Computer – Aided Design of the Critical Speed of Shafts AKPOBI

A computer aided design software for the analysis of the critical speed of shaft, is developed and presented in this work. The software was designed using the principles of object oriented programming, and implemented with the Microsoft Visual Basic Language. The package was tested on a number of benchmark design problems. The solutions obtained were highly accurate. Also, the software exhibited very high efficiency. To improve on the method of presenting these solutions, graphical features were incorporated. This enhances the ability to visualize results. @ JASEM All rotating shafts, even in the absence of external load, deflect during rotation. The magnitude of the deflection depends upon stiffness of the shaft and its supports, the total mass of shaft and attached part, the unbalance of the mass with respect to the axis of rotation, and the amount of damping in the system. As long as the deflections are minimal, the shaft can still operate satisfactorily. However, as the speeds increase, the shafts become unstable. Thus, there is the need to determine these critical speeds at which instability sets in. With the deflection, now considered as a function of speed, only the lowest (first) and occasionally and the second are of interest to the designer, the others will usually be so high as to be well out of range of the operating speed At the first critical speed, the shaft will bend to the simplest shape possible. At the second critical speed it will bend to the second simplest shape possible etc.for example a shaft supported at its end and having a large (compared to shaft) masses attached, will bend according to the configurations shown in Figures 1 and Figure 2 at the first and second critical speed respectively. Fig.1: Loading configuration at the first critical speed Fig. 2: A variant of the Loading configuration at the first critical speed We wish to state that to the best of our knowledge there is no documented computer-aided design work for resolving the critical speed of shafts. The authors would have conducted a comparative assessment of our results with any that was available. But the results presented here are highly accurate as they were solutions to bench mark examples and problems adopted from (Hall et al, 2000). This work is a continuation of efforts by the authors to develop software for the design of machine components. Earlier works had considered the design of: some types of gears (helical, and spur), design of Rolling Bearings, machine vibrations design see (Akpobi, 1998; Akpobi; Ardey, 2002; Akpobi; Airiohuodion 2005). And quite recently, software for the design of Flywheels, was developed and reported see (Akpobi; Lawani, 2006). The design for the critical speeds of shafts, is well expounded in standard machine design texts, see (Hall et al, 2000; Shigley; Mischke, 2001). In this work the problem we focused on, was to develop and implement a computer software that would greatly enhance an accurate and efficient design for the critical speeds of shafts. Also we addressed the problem of reducing the computational difficulties usually encountered in manually designing for the critical speeds of shafts. And then provided a visual display of the solutions, so as to easily and speedily interpret these solutions. The natural frequency of the shaft is very nearly the same as the critical speed and is usually taken as the same thing .There is a difference usually quite small due the gyroscopic action of the mass. DESIGN OF SOFTWARE MODEL In designing the software, all the parameters necessary for proper analysis of the critical speed of shafts, were carefully assembled, and used in developing the programme that forms the software in this present work. In the software, the allowable loads were defined and the corresponding induced obtained. The concept of object oriented programming technique was employed in the design of the software. This principle was then implemented using Microsoft Visual Basic language (Microsoft Computer–Aided Design of the Critical Speed of Shafts * Corresponding author: Akpobi, J.A. 80 Incorporated, 1998; Byron, 2002). These design parameters are presented as follows: SHAFT WITH SINGLE ATTACHED MASS If the shaft mass is small compares to the attached mass, the first critical speed (Wc) can be calculated approximately as: / c W k m = 1 Where k shaft spring constant, and m is mass Also / c W g δ = 2 where g is acceleration due to gravity (m/s)and δ is static deflection (m) SHAFT OF CONSTANT CROSS SECTION If the shaft is simply supported at the end, with no mass involved other than that of the Shaft itself, the first critical speed will be max 5 / 4( / ) c W g δ = 3 where nax δ is the maximum static deflection (m) SHAFT OF NEGLIGIBLE MASS CARRYING SEVERAL CONCENTRATED LOAD For shaft of negligible mass carrying several concentrated masses, the first critical speed is obtained using the Rayleigh Ritz equation (Equation 4).

All rotating shafts, even in the absence of external load, deflect during rotation.The magnitude of the deflection depends upon stiffness of the shaft and its supports, the total mass of shaft and attached part, the unbalance of the mass with respect to the axis of rotation, and the amount of damping in the system.As long as the deflections are minimal, the shaft can still operate satisfactorily.However, as the speeds increase, the shafts become unstable.Thus, there is the need to determine these critical speeds at which instability sets in.
With the deflection, now considered as a function of speed, only the lowest (first) and occasionally and the second are of interest to the designer, the others will usually be so high as to be well out of range of the operating speed At the first critical speed, the shaft will bend to the simplest shape possible.At the second critical speed it will bend to the second simplest shape possible etc.for example a shaft supported at its end and having a large (compared to shaft) masses attached, will bend according to the configurations shown in Figures 1 and Figure 2 at the first and second critical speed respectively.We wish to state that to the best of our knowledge there is no documented computer-aided design work for resolving the critical speed of shafts.The authors would have conducted a comparative assessment of our results with any that was available.But the results presented here are highly accurate as they were solutions to bench mark examples and problems adopted from (Hall et al, 2000).This work is a continuation of efforts by the authors to develop software for the design of machine components.Earlier works had considered the design of: some types of gears (helical, and spur), design of Rolling Bearings, machine vibrations design see (Akpobi, 1998;Akpobi;Ardey, 2002;Akpobi;Airiohuodion 2005).And quite recently, software for the design of Flywheels, was developed and reported see (Akpobi;Lawani, 2006).The design for the critical speeds of shafts, is well expounded in standard machine design texts, see (Hall et al, 2000;Shigley;Mischke, 2001).In this work the problem we focused on, was to develop and implement a computer software that would greatly enhance an accurate and efficient design for the critical speeds of shafts.Also we addressed the problem of reducing the computational difficulties usually encountered in manually designing for the critical speeds of shafts.And then provided a visual display of the solutions, so as to easily and speedily interpret these solutions.The natural frequency of the shaft is very nearly the same as the critical speed and is usually taken as the same thing .There is a difference usually quite small due the gyroscopic action of the mass.

DESIGN OF SOFTWARE MODEL
In designing the software, all the parameters necessary for proper analysis of the critical speed of shafts, were carefully assembled, and used in developing the programme that forms the software in this present work.In the software, the allowable loads were defined and the corresponding induced obtained.The concept of object oriented programming technique was employed in the design of the software.This principle was then implemented using Microsoft Visual Basic language (Microsoft Incorporated, 1998;Byron, 2002).These design parameters are presented as follows:

SHAFT WITH SINGLE ATTACHED MASS
If the shaft mass is small compares to the attached mass, the first critical speed (W c ) can be calculated approximately as:

SHAFT OF NEGLIGIBLE MASS CARRYING SEVERAL CONCENTRATED LOAD
For shaft of negligible mass carrying several concentrated masses, the first critical speed is obtained using the Rayleigh Ritz equation (Equation 4).The Rayleigh Rita equation above can be used for estimating the critical speed of shafts with distributed loads.The method involves breaking the distributed masses into a series of masses m 1 , m 2 , m 3 ……m n , after which each of the masses is taken to be a concentrated load having its weight acting at its center.

THE DUNKERLEY EQUATION
Another approximation for the first critical speed of a multiple mass system is

PROGRAMME DESCRIPTION
The program was designed and implemented using Microsoft Visual Basic object oriented programming language.The structure of the program is such that there are three stages.These stages include the input stage, analysis and output stage.

INPUT STAGE
At this stage, the user enters the required information about the Critical speed of the shaft, in the input interface form.The programme is well written in such a way that the software requires minimal input to carryout its analysis.

ANALYSIS
After the required data is in entered, the analysis carried out.Once the data is entered, the software computes all the parameters required in analyzing the critical speed of shaft in less than 2 seconds depending on the speed of the processor.

OUTPUT
The software is designed such that the critical speed parameters are outputted numerically and graphically with the accompanying description of the result.

PROGRAMME'S ALGORITHM OR PSEUDO-CODE
The software was designed using the following algorithm or the programme's pseudo-code: If     The software also produced as output: The critical speed due to flexible support is 394.464rad/s The flexibility of the support reduces the critical speed by15.0%

Conclusion:
We have presented in this work, a computer-aided design software that allows for easy design and analysis of the critical speed of shaft.The number of practical problems formulated and solved accurately using this software, shows that it is flexible, accurate, and robust.The graphical features incorporated in the software, also enhances the ability of the user to visualize and interpret solutions easily.

Fig. 1 :
Fig.1: Loading configuration at the first critical speed Fig. 2: A variant of the Loading configuration at the first critical speed

δ
acceleration due to gravity (m/s 2 )and δ is static deflection (m)SHAFT OF CONSTANT CROSS SECTIONIf the shaft is simply supported at the end, with no mass involved other than that of the Shaft itself, the first critical speed will be max is the maximum static deflection (m)

δ
is static deflection (m) at the nth mass n w is the weight of the nth mass and j is the total number of masses.
value = 3 Then If the uniform shaft carries no mass Then Input known parameters max δ Compute desired parameters using equation 3 Compute values for the critical speed of the shaft and generate a plot of Compute desired parameters using equation 4 Compute value for the critical speed of the shaft and generate a plot of critical speed against w 1 or w 2 .Else If the influence coefficient is known Then Input THE USE OF THE SOFTWARE To illustrate the effectiveness and numerical accuracy of the software, the following Examples were considered.EXAMPLE 1 Calculate the critical speed of rotating shaft carrying a mass M 1 of 50kg having a shaft spring constant of 250Nm, as shown in Fig. 4.

Fig. 4 :Fig. 5 :Fig. 6 :Fig. 7 :
Fig. 4: Loading configuration for Example 1SOLUTIONThe out put from the software for this analysis on inputting values for Mass of 50kg, spring constant of 250Nm is as follows:The critical speed is 2.236 rad/s The software also gave as output a graph of: Varying critical speed (rad/s) against mass (kg)The screen shot of the solution in shown Fig.5

Fig. 14 :
Fig.14: screenshot of the result for Example 4. EXAMPLE 5 The bearing support for the shaft is shown in Fig.12.It has flexibility equivalent to a spring constant of k 44MN/m in any direction perpendicular to the shaft

Fig. 15 SOLUTIONFig. 16 :
Fig. 15SOLUTIONThe out put from the software for this analysis on inputting values for Spring constant of 44MN/m, Deflection of 0.046mm, load of 1350N,Distance between support A and the mass is (L 1 ) = 250mm,while that between support B and mass (L 2 ) =500mm is as follows:

Fig. 17 :
Fig. 17: screenshot of the result for Example 5.Discussion: This software was tested for this paper using 5 Examples, in which the problems on critical for systems with single mass, multi-mass having rigid support or spring support and systems with known static deflections, known influence coefficient were analyzed.The results displayed by the software depend on the problem definition and provides the designer with the exact numerical values required to design for the critical speeds of shaft.The software also outputted graphical features for the following:Critical speed (rad/s) against mass (kg) Critical speed (rad/s) against maximum deflection (mm) Critical speed (rad/s) against weight 1 (N) Critical speed (rad/s) against weight 2 (N) The results for the various Examples shown in Figs. 5 -17 are highly accurate and even better than results obtain from the text(Hall, et al )  from which they were adopted.The accuracy obtained was due to the