into a large number of layers with equal thicknesses, for example, NL = 32. It has been found that the optimization algorithm treats the number of layers as an additional implicit variable. Sometimes the computer discards one or more layers by letting their thicknesses sink to the lower limits and sometimes makes some consecutive layers identical, that is, having the same fiber orientation and volume fraction. Such a situation was repeated for many cases of study. It was found that the appropriate number to be taken for the shaft problem under consideration is NL = 8. This would eliminate much of the numerical effort necessary for performing structural analysis in each optimization cycle and, consequently, reduces the computational time considerably.

symmetric and balanced laminates are given in Ref. [31]. The various parameters and variables are normalized with respect to known baseline design, which is constructed from cross-ply

Vf = Vm = 50%. Optimized shaft designs shall have the same transmitted power, length, outer diameter, boundary conditions, and material properties of those known for the baseline design. The different dimensionless quantities are defined in Ref. [31]. The optimal torsional buckl-

!

subject to mass limitation : <sup>M</sup>^ � <sup>1</sup> <sup>≤</sup> <sup>0</sup> (20)

≤ Vf ; θ; ^h � �

� 1:0 ≤ 0

<sup>2</sup> ; <sup>0</sup>:<sup>015</sup> � �

^hk ≤ 1:25

tional speed, respectively. The baseline design parameters are denoted by subscript "o."

τallow is the allowable shear stress that can be calculated according to the embedded material properties and volume fraction of the fiber [23]. This optimization problem may be thought as a search in an (3NL) dimensional space for a point corresponding to the minimum value of the objective function and such that it lies within the region bounded by subspaces representing the constraint functions. It must be noted that the outside dimensions (outer diameter and length) of

H � � is the maximum shear stress, Tmax is the maximum applied torque, and

]<sup>N</sup> with equal volume fractions of the fibers and matrix materials, that is,

<sup>¼</sup> Vf ; <sup>θ</sup>; ^<sup>h</sup> � �

k¼1, 2,…NL

k¼1,2,…NL

� � are the dimensionless critical torque and rota-

≤ 0:70;

π <sup>2</sup> ; <sup>0</sup>:<sup>20</sup> � �

, which minimizes the

(21)

(22)

laminates [0o

174 Optimum Composite Structures

objective function:

<sup>τ</sup>max <sup>¼</sup> Tmax=2πR<sup>2</sup>

/900

Figure 8. Shaft model and definition of reference axes.

ing problem is to find the design variables vector X

Minimize <sup>F</sup> ¼ �T^cr

Whirling : <sup>Ω</sup>^ max � <sup>Ω</sup>^ cr <sup>≤</sup> <sup>0</sup>

0:75 ≤

τallow � �

> X NL

k¼1

Torsional strength : <sup>τ</sup>max

Side constraints : <sup>0</sup>:30; � <sup>π</sup>

where <sup>T</sup>^cr <sup>¼</sup> Tcr=Tcro, <sup>Ω</sup>^ <sup>¼</sup> ð Þ <sup>Ω</sup> <sup>∗</sup> <sup>2</sup><sup>π</sup> <sup>=</sup> <sup>60</sup>ω1, <sup>o</sup>

The final attained optimal solution was a cross ply layup [900 /00 ]4 with the fiber volume fraction in the eight layers reached its upper value of 70%. The optimal dimensionless ply

Figure 9. <sup>T</sup>^cr—contours in (Vf1-θ) design space under mass constraint <sup>M</sup>^ <sup>¼</sup> 1. (Case of drive shaft with eight symmetric, balanced, carbon/ epoxy layers)

thickness was found to be [0.1994, 0.0967, 0.152, 0.019]s at which the shaft torsional buckling capacity was increased by 32.1% above that of the baseline design. However, the total structural mass has reached its baseline value, and whirling constraint became active at the achieved optimum design point.

where λ ¼ nπ=L, ω = circular natural frequency, and n = mode number. For each natural frequency of the nonrotating shaft, the rotational speed (Ω) develops gyroscopic moments, which cause the natural frequency to bifurcate into two. The higher of the two increases with Ω and is associated with forward precession, while the lower one decreases with Ω and is associated with backward precession. A critical instability occurs when the rotational speed coincides with the first backward-precision natural frequency, which is termed as the first critical speed. Two alternatives may be considered regarding the whirling optimization prob-

k¼1, 2,…NL

τmax <sup>τ</sup>allow � <sup>1</sup>:<sup>0</sup> <sup>≤</sup> <sup>0</sup>

Tmax Tcr, <sup>o</sup> 

Minimize <sup>F</sup> ¼ �Ω^ cr

Minimize <sup>F</sup> <sup>¼</sup> <sup>Ω</sup>^ cr � <sup>Ω</sup>^ <sup>∗</sup> <sup>2</sup>

The same set of constraints given in Eq. (23) is applied. The notation <sup>Ω</sup>^ <sup>∗</sup> is a dimensionless target rotational speed, which should be greater than the maximum permissible rotational speed by a reasonable margin (e.g., 10–20%). As a case study, a drive shaft with continuous material grading along the shaft axis is optimized considering the following power-law model:

Vfð Þ¼ <sup>x</sup>^ Vfð Þþ <sup>0</sup>:<sup>5</sup> Vfð Þ� <sup>0</sup> Vfð Þ <sup>0</sup>:<sup>5</sup> ð Þ <sup>1</sup> � <sup>2</sup>j j <sup>x</sup>^ <sup>n</sup> ð Þ<sup>P</sup>

where Vf (0) is the fiber volume fraction at the right or left end of the drive shaft, while Vf (0.5) is the fiber volume fraction at the middle of the shaft length. Figure 10 illustrates the level curves of the normalized critical speed augmented with the mass equality constraint. It is seen that there are four distinct zones separated by the contour lines <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:0. The upper left zone and lower right zone contain local maximum solutions. The best point (p, Δf) = (4.53, 0.3), corresponding to <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:045, is located inside the zone where the fiber taper ratio <sup>Δ</sup><sup>f</sup> <sup>¼</sup> Vfð Þ0 =Vfð Þ 0:5 is less than one. The upper empty zone contains infeasible solutions that violate the imposed constraints. Another case study considers the through-thickness grading pattern

, which minimizes the objective func-

http://dx.doi.org/10.5772/intechopen.82411

(29)

177

(30)

, (31)

Subject to <sup>M</sup>^ � <sup>1</sup> <sup>≤</sup> <sup>0</sup> (28)

Optimization of Functionally Graded Material Structures: Some Case Studies

� <sup>T</sup>^cr <sup>≤</sup> <sup>0</sup>

lem [34]:

tion:

(a) Direct maximization of the critical rotational speed

The other alternative of the objective function is defined by:

!

<sup>¼</sup> Vf ; <sup>θ</sup>; ^<sup>h</sup> 

Find the design variables vector X

(b) Placement of the critical speed

#### 4.2. Whirling optimization problem

The calculation of the critical speed, also referred to as whirl instability of a rotating shaft, is based on the work given in Refs. [32, 33]. The critical speed is defined as the point at which the spinning shaft reaches its first natural frequency. The shaft is modeled as a Timoshenko beam, which implies that first-order shear deformation theory with rotatory inertia and gyroscopic action was used. The shaft is assumed to be pinned at both ends with a Cartesian coordinate system (x, y, z), where x is measured along the longitudinal axis of the shaft. The displacements in the y and z directions are denoted by v and w, respectively, and Φ is the angle of twist. The cross-sectional area, second moment of area, and polar moment of area are denoted by A, I, and J, respectively. The equations of motion are derived by invoking Hamilton's principle, with the following results [32]:

$$\mathrm{C\_5B\frac{\partial^2 v}{\partial X^4} + \rho A \frac{\partial^2 v}{\partial t^2} - \left(\rho I + \frac{\mathrm{C\_B}}{\mathrm{C\_S}} \rho A\right) \frac{\partial^4 v}{\partial X^2 \partial t^2} - 2\rho I \Omega \left(\frac{\partial^3 w}{\partial x^2 \partial t} - \frac{\rho A}{\mathrm{C\_S}} \frac{\partial^3 w}{\partial t^3}\right) + \frac{\rho I \rho A}{\mathrm{C\_S}} \frac{\partial^4 v}{\partial t^4} + \left(\mathrm{C\_{B7}/2}\right) \frac{\partial^3 \phi}{\partial x^3} = 0 \tag{23}$$

$$\mathbf{C}\_{B}\frac{\partial^{4}w}{\partial x^{4}} + \rho A \frac{\partial^{2}w}{\partial t^{2}} - \left(\rho I + \frac{\mathbf{C}\_{B}}{\mathbf{C}\_{S}}\rho A\right)\frac{\partial^{4}w}{\partial x^{2}\partial t^{2}} + 2\rho I\Omega \left(\frac{\partial^{3}v}{\partial x^{2}\partial t} - \frac{\rho A}{\mathbf{C}\_{S}}\frac{\partial^{3}v}{\partial t^{3}}\right) + \frac{\rho I\rho A}{\mathbf{C}\_{S}}\frac{\partial^{4}w}{\partial t^{4}} + \left(\mathbf{C}\_{BT}/2\right)\frac{\partial^{3}\phi}{\partial x^{3}} = 0\tag{24}$$

$$(\mathbf{C}\_{\mathrm{BT}}/2)\left(\frac{\partial^3 v}{\partial \mathbf{x}^3} + \frac{\partial^3 w}{\partial \mathbf{x}^3}\right) - (\mathbf{C}\_{\mathrm{BT}}/2)\frac{\rho \mathbf{A}}{\mathbf{C}\_S} \left(\frac{\partial^3 v}{\partial \mathbf{x} \partial \mathbf{t}^2} + \frac{\partial^3 w}{\partial \mathbf{x} \partial \mathbf{t}^2}\right) + C\_T \frac{\partial^2 \phi}{\partial \mathbf{x}^2} - \rho \mathbf{J} \frac{\partial^2 \phi}{\partial \mathbf{t}^2} = \mathbf{0} \tag{25}$$

The symbol t denotes time and Ω the rotational speed. The rA, rI, and rJ terms account for translational, rotary, and torsional inertias, respectively, while the 2rI terms account for the gyroscopic inertia effects. It is assumed that the flexural and bending-twisting coupling rigidities (CB and CBT) associated with bending about the y and z axes are identical; likewise for the transverse shear stiffness (Cs) [32]. Bert and Kim [33] considered the case of simply supported shaft and assumed separable solution in space and time to solve the associated eigenvalue problem. The derived frequency equation is given by:

$$\begin{aligned} \left(\mathsf{C}\_{11}^{2} - \mathsf{C}\_{12}^{2}\right)\mathsf{C}\_{33} - 2\mathsf{C}\_{11}\mathsf{C}\_{13}\mathsf{C}\_{31} &= 0\\ \mathsf{C}\_{11} = \mathsf{C}\_{\beta}\lambda^{4} - \left(\rho I + \frac{\mathsf{C}\_{B}}{\mathsf{C}\_{S}}\rho A\right)\omega^{2}\lambda^{2} + \left(\frac{\rho I \rho A}{\mathsf{C}\_{S}}\omega^{2} - \rho A \omega^{2}\right) \end{aligned} \tag{26}$$

$$\begin{aligned} \mathbf{C}\_{12} &= 2\rho l \Omega \omega \Big(\lambda^2 - \frac{\rho A}{\mathbf{C}\_S} \omega^2\Big); \mathbf{C}\_{13} = (\mathbf{C}\_{\text{BT}}/2)\lambda^3\\ \mathbf{C}\_{31} &= \mathbf{C}\_{13} - (\mathbf{C}\_{\text{BT}}/2)(\rho A/\mathbf{C}\_S)\lambda \omega^2; \quad \mathbf{C}\_{33} = \mathbf{C}\_T \lambda^2 - \rho l \omega^2 \end{aligned} \tag{27}$$

where λ ¼ nπ=L, ω = circular natural frequency, and n = mode number. For each natural frequency of the nonrotating shaft, the rotational speed (Ω) develops gyroscopic moments, which cause the natural frequency to bifurcate into two. The higher of the two increases with Ω and is associated with forward precession, while the lower one decreases with Ω and is associated with backward precession. A critical instability occurs when the rotational speed coincides with the first backward-precision natural frequency, which is termed as the first critical speed. Two alternatives may be considered regarding the whirling optimization problem [34]:

(a) Direct maximization of the critical rotational speed

Find the design variables vector X ! <sup>¼</sup> Vf ; <sup>θ</sup>; ^<sup>h</sup> k¼1, 2,…NL , which minimizes the objective function:

$$\begin{array}{ll}\text{Minimize} & F = -\hat{\mathcal{Q}}\_{cr} \\ \text{Subject to} & \hat{M} - 1 \le 0 \end{array} \tag{28}$$

$$\begin{aligned} \left(\frac{\tau\_{\text{max}}}{\tau\_{\text{allow}}}\right) - 1.0 &\le 0\\ \left(\frac{T\_{\text{max}}}{T\_{cr,o}}\right) - \hat{T}\_{cr} &\le 0 \end{aligned} \tag{29}$$

(b) Placement of the critical speed

thickness was found to be [0.1994, 0.0967, 0.152, 0.019]s at which the shaft torsional buckling capacity was increased by 32.1% above that of the baseline design. However, the total structural mass has reached its baseline value, and whirling constraint became active at the

The calculation of the critical speed, also referred to as whirl instability of a rotating shaft, is based on the work given in Refs. [32, 33]. The critical speed is defined as the point at which the spinning shaft reaches its first natural frequency. The shaft is modeled as a Timoshenko beam, which implies that first-order shear deformation theory with rotatory inertia and gyroscopic action was used. The shaft is assumed to be pinned at both ends with a Cartesian coordinate system (x, y, z), where x is measured along the longitudinal axis of the shaft. The displacements in the y and z directions are denoted by v and w, respectively, and Φ is the angle of twist. The cross-sectional area, second moment of area, and polar moment of area are denoted by A, I, and J, respectively. The equations of motion are derived by invoking Hamilton's principle,

achieved optimum design point.

176 Optimum Composite Structures

with the following results [32]:

v ∂t

w ∂t

ð Þ CBT=<sup>2</sup> <sup>∂</sup><sup>3</sup>

<sup>2</sup> � rI þ

<sup>2</sup> � rI þ

v <sup>∂</sup>x<sup>3</sup> <sup>þ</sup>

CB CS rA ∂<sup>4</sup>

CB CS rA ∂<sup>4</sup>

∂3 w ∂x<sup>3</sup> 

problem. The derived frequency equation is given by:

<sup>C</sup><sup>11</sup> <sup>¼</sup> CBλ<sup>4</sup> � <sup>r</sup><sup>I</sup> <sup>þ</sup>

<sup>C</sup><sup>12</sup> <sup>¼</sup> <sup>2</sup>rIΩω λ<sup>2</sup> � <sup>r</sup><sup>A</sup>

C2 <sup>11</sup> � <sup>C</sup><sup>2</sup> 12

v ∂X<sup>2</sup> ∂t

w ∂x<sup>2</sup> ∂t

� ð Þ CBT <sup>=</sup><sup>2</sup> <sup>r</sup><sup>A</sup>

<sup>2</sup> � <sup>2</sup>rI<sup>Ω</sup> <sup>∂</sup><sup>3</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>rI<sup>Ω</sup> <sup>∂</sup><sup>3</sup>

CS

w ∂x<sup>2</sup> ∂t � <sup>r</sup><sup>A</sup> CS ∂3 w ∂t<sup>3</sup>

v ∂x<sup>2</sup> ∂t � <sup>r</sup><sup>A</sup> CS ∂3 v ∂t 3

∂3 v ∂x∂t <sup>2</sup> þ ∂3 w ∂x∂t 2

The symbol t denotes time and Ω the rotational speed. The rA, rI, and rJ terms account for translational, rotary, and torsional inertias, respectively, while the 2rI terms account for the gyroscopic inertia effects. It is assumed that the flexural and bending-twisting coupling rigidities (CB and CBT) associated with bending about the y and z axes are identical; likewise for the transverse shear stiffness (Cs) [32]. Bert and Kim [33] considered the case of simply supported shaft and assumed separable solution in space and time to solve the associated eigenvalue

<sup>C</sup><sup>33</sup> � <sup>2</sup>C11C13C<sup>31</sup> <sup>¼</sup> <sup>0</sup>

<sup>C</sup><sup>31</sup> <sup>¼</sup> <sup>C</sup><sup>13</sup> � ð Þ CBT=<sup>2</sup> ð Þ <sup>r</sup>A=CS λω<sup>2</sup>; C<sup>33</sup> <sup>¼</sup> CTλ<sup>2</sup> � <sup>r</sup>Jω<sup>2</sup>

<sup>ω</sup><sup>2</sup>λ<sup>2</sup> <sup>þ</sup> <sup>r</sup>Ir<sup>A</sup>

; C<sup>13</sup> <sup>¼</sup> ð Þ CBT=<sup>2</sup> <sup>λ</sup><sup>3</sup>

CS

<sup>ω</sup><sup>2</sup> � <sup>r</sup>Aω<sup>2</sup>

(26)

CB CS rA 

CS ω2  <sup>þ</sup> <sup>r</sup>Ir<sup>A</sup> CS ∂4 v ∂t

<sup>þ</sup> <sup>r</sup>Ir<sup>A</sup> CS ∂4 w ∂t

þ CT ∂2 ϕ <sup>∂</sup>x<sup>2</sup> � <sup>r</sup><sup>J</sup>

<sup>4</sup> <sup>þ</sup> ð Þ CBT <sup>=</sup><sup>2</sup> <sup>∂</sup><sup>3</sup>

<sup>4</sup> <sup>þ</sup> ð Þ CBT <sup>=</sup><sup>2</sup> <sup>∂</sup><sup>3</sup>

∂2 ϕ ∂t

ϕ <sup>∂</sup>x<sup>3</sup> <sup>¼</sup> <sup>0</sup>

ϕ <sup>∂</sup>x<sup>3</sup> <sup>¼</sup> <sup>0</sup>

<sup>2</sup> ¼ 0 (25)

(23)

(24)

(27)

CB ∂2 v <sup>∂</sup>X<sup>4</sup> <sup>þ</sup> <sup>r</sup><sup>A</sup> <sup>∂</sup><sup>2</sup>

CB ∂4 w <sup>∂</sup>x<sup>4</sup> <sup>þ</sup> <sup>r</sup><sup>A</sup> <sup>∂</sup><sup>2</sup>

4.2. Whirling optimization problem

The other alternative of the objective function is defined by:

$$\text{Minimize} \quad F = \left(\hat{\mathcal{Q}}\_{cr} - \hat{\mathcal{Q}}^{\*}\right)^{2} \tag{30}$$

The same set of constraints given in Eq. (23) is applied. The notation <sup>Ω</sup>^ <sup>∗</sup> is a dimensionless target rotational speed, which should be greater than the maximum permissible rotational speed by a reasonable margin (e.g., 10–20%). As a case study, a drive shaft with continuous material grading along the shaft axis is optimized considering the following power-law model:

$$\mathbf{V}\_f(\hat{\mathbf{x}}) = \mathbf{V}\_f(\mathbf{0.5}) + \left[\mathbf{V}\_f(\mathbf{0}) - \mathbf{V}\_f(\mathbf{0.5})\right] \left((\mathbf{1} - \mathbf{2}|\hat{\mathbf{x}}|)^\mathbf{n}\right)^P,\tag{31}$$

where Vf (0) is the fiber volume fraction at the right or left end of the drive shaft, while Vf (0.5) is the fiber volume fraction at the middle of the shaft length. Figure 10 illustrates the level curves of the normalized critical speed augmented with the mass equality constraint. It is seen that there are four distinct zones separated by the contour lines <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:0. The upper left zone and lower right zone contain local maximum solutions. The best point (p, Δf) = (4.53, 0.3), corresponding to <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:045, is located inside the zone where the fiber taper ratio <sup>Δ</sup><sup>f</sup> <sup>¼</sup> Vfð Þ0 =Vfð Þ 0:5 is less than one. The upper empty zone contains infeasible solutions that violate the imposed constraints. Another case study considers the through-thickness grading pattern

Figure 10. Normalized critical speed <sup>Ω</sup>^ cr augmented with the mass constraint (M^ <sup>¼</sup> <sup>1</sup>:0) in (p-Δf) design space.

given by Eq. (3). The corresponding design variable vector is defined by X ! <sup>¼</sup> Vfð Þ<sup>0</sup> ; Vf <sup>1</sup> 2 ; <sup>p</sup>; <sup>H</sup>^ with lower and upper limits XL ! = (0.3, 0.3, 0, 0.75) and XU ! = (0.7, 0.7, ∞, 1.25). The attained optimal design variable vector was calculated to be Xopt ! ¼ ð Þ 0:7; 0:3; 5:61; 0:955 at which the maximum critical speed increased by 14% above that of the baseline design with active mass constraint.

A last optimization strategy to be addressed here is to combine the two criteria in a single objective function subject to the mass, strength, and side constraints.

$$\begin{aligned} \text{Minimize} \quad & F = -\left(\hat{\mathcal{Q}}\_{cr} + \hat{T}\_{cr}\right) \\ \text{Subject to} \quad & \hat{M} - 1 \le 0 \\ & \left(\frac{\tau\_{\text{max}}}{\tau\_{\text{allow}}}\right) - 1.0 \le 0 \end{aligned} \tag{32}$$

5. Optimization of FGM wings against divergence

deformation described in terms of one space coordinate.

The chord distribution is assumed to have the form:

relation [35]:

section and applied airloads.

The use of the in-plane grading in aeroelastic design was first exploited by Librescu and Maalawi [6], who introduced the underlying concepts of using material grading in optimizing subsonic rectangular wings against torsional instability. Exact mathematical models were developed allowing the material physical and mechanical properties to change in the wing spanwise direction, where both continuous and piecewise structural models were successfully implemented. In this section, analytical solutions are developed for slender tapered composite wings through optimal grading of the material volume fraction in the spanwise direction. The enhancement of the wing torsional stability is measured by maximization of the critical flight speed at which aeroelastic divergence occurs. The total structural mass is maintained at a value equals to that of a known baseline design in order not to violate other performance requirements. Figure 11 depicts a slender wing constructed from Np panels with trapezoidal planform and known airfoil cross section. The wing is considered to be made of unidirectional fiber-reinforced composites with variable fiber volume fraction in the spanwise direction. The flow is taken to be steady and incompressible, and the aspect ratio is assumed to be sufficiently large so that the classical engineering theory of torsion can be applicable and the state of

The symbol Δ<sup>c</sup> denotes the chord taper ratio (= tip chord Ct/root chord Cr) and x (= x1/L) denotes the dimensionless spanwise coordinate. The equivalent shear modulus G of a unidirectional reinforced composite, thin-walled cross section can be determined from the

where f1 is a function that depends on the geometry and thickness ratio of the cross section (h/ C) and the ratio (G12/ G13), where G12 and G13 are the in-plane and out-of-plane shear moduli,

Figure 11. Trapezoidal wing planform and cross section geometry. (a) Multipanel, piecewise wing model, (b) airfoil

C xð Þ¼ Cr <sup>1</sup> � <sup>β</sup>c<sup>x</sup> , <sup>β</sup><sup>c</sup> <sup>¼</sup> ð Þ <sup>1</sup> � <sup>Δ</sup><sup>c</sup> (33)

Optimization of Functionally Graded Material Structures: Some Case Studies

http://dx.doi.org/10.5772/intechopen.82411

179

G ¼ f <sup>1</sup> G12 (34)

Eq. (22) assumes that whirling and torsional buckling instabilities are of equal relative importance. This model resulted in a balanced improvement in both stabilities with active mass constraint. The attained optimal solution was found to have a uniform distribution of the fiber volume fraction with its upper limiting value of 70% and wall thickness = 0.935. The corresponding optimal values of the design objectives were <sup>Ω</sup>^ cr <sup>¼</sup> <sup>1</sup>:135 and <sup>T</sup>^cr <sup>¼</sup> <sup>1</sup>:161, representing optimization gains 13.5 and 16.1%, respectively, as measured from the baseline design.
