Analysis and Validation of Kennedy Key


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Analysis and Validation of Kennedy Key

Aditya Natu

Production and Industrial Engineering Delhi Technological University Delhi, India

Abstract- This paper entails the design of Kennedy Key complying with the theory of machine design which is validated by Finite Element Analysis to provide a realistic solution to overcome the assumptions in the numerical solution. A comparative study is also conducted to observe the equivalent stress distribution, total deformation, shear stress, normal stress and factor of safety (fs) of the key based on variations in length and cross sectional area under same loading and constrain conditions. The study is done to analyse the key in dynamic conditions so as to ensure the key is designed to sustain the loads and avoid failure.

Keywords- FEA, fs, stress, Kennedy Key, Ansys

  1. INTRODUCTION

    A key is a machine element which is used to connect the transmission shaft to rotating machine elements like pulleys, gears, sprockets or flywheels. The primary function of the key is to transmit the torque from the shaft to the hub of the mating element and vice versa. The secondary function of the key is to prevent relative rotational motion between the shaft and the joined machine element. A recess or slot is machined either on the shaft or in the hub to accommodate the key is called keyway. The main drawback of a keyed joint is that the keyway results in stress concentration in the shaft and the part becomes weak. Keys are made of plain carbon steels in order to withstand the shear and compressive stresses resulting from transmission of torque. Many types of keys are available and there are a number of standards which specify the dimensions of the key. The selection of the type of key for a given application depends on factors like power to be transmitted, tightness of fit, stability of connection and cost [1].

    In this paper Kennedy Key is discussed and analysed in detail. The Kennedy key consists of two square keys. The hub is bored off the centre and the two keys force the hub and the shaft to a concentric position. Kennedy key is used for heavy duty applications. The analysis of the Kennedy key is based on two criteria which are failure due to shear stress and failure due to compressive stress.

    Since there are two keys, the torque transmitted by each key is one half of the total torque. The two equal and opposite forces are due to the transmitted torque. The forces are assumed to act tangential to the shaft diameter.

    Fig. 1. Cross-Section of Kennedy Key

  2. OBJECTIVE

    The study is conducted with the following objectives:

    • To validate numerical solutions through the well- developed finite element analysis solvers.

    • To study comparatively the effect of variations in length and cross sectional area of key on stress, factor of safety and deformation.

  3. CASE SPECIFICATIONS

    Simulation of key requires the designing of a key. The foundation of designing a key lay in its application. Hence, a general application has been considered for designing the required key [1].

    The data considered is shown in Table-1.

    TABLE 1. CASE SPECIFICATIONS

    Shaft Diameter

    40 mm

    Power

    35 kW

    RPM

    300

    Material

    Steel 45C8

    Cross-section of Key

    10 x 10 mm

    Factor of Safety

    3

  4. MATERIAL PROPERTIES

    The material selected for the Kennedy key is Steel 45C8. The mechanical properties and chemical composition of the material are given in Table-2 and Table-3 [2].

    TABLE 2. MECHANICAL PROPERTIES

    PROPERTY

    VALUE

    Density

    7.85 g/cm3

    Elastic Modulus

    210 GPa

    Poisons Ratio

    0.3

    Yield Strength

    380 MPa

    Ultimate Tensile Strength

    630 MPa

    TABLE 3. CHEMICAL COMPOSITION

    ELEMENT

    CONTENT

    Carbon

    0.4%

    Manganese

    0.6%

    Silicon

    0.15%

    Aluminium

    0.02%

  5. ANALYTICAL SOLUTION

    The key is designed by solving the considered application analytically complying with the theory of machine design. The key is analysed by its free body diagram. The forces acting on the Kennedy key are shown in Fig. 2. The shear stress and the compressive stress are given by =

  6. NUMERICAL SOLUTION

    The key is initially designed according to the specifications of application considered and using the analytical solution as well. The designing is done in SolidWorks which is a 3D CAD design software developed and marketed by Dassault Systèmes while the analysis is carried out in Ansys which is an engineering simulation software developed and marketed by Ansys, Inc. to obtain a numerical solution.

    and = 2 respectively [1].

    2

    A. Permissible compressive and shear stresses

    = = 380 = 126.67 /2

    3

    According to distortion energy theory of failure,

    = 0.577 = 0.577(380) = 219.26 /2

    = = 219.26 = 73.09 /2

    3

    Fig. 3. Kennedy Key fitted in keyway in a shaft

    1. Finite Element Analysis

      FEA is a method in which the model is discretized into small elements, properties of which are then evaluated using

    2. Torque transmitted by shaft

      [60 × 106 × ()] [60 × 106 × 35]

      general equations of motion and boundary conditions specified during a test. This involves solution of the equation:

      =

      =

      2

      2(300)

      [Reaction] = [Stiffness] [Displacement] + [Load]
    3. Key length

    = 1114084.6

    1. Boundary Conditions

      The boundary conditions in finite element analysis

      Using the formula of shear stress on the key, we get

      comprises of the loads and constraints applied over a region

      = =

      1114084.6

      = 26.95

      or part in which a set of differential equations needs to be

      solved.

      2 2(40)(10)(73.09)

      And using the formula of compressive stress, we get

      In the case of Kennedy key in application, shear and compressive loads are being applied while the movement of

      2 2(1114084.6)

      = =

      (40)(10)(126.67)

      = 31.10

      the key is constrained in particular directions due to the shaft and hub between which it is fitted. The loading diagram of the key is shown in Fig. 4.

      Thus, from the analytical solution obtained above we can

      estimate the length of the key to be approximately 30mm to get a factor of safety of 3.

      Fig. 2. Forces acting on Kennedy Key

      Fig. 4. Boundary Conditions applied on Key

    2. Mesh Generation

      This process involves the solid model of key being discretized into small elements and thus equations are solved on nodes resulting from meshing of elements. To obtain a more accurate solution, a fine mesh is generated in Ansys Mechanical which consists of elements of very small size as shown in Fig. 5. The mesh statistics are given in Table-4.

      Fig. 5. Mesh

      TABLE 4. MESH STATISTICS

      Nodes

      118712

      Elements

      27342

      Size Function

      Proximity and Curvature

      Relevance Center

      Fine

      Span Angle Center

      Fine

      Minimum Edge Length

      1.e-002 m

    3. Solution

    The mesh obtained is solved for the applied boundary conditions using finite element solvers to obtain an accurate solution in terms of equivalent stress, normal and shear stress and the factor of safety. The results are displayed below.

    Fig. 6. Equivalent Stress

    Fig. 7. Normal Stress

    Fig. 8. Shear Stress

    Fig. 9. Total Deformation

    Fig. 10. Factor of Safety

  7. EFFECT OF VARIATIONS IN LENGTH AND

    CROSS-SECTION

    To develop a better understanding of stress distribution, stress concentration and the resulting factor of safety, a comparative study is conducted by varying the length and cross section of the Kennedy Key and subsequently plotting graphs of equivalent stress, normal and shear stress, factor of safety versus the varying length and cross-section as shown in Fig. 11 and Fig. 12. Conclusions are based upon the interpretation of these graphs. Thus, from these graphs optimum length and cross-section of the Kennedy Key can be determined for a specific application for complying with the recommended minimum value of factor of safety to be kept while designing the key according to industrial standards.

    Fig. 11. Variations in stress with length for different cross-sections

    Fig. 12. Variations in factor of safety with length for different cross- sections

  8. CONCLUSIONS

    The Kennedy Key is designed based upon the analytical calculations and then analysed through the finite element analysis in Ansys Workbench with relevant boundary conditions. The numerical and analytical solutions differ by a considerable amount for the compressive stress, shear stress as well as the factor of safety. An increase in the magnitude of normal and shear stresses from the analytical results in an increase in equivalent stress and eventually a reduction in factor of safety from a value of 3 as per the key is designed to a value of 1.1 which is obtained through the FEA solution. This is due to the fact that in analytical solution the part is assumed to have uniform factor of safety throughout the body but it varies throughout the regions of part due to varying geometrical sizes and thickness. The region having lowest factor of safety is most likely to fail or fatigue during the part lifetime. Hence, the key should be designed such that the factor of safety in no region falls below 1.5 which is taken as reference to industrial standards. Hence, for the particular application considered, the dimensions of the Kennedy Key has to be changed to increase the minimum factor of safety from 1.1 to 1.5. This iterative process can be assisted by the comparative study in which variations in equivalent stress and factor of safety with length have been plotted on the graph for different cross-section areas. The graphs can be interpreted to select the optimum dimensions of cross-section and length of the Kennedy Key for obtaining required magnitudes of factor of safety required for the specific application.

  9. REFERENCES

  1. V.B. Bhandari, Design of Machine Elements, 3rd ed, Tata McGraw- Hill Education, 2010.

  2. IS: 1570-2. Schedules for Wrought Steels, Part-2: Carbon Steels, Indian Standards Institution, 1979

  3. ANSYS Workbench Users Guide, Release 12.1, ANSYS, Inc., 2009

  4. O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method. 5th ed, vol 1: The Basis, Butterworth-Heinemann, 2000.

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