## Abstract

The vacuum interrupter (VI) is the main part of the vacuum circuit breaker (VCB), and an essential component of its design is the bellows. In future high-voltage VCBs, faster switching speeds will lead to higher dynamic impact-type loads, and the oscillation of the bellows after the contact opening and closing could shorten the mechanical life of the bellows. This article provides a 1D computer-aided engineering (CAE) analysis of the strain of the bellows oscillation in a VI caused by the axial load from the impact of the contact. A formula for the oscillation is derived using the longitudinal vibration response of a mass-spring 1D model, calculated by the convolution integral of the input and the impulse response function of the vibration modes. The proposed 1D analysis was verified and shown to be effective under four different experimental conditions. This article also provides a 1D CAE analysis of the mechanical life of the bellows. These 1D analyses estimate the effect of stress reducing countermeasures on the bellows' lifetime without increasing its size.

## 1 Introduction

Electrical power transmission networks are protected and controlled by high-voltage circuit breakers. A circuit breaker is an electrical safety device designed to protect an electrical circuit from damage caused by current in excess of that which the equipment can safely carry (overcurrent). It works by breaking the flow of electrical currents in transmission lines. Its basic function is to interrupt current flow to protect equipment and to prevent fire.

Substation equipment often costs millions of dollars and the reliability and safety of these assets largely depend on power circuit breakers. When a circuit breaker fails, it can damage the surrounding equipment, contributing to costly outages and the endangerment of lives. Sulfur hexafluoride (SF_{6}) gas is commonly used as an insulating medium in high-voltage circuit breakers and other electrical equipment. It offers excellent electrical insulation properties and high thermal stability, making it suitable for use in these applications.

However, SF_{6} is the most potent greenhouse gas, with a potential for global warming roughly 24,300 times greater than that of CO_{2} [1]. In light of the drive toward reducing greenhouse gas emissions, the demand for alternatives to SF_{6} is growing, as are regulations to limit the use of, or ban, such fluorinated gases [2,3].

With the goal of reducing greenhouse gas emissions, we have been working on the development of vacuum circuit breakers (VCBs) [4] for SF_{6}-free gas-insulated switching solutions that replace this greenhouse gas with a pressurized synthetic air insulating medium.

Figure 1 shows an example of a VCB. A vacuum interrupter (VI) [5], which is a switch of electrical contacts, is the main part of the VCB, and the bellows is an essential component of the VI's design. Metal bellows provides an ultra-high vacuum seal while also allows the electrical contact to move in the VI chamber. If the bellows leaks, the vacuum of the VI is reduced or even lost [6]. Fatigue of the bellows due to the stress of opening and closing operations limits the mechanical life of the VI.

In Fig. 1, one end of the bellows is fixed to the movable flange, and the other end is fixed to the movable conductor. When the movable contact opens and closes, forced displacement is applied to the bellows, causing it to repeatedly contract and expand. If the opening speed is too slow, an arc will ignite, and the current cannot be cut. In order to prevent arc ignition and shorten arcing time, the opening of the VI must be fast. The closing speed must also be high enough to reduce the electrical wear caused by pre-breakdown of the contacts during the short-circuit closing process and avoid contact welding. When the opening or closing speed is too high, the vibration during operation becomes serious, and the bellows may even be damaged.

Figure 2 shows an example of bellows strain caused by the opening and closing of the VI. This strain time chart consists of a static motion response caused by the change in the length of the bellows due to the moving electrical contact and the transient responses generated by the dynamic loads in the contact opening and closing operations.

*p*[10]:

Here, *E* is Young's modulus, $tn$ is the nominal material thickness, *h* is the pitch of half a corrugation, *w* is the width of the bellows (the height of the corrugation), *N* is twice the number of active bellows discs, and *n* is the number of layers of the bellows. Strain $\epsilon $ can be obtained by dividing the stress $SR$ by Young's modulus *E*. The bellows designer considers design variables such as material type, thickness, number of convolutions, and their geometry to produce a reliable design with a suitable cycle life expectancy [7]. To reduce stress using Eq. (1) without changing the material while maintaining pressure resistance is to increase the size of the bellows, that is, *h*, *N*, *n*.

On the other hand, transient responses with large amplitudes are not accounted for in these standards or formulas. In future high-voltage VCBs, faster switching speeds will lead to higher transient responses and could increase the fatigue of the bellows. Experiments and simulations have been reported regarding the transient response of VI bellows. Liu et al. experimentally studied the mechanical endurance of VI bellows subjected to high gas pressure and high opening velocity [12].

Sellappan et al. proposed a finite element analysis (FEM) model for the reliable design of bellows [13]. Hong et al. used the FEM software ansys to calculate the mechanical endurance of bellows for VIs subjected to high gas pressure and high operating velocity [14]. Fan et al. used ansys to calculate the equivalent stress of bellows under several conditions to determine the most appropriate parameters of U-shaped VI bellows subjected to high operating velocity [15]. Taken together, these studies indicate that a numerical simulation by FEM is a powerful way to deal with several conditions. However, a numerical analysis is not helpful for understanding the physics, and hence, cannot be used to formulate a policy for countermeasures to reduce stress.

One of the objectives of this article is to understand the bellows oscillation phenomenon and help to design bellows' parameters by 1D modeling. It focuses on the phenomenon of transient responses consisting of longitudinal vibration modes. Here, factors that are not considered in the 1D model may lead to analysis errors. For example, the analysis accuracy will decrease if the bellows behaves nonlinearly. Errors may also occur when vibration phenomena of a calculated order or higher are dominant.

It is necessary for VCB to prove the quality through mechanical tests determined in International Electrotechnical Commission (IEC), Japanese Electrotechnical Committee (JEC), Institute of Electrical and Electronic Engineers (IEEE) [16–18]. The VI bellows must maintain its quality to withstand 10,000 opening and closing operations and must be designed to avoid large deformation that would cause nonlinear response. Additionally, the time interval between opening and closing required by this standard is longer than the bellows' natural period, so that vibrations are not amplified by high-frequency switching. Therefore, in this study, we chose a linear vibration of the bellows as a model. The proposed 1D model is based on an understanding of the bellows vibration phenomenon, so it is possible to precisely reproduce the actual phenomenon through analysis.

Another objective is to understand the fatigue life of the bellows. In general, the fatigue life can be calculated from time-history waveforms using a proven standard method like the rain flow cycle counting method [19] or the range pair counting method [20]. Since strain has a vibration-like waveform (Fig. 2), we propose a fatigue life estimation method using the natural frequency and damping rate that represent the dynamic characteristics of the bellows. Analyzing the contributions of these characteristics will make it easier to formulate countermeasures for extending the fatigue life.

## 2 Bellows' Strain Formula With Contact Motion

### 2.1 One-Dimensional Model of Motion and Vibration of Bellows.

First, let us discuss the forces that occur at the ends of the bellows as a result of static deformation. Figure 3(a) shows the change in the bellows' shape, and Fig. 3(b) shows bellows modeled as a spring. The force at either end of the bellows $Fo(t)$ at time *t* can be obtained by multiplying the spring displacement $xo(t)$ by the spring constant *k*.

Here, $xo(t)$ is the static force displacement applied to the bellows in the opening and closing operations. For static motion, the force generated at the end of the bellows is the product of the stiffness and displacement of the bellows in the contact moving direction.

Next, let us examine the bellows' transient responses caused by the dynamic loads in the contact opening and closing operations. The acceleration, which is the second derivative of the motion displacement with respect to time, generates inertia force on the bellows and makes it oscillate depending on the resonant longitudinal vibration in the contact moving direction. The acceleration is highest at the ends of the VI contact opening and closing operations, and higher switching speeds will lead to higher acceleration. For dynamic movements, the inertia of the bellows provides a resistance in addition to its stiffness in the contact moving direction. In the case of a continuum such as a bellows, multiple vibrational modes are excited when an impact force is applied. The relationship between force and displacement in each vibration mode is expressed by the corresponding mode stiffness and mode mass [21]. The bellows will fail at the ends where the stress is greatest. By calculating the displacement in the contact moving direction at both ends and multiplying it by the spring constant from the vibration mode node to the ends, the force generated at both ends can be determined.

Figure 4 shows the first longitudinal vibration mode of the bellows excited by the dynamic loads in the VI contact opening and closing operations. Figure 4(a) shows the vibration mode of the bellows' shape, and Fig. 4(b) shows the results for the bellows modeled as a spring and mass. In Fig. 4(b), the acceleration $a0(t)$ due to the motion of the VI acts on a concentrated mass *m* equal to the mass of the moving part of the bellows at the antinode of the vibration mode and generates the inertia force.

Figure 5 shows the second longitudinal vibration mode of the bellows excited by the VI contact operation. In Fig. 5(b), $a0(t)$ acts on concentrated masses $m/2$, half of the mass of the moving part of the bellows, located at the two antinodes of the vibration mode and generates the inertia forces. When the contacts move from the closed to open position, compressive force is generated on the contact side of the bellows and a compressive wave with a shorter wavelength than that of the first longitudinal vibration mode propagates toward the fixed side. Therefore, the forces at the ends are in-phase. The absolute value of the force at the ends of the bellows $|F2(t)|$ can be obtained by multiplying the spring displacement $|x2(t)|$ by the spring constant $4k$.

### 2.2 Formulation of Bellows' Dynamic Response.

*n*th natural angular frequency, and $\zeta n$ is the

*n*th mode attenuation ratio $\omega n$. Moreover, the response of the

*n*th mode $xn(t)$ by the external force $Lo(t)$ is expressed with the following equation using the convolution integral of $Lo(t)$ and $hn(t)$ [21]:

*n*th mode mass, which is equal to $m/n$. For example, in Fig. 5, $m2$ is $m/2$. Substituting Eq. (7) for $Lo(t)$ in Eq. (6), $xn(t)$ becomes

*n*th mode at the edge of the bellows $Fn(t)$ is proportional to the strain at either ends; it can be calculated from the product of the spring constant of the bellows $kn$ and the displacement $xn(t)$

*k*gives the displacement when $FC(t)$ is statically applied to the bellows. Since the strain generated at either end is proportional to the force, the strain at the contact side $\epsilon C(t)$ can be expressed as

Similarly, the strain of the fixed side $\epsilon F(t)$ can be expressed as

### 2.3 One-Dimensional Calculation Procedure of Bellows' Strain.

The strain calculation procedure is as follows:

Step 1: Calculate $a0(t)$ by taking the second derivative of the bellows' motion displacement with respect to time.

Step 2: Experimentally obtain the natural frequency and the mode attenuation ratio of the longitudinal vibration modes fixed at both ends of the bellows applied pressure when mounted on the VI.

Step 3: Experimentally obtain the relationship (Eq. (13)).

*n*th resonant frequency of the longitudinal vibration of the bellows $fn$ in Step 2 can also be calculated as [22]

The attenuation ratio $\zeta n$ without a theoretical value must be determined by the experiment, and the natural frequency $fn$ is also observed in the experiment.

### 2.4 Formulation of Bellows Fatigue Life.

The S–N curve of the material can be obtained from the Databook on the fatigue strength of metallic materials [23]. Figure 6 shows an example of the S–N curve of the material used in bellows. As shown in Fig. 6, the relationship between the strain and the fatigue life is a linear one when it is plotted on a log–log graph. The strain $\epsilon $ and fatigue life *N* can be expressed as

Here, *C* and $\alpha $ are constants determined by the material. In the case of Fig. 5, *C* is $92276$, and $\alpha $ is $0.345$, so when $\epsilon $ is reduced to 1/3, the life is extended by 30 times.

Figure 7 shows an example of the transient vibration response in a single-degree-of-freedom mode. This waveform is the result of extracting the component of the first vibration mode from the waveform of the bellows' strain shown in Fig. 3. $\delta n$ in Fig. 6 is the logarithmic decrement of the mode, which is $2\pi \zeta n$.

*i*th amplitude $\epsilon i$ in Fig. 7 contributes to a fatigue life value

*N*in accordance with Eq. (18) and Fig. 6. Therefore, the cumulative damage due to fatigue

*D*can be derived as [24]

As can be seen from the series of equations, the greater the damping, the faster the vibration amplitude converges with time and the longer the life.

## 3 Experimental Verification

### 3.1 Bellows and Test Apparatus Simulated Vacuum Circuit Breaker.

Figure 8 shows a diagram of the apparatus for the durability test that simulated the pressure conditions and the contact motions of the VCB. To measure the strain of the bellows, the tank of the test apparatus did not include a VI and only included the bellows and the moving and fixed contact. The moving contact was connected to the bellows edge on the contact side. Positive pressure was applied inside the bellows set in the tank of the test apparatus; this simulated the differential pressure of the bellows of a practical VCB. The motion of the contact was simulated in a multibody dynamics analysis of the VCB, and the target motion of the contact was achieved by adjusting the mechanical elements and the torsion-bar spring operating mechanism [25,26]. Strain gauges were attached at each crest of the corrugations near the ends of the bellows where the maximum strain is generated. Therefore, the strains of the corrugations at these points on the bellows were simulated and compared with the experimental data.

To verify the accuracy of the strain calculation procedure, four different kinds of test were performed. Table 1 shows the test conditions. Two sorts of bellows with different plate thicknesses and shapes were experimented on.

Test | Stroke | Bellows | Bellows liner |
---|---|---|---|

Test A | Stroke A | Bellows A | – |

Test B | Stroke A | Bellows B | – |

Test C | Stroke A | Bellows B | Attached |

Test D | Stroke B | Bellows B | Attached |

Test | Stroke | Bellows | Bellows liner |
---|---|---|---|

Test A | Stroke A | Bellows A | – |

Test B | Stroke A | Bellows B | – |

Test C | Stroke A | Bellows B | Attached |

Test D | Stroke B | Bellows B | Attached |

The temporal changes in the contact motions of the opening and the closing operations were determined by considering mechanical restrictions and the arc discharge related to the distance between the contacts. The opening operation of test D was designed to be slower than those of the other tests. In addition, the bellows contraction length during the opening operation was set longer than in the other tests, and the bellows extension length during the closing operation was set shorter. The bellows liner (Table 1) was a tubular resistor that causes friction through its contact with the bellows [27].

### 3.2 Strain Calculation Procedure.

The closing operation finished within 0.03 s, and the start and end of the motion were the only inflection points. The opening operation was designed to be more complex than the closing operation because the arc needs to be controlled near the contact point. In all of the tests of the opening operation, the contact reached a maximum speed at 0.0098 s from the start of the operation. After reaching the maximum speed, it decelerated until the end of the operation. Tests A, B, and C terminated the contact motion at 0.04 s, and test D terminated the contact motion at 0.06 s; therefore, in the period after the maximum speed was reached, the acceleration was smaller in test D than in tests A, B, and C.

Figure 9 shows $a0(t)$ determined from the second derivative with respect to time of the opening and closing motion displacements of the bellows. The vibration of the test apparatus due to the reaction of the opening and closing shock was transmitted to the displacement transducer and caused high-frequency noise in the displacement waveform. By differentiating the displacement signal twice in time, this noise was increasingly accentuated. However, this noise could be canceled in the calculation procedure of Eqs. (14) and (15) because the noise frequencies were sufficiently high compared with the resonant frequencies used in the calculation.

*n*th mode vibration amplitude attenuates as $e\u2212\omega n\zeta nt$, and the slope of the logarithm of the amplitude of the time-history waveform becomes $\u2212\omega n\u03c2n$. During the free vibration of the strain by the impulse excitation, the amplitude of the first mode was larger than those of the others; therefore, the attenuation ratio of the first vibration mode $\u03c21$ could be expressed using the first angular natural frequency $\omega 1$ and the slope of the logarithm of the amplitude of the history waveform $Bln1$:

The experimental results for the first natural frequency and the attenuation ratio are shown in Table 2. The natural frequency depended on the contact position, i.e., the bellows length. When the contact was open, the bellows shrunk from its natural length, and when the contact was closed, the bellows extended from its natural length. The reason why the natural frequency depends on the contact position is presumed to be the stiffness changes depending on the bellows' length. Therefore, it is preferable to ascertain the natural frequency by the experiment rather than relying on the theoretical formula (Eq. (16)). The attenuation ratio of bellows B was measured to be higher than that of bellows C; moreover, the attenuation ratios of test C and test D were measured to be higher than those of the other tests because of the attachment of the bellows liner.

Test | Contact position | First natural frequency | Attenuation ratio |
---|---|---|---|

Test A | Open | 73 Hz | 0.004 |

Close | 83 Hz | 0.002 | |

Test B | Open | 64 Hz | 0.008 |

Close | 69 Hz | 0.003 | |

Test C | Open | 68 Hz | 0.03–0.04 |

Close | 68 Hz | 0.003–0.004 | |

Test D | Open | 73 Hz | 0.03–0.05 |

Close | 63 Hz | 0.001–0.002 |

Test | Contact position | First natural frequency | Attenuation ratio |
---|---|---|---|

Test A | Open | 73 Hz | 0.004 |

Close | 83 Hz | 0.002 | |

Test B | Open | 64 Hz | 0.008 |

Close | 69 Hz | 0.003 | |

Test C | Open | 68 Hz | 0.03–0.04 |

Close | 68 Hz | 0.003–0.004 | |

Test D | Open | 73 Hz | 0.03–0.05 |

Close | 63 Hz | 0.001–0.002 |

The relationship between $\epsilon $ and $\xi $, i.e., Eq. (13), was experimentally obtained in a load test of tension and compression. Figure 10 shows the setup of the experiment on a universal testing machine. To expand and contract the bellows without bending, the installation angle of the bellows was adjusted by measuring the displacement at different points on the bellows. Four strain gauges were mounted every 90 deg around the second corrugation from the bottom to ensure uniform loading.

Figure 11 shows examples of the relation between the strain and the displacement of the bellows, which compare functional equation (13) obtained from the measurements and the calculation using the first term on the right side of Eq. (1). The thickness $tn$ of the root was larger than that of the crest, because of the manufacturing process of the bellows. As the first term on the right side of Eq. (1) is proportional to $tn$, the strain of the root in the calculation became larger than that of the crest, and the experimental results became similar. Moreover, the calculated values of $\epsilon $ were larger than the measured values. Kellogg's formula’ strain estimation thus seems to be sufficiently conservative for the load displacement [12], although we consider that experimental tests are needed, as the actual bellows do not have ideal shapes.

### 3.3 Experimental Validation of the Strain Calculation Method.

Figure 12 compares the experimentally measured strains [28] and the strains calculated using Eqs. (14) and (15). If the bellows expanded and contracted uniformly in one direction, the strains at each position around one corrugation should all have the same value. However, the data measured at each position of one corrugation showed a lot of variation and the bias values were different. The reason for this variation is considered that assembly errors of the test apparatus due to component tolerances were magnified by the impacts of the opening and the closing operations.

As shown in Fig. 12, the strain oscillation amplitudes in the opening operation were larger than in the closing operation. Therefore, the strain in the opening operation would be the dominant cause of fatigue failure in this case. From this point of view, two improvements were made to the opening operation motion in test D. The first improvement was to make the opening operation time longer than those of the other tests. The second improvement was to the opening operation time curve. As shown in Fig. 9(a), after the peak acceleration at 0.002 s, another peak appeared at 0.016 s in tests A, B, and C in the opening operation. This interval time, 0.014 s, was synchronized with the natural period of the bellows shown in Table 2, so we made this interval longer in test D so that the accelerations would not to be synchronized with the natural period of the bellows.

A comparison of the results of tests A and B indicates that the strain of bellows B is smaller than that of bellows A. The reason for this difference is presumed to be that the attenuation ratio changes depending on the bellows.

Moreover, comparing the results of test B and test C reveals that in almost all cases, the strain with the bellows liner attached is decayed faster than the strain with no liner. The likely reason for the amplitude on the contact side (Fig. 12(i)) being larger is that not all of the surface of the liner touches the bellows because of its shape tolerance. Thus, we increased the size of the bellows liner to homogenize the contact pressure in test D. Finally, after measures were taken as to the input acceleration, selection of bellows, and size of the bellows liner, all of the strain data in test D were small enough to ensure a sufficient life over the opening and the closing operations.

In Fig. 12, the error between the analysis and the experiment is especially evident for the high-frequency amplitudes after the third wave of the first resonance of the bellows. When the first resonance wave decays, the higher mode resonances stand out, because the attenuation ratios of the higher modes are smaller than that of the first one. To improve the accuracy, the attenuation ratio of the higher resonances should be measured. However, as shown in Fig. 6, this error is negligible from the point of view of a fatigue evaluation.

### 3.4 Experimental Validation of the Fatigue Life Calculation Method.

Table 3 compares the experimentally measured life and the life calculated by Eq. (22) using $Do$ and $Dc$ obtained from Eq. (20) with the strains. $Do$ and $Dc$ of calculation A in Table 3 are obtained from Eq. (20) for the maximum amplitude of the transient vibration response strain in the opening and closing operations in the experiment and the attenuation ratio shown in Table 2, and $Ds$ is obtained from the reciprocal of Eq. (18) with the static motion response strain measured in another experiment [28]. As shown in the table, the experimental and theoretical fatigue life in test A and test B were almost the same, which means that the fatigue life estimation formula (Eq. (22)) is valid. In particular, as shown in Figs. 12(a) and 12(b) in test A, the maximum strain analysis value and the experimental value matched, so both experimental and calculation results of the life showed the same number. We terminated test D at 20,000 cycles without failure of the bellows. Note that the IEEE standard requires no failure in 10,000 no-load mechanical operations [16–18].

Test | Experiment | Calculation A | Calculation B |
---|---|---|---|

Using experimental strain data | Using calculated strain data | ||

Test A | 2200 | 2578 | 2578 |

Test B | 5500 | 5650 | 9037 |

Test C | No tests | 46,199 | 84,635 |

Test D | More than 20,000 times (no fracture) | 126,950 | 183,877 |

Test | Experiment | Calculation A | Calculation B |
---|---|---|---|

Using experimental strain data | Using calculated strain data | ||

Test A | 2200 | 2578 | 2578 |

Test B | 5500 | 5650 | 9037 |

Test C | No tests | 46,199 | 84,635 |

Test D | More than 20,000 times (no fracture) | 126,950 | 183,877 |

$Do$ and $Dc$ of calculation B in Table 3 are obtained from Eq. (20) with the strains calculated using Eqs. (14) and (15). The calculated life was longer than the experimental result in test B. Because the predicted strain values in test B were lower than the experimental values, as shown in Fig. 12(f), the life calculated from the predicted values were longer than the experimental values. A more accurate strain estimation would improve the accuracy of the fatigue life estimation. It was found that the life extension effect of each countermeasure could be evaluated by 1D analysis in which the strain estimated from Eqs. (14) and (15) is used in Eq. (22) to calculate the life.

## 4 Conclusion

We derived Eqs. (14) and (15) to estimate the bellows' transient strain responses caused by the dynamic loads in the contact opening and closing operations, using the natural frequency and the mode attenuation ratio of the longitudinal vibration modes. Using these formulas, it became possible not only to estimate the strain responses without 3D simulations but also to formulate a policy for countermeasures to reduce strain of the bellows. In order to enable readers to use these formulas in their design work, we have presented test methods for determining the coefficients of formulas, and the strain calculation procedure. These formulas were verified by comparing its results with experimental data under four different experimental conditions as shown in Fig. 12, demonstrating their effectiveness.

Generally, in order to reduce the stress using a conventional formula (Eq. (1)) without changing the material while maintaining pressure resistance, the size of the bellows increases. On the other hand, the proposed analysis allowed us to derive countermeasures such as preventing the motion displacement of the contact from synchronizing with the natural period of the bellows and adding vibration damping. With these countermeasures, we terminated the bellows durability test at 20,000 cycles, which are more cycles than required by the IEEE standard [16–18] without increasing its size.

By focusing on the fact that the strain exhibits a vibration attenuation waveform as shown in Fig. 2, we proposed a fatigue life estimation method using Eqs. (20)–(22). Experimental results shown in Table 3 demonstrated that these formulas could be practical for predicting the life of bellows. Based on these formulas, it was discovered that reducing the maximum value of strain and increasing the damping ratio led to a longer life. Furthermore, experiments have also demonstrated the effectiveness of these countermeasures as shown in Table 3.

This analysis will be applied to a high-voltage VI under development.

## Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

## Data Availability Statement

The authors attest that all data for this study are included in the paper.