**4. Dynamic optimization of FGM bars in axial motion**

Elastic slender bars in axial motion can give rise to significant vibration problems, which assesses the importance of considering optimization of natural frequencies. These frequencies, besides being maximized, must be kept out of the range of the excitation frequencies in order to avoid large induced stresses that can exceed the reserved fatigue strength of the materials and, consequently, cause failure in a short time. Expressed mathematically, two different design criteria are implemented here for optimizing frequencies:

$$\text{Frequency-placement criterion: Minimize } \sum\_{i} W\_{f} \text{o}\_{i} \tag{14}$$

$$\text{Maximum-frequency criterion:} \quad \text{Maximize} \quad \sum\_{i} W\_{j\uparrow} \alpha\_{i} \tag{15}$$

In both criteria, an equality constraint should be imposed on the total structural mass in order not to violate other economic and performance requirements. Equation (10) represents a weighted sum of the squares of the differences between each important frequency i and its desired (target) frequency <sup>i</sup> \* . Appropriate values of the target frequencies are usually chosen to be within close ranges (called frequency windows) of those corresponding to a reference or baseline design, which are adjusted to be far away from the critical exciting frequencies. The main idea is to tailor the mass and stiffness distributions in such a way to make the objective function a minimum under the imposed mass constraint. The second alternative for reducing vibration is the direct maximization of the system natural frequencies as expressed by equation (11). Maximization of the natural frequencies can ensure a simultaneous balanced improvement in both of stiffness and mass of the vibrating structure. It is a much better design criterion than minimization of the mass alone or maximization of the stiffness alone. The latter can result in optimum solutions that are strongly dependent on the limits imposed on either the upper values of the allowable deflections or the acceptable values of the total structural mass, which are rather arbitrarily chosen. The proper determination of the weighting factors Wfi should be based on the fact that each frequency ought to be maximized from its initial value corresponding to a baseline design having uniform mass and stiffness properties. Reference [14] applied the concept of material grading for enhancing the dynamic performance of bars in axial motion. The associated eigenvalue problem is cast in the following:

30 Advances in Computational Stability Analysis

Optimum

[900/±200] layup [900/±200/900] layup

MPa

pcr,max=9.37x(10h/R)3 MPa pcr,max=36.634x(10h/R)3

value % gain Optimum value % gain

1/50 0.075 17.19 0.293 26.84 1/25 0.596 15.5 2.344 26.84 1/20 1.171 15.6 4.579 26.91 1/15 2.776 14.81 10.854 26.92

*)=* (0.25, 0.225, ±200) (0.2925, 0.235, ±200)

*)=* (0.705, 0.275, 900) (0.6835, 0.265, 900)

Elastic slender bars in axial motion can give rise to significant vibration problems, which assesses the importance of considering optimization of natural frequencies. These frequencies, besides being maximized, must be kept out of the range of the excitation frequencies in order to avoid large induced stresses that can exceed the reserved fatigue strength of the materials and, consequently, cause failure in a short time. Expressed mathematically, two different design criteria are implemented here for optimizing

*i*

*i*

In both criteria, an equality constraint should be imposed on the total structural mass in order not to violate other economic and performance requirements. Equation (10) represents a weighted sum of the squares of the differences between each important frequency i and its

*<sup>W</sup>* (14)

*<sup>W</sup>* (15)

**Table 4.** Optimum buckling design of multi-layered, AS-4, FGM composite cylinders

**4. Dynamic optimization of FGM bars in axial motion** 

**Frequency-placement criterion**: Minimize *fi i*

**Maximum-frequency criterion**: Maximize *fi i*

*(h/R)* 

**Optimum solutions Two helical layers**  *(Vf, h,* 

**Two hoop layers** *(Vf, h,* 

frequencies:

$$
\hat{E}\frac{d^2\hat{\mathcal{U}}}{d\hat{\chi}^2} + \frac{d\hat{E}}{d\hat{\chi}}.\frac{d\hat{\mathcal{U}}}{d\hat{\chi}} + \hat{\rho}\_{00}^2\hat{\mathcal{U}} = 0,\ \ 0 < \hat{\mathfrak{x}} < 1\tag{16}
$$

where ܷ ൌ ܷȀܮ is the dimensionless amplitude and ߱ෝ =߱ܮඥߩȀܧ dimensionless frequency. Both continuous and discrete distributions of the volume fractions of the selected composite material were analyzed in [14]. The general solution of Eq. (12), where the modulus of elasticity and mass density vary in the axial direction, can be expressed by the following power series:

$$\hat{\mathbf{U}}(\hat{\mathbf{x}}) = \sum\_{m=1}^{2} \mathbb{C}\_{m} \mathbb{A}\_{m}(\hat{\mathbf{x}}) \tag{17}$$

where *Cm's* are the constants of integration and *m's* are two linearly independent solutions that have the form:

$$\mathfrak{Z}\_{\mathbf{m}}(\hat{\mathbf{x}}) = \sum\_{n=m}^{w} a\_{m,n} \hat{\chi}^{n-1} \quad \text{ ( $\mathbf{n} \ge \mathbf{m}$ )}\tag{18}$$

The unknown coefficients *am,n* can be determined by substitution into the differential equation (12) and equating coefficients of like powers of *xˆ* . Table 5 summarizes the appropriate mathematical expressions of the frequency equation for any desired case, which can be obtained by application of the associated boundary conditions and consideration of nontrivial solutions.

Variation of the volume fractions in *FGM* structures is usually described by power-law distributions. Figure 10 shows both linear and parabolic models for material grading along the bar span. Results given in [14] showed that, for Fixed-Fixed and Fixed-Free boundary conditions, patterns with higher fiber volume fraction near the fixed ends are always


Stability, Dynamic and Aeroelastic Optimization of Functionally Graded Composite Structures 33

**Figure 11.** Dimensionless frequency isomerits of free-free bar under mass constraint

**5. Material grading for improved aeroelastic stability of composite wings** 

(b) Second frequency (Symmetrical mode).

(a) Fundamental frequency (Unsymmetrical mode)

Aircraft wings can experience aeroelastic instability condition in high speed flight regimes. A solution that can be promising to enhance aeroelastic stability of composite wings is the use of the concept of functionally graded materials (FGMs) with spatially varying properties. Reference [15] introduced some of the underlying concepts of using material grading in optimizing subsonic wings against torsional instability. Exact mathematical approach allowing the material properties to change in the wing spanwise direction was applied, where both continuous and piecewise structural models were successfully implemented. The enhancement of the torsional stability was measured by maximization of the critical flight speed at which divergence occurs with the total structural mass kept at a

**Table 5.** Frequency equations for different types of boundary conditions. *ˆ* o,i are the dimensionless natural frequencies of the baseline design ( *ˆ* o=0 corresponding to the first rigid body mode of a Free-Free bar). The notation *( )׳* means *d/d xˆ* .

favorable. The opposite trend is true for cases of Free-Free bars. Maximization of the fundamental frequency alone produces an optimization gain of about 14.33% for the linear model with 0% and 100% volume fractions at the ends of the optimized bars with different boundary conditions. However, a drastic reduction in the 2nd and 3rd frequencies was observed. Better solutions have been achieved by maximizing a weighted-sum of the first three frequencies, where the parabolic model was found to excel the linear one in producing balanced improvements in all frequencies. Results have also indicated that the Fixed-Fixed bars are recommended to have concave distribution rather than convex one. The latter produce poor patterns with degraded stiffness-to-mass ratio levels. The opposite trend was observed for the free-free bars, where the convex type is much more favorable than the concave type. Both concave and convex shapes can be accepted for a cantilevered bar. For piecewise models, the developed isomerits for the case of Free-Free bar built of four symmetrical segments made of carbon/epoxy composites are shown in Figure 11. The global maximum of the fundamental frequency is located at the lower region to the left of the design space having a value of *ˆ* <sup>1</sup>*, max* 3 45406 *.* at the optimal design point (*VA, ˆ L )k*=1,2 =(0.1885,0.1625), (0.650, 0.3375), which represents about 10% optimization gain.

**Figure 10.** Symmetrical shape models of volume fraction distribution along bar length

Fixed-Free Bar 0 10 *U( ) U ( )*

Boundary conditions Frequency equation **(** *ˆ* **o)i**

2 3

3

1 3

*<sup>n</sup> <sup>n</sup> <sup>a</sup> (n ) ( )*

2 2

*<sup>n</sup> <sup>n</sup> <sup>a</sup> ( )*

*<sup>n</sup> <sup>n</sup> <sup>a</sup> (n ) ( )*

1 3

*<sup>n</sup> <sup>n</sup> <sup>a</sup> ( )*

1 1 *<sup>n</sup> <sup>n</sup> <sup>a</sup> (n )*

1 1 2 *<sup>n</sup>*

2 0 *<sup>n</sup>*

1 0 2 *<sup>n</sup>*

2 1 *<sup>n</sup>*

*(π, 3π, 5 π*)

(2*π, 4π, 6π)*

*(π, 3π, 5π) /2* 

(2*π, 4π, 6π)*

*(π, 3π, 5 π)*

*L )k*=1,2

*ˆ ˆ* <sup>2</sup>

*ˆ ˆ* <sup>2</sup>

*ˆ ˆ* <sup>1</sup>

*ˆ ˆ* <sup>1</sup>

*ˆ ˆ* <sup>2</sup>

**Table 5.** Frequency equations for different types of boundary conditions. *ˆ* o,i are the dimensionless natural frequencies of the baseline design ( *ˆ* o=0 corresponding to the first rigid body mode of a Free-

favorable. The opposite trend is true for cases of Free-Free bars. Maximization of the fundamental frequency alone produces an optimization gain of about 14.33% for the linear model with 0% and 100% volume fractions at the ends of the optimized bars with different boundary conditions. However, a drastic reduction in the 2nd and 3rd frequencies was observed. Better solutions have been achieved by maximizing a weighted-sum of the first three frequencies, where the parabolic model was found to excel the linear one in producing balanced improvements in all frequencies. Results have also indicated that the Fixed-Fixed bars are recommended to have concave distribution rather than convex one. The latter produce poor patterns with degraded stiffness-to-mass ratio levels. The opposite trend was observed for the free-free bars, where the convex type is much more favorable than the concave type. Both concave and convex shapes can be accepted for a cantilevered bar. For piecewise models, the developed isomerits for the case of Free-Free bar built of four symmetrical segments made of carbon/epoxy composites are shown in Figure 11. The global maximum of the fundamental frequency is located at the lower region to the left of the design space having a value of *ˆ* <sup>1</sup>*, max* 3 45406 *.* at the optimal design point (*VA, ˆ*

0 12 0 *U( ) U ( / )*

0 12 0 *U( ) U( / )*

0 12 0 *U( ) U( / )*

0 12 0 *U ( ) U( / )*

=(0.1885,0.1625), (0.650, 0.3375), which represents about 10% optimization gain.

**Figure 10.** Symmetrical shape models of volume fraction distribution along bar length

means *d/d xˆ* .

Fixed-Fixed Bar Symmetrical modes Unsymmetrical modes

Free-Free Bar Symmetrical modes Unsymmetrical modes

Free bar). The notation *( )׳*

(a) Fundamental frequency (Unsymmetrical mode)

(b) Second frequency (Symmetrical mode).

**Figure 11.** Dimensionless frequency isomerits of free-free bar under mass constraint
