A scaling procedure for measuring thermal structural vibration generated by wall pressure fluctuation

Xiojin ZHAO , Hio CHEN , Junmin LEI,*, Bngheng AI

a School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

b CAS Key Laboratory of Mechanical Behavior and Design of Materials, University of Science and Technology of China,Hefei 230026, China

c China Academy of Aerospace Aerodynamics, Beijing 100074, China

KEYWORDS Aero-thermal;Hypersonic;Scaling procedure;Structure vibration;Turbulent boundary layer

Abstract This paper attempts to develop a scaling procedure to measure structural vibration caused simultaneously by wall pressure fluctuations and the thermal load of hypersonic flow by a wind tunnel test. However, simulating the effect of thermal load is difficult with a scaled model in a wind tunnel due to the nonlinear effect of thermal load on a structure. In this work, the temperature variation of a structure is proposed to indicate the nonlinear effect of the thermal load,which provides a means to simulate both the thermal load and wall pressure fluctuations of a hypersonic Turbulent Boundary Layer (TBL) in a wind tunnel test. To validate the scaling procedure,both numerical computations and measurements are performed in this work. Theoretical results show that the scaling procedure can also be adapted to the buckling temperature of a structure even though the scaling procedure is derived from a reference temperature below the critical temperature of the structure.For the measurement,wall pressure fluctuations and thermal environment are simulated by creating hypersonic flow in a wind tunnel. Some encouraging results demonstrate the effectiveness of the scaling procedure for assessing structural vibration generated by hypersonic flow. The scaling procedure developed in this study will provide theoretical support to develop a new measurement technology to evaluate vibration of aircraft due to hypersonic flow.

1. Introduction

Certain structural skin components of re-entry spacecraft and hypersonic cruise vehicles are subjected to serious wall pressure fluctuation induced by a Turbulent Boundary Layer (TBL) in an elevated thermal environment.High-intensity pressure fluctuation will generate severe vibration of an aircraft structure.The temperature variation generated by hypersonic flow will change the material properties and the stress states of the structure. Thus, it is critical to investigate the characteristics of the structure in such an extreme environment and understand these types of structural responses during the design and test phases of a design cycle.

Generally,thermal-acoustic tests are performed to simulate wall pressure fluctuation and thermal load simultaneously in a progressive wave tube1,2or a heated acoustic-testing chamber.3,4The maximum temperature simulated could reach 1480°C and the overall simulated sound-pressure level could reach 170 d B5in thermal-acoustic test equipment designed for hypersonic vehicles. The acoustic load is generated by an electro-pneumatic acoustic generator that modulates highpressure air from the plenum, and a horn loudspeaker can be used to increase the efficiency of the acoustic-load generation.6The thermal load is typically simulated by lamp filaments or graphite heating elements.2,7However, these methods have notable shortcomings in simulating the acoustic and thermal effects, e.g., only a uniform sound field could be simulated in a progressive wave tube or heated acoustic chamber. In addition, the coupling effect between the thermal environment and the acoustic load cannot be simulated. Finally,the hypersonic airflow by which the thermal and acoustic loads are generated cannot be produced with this equipment. Thus,this paper attempts to derive a scaling procedure for predicting structure vibration induced by wall pressure fluctuations in a thermal environment, which will provide theoretical support for developing an approach to measure vibration caused by hypersonic flow in a wind tunnel.

Investigation of the scaling procedure used for analyzing structural vibration has been performed for many years. The technology of predicting structure vibration with a scaled model was firstly proposed by De Rosa et al.8,and the scaling law was developed for multi-component analysis of a onedimensional rod with the Finite Element Method (FEM) and Statistical Energy Analysis (SEA). Tabiei9employed a similitude theory to predict the buckling behavior of laminated cylindrical shells subjected to lateral pressure, and a new prediction equation was developed to establish the behavior of a prototype for both complete and partial similarities according to the scaling laws. The structural similitude theory was also applied to investigate sandwich beams and columns in Ref.10by Frostig. To consider the influence of TBL excitation on a structure, De Rosa and Franco11developed Asymptotical Scaled Modal Analysis (ASMA) to reduce the physical size of the solution domain and decrease the computational cost of the dynamic response for an isolated system. Then, ASMA was used to predict the flexural vibration of a more complex structure, i.e., a two-plate and two-beam assembly, in Ref.12.Ciappi13proposed a dimensionless representation of the universal solution of the vibration of a plate under TBL excitation, and the scaling procedure was validated by measurements under different conditions in both wind tunnels and a towing tank. To develop a technology for measuring structural vibration induced by a TBL in a wind tunnel,Zhao14developed a scaling procedure for the structural response of a panel caused by a TBL. However, the research in Ref.14was limited to a flat plate with a simply supported boundary. Hence, Zhao and Ai15extended the scaling procedure to cover curved-plate vibrations generated by a TBL with unsupported boundary conditions. For comparison, a windtunnel test was performed to simulate the TBL excitation and measure the structural response for scaling-procedure validation with a scaled-plate aircraft structure in Ref.15.

Although many studies have investigated scaling methods,an effective scaling method that could investigate structural responses to wall pressure fluctuations produced by a TBL with a thermal load is lacking. To investigate the vibration responses of thermal structures, previous research in Refs.14,15should be extended by considering the effect of thermal load.In fact, more attention has been given to structures subjected to acoustic and thermal loads simultaneously generated by hypersonic flow in recent years. Hybrid methods including the finite element method and the boundary element method were applied to compute the vibro-acoustic responses of isotropic and composite plates under a thermal load in Refs.16,17.Yang et al.18employed numerical computations to predict the behavior of structural vibration induced by acoustic excitation in a thermal environment under different conditions.A review paper19by Ohayon et al. introduced reduced-order models in solving linear dissipative structural-acoustic and fluidstructure problems. In Refs.20,21, both theoretical and numerical methods were used to discuss the effect of thermal stresses on simply supported and fully clamped plates,and the temperature of the thermal environment investigated was typically far above the critical buckling temperature of a structure with a supported boundary. Geng et al.22employed both experimental and numerical approaches to investigate the vibro-acoustic response of a thermal post-buckled plate. Gupta23introduced a novel computational-fluid-dynamics-based numerical solution procedure to solve aero-thermo-acoustics problems, and a finite element idealization was applied in both fluid and structure domains to account for thermal effects.

The investigations listed above primarily focused on a simple model including a structure and TBL excitation for theoretical research. To investigate structural vibration caused by hypersonic flow in a complex environment, wind tunnel measurements were proposed to simulate hypersonic wall pressure fluctuation and aero-thermal effects simultaneously.However,simulating the effect of thermal load is difficult with a scaled model in a wind tunnel due to the nonlinear effect of thermal load on the structure.In this work,a scaling procedure to simulate the thermal effect is developed and validated by numerical computations and a measurement. The remainder of the paper is structured as follows.The scaling procedure is derived in Section 2, and some theoretical results are discussed in Section 3. A measurement is engineered and performed in Section 4. Finally, a summary of the work is presented in Section 5.

2. Scaling procedure derivation

2.1. Plate vibration

To simplify the derivation, the scaling procedure was developed using a thin, homogeneous, flat rectangular plate simply supported along its edges,as shown in Fig.1.The investigation focuses on the x Oy plane,and the frequency-response function on the plate at a defined point r for a force exciting the plate at an arbitrary point s can be obtained from the following expression24:

Fig. 1 Schematic of elastic plate excited by wall pressure fluctuations.

2.2. Thermal effect

Suppose that the initial temperature is T0and that the structure is in a stress-free state at this temperature.When the temperature of the structure is changed to T,the stress state of the plate will change due to the variation in the structure temperature. Assuming that the temperature variation across the thickness is uniform, the thermo-elastic property of the plate could be described by the plate stress state. Then, the stressdisplacement relations could be expressed as20,21

where u and v indicate the in-plane displacement. α is the thermal-expansion coefficient of the investigated structure,and ΔT=T-T0denotes the relative variation in the structural temperature.

Due to the constraint of the supported boundary,the plate cannot generate an in-plane displacement even when heated by a thermal load.Thus,Eqs.(2)-(4)for the constrained structure will be simplif ied to

Then the membrane forces of the structure can be obtained by integrating Eqs. (5)-(7) as

2.3. Scaling procedure

The solution for the structural vibration of a plate with a simply supported boundary could be presented in an analytical form25,26. According to the expression in Refs.25,26, the auto-spectral density of displacement at an arbitrary point on the plate can be presented as

and the power-spectral density with respect to the velocity of the plate is

Assume that a scaling coefficient σ allows the structure vibration to be scaled by the method shown in Refs.14,15.Then,each side of the plate is scaled as a=σa,b=σb, andh=σh.

The normalized mode shapes can be scaled as

The generalized mass coefficient can also be scaled as

where ‘ˉ’ denotes the parameters of the scaled model. Taking into account the effect of temperature variation on the structure, the natural radian frequencies20,26,27of the plate can be scaled as

and the natural frequencies can be scaled

If a curved plate is being investigated, the effect of the curvature on the natural radian frequency should be considered.Then the natural radian frequencies26can be calculated as follows:

which has the same expression as Eq. (16).

Since the structure is scaled, the external excitation on the structure should also be scaled. If the structural vibration is caused by the wall pressure fluctuations of the TBL,the external excitation can be scaled in a manner similar to that in Refs.14,15by the aerodynamics scaling laws in Ref.29as follows:

where d and d are the characteristic dimensions of the original and scaled models, respectively; V represents the flow speed over the full-scale model;V is the flow speed on the scaled model; Sppdenotes the power spectrum density of wall pressure fluctuations; q denotes the dynamic pressure. However,if the structural vibration is generated by the acoustics or some point-force, the scaled external excitation expression could be simplif ied to

where Sppindicates the sound pressure power spectrum density or the force power spectrum density.

Then

Following the derivation of Eqs. (16) and (17), it requires that

The variation of Poisson’s ratio is typically too small to be

3. Theoretical results and discussion

3.1. Modal analysis

According to Eq. (16), the effect of the thermal environment on the modes of the structure is shown in two aspects.Firstly,it makes the elastic modulus of the structure decrease as the structural temperature rises. The material attribute and the elastic modulus at different temperatures for Al2O3are listed in Tables 1 and 2, respectively. Secondly, additional thermal stress would be generated if the temperature of the restricted structure is changed. The scaling procedure for the structure developed in this paper considers both the effect of the elastic modulus variation and the effect of the thermal stress caused by the structural temperature. Basing on Eq. (16), the premise of the scaling procedure is that the temperature variation should remain constant when the structure is scaled. Only when the temperature variation is constant, can the following derived equation in Eq. (22) be satisfied:

Eq. (25) indicates the relationship between the structural modes of the scaled model and those of the prototype (original model). The scaling expression of the thermal-structural modes assumes that the temperature of the plate is below the critical buckling temperature. In fact, the temperature of the thermal environment often exceeds the critical buckling temperature of the structure in engineering. Thus, it is important to validate Eq. (25) to be able to adapt to the structure in the post buckling range. The critical buckling temperature22of the plate is

According to the scaling method introduced in Section 2,the critical buckling temperature of the structure could be expressed as

Hence, the critical buckling temperature of the structure keeps constant when the structure is scaled.

Table 3 shows the structural modes calculated under different temperature loads. According to Eq. (26), structure buckling is mainly determined by the additional thermal stress generated by the structure temperature change. Therefore,the elastic modulus of the structure used for modal analysis is kept at a constant value. The FEM software MD NASTRAN (MSC Software Corp., Santa Ana, CA, USA) is used to perform a modal analysis of the thermal structure because it has a special module to compute the natural modes of the structure in different thermal environments. The dimensions of the plate are a×b×h=1.6 m×1.0 m×0.02 m and the plate is composed of Al2O3.The total number of elements(grid cells)used for the modal analysis is 4250(85 elements longitudinally and 50 elements latitudinally).

According to Eq. (26), the critical buckling temperature of the plate is 66.6°C.Two temperature variations were used for the modal analysis in the comparison. According to Eq. (16),the effect of the thermal load on the modes of the structure is shown in two aspects, i.e., changes in the elastic modulus of the structure decrease and cause additional thermal stress.Because the elastic modulus has been replaced by the current value in Eq.(16),the results listed in Table 3 assume a constant elastic modulus to investigate the effect of additional thermal stress on the structural modes. The comparison in Table 3 shows that the reference temperature of the structure does not affect the structural mode without considering the variation in the elastic modulus. In addition, the structural modes are mainly determined by the temperature variation even when the temperature variation of the structure exceeds the critical buckling temperature. Thus, it is reasonable to simulate the effect of thermal stress on the structural vibration by controlling the temperature variation.

3.2. Validation of FE model

In this work, the FEM is used to solve the structural response for validating the proposed scaling procedure, and Finite

Table 1 Material properties.

Table 2 Elastic moduli of Al2O3 at different temperatures.

Table 3 Structure modes for different temperature loads (Hz).

Fig. 2 Velocity responses of a simply supported panel by different grid elements.

Element (FE) models should be validated firstly before being used for computing the structural response of both the scaled models and the full-scale model. In this section, an approximate analytical solution (Eq. (11)) is applied to validate the FE model. In addition, finite element computations with different grid elements are performed by the structural analysis software VA-One (ESI Group, Paris, France) to disclose the effect of the computational mesh on the numerical results.

In this work, a simply supported plate is used for scaling procedure validation, and the dimensions of the plate are a×b×h=1.6 m×1.0 m×0.02 m. The coordinates for the four points of the full-scale plate are (x=0 m, y=0 m),(x=1.6 m, y=0 m), (x=1.6 m, y=1.0 m), and (x=0 m,y=1.6 m). Structural responses at two points on the plate are solved and compared. Assume that the plate is made of Aluminum (E=7.0×1010Pa;ν=0.33; ρ = 2700 kg/m3),and the amplitude of wall pressure fluctuation acted on the plate is 1 Pa2/Hz. Three different numbers of grid elements are applied for computation, consisting of 1075 elements (43 elements longitudinally and 25 elements latitudinally), 4250 elements (85 elements longitudinally and 50 elements latitudinally), and 9600 elements (128 elements longitudinally and 75 elements latitudinally), respectively. Result comparisons based on different grid elements and methods are presented in Fig.2,which discloses that the FEM could obtain an exactly consistent result as the analytical method. Meanwhile, Fig. 2 also releases that the more grid elements are used for numerical computation, the better results could be obtained. However,4250 elements(85 elements longitudinally and 50 elements latitudinally) are enough for computing the structural vibration of this plate,which could be seen from Fig.2.Hence,4250 elements (85 elements longitudinally and 50 elements latitudinally) are used for numerical computations in Section 3.3.

3.3. Structure response by FEM

Three different temperature loads are listed in Table 4. Fig. 3 shows the computed and predicted velocity PSD responses for the full-scale plate under the temperature load of Condition 1.The solid red line shows the result obtained using the FEM with the full-scale plate (original model under Condition 1),the dashed black line shows the result obtained by scaling the response given by the FEM for the scaled model (Condition 1) using the scaling procedure, and the dash-dot blue line indicates the result by scaling the response of the scaled model without considering the thermal load.Fig.4 shows the velocity PSD responses of the full-scale plate, computed based on the responses of the scaled model under three different thermal loads.The structural response of the full-scale model (original model) is computed under Condition 3, and the results of the scaled model are obtained under three different thermal loads.

The plate has the same dimensions as those in Section 3.2,and the scaling coefficient for the scaled models is chosen as 0.1. The coordinates for the four points of the full-scale plate are (x=0 m, y=0 m), (x=1.6 m, y=0 m), (x=1.6 m,y=1.0 m), and (x=0 m, y=1.6 m). Both the full-scale structure and the scaled model are assumed to be made of Al2O3. The amplitude of wall pressure fluctuation caused by a TBL is 1 Pa2/Hz which is distributed uniformly on the full-scale structure. According to the scaling laws of aerodynamics expressed in Eq. (19), the amplitude of wall pressure fluctuation for the scaled structure is 0.1 Pa2/Hz. The structural analysis software VA-One (ESI Group, Paris, France)aims at computing structural responses excited by different acoustic loads. Therefore, VA-One is employed here to compute the structural response of the plate caused by wall pressure fluctuations. However, the natural modes of the thermal structure still need to be computed by MD NASTRAN due to its superiority in computing thermal structural modes.The FEM grid size used for the computations is 4250 (85 elements longitudinally and 50 elements latitudinally), which ensures that a flexural wave is described by at least 8 nodes.Generally, aerodynamic similarity requires velocity and dynamical pressure variations to have the same inflow Mach number Ma, which cannot be achieved with commercial software.Hence,the velocity and dynamical pressure of the scaledmodel are assumed to be the same as those of the full-scale model here. Then, the frequencies and wall pressure fluctuations used for the scaled model computation are scaled by Eqs. (18) and (19), respectively.

Table 4 Temperature loads for the scaled model.

Fig. 3 Velocity responses of a simply supported panel for Condition 1.

Fig. 4 Velocity responses of a simply supported panel for different conditions.

In Figs.3 and 4,predictions based on the scaled-model data are compared with those obtained from the full-scale plate,verifying the validity of the scaling procedure of thermalstructural vibration induced by wall pressure fluctuations. In addition, it could be concluded that it is not necessary to simulate the absolute temperature but the temperature variation of the original model when using the scaling procedure of thermal structure vibration generated by wall pressure fluctuations. A thermal environment usually makes the natural frequencies of a structure decrease, and an obvious frequency offset could be generated at each natural frequency if the thermal effect is not considered (Fig. 3). Thus, it is important to simulate the effect of thermal environment on the structure vibration induced by hypersonic wall pressure fluctuations.

In addition,the results shown are all computed by the FEM which requires more grids and computational resources for high-frequency vibration prediction. Thus, to spare time cost,only vibration results below 500 Hz are provided. However,this has no effect on validation of the scaling procedure developed.

4. Measurement validation

4.1. Measurement design

Numerical computations have been used to validate the scaling procedure of thermal-structural vibration generated by the wall pressure fluctuations of a TBL in theory. In this section,a measurement was designed and performed to verify the scaling procedure developed.The measurement was intended to be performed in a hypersonic wind tunnel that could simulate the thermal environment and wall pressure fluctuation caused by hypersonic flow simultaneously.

The wind tunnel used is an intermittent wind tunnel located at the Chinese Academy of Aerospace Aerodynamics(CAAA)in China.The test section of the wind tunnel is 0.5 m×0.5 m.The range of achievable flow Mach numbers for the wind tunnel is from Ma=5.0 to 8.0.For this measurement,Ma=8.0 was simulated in the wind tunnel to create an aero-thermal environment and wall pressure fluctuations caused by hypersonic inflow over the structure. The total temperature of the airflow in the wind tunnel was 745.0 K,and the dynamic pressure simulated was approximately 3.7 k Pa.The flow speed simulated in the wind tunnel was 1181.4 m/s.

A scaled model with a scaling coefficient of 0.2 was designed to be the layout of an HTV2 aircraft according to the information found on the internet.The test model is made of 30Cr MnSiA, and Fig. 5 shows the test model designed for the wind-tunnel test. Different kinds of sensors, including accelerometers and thermocouples used for the temperature test, were all mounted on the inner wall of the test panel to protect them from the high-speed airflow. The thickness of the entire model was more than 5 mm, except the test panel shown in Fig. 5. The test panel was designed with a thickness of 2 mm to simulate plate vibration caused by wall pressure fluctuations. An opening cover on the low wall was designed to install sensors on the internal surface of the model from outside.

Fig. 5 Wind-tunnel test model.

Fig. 6 Dimensions and locations of the measurement points on the test model.

The test points and the main dimensions of the model are all shown in Fig. 6. P1 indicates the thermocouple that was used to measure the temperature of the model when a windtunnel test was performed. Accelerometers (PCB, 333C33)were placed at points P2 and P3 to measure structural vibrations generated by wall pressure fluctuation. Fluctuating pressure sensors (Kulite XCL-100, USA) were fixed at P4 and P5 to measure wall pressure fluctuations used for computing the structure response.Signals from the sensors,consisting of the accelerometers,the fluctuating pressure sensors,and the thermocouple, were all gathered by a data-acquisition system(VXI-16026A) and then transferred to a PC during a test.

4.2. Comparisons

For this wind-tunnel test30,31(shown in Fig. 7), the temperature range measured was from 22.5°C to 180°C,and three different temperatures were chosen for structure vibration analysis, namely, 30°C, 90°C, and 165°C. The test model is made of 30CrMnSiA, and the elastic moduli of 30CrMnSiA for different temperatures are shown in Table 5. T0denotes the initial reference temperature,and ΔT indicates the temperature variation.

Fig. 7 Test model installed in the wind tunnel.

Table 5 Elastic moduli of 30Cr MnSiA for different temperature loads.

To validate the scaling laws developed for predicting structure vibration induced by hypersonic wall pressure fluctuation,numerical computations were used to calculate the structural response of the full-scale model for comparison with the measurement performed on the scaled model. As in the method introduced in Section 3, numerical computation applied MD NASTRAN to compute the natural modes of the thermal structure and the FEM solver in VA-ONE to compute the structural response. The total number of elements (grid cells)used was 6660 which ensured that a flexural wave could be represented with at least 8 nodes. Response computation for the full-scale model made use of wall pressure fluctuations measured in the wind tunnel. Although the flow speed or the dynamical pressure could not be simulated by the commercial software MD NASTRAN or VA-ONE directly, wall pressure fluctuations measured in the wind tunnel could be scaled by Eqs. (18) and (19) to simulate the aerodynamic effect before being used for structure response computation. Measurement results of the full-scale structure response could be obtained by scaling the structure response measured on the scaled model using the scaling procedure in Eq. (23). The flow speed and dynamical pressure for the scaled model that has been introduced in Section 4.1 were measured directly from the wind tunnel. However, the flow speed and dynamical pressure for the full-scale model, which is too big to be installed in the wind tunnel, could be assumed to be any reasonable constants,which would not influence the results comparison according to Eq. (23). This is because the ratio determined by the flow speed and dynamical pressure would be cancelled out as the structure response of the full-scale model was obtained by Eq. (23) based on the structure response of the scaled model measured directly in the wind tunnel.However,the flow speed and dynamical pressure for the full-scale model could be provided exactly if the wall pressure fluctuations used for computing the full-scale structure response were measured on the fullscale model. Generally, a flight test can be used to directly measure the full-scale structure. However, the high cost of a flight test makes this difficult to realize in reality. Thus, only the scaled model in the wind tunnel was measured and investigated in this study.

Fig. 8 Velocity-response comparison between measurements and numerical computations at 30°C.

Figs.8-10 show comparisons of results obtained by measurement and numerical computation at three different current temperatures.In Figs.8-10,the frequency ranges of the structural responses shown are limited to 200 Hz-800 k Hz.The reason is that the full-scale structure was designed to be suff iciently‘‘hard”, and the natural frequencies of the structure are in the frequency range above 200 Hz.The solid red line indicates the velocity response predicted using the scaling procedure and the data measured on the scaled model.The dashed black line shows the velocity response obtained by computing the response of the full-scale model directly.Before being compared with the numerical results of the full-scale structure, the analysis frequency of the measurement on the scaled model should be converted to be consistent with that of the full-scale structure in Eq.(18). Result comparisons shown in Figs. 8-10 present that the numerical results match well with the measurement results at P2.However,remarkable errors are revealed in the comparisons at P3,particularly for a low frequency.

Fig. 9 Velocity-response comparison between measurements and numerical computations at 90°C.

Fig. 10 Velocity-response comparison between measurements and numerical computations at 165°C.

This error shown in the f igures could be attributed to several factors.Firstly,the numerical results were obtained under an ideal assumption that the temperature distribution on the structure was uniform. However, the temperature distribution on the structure was not always uniform,and a sharp gradient of heat flux or temperature mostly appeared on the front edge of the structure32-34. Thus, the response error at P3, which is closer to the front edge, is more remarkable than that at P2.However, this kind of error could be avoided if the structural response of the full-scale model was also obtained by the windtunnel test, because the temperature distribution on the test model was determined by the actual hypersonic flow over the structure, which was not restricted by the ideal assumption of a uniform temperature distribution.

The error generated by manufacturing and installing the test model is another important factor causing the difference between measurement and numerical computation. Generally,the internal dimensions of the structure around the front edge are difficult to guarantee. Thus, the front edge of the model only simulates an approximated simple support boundary of the test panel shown in Fig. 5, while numerical computation always applies an ideal support boundary. Investigations in Refs. [14,15] have disclosed that low-frequency vibration is more easily influenced by the difference of the simulated support boundary. Hence, the response error between two different methods at P3,which lies closer to the boundary of the test panel, is more remarkable than that at P2, especially for the low-frequency response. Result comparisons in Figs. 8-10(b)all indicate this phenomenon.

5. Conclusions

In this work, a scaling procedure was developed to measure structure vibrations due to hypersonic wall pressure fluctuations in a wind tunnel test.According to the scaling procedure proposed,the most important point of the scaling procedure is to simulate the temperature variation of a structure for the nonlinear effect of the thermal load.

To validate the scaling laws, both an FEM computation and a wind tunnel test were performed to investigate the natural modes and the vibration response of the thermal structure.Some encouraging results are shown in that the initial temperature of the structure does not affect the structural modes if the elastic modulus is constant, and the structural modes are still determined by the temperature variation even when the temperature variation of the structure exceeds the critical buckling temperature. For the wind tunnel test, the difference between measurement and numerical computation is mainly due to the assumption of the temperature being distributed uniformly and errors in model manufacturing and installation.However,this difference between the two different methods could be eliminated through replacing numerical computations by measuring the structure response of the full-scale structure directly and employing better structure manufacturing and installation simultaneously.

This investigation shows important factors when predicting thermal structure vibration generated by wall pressure fluctuations of a hypersonic TBL.It provides a theoretical support for developing a new approach to evaluate structure vibration of aircraft caused by hypersonic flow.

Acknowledgement

The authors gratefully acknowledge the f inancial support of the Equipment Priority Research Field Foundation of China(No. 6140246030216ZK 01001).