Stress-wave balance for measuring the aerodynamic drag force in impulse hypersonic facilities

Abstract

In the present study, a stress-wave balance and its shield are developed for measuring the aerodynamic drag force of diverse models in the shock tunnel, while the test environment is accompanied by strong vibration interferences and the test time is as short as a few milliseconds. In order to avoid the inertial force interference caused by vibration of the sting support, the balance is suspended in the shield by an innovative polymer wire system. Several commonly used deconvolution algorithms are quantitatively compared, and errors of each algorithm are obtained accordingly. Moreover, the influence of vibration interference of the balance itself on the results of deconvolution calculation is studied. Then, a constraint criterion for the signal length is proposed in the deconvolution calculation process to resolve this problem. Shock tunnel tests are carried out by using a solid hemisphere model and a hollowed Apollo re-entry capsule model. The obtained results show that the deviation between experimental results and reference values is less than 5%.

Appendices

Appendix 1

Modal analysis

Since the arrival of the flow field will stimulate the vibration of the balance, which will cause interference to the effective signal, we carried out the finite element simulation analysis by ANASYS software on the vibration modes of several balance structures.

Table 3 Modal frequencies of balance with equal-section sting

Fig. 20

Vibration shapes of balance with equal-section sting

Table 4 Modal frequencies of balance with the conical sting and the solid model

Fig. 21

Vibration shapes of balance with the conical sting and the solid model

• (a) Balance structure with an equal-section sting

This structure was mentioned by Sanderson and Simmons [14] and Sahoo et al. [3], respectively, in the papers on stress-wave balance research. Sanderson and Simmons [14] mentioned the length of the sting in the paper (page 1, paragraph 5) published in AIAA Journal in 1991: “The strain gauges are placed near the model end of the sting in order to maximize the observation time before the onset of complications caused by the return of stress waves reflected from the free (downstream) end of the sting.” Based on this, the length of the balance can be calculated from the propagation speed of the stress wave in the material (brass, 3080 m/s) and the effective test time. For our experimental flow field (6-ms effective test time), the length is almost 9.24 m. Since the shape of the model does not affect the results of the modal analysis, we simplified the model to a cylinder with mass, and the modal analysis results are given in Table 3 and Fig. 20.

Mode 22 is the axial vibration mode of the model–balance structure, and the calculated frequency is similar to the frequency obtained by Sahoo et al. [3]. According to the calibration data of the balance, because the balance is used for the measurement of axial force, the balance is also sensitive to the axial vibration mode of the balance.

• (b) Balance structure with the conical sting and the solid model

The conical sting was used to enhance the dispersion effect of stress wave in the process of propagating downstream, so as to weaken the intensity of the reflected stress wave from the sting end and shorten the length of the balance. The modal analysis results are given in Table 4 and Fig. 21.

• (c) Balance structure with the conical sting and the hollowed model

The modal analysis results are given in Table 5 and Fig. 22.

Table 5 Modal frequencies of balance with the conical sting and the hollowed model

Fig. 22

Vibration shapes of balance with the conical sting and the hollowed model

It can be seen that the structure of the balance has an effect on the dynamic characteristics of the balance and the output of the balance. The dumbbell-shaped balance structure highlights the axial vibration mode of the balance (from mode 22 to mode 14) and means that the vibration signal contained in the output signal of the balance is more accurate. The shorter balance length makes it easier to test in shock tunnels.

Appendix 2

Dynamic analysis

The quality parameters of the two model–balance systems (rough calculation value) are given in Table 6.

Due to the dumbbell shape of the model–balance structure and its constraint condition (suspended), the model–balance system can be regarded as a single-degree-of-freedom double-oscillator vibration system, as shown in Fig. 23, where the mass of the oscillators is $m_{1}$ and $m_{2}$, the center of the mass is C, and the distance of the oscillators from the center of the mass is $l_{1}$ and $l_{2}$.

The vibration frequency of system is deduced as follows:

$$\begin{aligned}&m_{1} \times l_{1}=m_{2} \times l_{2},\\&l_{1}+l_{2}=l,\\&l_{1}=m_{2} \times l /( m_{1}+m_{2}),\\&l_{2}=m_{1} \times l /( m_{1}+m_{2}). \end{aligned}$$

Suppose the stiffness of $l_{1}$ is equal to $k_{1}$ and the stiffness of $l_{2}$ is equal to $k_{2}$:

$$\begin{aligned}&k_{1} \times l_{1}=k_{2} \times l_{2}=k \times l,\\&k_{1}=k \times l / l_{1}=k \times ( m_{1}+m_{2}) / m_{2},\\&k_{2}=k \times l / l_{2}=k \times ( m_{1}+m_{2}) / m_{1}. \end{aligned}$$

The vibration frequency of the system is:

$$\begin{aligned}&T_{1}=2 \times \pi \times \hbox {sqrt}(m_{1}/k_{1})\\&\quad =2 \times \pi \times \hbox {sqrt}(m_{1} \times m_{2}/((m_{1}+m_{2}) \times k))=T_{2}=T. \end{aligned}$$

Among the equations, $k=k_{1} \times l_{1}/(l_{1}+l_{2})=k_{2} \times l_{2}/(l_{1}+l_{2})$.

The frequency $f=1/T=1/(2 \times \pi \times \hbox {sqrt}(m_{1}\times m_{2}/((m_{1}+m_{2}) \times k)))$ (Table 7).

Table 6 Quality parameters of the two model–balance systems

Fig. 23

Double-oscillator vibration system model

Table 7 Equivalent mass of the two model–balance systems

The equivalent mass of the solid model system is less than that of the hollowed model system, so the vibration frequency of the solid model system is higher under the condition of the same stiffness. At the same time, it is found that when the mass of the model and the conical sting is similar, the system will obtain greater damping, which makes the vibration signal decay faster. This further shows the influence of balance structure on dynamic characteristics of the balance.