Reliability and structural integrity evaluation of rotating machinery: a case study on turbo-compressors with ANSYS workbench

2.1. Euler-Bernoulli theory

Primarily, in actual circumstances, the TB is affixed to the ground using six holes and supported by bolts (cyan circles in Fig. 2(a)). Assuming a two-dimensional diagram, TB has combined two simply supported beams (ss-beams). Fig. 3 depicts a double-ss-beam with a point load positioned at its center.

Fig. 3Analogy model of the testbed to simply supported beam

The Euler-Bernoulli theory is a key theorem used to analyze the mechanical behavior of beam structures. Therefore, in the Euler-Bernoulli beam theory, it is assumed that there is no shear stress in beam deflection. As a result, the displacement is denoted as $w\left(x\right)$ and the bending moment is denoted as $M\left(x\right)$. The displacement of a simply supported beam, $w\left(x\right)$, is described as follows:

By putting the middle load of the beam (as shown in Fig. 3) into Eq. (2), that calculates the maximum deflection of the simply supported beam at the midpoint of the beam:

In order to assess Eqs. (1) and (2), it is necessary to consider the parameters listed in Table 2, which may be determined during the procedure. Fig. 4 depicts the cross-section of the TB structure, where the black region represents the actual cross-section, while the red rectangle area represents a simplified version used for calculating Eq. (3).

It is important to know the safety factor (SF) when judging how reliable a planned model is. The SF is found by comparing the planned values to the experimental values as a ratio. An SF greater than 1 is considered acceptable by engineers, while an SF less than 1 means the model failed. The SF is shown in two formats: ultimate strength and yield strength:

Indices u and y define the ultimate and yield statements. They represent the tensile ultimate and/or yield strengths. Su and Sy are the maximum values of the ${\sigma }_u$ and ${\sigma }_y$ strengths, respectively. Furthermore, the quantity of $S_u$ or $S_y$ is derived using empirical standard testing. Additionally, based on the mechanical characteristics of the material considered in the test bed construction, the tensile ultimate strength is 600 MPa and the yield strength is 370 MPa.

Fig. 4Simplification cross-section of the testbed

2.2. Methodology

The FEM is one of the most practical numerical methods used for modeling and analyzing structures in various engineering research fields. ANSYS, a commercial FEM software, efficiently analyzes mechanical behaviors under different physical conditions, geometries, and materials. Based on the three-dimensional model (Fig. 2(c)), which is imported into ANSYS Workbench in .stp, .x_t, or .igs format, the analysis follows a step-by-step approach across multiple studies, including modal, static, harmonic, and transient modeling to evaluate the mechanical properties of the TB. Considering the boundary conditions, such as supports at different locations and automatic meshing of the structure, the modal analysis is first performed to extract the natural frequencies of the structure. Subsequently, distributed loads are applied to the top face of the structure, preparing the TB for further analyses. For instance, the static study determines the Von Mises Stress (VMS) to evaluate the ultimate safety factor. Based on the extracted natural frequencies, the harmonic analysis is then conducted to examine VMS under cyclic loads, helping to compute the yield safety factor. Finally, the transient analysis monitors the VMS of the TB at the initial loading phase, where the operational load is applied as a step force (see Fig. 5).

Fig. 5FE procedures in ANSYS workbench: a) configuration model, b) loading, c) meshing

a)

b)

c)

3.1. Simulation process

The table below presents numerical data obtained from ANSYS Workbench 2019-R3, including static stress (SS), dynamic stress (or harmonic analysis) (DS), and transient stress (TS). These results are used to improve the reliability of the design. The selection models operate according to the following methodology:

1) Static Analysis: This analysis evaluates the long-term effects of the continuous application of force by the rotor on the structure of the TC.

2) Dynamic Analysis: This type of analysis looks at how the system responds to harmonics, which means that the angular speed of the TC changes over time while it is working.

Analysis Steps:

Step 1: Modal Analysis. To find resonance events, modal analysis simulates all the proposed models in ANSYS and compares them to a reference model that doesn't have any support. The natural frequencies of the proposed models show significant differences from those of the reference model, as illustrated in Fig. 7 and Table 3.

Step 2: Static Analysis. Static analysis checks to see if the proposed models’ material strengths are realistic and stay below standard values, like ultimate tensile and yield strengths. This information is provided in the table.

Step 3: Safety Factor for Static Analysis ($S{F}_u$). Eq. (5) is used to find models that can be used for further analysis in ANSYS by calculating the $S{F}_u$ values of the static analysis. Proposed models with an $S{F}_u$ value below 1, such as models 4 and 7, are rejected.

Step 4: Harmonic Analysis. Harmonic analysis looks at how the TC reacts to the forces that the rotor puts on its structure at different angular speeds.

Step 5: Safety Factor for Harmonic Analysis ($S{F}_y$). Eq. (6) is used to figure out which proposed models should be left out of the next steps by finding their $S{F}_y$ values for harmonic analysis. Models 7 is rejected based on these calculations.

Step 6: Transient Analysis. In ANSYS, transient modeling looks at the system’s starting conditions, which could be changed by unknowns like electrical or mechanical shocks or impulses.

Step 7: Final Model Selection. The ultimate selection narrows down to models 1 to 6. Among these, model 5 demonstrates superior performance and is strongly recommended for implementation under real operating conditions.

In order to avoid resonance occurrences, it is crucial to take into account the natural frequencies of all models. The Fig. 7 and Table 3 show that when models 1 through 7 are looked at as a whole, their natural frequencies are very different from the test bed’s natural frequency structure, in Table 3.

According to the Table 3, the natural frequencies of all the models are significantly shown different from those of the original model (Fig. 7), which is an outdated form of the structure. Therefore, models one to seven are eligible for usage under TC in the lab. However, it must need to choose the most optimal model using the trial-and-error approach. Consequently, to suggest using supplementary simulation methods to assess the material durability of all models, each with distinct support positions. To clarify, this approach involves using basic analytical techniques such as static and harmonic approaches to represent many proposals and determine the most optimal choice. The forces applied to the structure are dispersed according on the lowest and maximum angular velocity of TC’s rotor in our lab. In Table 4, these forces are 50 to 150 tons per meter and are spread over the surface of the structure in a normal distribution, based on the weighted force formula $F=mg$.

Fig. 7Natural frequencies of the 7 models

Fig. 8 displays the bending moments and deflections of ss- beam with different inputs, namely point loads specified in Table 4 (third column), as determined by Eqs. (1) and (2).

Above all the mentioned steps, the FEM’s results derived from Table 4 (Fig. 5) include seven models, as shown in Figs. 9-14, which illustrate the VMSs in different analyses: static (Fig. 9), harmonic (Fig. 10), and transient (Fig. 11) compare the maximum mechanical stress in different models at each node. Similarly, based on the accepted six models in Table 4, Figs. 12 and 13 clearly compares the ultimate and yield safety factors, determining which model provides the best performance. Finally, model 5 stands out as one of the best designs, demonstrating high material and structural strength across different analyses and operating conditions.

Fig. 8Comparison results: a) maximum torque and displacement of ss-beam based on TB’s properties, b) mechanical stress in static forces

a)

b)

Fig. 9Maximum stress of nodes in static analysis

In detail, Fig. 14 the numerical results (FEM models in ANSYS) for six acceptable models based on Table 4. These results include static, harmonic, and transient analyses. Models 1 to 6 demonstrate reasonable strength under various load conditions. Hence, the safety factors ($S{F}_u>$ 1 and $S{F}_y>$ 1) indicate that the maximum mechanical stresses in models 1 to 6 exceed their ultimate and yield strengths under different loads. Therefore, these models are analyzed with different support configurations based on the boundary conditions of the TBs in various directions. Elaborately, Fig. 14 compares various maximum stress, safety factors from previously accepted models and highlighting the most notable ones.

Fig. 10Maximum stress of nodes in harmonic analysis

Fig. 11Maximal stress in transient analysis

Currently, it is deemed acceptable to thoroughly study all seven models in order to proceed with the next phases. Utilizing ANSYS, the FEM of the fifth recommended model was analyzed using four methods: modal, static, harmonic, and transient studies. The contour spectrum in the following illustrations represents three-dimensional displacements and corresponding stress (see Figs. 8-11). Based on Fig. 5, four types of analyses are conducted in the tree design.

Fig. 12Ultimate safety factor of the 7 models

Fig. 13Yield safety factor of the 7 models

In the modeling phase, a triangular mesh (mechanical type) is automatically and adaptively generated, as shown in the mesh information section of Fig. 5. For example, the number of nodes and elements are 10,426 and 5,612, respectively. At this stage, the model is ready for modal analysis. Secondly, fixed support is assumed at three positions at the bottom of the holes. Thirdly, the load distribution is applied to the left and right top faces of the structure with varying values of 50, 80, 100, and 150 tons per meter. The model is then utilized for static, harmonic, and transient analyses. In the harmonic analysis, the lower and upper frequency bands are determined based on the minimum and maximum natural frequencies extracted from the modal analysis. In the transient analysis, the applied load is assumed to be a step force with a time step of 0.01 seconds. As a consequence, the corresponding results of the shape modes, three-dimensional displacements, and mechanical stress for different loads in model 5 are shown in Fig. 15, Tables 5, 6, and 7 illustrate the spectrum, where the critical zones are indicated by maximum values in red and minimum values in blue.

Fig. 14Comparison maximum stress and safety factors of the 6 models

Fig. 15Ten mode shapes of the fifth model

Table 5Three-dimensional displacements of distributed forces for the best model

80 ton/m	50 ton/m
		$X$
		$Y$
		$Z$
150 ton/m	100 ton/m
		$X$
		$Y$
		$Z$

In short, Table 8 provides a brief description of each procedure in SOLIDWORKS and ANSYS simulations for understanding why the optimized model is considered the best model in this research. Finally, the fifth model is nominated as the optimal design, as shown with the size of supports in Fig. 16.

In short, the entire process of this research is summarized in Table 7.

Fig. 16Support size in model 5

Table 6Stress contours of the best model in static, harmonic, and transient analyses for model 5

Static	Harmonic	Transient
50 (ton/m)
80 (ton/m)
100 (ton/m)
150 (ton/m)

Table 7Three-dimensional displacements of harmonic analyses with distributed forces for model 5

80 ton/m	50 ton/m
		$X$
		$Y$
		$Z$
150 ton/m	100 ton/m
		$X$
		$Y$
		$Z$