**5. Research on structural thermal modals**

Aeroelastic problems have been the key in vehicle design, which has gradually become a notable obstacle to better vehicle performance with the development of vehicle. Under the effect of aeroheating, the temperature rise of a structure leads to variation in physical parameters of its material. Also, non-uniform temperature field within the structure causes prominent temperature gradients, which produces subsidiary thermal deformation and thermal stress, greatly alters the structural rigidity and thus changes the natural vibration performance of the structure. The variation of natural vibration performance due to thermal load significantly affects the trim, flutter and control characteristics of the vehicle and these effects tend to be unfavorable.

*5.2.1. Thermal modal analysis along the static trajectory*

prominent.

the case with zero wing AOA and that with non-zero wing AOA.

**Figure 16.** The thermal modal analysis strategy based on multi-physics coupling integration.

Set the total cruise flight time to 100 s and the flight environment to be 30 km within standard atmosphere. Calorically perfect gas model is adopted. The thermal modal is analyzed in both

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**Figure 18** presents the first six modal shapes of the hypersonic wing with zero AOA at time *t* = 100 s. It can be seen that the modal shape of each order at time *t* = 100 s has local changes to varying degrees compared to the modal shape at initial time. The first four modes generally retain the original modal shapes, while the fifth and the sixth change a lot in their modal shapes with the trend towards bending-torsion coupling that usually leads to vibration of the wing. Therefore, sustained aeroheating has effect on modal shapes of higher order more easily for wings that are fixed-supported at root. As AOA is gradually increased, the aerodynamic forces and thermal load imposed on lower wing surface becomes larger than those imposed on the upper wing surface, which changes the thermodynamic state within the wing and thus influences the natural vibration characteristics of the wing. **Figure 19** presents the first six modal shapes of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. It can be seen from the figure that compared with the modal shapes in the zero AOA case, modal shapes of all orders have little change, which indicates that modal shape is not very sensitive to variation of AOA. It is necessary to point out that the analysis above only involves flight time of 100 s. Actually, time of sustained flight of vehicle is an influencing parameter of great importance. The fluid-thermal-structural coupling will finally reach thermodynamic equilibrium. In the process, the modal frequency and modal shape of the wing will continue to change and at the same time the effect of AOA on modal frequency and modal shape will become more

## **5.1. Thermal modal analysis strategy based on multi-physics coupling**

The thermal modal analysis strategy based on multi-physics coupling integration method is shown in **Figure 16**. It can be summarized as: (1) employing the multi-physics coupling integration method HyCCD based on CFD, CTD and CSD, transient temperature field and stress field within the solid structure along its static or dynamic trajectory are obtained by doing hypersonic fluid-thermal-structural coupling analysis according to the coupling analysis strategy described above and (2) thermal rigidity matrix can be constructed by taking parameters of thermodynamic state within the solid structure of each time point for coupling calculation. Then, the thermal modal characteristics at each time point for coupling calculation of the solid structure are obtained by solving the generalized eigenvalue problem by means of conventional mode analysis method.

## **5.2. Thermal modal characteristic analysis of a typical hypersonic wing**

The thermal modal characteristic analysis under sustained flight of a typical three-dimensional low-aspect-ratio hypersonic wing model is presented in **Figure 17**. It is a symmetrical double edge with the leading edge blunted and small thickness of the trailing edge retained to avoid sharp edges. The material has mass density of 4539 kg/m<sup>3</sup> and Poisson number of 0.32 with all the other physical properties varying with temperature. The temperature at initial time is 300 K with zero initial stress and the reference temperature of thermal stress is 300 K. Fixed support is adopted at the wing root.

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**Figure 16.** The thermal modal analysis strategy based on multi-physics coupling integration.

#### *5.2.1. Thermal modal analysis along the static trajectory*

In general, the aerodynamic force and thermal load have a great impact on the inlet cowl leading edge, which suffices to cause thermal and dynamic damage to thermal protection structures despite the short imposing time in actual flight, presenting severe challenge for material selection and structure design of thermal protection. As for the air-breathing hypersonic vehicles, the impact of shock interaction is common in the surrounding flowfield. Therefore, thermal protection design of local leading edges (structures such as the tail and rudders) on the windward side wrapped by the nose shocks should be done carefully besides the nose and

Aeroelastic problems have been the key in vehicle design, which has gradually become a notable obstacle to better vehicle performance with the development of vehicle. Under the effect of aeroheating, the temperature rise of a structure leads to variation in physical parameters of its material. Also, non-uniform temperature field within the structure causes prominent temperature gradients, which produces subsidiary thermal deformation and thermal stress, greatly alters the structural rigidity and thus changes the natural vibration performance of the structure. The variation of natural vibration performance due to thermal load significantly affects the trim, flutter and control characteristics of the vehicle and these effects

The thermal modal analysis strategy based on multi-physics coupling integration method is shown in **Figure 16**. It can be summarized as: (1) employing the multi-physics coupling integration method HyCCD based on CFD, CTD and CSD, transient temperature field and stress field within the solid structure along its static or dynamic trajectory are obtained by doing hypersonic fluid-thermal-structural coupling analysis according to the coupling analysis strategy described above and (2) thermal rigidity matrix can be constructed by taking parameters of thermodynamic state within the solid structure of each time point for coupling calculation. Then, the thermal modal characteristics at each time point for coupling calculation of the solid structure are obtained by solving the generalized eigenvalue problem by

The thermal modal characteristic analysis under sustained flight of a typical three-dimensional low-aspect-ratio hypersonic wing model is presented in **Figure 17**. It is a symmetrical double edge with the leading edge blunted and small thickness of the trailing edge retained

0.32 with all the other physical properties varying with temperature. The temperature at initial time is 300 K with zero initial stress and the reference temperature of thermal stress is

and Poisson number of

**5.1. Thermal modal analysis strategy based on multi-physics coupling**

**5.2. Thermal modal characteristic analysis of a typical hypersonic wing**

to avoid sharp edges. The material has mass density of 4539 kg/m<sup>3</sup>

the engine cowl leading edge.

126 Advances in Some Hypersonic Vehicles Technologies

tend to be unfavorable.

**5. Research on structural thermal modals**

means of conventional mode analysis method.

300 K. Fixed support is adopted at the wing root.

Set the total cruise flight time to 100 s and the flight environment to be 30 km within standard atmosphere. Calorically perfect gas model is adopted. The thermal modal is analyzed in both the case with zero wing AOA and that with non-zero wing AOA.

**Figure 18** presents the first six modal shapes of the hypersonic wing with zero AOA at time *t* = 100 s. It can be seen that the modal shape of each order at time *t* = 100 s has local changes to varying degrees compared to the modal shape at initial time. The first four modes generally retain the original modal shapes, while the fifth and the sixth change a lot in their modal shapes with the trend towards bending-torsion coupling that usually leads to vibration of the wing. Therefore, sustained aeroheating has effect on modal shapes of higher order more easily for wings that are fixed-supported at root. As AOA is gradually increased, the aerodynamic forces and thermal load imposed on lower wing surface becomes larger than those imposed on the upper wing surface, which changes the thermodynamic state within the wing and thus influences the natural vibration characteristics of the wing. **Figure 19** presents the first six modal shapes of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. It can be seen from the figure that compared with the modal shapes in the zero AOA case, modal shapes of all orders have little change, which indicates that modal shape is not very sensitive to variation of AOA. It is necessary to point out that the analysis above only involves flight time of 100 s. Actually, time of sustained flight of vehicle is an influencing parameter of great importance. The fluid-thermal-structural coupling will finally reach thermodynamic equilibrium. In the process, the modal frequency and modal shape of the wing will continue to change and at the same time the effect of AOA on modal frequency and modal shape will become more prominent.

**Figure 17.** The three-dimensional low-aspect-ratio hypersonic wing.

*5.2.2. Thermal modal analysis along the dynamic trajectory*

6, *ω*<sup>6</sup> = 107.44 Hz.

variation with dynamic variation of the flight trajectory.

A simple flight trajectory is assumed referring to **Figure 5** to further analyze variation of thermal modal characteristics in complicated flight. **Figure 20** presents that dynamic variation of flight trajectory leads to the variation of the thermodynamic state within the wing as well as its modal frequency. From the initial time to the end of cruise, the modal frequency of each order gradually declines; the decline tends to be gentle and then shows a recovery as the vehicle descends. It turns out that the first modals have little change along the flight trajectory, generally retaining the original modal shapes while major variation takes place in the fifth/sixth-order mode, especially the sixth-order mode in which case bending-torsion coupling tends to occur. If the climbing, cruise and descending phase, especially cruise, last long enough, the modal frequency and modal shape of each order can have more significant

**Figure 19.** The first six modals of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. (a) modal 1, *ω*<sup>1</sup> = 18.85 Hz, (b) modal 2, *ω*<sup>2</sup> = 40.04 Hz, (c) modal 3, *ω*<sup>3</sup> = 63.44 Hz, (d) modal 4, *ω*<sup>4</sup> = 77.84 Hz, (e) modal 5, *ω*<sup>5</sup> = 97.63 Hz, and (f) modal

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The modal frequencies of each order show a downward trend over time and the decreasing rates vary from mode to mode. Modes of all orders get nearer or farther one another to varying degrees over time, which might impose severe impacts on the natural vibration characteristics of the structure. The results indicate that sustained aeroheating has effect more easily on modal shapes of higher order. Therefore, for hypersonic vehicles with large thin-walled

**Figure 18.** The first six modals of the hypersonic wing with zero AOA at time *t* = 100 s. (a) modal 1, *ω*<sup>1</sup> = 18.98 Hz, (b) modal 2, *ω*<sup>2</sup> = 40.28 Hz, (c) modal 3, *ω*<sup>3</sup> = 63.74 Hz, (d) modal 4, *ω*<sup>4</sup> = 78.41 Hz, (e) modal 5, *ω*<sup>5</sup> = 98.58 Hz, and (f) modal 6, *ω*<sup>6</sup> = 108.24 Hz.

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**Figure 19.** The first six modals of the hypersonic wing at time *t* = 100 s under the AOA of 10 deg. (a) modal 1, *ω*<sup>1</sup> = 18.85 Hz, (b) modal 2, *ω*<sup>2</sup> = 40.04 Hz, (c) modal 3, *ω*<sup>3</sup> = 63.44 Hz, (d) modal 4, *ω*<sup>4</sup> = 77.84 Hz, (e) modal 5, *ω*<sup>5</sup> = 97.63 Hz, and (f) modal 6, *ω*<sup>6</sup> = 107.44 Hz.

#### *5.2.2. Thermal modal analysis along the dynamic trajectory*

**Figure 18.** The first six modals of the hypersonic wing with zero AOA at time *t* = 100 s. (a) modal 1, *ω*<sup>1</sup> = 18.98 Hz, (b) modal 2, *ω*<sup>2</sup> = 40.28 Hz, (c) modal 3, *ω*<sup>3</sup> = 63.74 Hz, (d) modal 4, *ω*<sup>4</sup> = 78.41 Hz, (e) modal 5, *ω*<sup>5</sup> = 98.58 Hz, and (f)

modal 6, *ω*<sup>6</sup> = 108.24 Hz.

**Figure 17.** The three-dimensional low-aspect-ratio hypersonic wing.

128 Advances in Some Hypersonic Vehicles Technologies

A simple flight trajectory is assumed referring to **Figure 5** to further analyze variation of thermal modal characteristics in complicated flight. **Figure 20** presents that dynamic variation of flight trajectory leads to the variation of the thermodynamic state within the wing as well as its modal frequency. From the initial time to the end of cruise, the modal frequency of each order gradually declines; the decline tends to be gentle and then shows a recovery as the vehicle descends. It turns out that the first modals have little change along the flight trajectory, generally retaining the original modal shapes while major variation takes place in the fifth/sixth-order mode, especially the sixth-order mode in which case bending-torsion coupling tends to occur. If the climbing, cruise and descending phase, especially cruise, last long enough, the modal frequency and modal shape of each order can have more significant variation with dynamic variation of the flight trajectory.

The modal frequencies of each order show a downward trend over time and the decreasing rates vary from mode to mode. Modes of all orders get nearer or farther one another to varying degrees over time, which might impose severe impacts on the natural vibration characteristics of the structure. The results indicate that sustained aeroheating has effect more easily on modal shapes of higher order. Therefore, for hypersonic vehicles with large thin-walled

**Acknowledgements**

**Author details**

Fang Chen<sup>1</sup>

**References**

2014;**28**(4):635-646

11672183, 91641129 and 91441205).

\*, Shengtao Zhang1,2 and Hong Liu<sup>1</sup>

2 AECC Commercial Vehicle Engine Co., Ltd, Shanghai, China

going. Progressing in Aerospace Sciences. 2003;**39**(6):511-536

\*Address all correspondence to: fangchen@sjtu.edu.cn

1 Shanghai Jiao Tong University, Shanghai, China

This study was supported by the National Natural Science Foundation of China (Nos.

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**Figure 20.** The first six modal frequencies along the flight trajectory.

control surfaces, sustained aeroheating has great effect on the natural vibration characteristics. In general, the strategies and methods for multi-physics field coupling integration analysis developed can effectively predict and analyze the variation of natural vibration characteristics (natural frequency and natural vibration shape), which lays a good foundation for further research on aerothermoelastic problems.
