Experimental research and numerical simulation on vibration characteristics of a rectangular plate structure in fast time-varying thermal environments

2.1. Specimen

As shown in Fig. 1, the specimen is a rectangular plate structure made of high-temperature nickel-based stainless steel 1Cr18Ni9Ti, with the length of 340 mm, width of 220 mm and thickness of 7.5 mm. The shorter edge of specimen is completely wielded to a mounting bracket perpendicular to the specimen. There are 9 holes on the specimen. Eight of these holes are distributed on the longer edges and used to install the high-temperature-resistant extended measurement components of the vibration signals, with a diameter of 5 mm. The other one hole is distributed on the free edge and used to attach the excitation guiding rod, with a diameter of 8 mm.

Fig. 1Schematic of the specimen (units: mm)

2.2. Experimental apparatus.

The thermal-vibration test apparatus is shown in Fig. 2. The rectangular plate is horizontally installed and fixed on the vertical beam through the bolts and bracket. A heat-insulation metal plate having water cooled tubes is designed between the mounting bracket and vertical beam. In the test, flowing liquid is running through the tubes to cool down the heat-insulation metal plate. It creates a temperature buffer zone between the specimen’s root and vertical beam, and thus protects the loading setup in high-temperature environments.

Fig. 2Experimental apparatus

The exciter is set under the free end of the specimen and exerts excitation force to the specimen through the metal pole from the outside of the high-temperature thermal field. Acceleration sensors used to pick the vibration signals are fixed at the lower end of the extended poles. These poles are hollow and made from corundum material, with light mass, high stiffness and strong anti-deforming capability, which all contribute to transmit the vibration signals of the specimen effectively. Being installed out of the thermal environment, the ordinary acceleration sensors are able to be used in the test of this paper to obtain the vibration signals of the specimen in high-temperature environments which are difficult to measure.

2.3. Thermal environments in the test

In this study, 6 kinds of thermal environments with different linear heating rates are preset and the final heating temperatures are 100 °C, 200 °C, …, 600 °C, respectively. During the test, the thermal control system heats both sides of the specimen simultaneously to the end temperatures within 200 s.

2.4. Modal frequency

Fig. 3 shows the smoothed curves of the dynamic changes of the first-, second-, and third-order natural frequencies versus time at different heating conditions, in which the black solid line represents the modal frequency at room temperature. Fig. 3 indicates that the modal frequency of each order tends to decrease during heating period. The higher is the heating rate, the faster the changes in the natural frequency and the greater the amplitude of the decrease.

Fig. 3Modal frequencies of the specimen under different heating conditions

a) First-order frequency

b) Second-order frequency

c) Third-order frequency

3.1. Boundary conditions

According to the actual installation of the specimen, clamped boundary condition is set on the root edge of the rectangular plate in the numerical calculation.

During the test, non-uniform temperature field forms in the specimen because flowing liquid in the water-cooling device at the root of the specimen cools down the fixed bracket. Thermal stresses introduced by non-uniform temperature field will change the vibration characteristics of the specimen. Therefore, the temperature distribution of the specimen must be taken into consideration in the numerical simulation.

The temperature distribution on the specimen was obtained by measuring during the test. 23 temperature sensors are installed on the upper surface of the specimen and their locations and serial numbers are shown in Fig. 4. One temperature sensor (1#) is installed at the welding position between the rectangular plate and the fixed bracket. Two sensors (23# and 24#) are installed at the center portions of the upper and lower surfaces of the plate, respectively, to control the surface temperature. Seven temperature sensors (1#, 4#, 9#, 14#, 19#, 22#, and 23#) at the center line are used to measure the temperature variation along the x direction of the plate while the 4 columns of temperature sensors are used to get the temperature variation along the y direction near the root of the specimen.

Table 1 and 2 list the temperature data at 100 s (final heating temperature 600 °C) obtained from several typical sensors. It can be seen that the temperature changes much in the $x$ direction (Table 2) while keep roughly uniform in the $y$ direction (Table 2). For example, along the $x$ axis, the temperature at point 23 is 256 % higher than that at point 1; in the y direction, the temperature at point 16 is approximately the same as the temperature at point 14 with relative error 2.1 %. In addition, it is evident that the temperature in $x$ direction has no change when $x>$165 mm. Therefore, to simplify the transient temperature field on the specimen surface, this temperature field can be regarded as a function of time t and position coordinate $x$ and the coordinate y can be neglected in the numerical simulation.

Fig. 4Temperature sensors on the surface of the specimen

The specimen made of nickel-based stainless steel has a good thermal conductivity and a very thin thickness, so the inner temperature of the specimen is in close agreement with the surface temperature (namely, the temperature doesn’t change along the specimen’s thickness) when the both sides of the specimen are heated simultaneously. Hence, in the numerical calculation, shell element can be employed and the temperature field of the whole specimen can be represented by the surface temperature.

3.2. Comparison between calculated and test results

Fig. 5 shows the comparison between the calculated and test results of the first-, second-, and third-order modal frequencies of the specimen under two heating conditions of the end temperatures 300 °C and 600 °C (due to limited space, comparisons under other heating conditions are not put out). The curves in Fig. 5 show the difference between the calculated and test result is not very significant and their change trend is similar. Table 3 lists the calculated and test data of the first three orders modal frequencies of the specimen under the heating condition of final temperature 600 °C. As indicated by the data in Table 3, during the whole heating process, the relative errors between the numerical and test results of the first- and third-order modal frequencies are less than 6.0 % and those of the second-order modal frequencies are less than 8.0 %, suggesting that the numerical results coincide favorably with the test results for the specimen under rapid heating conditions.

From the expression for initial stiffness matrix $K_T$ and additional stress stiffness matrix $K_{\sigma }$, it can be known that the data size of the former is mainly related to the elasticity matrix [$D_T$] while that of the latter is to the thermal stress matrix [$S$]. Calculated results show that the highest magnitude of thermal stresses in this test is 10⁹, much smaller than the magnitude of the elasticity matrix because the magnitude of the material’s elastic modulus is 10¹¹. Therefore, the main reason why the modal frequency of the specimen changes during the heating process is that the elastic performance of the structure changes in the high-temperature environments.

Fig. 5Comparison between the calculated and test modal frequency

a) Final heating temperature 300 °C

b) Final heating temperature 600 °C