Dynamic response analysis and optimization of orbital support structure

2.1. The establishment of finite element model

The principle of the guideway transport system is shown in Fig. 1(a), where it can be seen that the orbital support structure, as a key load-bearing component, is always subjected to both gravity loads and complex dynamic loads. As shown in Fig. 2(a), the orbital support structure consists of support leg, guide rail, reinforcing rib, and other components. The dimensions of these components are critical to the modal characteristics and strength of the model. To facilitate the multi-objective optimization of the model, parameterized modeling is requisite. In the process of optimization analysis, the selection of design variables should not affect assembly dimensions and should not interfere with working limit conditions. Grounded on the structural characteristics of the model, the width of the support leg $t$, the length of the intermediate support rod $h$, and the lateral length of the rib $l$ are regarded as the parameterized dimensions. The range of values for design variables is shown in Table 1. The three selected design variables not only do not affect the overall assembly dimensions, but also do not exceed their respective constraint boundaries, leading to unreliable or even failed designs, which has a higher engineering feasibility. In order to ensure the accuracy and efficiency of the calculation, unnecessary features such as chamfers and fillets that have a minimal impact on the analysis results need to be removed. In terms of load and constraint settings, complex models such as the drive motor are removed and their mass is converted into a load applied to the corresponding contact surface. After simplifying the model, material types and related properties need to be set, as shown in Table 2.

Fig. 1Structure and modeling of orbital support

a) Structure composition and principle

b) Structure of orbital support

2.2. Analysis and discussion of results

In order to provide effective base data for multi-objective optimization, a coupled module was established based on the ANSYS Workbench platform. The modal module was added to the static structure module in the solution, and the modal analysis was calculated using the subspace method. Additionally, since the vibrations generated by the driving motor can have a significant impact on the entire support structure, a harmonic response analysis was also conducted, and the unsuitable working frequency range of the motor was determined based on the analysis results, in order to avoid resonance as much as possible. Through continuous iterative calculations, the first eight natural frequencies and the first four modal shapes of the model can be obtained, as shown in Table 3 and Fig. 2.

According to the simulation calculation results, the first-order natural frequency is 5.89 Hz, which corresponds to the main modal vibration pattern of rotating around the $Z$ axis. Since the maximum working speed of the driving motor is 300 r/min, which is the excitation frequency of 5 Hz, it is close to the first-order vibration mode, it is necessary to optimize the first-order natural frequency. If it works in the resonance frequency band, it may cause the end effector to collide and lead to failure of material transport.

Fig. 2The first six effective modal shapes

a) The firs order

b) The second order

c) The fourth order

d) The fifth order

Harmonic response analysis reflects the dynamic characteristics of the structure under different frequency sinusoidal (harmonic) loads, thus verifying whether the design can overcome resonance, fatigue, etc. The driving motor may have a bias due to processing, assembly errors, etc., and an additional external load will be caused by the bias during rotation. By adding the harmonic response module to the modal module, the harmonic response analysis is performed separately for the motor during operation using modal superposition method based on the modal analysis results. According to the actual situation and modal analysis results, the frequency range of 0-50 Hz and frequency interval of 1 Hz are selected, and the displacement-frequency curve at the center of mass is obtained, as shown in Fig. 3. It can be seen that when the drive motor works at frequencies of 5-7 Hz, 13-17 Hz, 30-33 Hz, and 43-46 Hz, there are extreme values of displacement in the $X$, $Y$, and $Z$ axes, indicating that the drive motor working at these frequencies will cause the actuator connected to the end of the column to resonate and produce positioning accuracy errors. These frequencies basically include the first few natural frequencies obtained from the modal analysis.

3.1. Construction of agent model

One of the crucial aspects in establishing a response surface surrogate model lies in selecting the appropriate number of sample points in the experimental design and arranging them rationally within the design space, which entails the choice of an experimental design approach. Due to the relatively smooth and regular structure of the model, a quadratic polynomial function is employed to construct the response surface, which can furnish significant information regarding the design variables and experimental error with the minimum number of experiments and boasts the merits of simple design, excellent predictability, and high modeling efficiency. For the design of sample parameters, based on the Latin square method, a discrete data set of design variables and optimization objectives is obtained as presented in Table 4. The objective function and its parameters fail to demonstrate a continuously linear response, thereby posing a challenge for accurately expressing the first-order response. In this research, we select a quadratic response formulation to represent the response surface function, considering that the optimization scale of the partial structure is relatively narrow and the variation of the response surface with respect to the independent variables is rather subtle.

Fig. 3Results of harmonic response analysis of center of mass

Once the second-order polynomial response surface surrogate model for weight, maximum equivalent stress, and first-order natural frequency has been constructed, it is imperative to undertake a precision validation of the established response surface surrogate model to ascertain whether it possesses adequate accuracy and can substitute the actual engineering model for the subsequent lightweight design. In numerous practical engineering design issues, it is frequently indispensable to concurrently fulfill multiple design criteria to attain an optimal condition, which constitutes an extremely challenging and intricate problem. Such problems frequently emerge in diverse domains of engineering design. To address this complex problem, multi-objective optimization approaches are requisite for obtaining the optimal solution. This approach demands simultaneous consideration of various design objectives and their integration to achieve a state of simultaneous optimality for multiple objectives. n the realm of multi-objective optimization design, it is of paramount importance to admit that an inevitable certain degree of disparity exists between the response surface model and the actual response values. This variance can exert a significant influence on the reliability of the optimization procedure. To guarantee the accuracy of the fitting function, it is indispensable to verify its precision. If this error lies within an acceptable limit, it implies that the optimization function is feasible. The results of error verification are shown in Table 5, where it can be seen that the errors of the three optimized objectives all meet the requirements.