Startseite A study of the vibrations of a rotor bearing suspended by a hybrid spring system of shape memory alloys
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A study of the vibrations of a rotor bearing suspended by a hybrid spring system of shape memory alloys

  • Laith H. Mohammed EMAIL logo und Qasim A. Atiyah
Veröffentlicht/Copyright: 6. Februar 2024
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Abstract

The study of rotor-bearing vibration is crucial across various fields, encompassing applications such as rotating machinery, wind turbines, washing machines, and elevators. However, operational challenges can arise from these machines’ propensity to vibrate under specific conditions. To address this issue, the current research investigates the utilization of shape memory alloy (SMA) springs as intelligent materials in the rotor suspension system. SMA’s unique property of changing stiffness with temperature-induced phase shifts is harnessed, leading to the construction of a test rig for validating vibration characteristics. A novel hybrid bearing is devised to effectively manage vibrations, particularly in resonance zones. To the authors’ knowledge, this new model is never reported in the literature. Then, an accelerometer is employed to measure the rotor shaft response corresponding to disc position vibration signals. Furthermore, a numerical model is developed to validate experimental results, taking into account the phase change of SMA springs and the disc’s influence on the rotor bearing’s natural frequency. Temperature variations from 20 to 80°C resulted in a 13% change in natural frequency for the hybrid spring configuration. The experimental findings align closely with ANSYS simulations, displaying an acceptable error ratio, with the highest error remaining within 20% thresholds.

1 Introduction

Rotor-bearing vibration may be incredibly dangerous and expensive. It may result in equipment failure, property damage, and safety issues. It is important to control vibration levels in rotating equipment to minimize its effects. One may achieve this through a variety of approaches. Using smart materials is one of these methods. These materials can alter their phase due to temperature changes, which affects their characteristics, particularly the modulus elasticity [1,2,3,4]. The use of shape memory alloys (SMAs) for this purpose has attracted many researchers in recent years. When heated over a given temperature, SMAs can change from the martensite phase, which has a certain shape, to the austenite phase, which has a different shape. Therefore, the rotor of a rotating machine can be suspended at the correct place using SMA wires that carry electric current for heating to alter their phase [5,6,7]. Several studies have thoroughly investigated how SMAs affect the vibration behavior of rotating bearings. For example, Yogaraju et al. [8] developed a novel semi-active experimentally and numerically journal bearing using SMAs to provide needed force. Their experimental results found that the ovality ratio can impact the dynamic response, where when it increased, this would increase dynamic stiffness and damping. Garafolo and McHugh [9] studied the effect of SMA wire on the reduced vibration of flexible clamped–free beams embedded by SMA wire, where the accelerometer sensor was connected at the tip of the beam, and the natural frequency was determined using a frequency response function at various temperatures. According to their experimental findings, the natural frequency shift was 44.7%, and the amplitude at the beam’s tip dropped by 34%. Likewise, the unactivated SMA had a 44.6% amplitude reduction and a 160% frequency shift, whereas the activated SMA had a 258% amplitude reduction and a 63.6% frequency shift. Borges et al. [10] investigated that the first natural frequency, which is based on spring stiffness increasing as temperature increases and was empirically measured, might be reduced by utilizing helical SMA springs. A fuzzy controller is used to raise the temperature from 30 to 70°C in 25 s. Their results showed that the resonance zone’s amplitudes decreased by up to 60%. Xu et al. [11] utilized, experimentally and theoretically, the rotor seal system and the dynamic vibration absorber to reduce unbalanced or critical vibration. Their findings demonstrated that the dynamic vibration absorber alters the instability threshold and instability vibration frequency. Atiyah et al. [12] looked into the effect of the SMA wire number on the composite cantilever beam’s inherent frequency. According to their findings, increasing the number and diameter of SMA wires in the martensite phase caused a drop in the beam’s natural frequency, but increasing them in the austenite phase caused an increase. Also, it was discovered that a decrease in the beam’s inherent frequency might have occurred due to lengthening the beam. Braga et al. [13], utilizing a passive controller, examined numerically and experimentally the effect of SMA wire at different temperatures (30, 45, and 60°C) on the dynamic response of spinning. The results supported computational predictions that the SMA may reduce vibration amplitudes. Oliveira et al. [14] employed the Jeffcott modified approach, which operates at frequency across the resonance area, to examine the effect of stiffness change brought on by temperature rise on the performance of helical SMA springs of smart bearings. With a response time of 12–15 s, an amplitude reduction of 63% (RMS) and a maximum reduction of 73% were achieved. Rahmana et al. [15] investigated the whirling effect of the Jeffcott rotor formed from an SMA theoretically and experimentally. Based on the findings, they concluded that more damping greatly lowers the whirling amplitude (r/e). The maximum r/e for synchronous whirls occurs at unity spin ratios due to resonance. Yet, at high spin ratio values, the value of r/e approaches unity. It is noteworthy that the shaft reaction is notably different for a certain spin ratio of 1.414, all Pd/mg vs on both sides of the crossing point. Abdulkadhim et al. [16] used a proportional–integral–derivative controller on the two-dimensional rotor system suspended by helical SMA springs, and it was discovered that the controller had a high tracking and set point precision. Senko et al. [17] used experiments to analyze the mechanical performance of a unique bending spring shape composed of superelastic SMA (SMA-SE). The findings showed that, compared to cases without and with 1.5 mm of preload, employing the innovative M-shape spring of SMA-SE might reduce vibration by up to 23 dB. It has been demonstrated how well the M-shape device works to reduce mechanical vibration in rotor systems. Tuaib et al. [18] studied the effect of adding active magnetic bearings on rotating machines to minimize vibration and vibration behavior in traditional bearing systems. Their article uses simulation tools to analyze the vibrations in the rotary bearing system to determine their natural frequencies and performance-affecting factors. As a result of adding active magnetic bearings to rotating bearing shafts, vibration amplitude was reduced by around 60%. According to their study, the spinning bearing shaft is more solid and stable with the active magnetic bearing attached. Furthermore, the natural frequency was inversely proportional to the shaft length. The research was done on the effects of adding active magnetic bearings to rotating machines to reduce vibration and the vibration behavior in conventional bearing systems. They analyzed theoretically and analytically the vibrations in the rotary bearing system using simulation tools to identify the natural frequencies and performance-influencing elements. According to their results, adding active magnetic bearings to the rotating bearing shafts reduced vibration amplitude by almost 60%. This research suggests that adding the active magnetic bearing to the spinning bearing shaft makes the system more robust and solid. The natural frequency was also inversely related to the diameter of the spinning shaft, as seen in previous studies [19,20,21].

Different from the mentioned literature, this research fills many gaps. The main objective of this work is to monitor the variation and the natural frequency of the rotor system using SMA suspensions and compare the experimental with numerical results. A hybrid spring that combines helical and leaf springs is used to study rotor-bearing dynamics. To the authors’ knowledge, utilizing helical spring has never reported in the literature before in similar studies. Also, the effect of disc location on the dynamic reaction of the rotor bearings, as well as their dynamic behavior at the second, third, and fourth critical speeds, is investigated. The general objectives can be summarized as follows:

The dynamic response of a new design of support consisting of the combination of the leaf and helical springs of rotor bearings (hybrid type) is studied.

  • Designing and manufacturing molds for the SMA springs.

  • Manufacturing an experimental rig with a flexible rotor shaft and bearings with supports.

  • Implementing the controller of vibration and heating.

  • Designing and implementing electrical circuits for heat generation.

  • Develop an analytical model and compare it with the experimental and numerical by using the ANSYS 18.1 in simulation.

The research methodology is summarized in Figure 1.

Figure 1 
               Research methodology flowchart.
Figure 1

Research methodology flowchart.

The article is organized as follows: Section 2 presents the theoretical investigation. The experimental part is presented in Section 3. Section 4 discusses the results. Finally, Section 5 concludes the article.

2 Theoretical investigation

The built-in rig was numerically modeled using the ANSYS tool in the majority of earlier works that could be found in the literature in order to analyze the natural frequency behavior at martensite and austinite phases. Different from other literature, utilizing SMA suspensions, track the rotor system’s variation in natural frequency and contrast the experimental and numerical results. To analyze rotor-bearing dynamics, helical and leaf springs are coupled to create a hybrid spring. Noting that, this novel design is never reported in the literature. Additionally, the dynamic behavior of the rotor bearings at the second, third, and fourth critical speeds is examined, as well as the impact of disc position on these factors. This study utilizes the ANSYS workbench version 18.1 software. In this section, creating the geometry using the AUTOCAD 2021 program, which is interoperable with various simulation modeling tools, the geometry for each module was drawn. Then, as shown in Figure 2, it was loaded into the ANSYS program. Then, the identical instances explored experimentally are modeled and numerically resolved. This study uses an eight-node SOLID186 element to mesh a 3D rotor system model, as illustrated in Figure 3. The default mesh size is also chosen. To conduct a free vibration analysis, finite-element quantities are compared to test ones. To keep the layers from growing genetically related to one another, requirements link layers and skins of the rotor plate at their contact points. Drawing the structure after choosing the element types, adding the mechanical properties of the materials for the design, supporting the system following the supported type for the rotor structure, and finally using modal analysis to determine the free vibration characteristics for the rotor were all part of the numerical solution. Note that in this work, the properties of the shaft are assumed to be constant such as cross section, Young’s modulus, and moment of inertia. These assumptions do not affect the current study. On the other side, the internal and the external damping of the shaft are neglected. Since the study is involved in studying and investigating the damping effect of the hybrid spring, the boundary conditions are assumed firm and articulated for the clamped side (no spring side) and the hanged side (the spring side) (Figure 4).

Figure 2 
               The SMA springs with shaft and disc modeled in AutoCAD.
Figure 2

The SMA springs with shaft and disc modeled in AutoCAD.

Figure 3 
               Boundary conditions used.
Figure 3

Boundary conditions used.

Figure 4 
               Demonstration of the first mode shape.
Figure 4

Demonstration of the first mode shape.

3 Experimental work

In this article, a test rig is built to conduct an experiment on which the dynamics of an asymmetric journal-bearing supported rotor system is to be examined. The base for the experiment is a stiff cylindrical shaft supported by an antifriction bearing. On the shaft side, a single disc is attached at various distances from the bearing and spring arrangement. The translational vibrations of the pedestal were measured using the accelerometer sensor, which was mounted on the end of the rotating shaft. When the shaft rotates at a slow rotational speed, a continuous synchronous vibration with a small amplitude is observed. The system reaches the initial resonance with a significant vibration amplitude at the first natural frequency as the rotational speed rises. The vibration resumes its regular state after the resonance. As the shaft emerges, the vibration gets stronger and faster.

Results for the chemical composition of the rotor shaft are shown in Table 1, while the mechanical properties of the shaft are shown in Table 2. Tables 3 and 4 show the mechanical characteristics of springs, while Table 5 shows the chemical components of springs (SMAs) (Figures 5 and 6).

Table 1

Chemical composition of the steel shaft

Sample C% Si% Mn% P% S% Cr% Mo% Ni% Al% Cu% Fe%
Diameter (8 mm) 0.128 0.364 12.53 0.0369 0.0056 13.24 0.0162 1.35 0.0057 0.593 Bal.
Table 2

Steel shaft specifications (experimentally)

Density (kg/m3) Yield strength (MPa) Ultimate strength (MPa) Elongation (%) Modulus of elasticity (GPa)
7,850 812.31 1067.9 40 204
Table 3

Specification of SMA helical spring wire

IT. Properties Values
1 Melting point (°C) 1,300
2 Density (kg/m3) 6,450
3 Modulus of elasticity (GPa) (martensite) 36
Modulus of elasticity (GPa) (austenite) 87
4 Poisson’s ratio 0.33
5 Yield strength (MPa) (martensite) 392
Yield strength (MPa) (austenite) 759
6 Austenite start temperature (A s) (°C) 34.5
7 Austenite finish temperature (A f) (°C) 66.8
8 Wire diameter (mm) 2
9 Inner diameter (mm) 25
10 No. of coils 3
11 Pitch length (mm) 20
12 Exp. spring constant (N/m) (martensite) 553
13 Exp. spring constant (N/m) (austenite) 1,331
Table 4

Specification of SMA leaf spring

IT. Properties Values
1 Melting point (°C) 1,300
2 Density (kg/m3) 6,450
3 Modulus of elasticity (GPa) (martensite) 36
Modulus of elasticity (GPa) (austenite) 87
4 Poisson’s ratio 0.33
5 Yield strength (MPa) (martensite) 392
Yield strength (MPa) (austenite) 759
6 Austenite start temperature (As) (°C) 34.5
7 Austenite finish temperature (Af) (°C) 66.8
8 Thickness (mm) 2
9 Width (mm) 5
10 R1 (mm) 35
11 R2 (mm) 70
12 Exp. spring constant (N/m) (martensite) 1,189
13 Exp. spring constant (N/m) (austenite) 2,560
Table 5

Chemical composition of SMA

C% Co% Cu% Cr% H% Fe% No% Nb% Ni% Ti%
0.05 0.05 0.01 0.01 0.005 0.05 0.0162 0.025 55 Bal.
Figure 5 
               Experiment setup.
Figure 5

Experiment setup.

Figure 6 
               Vibration rig.
Figure 6

Vibration rig.

4 Results

In this work, the experimental investigation is carried out to analyze the free vibration problem of the rotor system with SMA metal. The evaluated results include the amplitude, natural frequency, and temperature profile of the clamped hanged supported rotor journal bearings with various parameters, temperature, and geometrical properties. Numerical investigation using ANSYS software for verification purposes is also employed. The obtained results are tabulated and drawn with multiple curves. Figures 710 show the critical speed relationship in the XY directions for free vibration analysis with springs only and with the disc, and mass at 20 and 80°C, where we note that:

  • The areas of increasing amplitudes along the ascending frequency line, where these points represent the critical speed, as within the used range, more than one essential point appeared, and this was previously planned when choosing the length and diameter of the rotating shaft.

  • The increase in stiffness caused by the modules of elasticity increasing as a result of phase transfer resulted in an increase in natural frequencies. Due to the increased gravity impact, progress is primarily seen on the frequencies in the Y-direction rather than the other axis (X-direction).

  • When operating at critical speeds, natural frequencies are displayed within the applied speed range. It was noted that the amplitudes of the two axes for the same crucial velocity were not the same. As a consequence of the stiffness asymmetry in the two axes (orientation of leaf spring installation), the critical speed in the Y-axis was determined earlier than in the X-axis. Gravitational effects also contribute to the speed along the vertical axis (Y-axis). As a result of the lack of symmetry, the increased stiffness resulted in a delay in the crucial time on the horizontal axis (X-axis). In addition, there is a slight difference between the first and second critical speeds, but it is evident at the third and fourth necessary speeds, where the speed has been higher due to the different mode shapes.

Figure 7 
               Vibration amplitudes of hybrid springs SMA at 20°C.
Figure 7

Vibration amplitudes of hybrid springs SMA at 20°C.

Figure 8 
               Vibration amplitudes of hybrid springs SMA at 80°C.
Figure 8

Vibration amplitudes of hybrid springs SMA at 80°C.

Figure 9 
               Experimental dynamic response hybrid springs SMA in X–Y directions at temperature 20°C, when disc position at 40 cm.
Figure 9

Experimental dynamic response hybrid springs SMA in XY directions at temperature 20°C, when disc position at 40 cm.

Figure 10 
               Experimental dynamic response of hybrid springs SMA in X–Y directions at temperature 80°C, when disc position at 40 cm.
Figure 10

Experimental dynamic response of hybrid springs SMA in XY directions at temperature 80°C, when disc position at 40 cm.

The initial natural frequency of leaf and helical springs is compared in Table 6 using computational (by ANSYS software) and experimental results. Results showed that the Austinite phase is more important than the Martensite phase, and the natural frequencies drop with increasing disc distance. The rotor’s high rigidity in the Austinite alloy is thought to be the cause. The Second Natural Frequency of Leaf and Helical Springs data are shown in Table 7, with the largest disparity (24%) occurring at a disc distance of 60 mm. The third and fourth frequency findings for the identical rotor system are displayed in Tables 8 and 9. Regarding disc distance and alloy, the system behaves identically in all cases. The difference between the experimental and numerical solutions is within acceptable bounds, with the third and fourth frequencies experiencing the largest divergence (26%) overall (Figures 1114).

Table 6

First critical speed (RPM) of hybrid springs

Disc position (cm) Martensite ANSYS Martensite exp. Error% Austinite exp. Austinite ANSYS Error%
Without disc 997.74 1,500 33 1579.00 1484.04 6
10 995.73 1,412 29 1478.00 1378.12 7
20 983.31 1,348 27 1428.00 1350.90 5
30 948.25 1,207 21 1280.00 1290.83 1
40 894.36 1,102 19 1150.00 1229.33 6
50 841.27 1,024 18 1100.00 1158.34 5
60 800.62 1,101 27 1158.00 1129.78 2
70 773.19 1,138 32 1208.00 1136.94 6
Table 7

Second critical speed (RPM) of hybrid springs

Disc position (cm) Martensite ANSYS Martensite exp. Error% Austinite exp. Austinite ANSYS Error%
Without disc 1241.01 1,554 20 1,636 1328.12 19
10 1241.57 1,441 14 1,550 1326.21 14
20 1231.74 1,400 12 1,444 1290.38 11
30 1114.93 1,251 11 1,290 1192.64 8
40 979.96 1,134 14 1,152 1077.04 7
50 923.27 1,098 16 1,110 1001.46 10
60 875.35 1,196 27 1,280 972.99 24
70 1081.95 1,245 15 1,315 1154.33 12
Table 8

Third critical speed (RPM) of hybrid springs

Disc position (cm) Martensite ANSYS Martensite exp. Error% Austinite exp. Austinite ANSYS Error%
Without disc 2622.90 2116.00 19 2280.00 2722.93 16
10 2513.79 1978.00 21 2195.00 2675.16 18
20 2382.42 1732.00 27 1846.00 2502.80 26
30 2066.37 1655.00 20 1790.00 2233.85 20
40 1998.63 1697.00 15 1800.00 2224.97 19
50 2139.94 1770.00 17 1900.00 2381.94 20
60 2431.05 1850.00 24 2085.00 2646.78 21
70 2511.97 1947.00 22 2178.00 2665.61 18
Table 9

Fourth critical speed (RPM) of hybrid springs

Disc position (cm) Martensite ANSYS Martensite exp. Error% Austinite exp. Austinite ANSYS Error%
Without disc 2904.01 2190.00 25 2228.00 2967.99 25
10 2806.24 2055.00 27 2162.00 2927.29 26
20 2472.71 1892.00 23 1990.00 2602.64 24
30 2094.17 1788.00 15 1900.00 2330.45 18
40 2034.36 1820.00 11 1920.00 2344.39 18
50 2192.10 1917.00 13 2024.00 2511.50 19
60 2559.36 1988.00 22 2088.00 2704.39 23
70 2736.69 2115.00 23 2312.00 2870.35 19
Figure 11 
               First natural frequency behavior due to disc position variation at different temperatures for hybrid springs SMA.
Figure 11

First natural frequency behavior due to disc position variation at different temperatures for hybrid springs SMA.

Figure 12 
               Second natural frequency behavior due to disc position variation at different temperatures hybrid springs SMA.
Figure 12

Second natural frequency behavior due to disc position variation at different temperatures hybrid springs SMA.

Figure 13 
               Third natural frequency behavior due to disc position variation at different temperatures hybrid springs SMA.
Figure 13

Third natural frequency behavior due to disc position variation at different temperatures hybrid springs SMA.

Figure 14 
               Fourth natural frequency behavior due to disc position variation at different temperatures for hybrid springs SMA.
Figure 14

Fourth natural frequency behavior due to disc position variation at different temperatures for hybrid springs SMA.

The Campbell diagram is examined at various parameters in Figures 1522. One can infer from the graphics that an operating system’s Campbell diagram illustrates the vibration excitation zone that takes place across the rotor system. Additionally, Campbell diagrams may be produced using the machine’s design criteria or operational information. The frequency responsiveness decreases with temperature, as seen by the link between the engine’s rotating speed along the X-axis and the system’s frequency on the Y-axis. Furthermore, as disc distance increases, the same pattern can be seen. When deciding if a running frequency, its harmonics, or subharmonics are to blame for the excitation of a natural frequency, the rotor system must perform this design analysis (Tables 1013).

Figure 15 
               Campbell diagram of the system at 20°C without disc.
Figure 15

Campbell diagram of the system at 20°C without disc.

Figure 16 
               Campbell diagram of the system at 80°C without disc.
Figure 16

Campbell diagram of the system at 80°C without disc.

Figure 17 
               Campbell diagram of the system at 20°C with disc position 20 cm.
Figure 17

Campbell diagram of the system at 20°C with disc position 20 cm.

Figure 18 
               Campbell diagram of the system at 80°C with disc position 20 cm.
Figure 18

Campbell diagram of the system at 80°C with disc position 20 cm.

Figure 19 
               Campbell diagram of the system at 20°C with disc position 40 cm.
Figure 19

Campbell diagram of the system at 20°C with disc position 40 cm.

Figure 20 
               Campbell diagram of the system at 80°C with disc position 40 cm.
Figure 20

Campbell diagram of the system at 80°C with disc position 40 cm.

Figure 21 
               Campbell diagram of the system at 20°C with disc position 70 cm.
Figure 21

Campbell diagram of the system at 20°C with disc position 70 cm.

Figure 22 
               Campbell diagram of the system at 80°C with disc position 70 cm.
Figure 22

Campbell diagram of the system at 80°C with disc position 70 cm.

Table 10

First critical speed (RPM) of hybrid springs at different disc positions and temperatures

Disc position (cm) 20°C 40°C 50°C 60°C 80°C Increasing RPM Increasing%
Without disc 1,500 1,520 1,530 1,540 1,579 79 5
10 1,412 1,400 1,434 1,465 1,478 66 5
20 1,348 1,340 1,350 1,411 1,428 80 6
30 1,207 1,240 1,268 1,274 1,280 73 6
40 1,102 1,118 1,141 1,145 1,150 48 4
50 1,024 1,033 1,042 1,069 1,100 76 7
60 1,101 1,110 1,115 1,129 1,158 57 5
70 1,138 1,140 1,174 1,195 1,208 70 6
Table 11

Second critical speed (RPM) of hybrid springs at different disc positions and temperatures

Disc position (cm) 20°C 40°C 50°C 60°C 80°C Increasing RPM Increasing%
Without disc 1,554 1,572 1,585 1,591 1,636 82 5
10 1,441 1,475 1,520 1,542 1,550 109 8
20 1,400 1,410 1,420 1,434 1,444 44 3
30 1,251 1,271 1,277 1,286 1,290 39 3
40 1,134 1,137 1,144 1,146 1,152 18 2
50 1,098 1,090 1,100 1,104 1,110 12 1
60 1,196 1,200 1,255 1,275 1,280 84 7
70 1,245 1,261 1,291 1,309 1,315 70 6
Table 12

Third critical speed (RPM) of hybrid springs at different disc positions and temperatures

Disc position (cm) 20°C 40°C 50°C 60°C 80°C Increasing RPM Increasing%
Without disc 2,116 2,243 2,272 2,275 2,280 164 8
10 1,978 2,070 2,100 2,154 2,195 217 11
20 1,732 1,765 1,780 1,810 1,846 114 7
30 1,655 1,665 1,695 1,730 1,790 135 8
40 1,697 1,690 1,750 1,772 1,800 103 6
50 1,770 1,794 1,838 1,872 1,900 130 7
60 1,850 1,960 2,000 2,075 2,085 235 13
70 1,947 2,016 2,056 2,105 2,178 231 12
Table 13

Fourth critical speed (RPM) of hybrid springs at different disc positions and temperatures

Disc position (cm) 20°C 40°C 50°C 60°C 80°C Increasing RPM Increasing%
Without disc 2,190 2,206 2,285 2,292 2,328 138 6
10 2,055 2,066 2,095 2,100 2,162 107 5
20 1,892 1,892 1,900 1,948 1,990 98 5
30 1,820 1,832 1,841 1,904 1,920 100 5
40 1,788 1,797 1,800 1,875 1,900 112 6
50 1,917 1,920 1,937 2,000 2,024 107 6
60 1,988 2,000 2,056 2,070 2,088 100 5
70 2,115 2,139 2,213 2,272 2,312 197 9

It can be noticed that the gravitational acceleration is assumed to be constant, and it is equal to 9.81 m/s.

5 Conclusions

The most significant findings and observations from this work are presented in this section, where it is revealed that the apparent impact of temperature on the dynamic response was caused by a phase transition that altered the SMA springs’ elasticity modulus. Additionally, there was a clear mass effect, which peaked when the disc was positioned in the center of the revolving shaft (proving the Jeffcott approach). Additionally, increasing the stiffness led to an increase in the system’s natural frequencies. The suspension system’s natural frequency changed by 235 RPM (13%), which was the largest change as a result of temperature changes from 20 to 80°C. Additionally, the experimental results supported the ANSYS results with a tolerable error ratio. For future work, the authors intend to use optimization methods to predict the properties of SMA springs and natural frequencies.

Acknowledgment

The authors thank the head and staff members of the Mechanical Engineering Department at the University of Technology for their efforts in solving the difficulties that they faced during this work.

  1. Funding information: This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

  2. Author contributions: All authors contributed equally to this work.

  3. Conflict of interest: The authors declare no conflict of interest.

  4. Data availability statement: The data supporting this study’s findings are available on request from the corresponding author.

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Received: 2023-05-25
Revised: 2023-08-02
Accepted: 2023-08-08
Published Online: 2024-02-06

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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