Using Modern Engineering Tools to Efficiently Solve Challenging ...

Using Modern Engineering Tools to Efficiently Solve Challenging Engineering Design Problems: Analysis of the Stepped Shaft H.T.X. Truong1, E.M. Odom1, C.J. Egelhoff2 and K.L. Burns2 1

2

University of Idaho, Moscow, ID United States Coast Guard Academy, New London, CT

Abstract There is an inarguable elegance to the efficiency gains from the gear-driven compressor of an aircraft engine presently being developed or to a natural gas-fired power plant with a bottoming cycle that achieves 45% thermodynamic efficiency. But in considering the design as a whole it is easy to forget the most basic components. The shaft is an important rotating component used for the transmission of power and motion. The design considerations of a shaft can be broken down into three areas, fatigue, deflection, and critical frequency. During operation it can be subject to minimum and maximum axial, transverse and torsional loads leading to mean and alternating stress states. The effects of such stresses can be addressed during a fatigue analysis, a topic which is well covered in texts on machine component design and in governing standards. Critical frequency prediction is reasonably straightforward once the deflection of the shaft is known along with the attendant masses. As long as the loading is not complicated and the shaft has a constant diameter, determining the deflections of a shaft is simple and well covered in texts on mechanics of materials and machine component design. However, when the shaft cross section becomes practical it includes changes of diameter to provide steps that can be used to accurately mount bearings and gears. It can have overhanging ends and tapered cross sections. The determination of the deflection of these practical shafts has been historically cumbersome and intractable. Until now. A method of solution for these practical shaft geometries and loadings is presented. The method stays generalized, using an engineer’s knowledge of free body diagrams, writing moment equations, and Castigliano’s theorem to set up the problem solution into a form that is solved in an engineer’s favorite computer program. Thus far we have evidence that it is straightforward to implement this solution method in programs such as EES©, TK Solver, and Matlab®. An obvious alternative solution approach is to use a commercially available Finite Element Analysis (FEA) program such as ABAQUS. In this paper, several examples are presented to show the advantages of each solution approach but more importantly, it is shown how these solution methods can be complementary. For example, the equation-solver approach works nicely for overall sizing while the FEA solution shows the stress distributions visually. Each approach provides verification of the other. In the event they differ, the engineer needs to seek resolution for complete understanding and confidence in the solution results.

Copyright ASEE Middle Atlantic Regional Conference, April 29-30, 2011, Farmingdale State College, SUNY

Introduction There is an inarguable elegance to the efficiency gains from the gear-driven compressor of an aircraft engine presently being developed or to a natural gas-fired power plant with a bottoming cycle that achieves 45% thermodynamic efficiency. But in considering the design as a whole it is easy to forget the most basic components. The shaft is an important rotating component used for the transmission of power and motion. The design considerations of a shaft can be broken down into three areas, fatigue, deflection, and critical frequency. During operation it can be subject to minimum and maximum axial, transverse and torsional loads leading to mean and alternating stress states. The effects of such stresses can be addressed during a fatigue analysis, a topic which is well covered in texts on machine component design and in governing standards. Critical frequency prediction is reasonably straightforward once the deflection of the shaft is known along with the attendant masses. As long as the loading is not complicated and the shaft has a constant diameter, determining the deflections of a shaft is simple and well covered in texts on mechanics of materials and machine component design. However, when the shaft cross section becomes practical it includes changes of diameter to provide steps that can be used to accurately mount bearings and gears. It can have overhanging ends and tapered cross sections. Before computers were ubiquitous, engineers designed stepped and tapered shafts using elegant but time-consuming graphical techniques [1-2]. With the availability of scientific calculators and mainframe computers, semi-graphical and computerprogrammed techniques appeared [3-5]. In today’s undergraduate Machine Design textbooks, we see three general approaches to the solution of deflection for stepped or tapered shafts. One approach is graphical [6]. A second approach is the use of discontinuity functions to write a moment equation, M(x), or modified M(x) moment equation, , which is integrated twice and constants of integration are determined EI(x) using boundary conditions [7]. A third approach bypasses a first-order solution and purports to begin analysis/design by conducting Finite Element Analysis (FEA) using commercially available software [8-10]. In this paper, we demonstrate the use of two complementary approaches for the design of stepped and tapered shafts and we compare results to a classical work [11]. First we analyze the shaft deflection using a novel energy method and second we use an FEA method.

Novel Energy Method to Solve for Deflection The determination of the deflection of practical shafts has been historically cumbersome and intractable. Until now. We present an energy method based solution for these practical shaft geometries and loadings. The method stays generalized, using an engineer’s knowledge of free body diagrams, writing moment equations, and Castigliano’s theorem to set up the problem solution into a form that is solved in an engineer’s favorite computer program. The novel Copyright ASEE Middle Atlantic Regional Conference, April 29-30, 2011, Farmingdale State College, SUNY

approach shown in this paper, is to use Heaviside step functions to write the moment equation, a virtual axis identified by the variable ξ which specifies the point of interest for finding a deflection, and finally a-up look function to specify the diameter along the shaft during numerical integration. Thus far we have evidence that it is straightforward to implement this solution method in programs such as EES©, TK Solver, and Matlab®. This energy method for shaft deflection was introduced in the June 2010 issue of Machine Design magazine [12]. By Castigliano’s theorem the deflection is based on the strain energy, U, stored in an elastic beam loaded in bending. The strain energy is given as: (1)

2 The deflection, , at a location, i, is given as the partial derivative of the strain energy with respect to a load at the “i” location.

(2) The solution process to evaluate the integral can be described using five steps as follows: Step 1: Draw free body diagram (FBD) with dummy load Q, at secondary axis location, , and solve for reaction forces using statics. Step 2: Write the moment equation M(x,) for the entire length using discontinuity terms coupled with a Heaviside function H(x,). Step 3: Take the partial derivative with respect to the dummy load,

M(x, ) . Q

Step 4: Set Q=0 and write M(x,) for Q=0.

1 L M ( x,  )Q0 M ( x,  ) dx , and solve for any  using any . E 0 I ( x) Q of a number of commercially available equation-solving software tools. In this work, we have used EES©, TKSolver, and Matlab®.

Step 5: Write the integral  

To demonstrate the utility of this approach, a challenging shaft geometry and loading by [11] was selected. This shaft is shown in Figure 1. This rendered image is shown as well as the concomitant free body diagram for use in the five-step energy method process. It should be noted that this is a shaft that has seven changes in cross section, five loads and two moments. The solution will show the steps to calculate the deflection anywhere (i.e. everywhere) along the shaft.


a. Rendered Umasankar shaft.

b. Umasankar shaft FBD. Figure 1. Stepped Shaft with Applied Loads and Moments, and with Dummy-Load, Q. Nodes used for FEA with Locations Beneath (after Umasankar & Mischke) [11] Step 1: Draw FBD and solve statics We assume all applied forces and moments are in positive directions: positive force is vertically upward and positive moment is counterclockwise as shown in Figure 1b. For any given force on the shaft that is downward, the direction is still assumed to be positive but then the force is assigned a negative value. This is done to facilitate keeping track of signs and directions when writing statics equations. The origin is set at node 1 of the shaft as shown for ease of writing the moment equations about that node. A dummy, Q, is applied at an arbitrary distance, ξ, from the origin, in the negative direction as shown (Figure 1b). Table 1 below shows a summary of the known and unknown variables and values for this classical shaft.


Table 1. Summary of given variables from the Umasankar shaft figure Li (mm) 25 180 300 365 435 

Node i 2 5 8 10 12 Arbitrary

Fi or Ri (kN) -1 Unknown R5 5 Unknown R10 -1 Q

Mi (kN-mm) 100 -20

Comment Applied force Reaction Applied force and moment Reaction Applied force and moment Dummy load

To solve for reaction forces, the equilibrium equations (F=0; M=0) are used.

F

F2  R5  F8  R10  F12  Q  0

(3)

 F2 L2  R5 L5  F8 L8  M 8  R10 L10  F12 L12  M 12  Q  0

(4)

y

M

1

Using (3) and (4), we solve explicitly for the support reactions at node 5 and node 10 as follows:

R5 

F2 ( L2  L10 )  F8 ( L8  L10 )  F12 ( L12  L10 )  Q( L10   )  M 8  M12 L10  L5

(5)

F2 ( L2  L5 )  F8 ( L8  L5 )  F12 ( L12  L5 )  Q( L5   )  M 8  M12 L10  L5

(6)

R10  

Step 2: Write the moment equation M(x,) for the entire shaft using a Heaviside function H(x, ). The Heaviside function H(x, ) is defined to have a value of zero if x  , and a value of unity otherwise. It serves as a “switch” to turn “on” or turn “off” each term in the following moment equation. M ( x,  )  F2 ( x  L2 ) H ( x, L2 )  R5 ( x  L5 ) H ( x, L5 )  F8 ( x  L8 ) H ( x, L8 )  M 8 H ( x, L8 )  R10 ( x  L10 ) H ( x, L10 )  F12 ( x  L12 ) H ( x, L12 )  M 12 H ( x, L12 )  Q( x   ) H ( x,  )

(7)

Step 3: Substitute in the reaction forces found from equations (5) and (6) into equation (7) above, then take the partial derivative with respect to the dummy load, Q, obtaining (8) below.


M(x, ) L10     L5  (x  L5 )H(x,L5 )  (x   )H(x, )  (x  L10 )H(x,L10 ) Q L10  L5 L10  L5

(8)

Step 4: Set Q=0 and write M(x,) for Q=0. The result is shown in (9). M ( x,  ) Q 0  F2 ( x  L2 ) H ( x, L2 )  R5 ( x  L5 ) H ( x, L5 )  F8 ( x  L8 ) H ( x, L8 )  M 8 H ( x, L8 )  R10 ( x  L10 ) H ( x, L10 )  F12 ( x  L12 ) H ( x, L12 )  M 12 H ( x, L12 )

(9)

Step 5: Substitute in the expressions from Equations (8) and (9) into (2)

 

1 L M ( x,  )Q  0 M ( x,  ) . dx Q E 0 I ( x)

1 L 1 [ F2 ( x  L2 ) H ( x, L2 )  R5 ( x  L5 ) H ( x, L5 )  F8 ( x  L8 ) H ( x, L8 ) E 0 I ( x)  M 8 H ( x, L8 )  R10 ( x  L10 ) H ( x, L10 )  F12 ( x  L12 ) H ( x, L12 )  M12 H ( x, L12 )] 

.[

(10)

L10     L5 ( x  L5 ) H ( x, L5 )  ( x   ) H ( x,  )  ( x  L10 ) H ( x, L10 )]dx L10  L5 L10  L5

then solve for deflection using an equation solver of personal choice. The I(x) term is the d(x) 4 moment of inertia of the shaft and defined as equal to , where d(x) is the diameter of 64 different sections along the given stepped shaft. Values for diameter, d, are defined by a look-up table or from if-then statements, depending on the program being used to numerically integrate Equation (10). Results and Comparisons Table 2 shows the results obtained from the Castigliano-Heaviside energy method (solved by EES©) compared with the results published by Umasankar and Mischke [11]. The comparison indicates that the Castigliano-Heaviside numerical approach yields remarkably similar results compared to the solutions using the Umasankar-Mischke method. To illustrate the interchangeability of the programs used to numerically integrate Equation (10), Figure 2 below shows the solution for the deflection of the shaft with data points taken from three different equation-solving software tools: EES©, TK Solver, and Matlab®. The incredible agreement among these solutions is convincing evidence that the method of solution presented here is compatible with essentially any engineer’s favorite equation-solving program.


Table 2 Comparison of Energy Method (Castigliano-Heaviside) and Umasankar-Mischke Results at Points of Interest Along the Shaft. ξ 0 25 50 125 180 220 275 300 325 365 410 435 460

Castigliano-Heaviside δB (mm) -0.1287 -0.1047 -0.08 -0.02563 0 0.01189 0.02111 0.02197 0.0188 0 -0.03474 -0.06 -0.08748

Umasankar δB (mm) -0.126 -0.1028 -0.08017 -0.02559 0 0.0118 0.02097 0.02215 0.01901 0 -0.03493 -0.05944 -0.08618

Figure 2. Comparisons of deflection results from three different equations solvers

Equation-solving software is not the only way to determine deflection for a shaft. An obvious alternative solution approach is to use a commercially available Finite Element Analysis (FEA) program such as ABAQUS. FEA static analysis was performed on the Umasankar shaft [11] for the 2-dimensional (2D) case, using ABAQUS. This model consisted of 92 beam elements. Nodes at Points 5 and 10 (see Figure 1 for node numbers) were fixed and the remainder of the loads and concentrated moments were applied. The resulting deflected shape is shown in Figure 3 where the color indicates calculated von Mises stress levels (red is high, blue is low). Comparing the FEA shaft end points deflections and the general deflected shape to the results shown in Figure 2, it can be concluded that the two solution routes, i.e., energy methods and FEA are comparable.


Figure 3. Von Mises stress of deformed Umasankar Fig 10 shaft - 2D wire FEA model To further demonstrate the close agreement between the energy method and FEA, Figure 4 is provided. As this figure indicates, the data from the energy methods approach, FEA and Umasankar are essentially identical. A point worth noticing here is that the Castigliano‐ Heaviside energy method gives a continuous solution at “every” point along the shaft, while the Umasankar‐Mischke and FEA methods give discrete results. The FEA solution gives deflection at the assigned nodes only. Thus, it is the number of elements in the model’s mesh and the elements’ size that determines how close the FEA output data points approach one another and how close the overall FEA results approach the Castigliano‐ Heaviside solution. As observed, when more elements are assigned for this FEA model, the closer the data points are to each other and to the numerical solution from the presented Castigliano‐Heaviside method.

Figure 4. Comparisons of deflection results from FEA, Castigliano-Heaviside and Umasankar methods, shown against the original, un-deflected position.


Conclusion The shaft is an important rotating component used for the transmission of power and motion. As long as the loading is not complicated and the shaft has a constant diameter, determining the deflections of a shaft is simple and the solution process is well covered in texts on mechanics of materials and machine component design. However, when the shaft cross section becomes practical it includes changes of diameter to provide steps that can be used to accurately mount bearings and gears. It can have overhanging ends and tapered cross sections. The determination of the deflection of these practical shafts has been historically cumbersome and intractable. We have presented an energy method of solution for these practical shaft geometries and loadings. The method uses generalized equations developed using an engineer’s knowledge of free body diagrams, writing moment equations, and Castigliano’s theorem to set up the problem solution into a form that is solved by means of an engineer’s favorite computer program. We showed results from three equation-solvers which are essentially identical. We used FEA software to solve for deflection, which essentially matched the classic published solution and the energy method results. Additionally, we used the FEA method to determine stress everywhere along the subject shaft and we argue that engineering tools such as equation solvers and FEA software can be used in complementary roles for engineering analysis.

References 1. Shroeder, Walter, “Beam Deflections,” Machine Design, p. 85-90, January 1947. 2. Cowie, Alexander, “A tabular method for Calculating Deflections of Stepped and Tapered Shafts,” Machine Design, p. 111-118, August 9, 1956. 3. Margolin, Lawrence L., “Calculating Deflections in Variable-Section Beams,” Machine Design, p.173-181, March 16, 1961. 4. Halasz, Sandor T., “Minicalcs find Stepped-Shaft Deflections,” Machine Design, p. 78-82, June 27, 1974. 5. Hopkins, Bruce R., Design Analysis of Shafts and Beams, McGraw-Hill Book Company, New York, NY, 1970. 6. Spotts, Merhyle F., Terry E. Shoup and Lee E. Hornberger, Design of Machine Elements, 8th Edition, Pearson Prentice Hall, Upper Saddle River, NJ, 2003. 7. Collins, Jack A., Henry Busby and George Staab, Mechanical Design of Machine Elements and Machines, 2nd Edition, John Wiley & Sons, Hoboken, NJ, 2010. 8. Mott, Robert L., Machine Elements in Mechanical Design, 4th Edition, Pearson Prentice Hall, Upper Saddle River, NJ, 2004. 9. Juvinall, Robert C. and Kurt M. Marshek, Fundamentals of Machine Component Design, 4th Edition, John Wiley and Sons, Inc, Hoboken, NJ, 2006. 10. Budynas, Richard G. and J. Keith Nisbett, Shigley’s Mechanical Engineering Design, 8th Edition, McGraw-Hill, New York, NY, 2008. 11. Umasankar, G. and C.R. Mischke, “A simple Numerical Method for Determining the Sensitivity of Bending Deflections of Stepped Shafts to Dimensional Changes,” Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 107, p 141-146, 1985. Copyright ASEE Middle Atlantic Regional Conference, April 29-30, 2011, Farmingdale State College, SUNY

12. Odom, E.M. and Egelhoff, C.J., “Stepping Through Shaft Deflection Calculations,” Machine Design, pp. 58-60, June 10, 2010.
