International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 155
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
Stepped Shaft Automatic Design Considering a
New Integrated Technique of Stress, Deflection, and
Critical Speed Analysis
Hesham A. Abdou*, Sara A. El-Bahloul, Tawakol A. Enab, and N. Fouda
Production Engineering & Mechanical Design Department, Faculty of Engineering, Mansoura University
El-Gomhoureya Street, Mansoura 35516, Dakahlia, Egypt
* E-mail: [email protected]
Abstract-- Since the shaft is an important part of all
machines, it is deemed very necessary to design it effectively.
The whole shaft design involves several iterative steps
beginning with the determination of the shaft's preliminary
dimensions. This paper introduces software that is based on a
new integrated method of stress, deflection and critical speed
analysis to determine the optimal dimensions of different
stepped shafts. This software was developed with the help of a
graphical user interface and visual aids using MATLAB as a
programming language. In order to determine the optimal
shaft diameter, the fatigue stress analysis is conducted at the
most critical locations automatically. Analysis of deflection is
then performed to check whether the resulting dimensions are
safe or not. The study of vibration is also conducted to test
whether the shaft runs safely away from critical ones.
Verification is also carried out on the basis of a comparison
between the proposed design and the Shigley design on a given
case study, which shows that the new approach used in the
proposed design leads to more accurate results. The proposed
software greatly serves any user who has no experience or
knowledge of different design techniques and strategies,
resulting in time, effort, and cost savings.
Index Term-- Automatic design, stepped shaft, fatigue stress,
deflection, critical speed, and stress concentration.
1. INTRODUCTION
A shaft is a rotating machine part that is typically a circular
cross-section used for power transmission. In order to
transfer the necessary torque, a shaft rests on bearings and
holds other machine parts such as gears, pulleys, flywheels,
sprockets, and cranks. Such machine parts produce loads on
the shaft of bending, axial, and torsion. Additionally, Most shafts are usually stepped and having keyways, sharp
corners, etc., which introduce fatigue cracks that eventually
propagate in some highly stressed regions until a final
fracture occurs. Therefore, the shaft design must maintain
the static and dynamic loads produced in order to achieve
durability [1-3].
In order to obtain the most durable shaft design, extensive
research has been carried out. In the analysis of stepped
shaft, Egelhoff et al., used modern engineering tools to solve
a structured problem using energy techniques [4].
Marudachalam et al., used the Goodman approach to
improve shaft design under fatigue loading [5]. Gujar et al.,
designed a multi cross-section shaft using only the adapted
Goodman principle and compared with Finite Element Analysis (FEA) [3]. Shivakumar et al., made a FEA of lathe
spindle using ANSYS [6]. Ravinder et al., studied the gear
shaft by analyzing the stresses, torques and bending
moments by developing a Computer-Aided Design (CAD)
model in Pro-E software, and conducting a FEA utilizing
ANSYS software [7]. Tavares and de Castro compared the
fatigue analysis methodologies of shafts based on the
Soderberg criterion [8]. Armah used the ASME code
methodology of fatigue strength analysis to construct a
power transmission shaft at a constant speed during fatigue
loading [1]. From the literature review it is apparent that less
discussion and resreaches have been performed on stepped shaft automatic full design based on stress, deflection, and
critical speed analysis together.
The aim of this research is to automatically determine the
optimal design of a stepped shaft based on a new integrated
approach of stress, deflection, and critical speed analysis.
This software is developed with the help of a graphical user
interface and visual aids using MATLAB as a programming
language. Additionally, to aid with effective visualization, a
graph of the optimal shaft layout is accomplished. The
proposed software will greatly serves any user who has no experience or knowledge of different design techniques and
strategies, resulting in time, effort, and cost savings.
2. AN INTEGRATED PROCEDURE FOR STEPPED SHAFT
DESIGN
Figure 1 shows the integrated stepped shaft design
technique that will be used to automatically calculate the
optimal dimensions of stepped shafts, based on new
integrated approach of stress, deflection, and critical speed
analysis.
Firstly, it is important to provide a general labeled
layout for the shaft with dimensions. The second step is to
identify any unknown loads to estimate the reactions at the
supports using equations of static equilibrium.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 156
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
Fig. 1. The integrated procedure for the shaft design process
Accordinally, the shear force diagram (SFD), bending
moment diagram (BMD) and torque diagram (TD) are
drawn in two planes and then combined to identify the most critical location. The third step is to identify the shaft
material properties and the factor of safety. If the shaft
material is undetermined, a material of lower strength is
applied first. The fatigue strength (Se) is determined for an
infinite life shaft. For the factor of safety (FS), a minimum
desired one is specified first. Afterward, the goal is to
determine automatically the values of modification factors
used in the calculation for all potential critical locations
where higher stress exists. The values of surface factor (ka),
loading factor (kc), temperature factor (kd), reliability factor
(ke) and miscellaneous factor (kf) will be determined once and there is no need to change them later during iterations.
But the values of Size Factor (kb) and fatigue stress
concentration factor (Kf) will be determined first with
typical values and will be refined later during iterations for
more accurate design. To get preliminary shaft dimensions,
appropriate fatigue design equations are applied for the
most critical point to express diameter size in terms of Kf
and kb. [1]
Safe
Not Safe
Change
material
Load Analysis
Material Properties and Factor of Safety
Stress Analysis
Design Equation
Solution to get preliminary shaft dimensions
Check of other shaft critical points
Deflection Analysis
Vibration Analysis
Shaft Layout
Refine
constant
s
Safe
Not Safe
Change
dimension
s
Final result
Safe
Not Safe
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 157
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
To refine the decision choice, a deflection analysis is
performed. Singularity functions are used to determine
deflection as they are excellent for managing discontinuities.
If the deflection analysis is unsafe, the values of diameters
are changed by multiplying them with the largest
magnification factor resulted, consequently other related
dimensions change through appropriate proportions. By
determining the FS for all critical locations, the design is
safe at a certain location if its FS is larger than the desired
value. If the FS is less than the desired value, then failure
will happen before the desired number of cycles (N).
Therfore, higher strength material can be used in this case.
[2]
The shaft critical speeds are determined by performing
vibration analysis. Critical speeds are associated with
uncontrolled large deflections, which occur when inertial
loading on a slightly deflected shaft exceeds the restorative
ability of the shaft to resist. Shafts must operate well away
from such speeds. Since most shafts are of variable
diameter, Rayleigh’s equation will be more useful for estimating the first critical speed, treating simultaneously
the contributions of concentrated masses (e.g., gears,
pulleys, sprockets, cams, etc.) and the distributed shaft mass
as well. Additionally, Singularity functions method is used
for determining the deflection of the shaft to be used in
critical speed calculations. [2, 9]
3. PROPOSED SOFTWARE INTERFACE
The proposed software automatically determine the optimal
design of a stepped shaft based on a new integrated
approach of stress, deflection, and critical speed analysis.
This software is developed with the help of a graphical user interface and visual aids using MATLAB as a programming
language. The software greatly serves any user who has no
experience or knowledge of different design techniques and
strategies, resulting in time, effort, and cost savings. Figure
2 shows the performed software input interface where the
user can easily enter all needed input values such as axial
dimensions, acting forces, shaft speed, gear and bearing
dimensions and types, gear specific weight, desired safety
factor, temperature, reliability, surface finish, and starting
material properties. Then, the user can easily press the
CALCULATE button to begin the automated shaft design technique according to the integrated procedure for shaft
design previously discussed and illustrated in Fig. 1. It is
obvious that the software interface is designed to facilitate
the interaction of the user with the software.
4. SOFTWARE IMPLEMENTATION
In order to show in details the processing manner of the
software according to the previous procedure, a case study
taken from ‘Shigley's mechanical engineering design’ is
implemented. A shaft is used as a part of a double reduction
gearbox. The general layout and axial dimensions of the
shaft are illustrated in Fig. 3. The shaft is running at a
constant speed of 3000 rpm. It carries two spur gears at
points (G) and (J) and is supported simply by bearings (A) and (B). It is required to determine the appropriate
diameters for each section of the shaft for infinite life of it,
by selecting a suitable material and desired FS equal to 1.5.
[2]
The user can easily enter the required inputs in the software
interface as shown in Fig. 2. A cold drawn (CD) steel
( AISI 1020) material is initially chosen with Ultimate
tensile strength (Sut) = 68 kpsi and the desired FS = 1.5. The
material selected has an elasticity modulus (E) equal to 30
Mpsi.
4.1. Preliminary shaft dimensions calculations
The SFD, BMD, TD, and total BMD are automatically
performed using equations of static equilibrium as ilustrated
in Fig. 4. From the total BMD, it is clear that the critical location without considering the effect of stress
concentrations is found at point (J).
The Goodman method is used at point (I) to get the
preliminary shaft dimensions then making iterations to
refine them. Modification factors are determined
automatically as follows: ka=0.883, kc=1, kd =1, and
ke=0.753. For the constant kb, it is assumed first as 0.9, and
then it is refined later when the diameter is known. At
location (I), the software calculates automatically the
remaining constants as follow: theoretical stress
concentration factor for normal stress (Kt) = 1.5987 and for
shear stress (Kts) = 1.3304 and notch sensitivity for normal stress (q) = 0.7868 and for shear stress (qs) = 0.8314.
Accordinally, the fatigue stress-concentration factor for
normal stress (Kf) = 1.4711 and for shear stress (Kfs) =
1.2747. The first estimation of the small diameter of the
shoulder at point (I) is 1.6096 in. The iterations for the
solutions will include the parameters (kb, q, Kt, qs, Kts, Kf,
Kfs, and Se) to improve the results by refining the
calculations until achieving constant diameter for three
decimal places. After the final iteration (when the diameter
difference is smaller than 0.001) the dimensions at location
(I) will be as follow: the small diameter of the shoulder (D5)
= 1.62 in, fillet radius (r5) = 0.162 in and the large diameter of the shoulder (D4) = 1.94 in. The obtained preliminary
stepped shaft diameters on the basis of fatigue stress
analysis are: D1=1.13 in, D2=1.35 in, D3=1.62 in, D4=1.94
in, D5=1.62 in, D6=1.35 in, and D7=1.13 in.
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 158
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
Fig
. 2
. T
he p
rop
ose
d s
oft
war
e i
nte
rfac
e
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 159
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Fig. 3. General shaft layout (Dimensions are in inches)
Fig. 4. Shaft load analysis diagrams
4.2. Deflection analysis
After determining the preliminary shaft dimensions,
deflection analysis is applied to check the deflections and
slopes at gears and bearings location in order to control the
transverse deflection and the angular deflection (slope)
within an acceptable range. Otherwise, changes in the
dimensions should be performed to resolve this problem.
As the shaft hasn’t the same cross-section along its length,
a detailed analysis using the singularity functions method
must be used to give more accurate results compared to
the simplified analysis.
The simplified analysis can be performed by assuming a
uniform cross-section of the shaft with minimum diameter.
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The shaft slopes and deflections are obtained by modeling
the shaft twice with loads in two orthogonal planes using
equations (1) and (2) for xy-plane and equation (3) and (4)
for xz-plane. Then the results can be combined through an
orthogonal vector. Details like grooves and keyways are
ignored as they will have an insignificant effect on the deflections. It has been noticed that the stepped shaft will
lead to different second moment of inertia at the transition
sections, so deflection and slope have two unreal results at
the same location. Consequently, the results are not
accurate so one can either simplify the diameter to only one
value (smallest one), acordinlly; the second moment of
inertia is the same. Otherwise, use the singularity functions method to overcome this problem.
dy
dx =
(1
2∗356.725∗⟨x⟩2)−(
1
2∗197∗⟨x−2⟩2)−(
1
2∗885∗⟨x−7.75⟩2)−(4.0963∗103)
30∗106∗I1 (1)
y = (
1
6∗356.725∗⟨x⟩3)−(
1
6∗197∗⟨x−2⟩3)−(
1
6∗885∗⟨x−7.75⟩3)−(4.0963∗103∗x)
30∗106∗I1 (2)
dy
dx =
(1
2∗114.975∗⟨x⟩2)+(
1
2∗540∗⟨x−2⟩2)−(
1
2∗2431∗⟨x−7.75⟩2)−(6.0627∗103)
30∗106∗I1 (3)
y = (
1
6∗114.975∗⟨x⟩3)+(
1
6∗540∗⟨x−2⟩3)−(
1
6∗2431∗⟨x−7.75⟩3)−(6.0627∗103∗x)
30∗106∗I1 (4)
By using the singularity functions method, the bearing
reaction forces can be calculated to get the load intensity
equation, and numerically integrated to generate equations
for shear force, bending moment, slope and deflection
respectively. Figure 5 shows the diagrams of shear force,
bending moment, slope and deflection equations for xy-
plane and similarly for xz-plane. For the xy-plane, as (Ii) is the second moment of inertia for (ith) shaft section is not the
same (where i = 1:7), so the equation of (moment/inertia)
[M/I] can’t be used directly. Hence, some modifications
must be performed firstly. In the current case study, the
stepped shaft under consideration has six steps (w = 1:6)
according to diameter. Each step far of the datum by a
distance lw. For the first step (w =1), the values of (M/I) at
the points (b) and (c), and the step change [Δ(M/I)w=1] are
calculated. Also, the slope (m) for lines (ab) and (cd) are
mab and mcd respectively.The slope change for first step is
[(Δm)w=1] which can be calculated as the slope difference
between mab and mcd. Similarly, [Δ(M/I)w] and [(Δm)w] can
be calculated for the other five steps. Then the modification
is done by dividing the [M/I] equation terms by their (Ii)
and by adding steps [Δ(M/I)w] and ramps [(Δm)w].
Figure 6 shows the plotting of the (moment/inertia) equation
[M/I]. By integrating the equation of (Moment/inertia)
twice with adding previous modification, the detailed
equations of slope and deflection are obtained for xy-plane
using equation (5) and (6) and for xz-plane using equation
(7) and (8) respectively. Accodinly, the slope and
deflection diagrams are plotted for simplified and detailed
analysis methods of both planes as shown in Fig. 7.
dy
dx =
1
30∗106 {
356.725
2∗I1⟨x⟩2−
197
2∗I2⟨x − 2⟩2 −
885
2∗I5⟨x − 7.75⟩2 +
∑ Δ (M
I)
w∗ ⟨x − lw⟩1
w=1:6 + ∑ (1
2∗ Δmw⟨x − lw⟩2
w=1:6 ) − (9.8361 ∗ 103)} (5)
y = 1
30∗106 {
356.725
6∗I1⟨x⟩3−
197
6∗I2⟨x − 2⟩3 −
885
6∗I5⟨x − 7.75⟩3 +
∑ (1
2∗ Δ(M/I)w ∗ ⟨x − lw⟩2
w=1:6 ) + ∑ (1
6∗ Δmw⟨x − lw⟩3
w=1:6 ) − (9.8361 ∗ 103 ∗ x)} (6)
dy
dx =
1
30∗106 {
114.975
2∗I1⟨x⟩2+
540
2∗I2⟨x − 2⟩2 −
2431
2∗I5⟨x − 7.75⟩2 +
∑ Δ (M
I)
w∗ ⟨x − lw⟩1
w=1:6 + ∑ (1
2∗ Δmw⟨x − lw⟩2
w=1:6 ) − (1.2753 ∗ 104)} (7)
y = 1
30∗106 {
114.975
6∗I1⟨x⟩3+
540
6∗I2⟨x − 2⟩3 −
2431
6∗I5⟨x − 7.75⟩3 +
∑ (1
2∗ Δ(M/I)w ∗ ⟨x − lw⟩2
w=1:6 ) + ∑ (1
6∗ Δmw⟨x − lw⟩3
w=1:6 ) − (1.2753 ∗ 104 ∗ x)} (8)
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 161
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(b)
xz-p
lan
e
Fig
. 5
. S
hea
r fo
rce, b
end
ing
mo
men
t, d
efle
ctio
n a
nd
slo
pe
dia
gra
ms
ov
er s
haf
t le
ng
th f
or
(a)
xy-p
lan
e a
nd
(b
) xz
-pla
ne
(a)
xy-p
lan
e
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Fig. 6. Plotting of (Moment/inertia) equation [M/I]
Figure 7 illustrates a comparison between the resultant slope
and deflection according to detailed and simplified analysis methods. From the comparison it is clear that the detailed
method results are more accurate, so it is implemented in
the proposed software.
Fig. 7. Slope and deflection according to detailed and simplified analysis methods
The deflection check is carried out to control deformation at
locations of power transmitting elements (gears) and
angular deformation (slope) at the location of bearings at
an acceptable range. Whether these values are acceptable or
not, it depends on the specific bearings and gears selected,
as well as the level of performance expected. The deflection check shows that all bearing slopes are well
below typical limits for bearings. The gear slopes and
deflections satisfy the recommended limits. Once
deflection or slope at a point is larger than the allowable
value, larger shaft diameter is required. A new diameter can
be found by determining the largest magnification factor
then multiplying the diameters by this ratio, consequently,
the remaining dimensions change according to suitable
ratios. Since deflection analysis has been conducted
safely within accepted ranges, there is no need for larger dimensions. Aditionaly, the FS is calculated at these
critical locations and compared with the minimum accepted
value. If the check leads to the unsafe case, the material
will be replaced by higher strength one.
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4.3. Critical speed analysis
Vibration analysis is performed to check the critical speed
of the shaft. Rayleigh’s equation will be used to calculate
the first critical speed. Singularity functions method is also
used to reduce the difficulty of finding transverse
deflections of a stepped shaft caused by the weight of the shaft and the attachments. This can be performed by
dividing every (ith) shaft step into (kth) segments (except the
bearing steps). The weight of each segment (ik) is applied as
a concentrated force (fik) at the segment centroid. Also, the
gear weights are involved as concentrated forces (Wg1, Wg2)
as shown in Fig. 8. The shaft speed is compared to the
calculated critical speed to make sure that at least the shaft
is running at speed less than the half value of first critical speed. Hence, the shaft is safe; otherwise, the material
should be modified. [2, 9]
Fig. 8. Shaft loads caused by its weight and attachments
5. RESULTS AND DISCUSSION
According to the performed software, after the user enters
the required input data in the software interface shown in
Fig. 2, the user presses the CALCULATE button to start the
calculation process as described previously. The software output interface shown in Fig. 9 is then opened containing
all the results of the optimal design for the considered
stepped shaft based on the new integrated technique of
stress, deflection, and critical speed analysis. Additionally,
to aid with effective visualization, a two dimensional layout
of the optimal shaft is accomplished as shown in Fig. 10.
Also, the deflection analysis curves are plotted and can be
displayed to the user if needed.
Table 1 illustrates a comparisson between the proposed
software and Shigley design results for the implemented
case study. By comparing these results, it can be noticed that the high accuracy of the software results due to the
accuracy in determining the calculation constants compared
to the manual manner, which requires more assumptions
leading to less accuracy. Aditionaly, it is obvious that the
proposed software greatly serves any user who has no
experience or knowledge of different design techniques and
strategies, resulting in time, effort, and cost savings.
6. CONCLUSION
This paper introduces a new method based on an integrated
approach of stress, deflection and critical speed analysis to assess the optimal design of stepped shafts automatically.
The proposed software is performed with the help of
MATLAB as a programming language using a graphical
user interface and visual aids. In order to determine the
optimal shaft diameter, the fatigue stress analysis is
conducted at the most critical locations automatically.
Analysis of deflection is then performed to check whether
the resulting dimensions are safe or not. The study of
vibration is also conducted to test whether the shaft runs
safely away from the critical ones. Verification is also
carried out on the basis of a comparison between the proposed design and the Shigley design on a given case
study, which shows that the new approach used in the
proposed design leads to more accurate results. The
proposed software supports any user who has no experience
or knowledge of different design methods and strategies,
resulting in time, effort, and cost savings
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 164
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
Fig
. 9
. T
he o
utp
ut
inte
rfac
e o
f th
e p
rop
ose
d s
oft
war
e f
or
shaft
desi
gn
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:20 No:01 165
202801-5757-IJMME-IJENS © February 2020 IJENS I J E N S
Fig. 10. A two dimensional layout of the optimal shaft
Table I
A comparisson between the proposed software and Shigley design results
D1 (in) D2 (in) D3 (in) D4 (in) D5 (in) D6 (in) D7 (in) Sut (kpsi)
Proposed software 1.13 1.35 1.62 1.94 1.62 1.35 1.13 78
Shigley design 1 1.4 1.625 2 1.625 1.4 1 100
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Shaft under Fatigue Loading Using ASME Code. American
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[2] Budynas, R.G. and J.K. Nisbett, Shigley's mechanical
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mechanical engineering. 2011, New York: McGraw-Hill. xxi,
1082 p.
[3] Gujar, R. and S. Bhaskar, Shaft design under fatigue loading
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1066.
[4] Egelhoff, C., E. Odom, and B. Wiest. Application of modern
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(FIE). 2010. IEEE.
[5] Deepan Marudachalam, M., K. Kanthavel, and R. Krishnaraj,
Optimization of shaft design under fatigue loading using
Goodman method. International Journal of Scientific &
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[6] Shivakumar, S., N.K. Anupama, and V. Khadakbhavi,
ANALYSIS OF LATHE SPINDLE USING ANSYS.
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[7] Ravinder, S. and R. Banothu, Design and analysis of Gear
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[8] Tavares, S. and P. de Castro, A comparison of methodologies
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[9] Joseph E. Shigley, Charles R. Mischke, and T.H.B. Jr.,
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