Electrical machinery with rotating or linear mass requires a support system that can resist dynamic forces and the resulting vibrations. The excess of such vibrations may be detrimental to the machinery and its support system. Especially, it becomes more and more important for dynamic equipment foundations due to various design criteria and methods and procedures of analysis, design and construction. For stable operation of the equipment foundations, kinetics is an important tool in understanding the motion of objects whether translational, oscillatory, or circular. In case of translating or linear oscillating masses, the behavior of variables such as speed and acceleration (deceleration) must be described by the equations of motion. Of importance is the mechanical robustness, because a linear electric machine has to withstand high accelerations resulting from the oscillating masses. This paper deals with the design of shaft and bolt is achieved by fatigue analysis based on the kinetic equation considering mechanical load. The results of the study will give elaborate information about stable and effective movement data of linear electric machine.

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## 1. Introduction

The study accomplishes the establishment of kinetic equation taking mechanical load
into account. It will not only help to model and evaluate the performance of linear
oscillation motion, but also be beneficial for the design of shaft for fatigue strength.
Generally, the designer of electric machine has overlooked the modelling of shaft
considering dynamic behavior by mechanical load. As a result, it cannot give useful
information to design of linear oscillation movement. In the past, the fatigue strength
has been determined for axial, bending and torsional load independently^{[1]}. But a general approach to design with combined loads requires a fatigue strength
that does not depend on the type of loading. Measured differences in the fatigue strength
are attributed to the size of a shaft. In particular, shaft bending appears prominently
in this linear system because there is no support apparatus at the center of the shaft.
The reliable design of power transmitting shafts is predicated on three major elements.
First, the fatigue (stress-life) characteristics in expected service environment must
be established. This can be accomplished from full-scale component fatigue test data
or approximated, using test specimen data. The expected load-time history of the shaft
must be obtained or assumed from field service data and then properly simulated analytically.
At last, a reliable mathematical model is needed which rationally considers both the
fatigue characteristics of the shaft and its loading history to arrive at the proper
shaft diameter for the required service life and reliability. Furthermore, bolted
joints are treated which have to transit mechanical loads connected with the shaft
and which are designed with high duty bolts. The bolts must be designed such that
the shaft fulfills its allotted function and withstands the mechanical loads occurring.
The first studies of fatigue by this process was done by Woehler^{[2]}, observing that metals failed under repeated loading despite stress levels far below
the elastic limit. The Woehler curve represents maximum allowable stress amplitude
versus number of cycles at fracture for a given mean stress.

The principal objective of this investigation is to develop an approach to shaft design and to bolt selection from a strength standpoint. It will be accomplished for cylindrical linear machine taking the moving displacement (stroke) and mechanical load condition into account.

## 2. Kinetic Equation considering Mechanical Load

In order to determine the correct machine for particular application, it is necessary to be familiar with the following dynamic relations by mechanical load.

### 2.1 Numerical Calculation

It describes the basic principle of kinetic characteristics based on the physical theory by linear motion.

Displacement

Displacement is defined as the distance between initial point and final point in a straight line and is vector quantity. It can be expressed by equation (1) using the function of time, $t$.

where, $\omega =2\pi f$, and frequency $f$ indicates the number of travel per minute.

Velocity

Velocity is defined as the rate of change of displacement and is vector quantity.

where, $\Delta s$ and $\Delta t$ indicate a change of displacement and time, respectively.

Acceleration

Acceleration is defined as the rate of change of velocity and is a vector quantity. It is produced when a force acts on a mass. The greater the mass (of the object being accelerated), the greater the amount of force needed (to accelerate the object).

where, $\Delta v$ indicates a change of velocity.

Force by Newton’s Second law

Newton’s Second Law provides an exact relationship between force, mass and acceleration. As the force acting upon an object is increased, the acceleration of the object is increased. Also, as the mass of an object is increased, the acceleration of the object is decreased. It can be expressed as a mathematical equation (4).

##### (4)

$F=m· a=m·\left[-\hat{S}·\omega^{2}·\sin(\omega t)\right]$

$\hat{F}=m·\left(-\hat{S}·\omega^{2}\right)$

where, $m$ represents the mass of the object.

### 2.2 Analytical Calculation

Based on the above analysis, the equivalent expression considering moving displacement and mechanical load is as in the following :

Displacement, Velocity, and Acceleration

Since a is uniform, its magnitude is

In this case, the average speed will be the speed at $t/2$ and $\nu_{0}$ is an initial speed. Hence :

##### (6)

$\nu_{ave}=\dfrac{\nu_{0}+\nu}{2}$

$S =\nu_{ave}· t=\dfrac{\left(\nu_{0}+\nu_{0}+a· t\right)}{2}· t$

$=\nu_{0}· t+\dfrac{1}{2}a· t^{2}=\dfrac{\left(\nu_{0}+\nu\right)}{2}·\dfrac{\left(\nu -\nu_{0}\right)}{a}$

The $\nu_{ave}$ and $a_{ave}$ describe the average velocity and average acceleration, respectively.

Time functions

Assuming the velocity-time curve of a moving object is uniform, the acceleration time in specific frequency is given by

The function of displacement according to time is expressed by (8).

The symbol $S_{0}$ is the initial displacement.

Force Calculation considering Mechanical Load

The calculation of effective force considering mechanical load can be accomplished by double of mass and acceleration through Newton’s Second Law. The total mass consists of the magnet and back-iron core with the mover $\left(m_{m}\right)$, and mechanical load $\left(m_{l}\right)$. The average force is expressed as two divided by π of the maximum force.

## 3. Determination of Shaft Diameter

A shaft is the component of a mechanical device that transmits linear or rotational
motion and power. It is integral to any mechanical system in which power is transmitted
from a prime mover, such as an electric machine or an engine, to other parts of the
system. The shaft of transmission equipment has suffered from the fatigue such as
axial, bending and torsion stress depends on its application field. The fatigue in
linear or rotating shafts is a phenomenon that has been known and studied for nearly
a century^{[1]}. Thus, it is necessary to analyze fatigue performance for long life cycles of utilization.

### 3.1 Design Procedure

Only solid circular steel and steel alloy shafts are considered. In steel shafts with relatively long lifetimes, the fatigue strength approaches a value known as the endurance limit. Generally, the procedure for shaft design means the decision of shaft diameter by three stresses such as axial, bending, and torsional fatigue strength considering the relationship of the load moment in specific material. In sequence, the appropriate stresses must be calculated, and they must be determined by an adequate factor of safety and the endurance limit of the material. Additionally, it can be made for stress concentrations, temperature and surface condition by forming a coefficient for each applied stress. At last, the shaft is adequately designed to prevent fatigue failure for the specified operating conditions.

### 3.2 Fatigue Failure

Failure from fatigue is statistical in nature as much as the fatigue life of a particular
specimen cannot be precisely predicted but rather the likelihood of failure based
on a large population of specimens^{[2]}. This fatigue is caused by repeated cycling of the loads. The fatigue strength is
dependent on the type of loading (axial, bending, or torsion) and it has resulted
in three separate fatigue strengths being defined. In particular, this linear motion
is implemented by designing shafts using axial and bending stress among the three
fatigue strengths.

The state of stress to be considered is caused by force (torque) transmitted to the shaft, axial forces imparted to the shaft, bending of the shaft due to its weight or loads, and torsion by rotating of shaft. The characteristics of these three basic types or cases are given here : axial strength $\left(\sigma_{a}\right)$, bending strength $\left(\sigma_{b}\right)$, and torsion strength $\left(\sigma_{t}\right)$.

Axial Strength

Axial stress is referred to as normal stress since it acts in line with the material. Most simple form is tension stress since its effect is independent of the length of the material. The formula for definition of axial stress can be expressed as follows.

where, $A_{rea}$ is cross-section area of the shaft, and $F_{\max}$ represents the maximum force considering mechanical load given by equation (9).

Bending Strength

The first step in shaft design is to draw the bending moment diagram for the loaded shaft or the combined bending moment diagram. The bending stress is larger than the direct stress by axis when a shaft is easily subjected to bending moment in such a linear motion case. From the bending moment diagram, the points of critical bending stress can be determined.

The force and moment acting on shaft indicates equation (13) and (14), respectively.

where, $g$ is gravity acceleration.

The effective nominal stress is ^{[3]}.

Torsion Strength

Torsion occurs when any shaft is subjected to a torque in case of a rotating motion in general. This is true whether the shaft is rotating (such as drive shafts on engines, motor and generators) or stationery (such as with a bolt or screw). This torque makes the shaft twist and one end rotates relative to the other inducing shear stress on any cross-section.

It represents as the equation (16) for circular solid^{[4]}.

where, $T_{m}$ is close to zero, which represents the twisting moment. In this study, the torsional strength is excluded because it has little impact on the shaft due to the linear motion.

Total Strength

The total strength acting on the shaft is expressed by sum of the axial and bending strength.

Woehler Curve

First systematic study to characterize the fatigue behavior of materials cyclic stress
range was conducted by Woehler^{[5]}. In high-cycle fatigue situations, materials performance is commonly characterized
by an S-N Curve, also known as a Woehler curve. This is a graph of the magnitude of
a cyclic stress against the logarithmic scale of cycles to failure.

Alternating Strength by Woehler Curve

By Woehler curve, we can find the endurance limit value of selected material; it is marked as $\sigma_{en}$ in failure cycle $10^{6}< N < 10^{7}$.

The safety factor, $\zeta$, is a ratio of maximum strength to intended loads for the
actual item that was designed. The value to use for safety factor is based on judgment.
It depends on the consequences of failure, that is, cost, time, safety, etc. Some
factors to consider when selecting a value for safety factor are how well the actual
loads, operating environment, and material strength properties are known, as well
as possible inaccuracies of the calculation method. Values typically range from 1.3
to 6 depending on the confidence in the prediction technique and the criticality of
the application. Unless experience or special circumstances dictate it, the use of
safety factor values of less than 1.5 is not normally recommended^{[3]}.

Summary

Its features for this study may be summarized as follows ;

The Woehler curve shows fatigue life corresponding to a certain stress amplitude

The Woehler diagram can be used to design for finite (and infinite) life

For steel, the fatigue limit corresponds to $10^{6}< N < 10^{7}$

### 3.3 Determination of Shaft Diameter

Through the earlier analytical calculation, we can attain the total strength acting on the shaft, $\sigma_{sh}$. It must not be greater than alternating strength $\left(\sigma_{al}\right)$ by the endurance limit value of selected material and safety factor as expression (19). Finally, the diameter of shaft is decided by satisfying maximum value below the calculated alternating strength.

## 4. Machine Elements Assemblies with Shaft

Virtually all machines involve the transmission of power and/or motion from an input source to an output work site. The input source, usually an electric machine or internal combustion engine, typically supplies power as a linear driving force to input shaft of the machine under consideration. The power transmission to or from linear moving shaft is accomplished either by coupling the liner moving shaft end-to-end with a power source, or by attaching power input or output components such as pistons or connecting rods to the shaft in linear systems. For the transfer of power bolted or screw connections are intended to fasten together machine components. The joints and connections between parts must be given special attention by the designer because they always represent geometrical discontinuities that tend to disrupt uniform force flow. In this study, it deals with the analysis about threaded joint by means of a threaded fastening such as a bolt or a screw.

### 4.1 Bolt of Uniform Strength

Bolts are subjected to shock and impact loads in certain applications. The bolts of cylinder head of an internal combustion engine or the bolts of connecting rod are the examples of such applications. In such cases, resilience of the bolt is important design consideration to prevent breakage at the threads. Resilience is defined as the ability of the material to absorb energy when deformed elastically and to release this energy when unloaded. A resilience bolt absorbs shocks and vibrations like leaf springs of the vehicle. In other words, the bolt acts like a spring.

Figure 2.a shows an ordinary bolt with usual shape. The major diameter of thread as well as the diameter of the shank is $d_{bs}$. The core diameter of the threads is $d_{bs-1}$. When this bolt is subjected to tensile force, there are two distinct regions of stress.

The diameter of threaded portion $d_{bs-1}$ is less than the shank diameter $d_{bs}$. The threaded portion is also subjected to stress concentration. Therefore, stress induced in the threaded portion is more than the stress in the shank portion. The energy absorbed by each unit volume of bolt material is proportional to the square of the stress. Hence, a large part of the energy is absorbed in the threaded portion of the bolt.

The diameter of the shank is more than the core diameter of the threaded portion. There is no stress concentration in the shank. Therefore, when the bolt is subjected to tensile force, the stress in the shank portion is less than the stress in the threaded portion. The energy absorbed in the shank, which is proportional to the square of the stress, is less than the energy absorbed in the threaded part.

The resilience of the bolt can also be increased by increasing its length^{[6]}. The strain energy absorbed by the shank is linearly proportional to its length.
The ideal bolt will be one which is subjected to same stress level at different cross-sections
in the bolt. It is called the bolt of uniform strength.

There are two ways to reduce the cross-sectional area of the shank and convert an ordinary bolt into a bolt of uniform strength. They are illustrated in Figure 2.b and c. One method is to reduce the diameter of the shank as shown Figure 2.b. In this method, the diameter of the shank is usually reduced to the core diameter of the threads. Therefore, the cross-sectional area of the shank is equal to the cross-sectional area of the threaded portion. When this bolt is subjected to tensile force, the stress in the shank and the stress in the threaded portion are equal. In the second method, the diameter of the hole ($d_{bs-2}$) is obtained by equating the cross-sectional area of the shank to that of the threaded part. In another method, the cross-sectional area of the shank is reduced by drilling a hole, as illustrated in Figure 2.c. Both methods reduce cross-sectional area of the shank and increase stress and energy absorption.

##### (20)

$\dfrac{\pi}{4}· d_{bs}^{2}-\dfrac{\pi}{4}· d_{bs-2}^{2}=\dfrac{\pi}{4}· d_{bs-1}^{2}$ or

$d_{bs-2}=\sqrt{d_{bs}^{2}-d_{bs-1}^{2}}$

Machining a long hole is difficult operation compared with turning down the shank
diameter. Drilled hole results in stress concentration. Therefore, a bolt with reduced
shank diameter is preferred over a bolt with an axial hole^{[6]}.

### 4.2 Tensile and Compressive Strength in Forms of Normal Stress

The needed strength and rigidity of this connection is achieved, in the first place, by tensile strength and compressive strength of the connected parts against each other. The magnitude of the forces should be enough to prevent joint separation in the connection. Both the tensile strength and the compressive strength are in parallel with the surface normal vector, and thus perpendicular to the cutting plane of a workpiece. Thus, both types of strength are normal stress. As mentioned above in 3.2 Fatigue Failure, an axial strength can be reinterpreted to a tensile strength and compressive strength by direction of the applied force.

The selected material for joint components is the steel grade ST series (DIN 17100). It is low carbon, high strength structural steel which can be readily welded to other weldable steel. With its low carbon equivalent, it possesses good cold-forming properties. Among the many types of rigid connections, the most commonly used is considered here; the connection of bolt or screw with a shaft.

### 4.3 Choice of Bolt

Tensile Strength

For the calculation of permissible tensile strength shown Figure 3.a, the following procedure is given.

where, $\sigma_{yi}$ is the yield strength in the bolt, and varies slightly over 285~355$\left[N/mm^{2}\right]$
by its thickness in DIN 17100 steel sheet^{[7]}. The yield strength in shear is equal to half of the yield strength in tension^{[6]}.

Compressive Strength

Given the geometry of the joint component, and force applied as a compressive strength to the component. The procedure is completely the same as in calculating the tensile strength; it has only difference that the mechanism of the pressure has a negative sign as shown Figure 3.b.

Force on the Bolt

The strength of the bolt or screw in tension is given by

Decision of the Size

The size of bolt can be obtained from the equation (24).

Generally, the approximate relationship between $d_{bs}$ and $d_{bs-1}$ can be used
^{[6]}.

From a specification, the standard size of the bolt is selected.

Summary

Assuming infinite thickness of the shaft, the shear stress can be left out of consideration in the connection of coupling components with shaft. The bending stress can also be ignored because the length of bolt is considerably shorter than that of shaft. Besides the effects in the types of bolt such as machine screws, thread cutting screws, and hex bolts with different heads do not take into account.

## 5. Conclusion

A linear electric machine’s shaft is a cylindrical part that protrudes from the machine and housing. The shaft’s function is to transfer energy from the machine to the intended use. Operating in relation to speed and force are precision pins and shafts. A shaft in linear electric machine is a mechanical section for transforming the movement and force. Shaft size significantly influences the force in these devices. Therefore, the precise modeling and prototyping of shafts are essential for all applications. In addition to the need for mounting the mover and a variety of attachments, the model of the mover shaft is very much based on the cooling concept selection of the electric device. Particularly with greater electric machines and thus longer and larger machine shafts, a hollow shaft enables new model matters both for lightweight construction and for the cooling system.

The dynamic analysis of cylindrical linear electric machine is performed by using the combination of electric and kinetic equation considering stroke and mechanical load. It provides useful information to analysis the dynamics characteristics of the machine as well as to design of the shaft. Based on this characteristic, an empirical design procedure for shafts with fatigue loadings is presented. The shaft design consists of the determination of the correct shaft diameter to ensure satisfactory strength and rigidity when the shaft is transmitting power under various operating and loading conditions. The shaft diameter is determined by using the axial and bending strength of the selected material by Woehler curve. In addition, the consideration of the bolt coupling is achieved by influences such as uniform and normal strengths, load, and others for ideal alignment of the shaft, which is made of steel and therefore very rigid, to some intermediate resilient element that can be deformed by relatively small force and decrease the additional load of the shaft. In conclusion, the shaft design and machine element is closely related with the dynamic performances by mechanical load conditions.

### Acknowledgements

This study was conducted by research funds from Gwangju University in 2024.

This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. RS-2022-00166724).

### References

## 저자소개

He received B.S. and M.S. degrees in Electrical Engineering from Dongguk and Hanyang University, South Korea, respectively. And then he was responsible for the development of electrical machine and its drive at Samsung Heavy Industry, Samsung Electronics, and Daewoo Electronics, in order. After he received Dr.-Ing. degree from Technical University Braunschweig, Germany, he was in the Daelim Motor, South Korea. Since March 2018, he has joined Gwangju University, where he is currently a professor in the dept. of electrical engineering. His research and development field and interest included design, analysis and drive of electric machine for seamless e-mobility.