Simulation of Stress Distribution in Leaf Spring Under Variable Parametric and Loading Conditions

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Simulation of Stress Distribution in Leaf Spring Under Variable Parametric and Loading Conditions

Jonnala Subba Reddy

Department of Mechanical Engineering Lakireddy Balireddy College of Engineering Mylavaram, India

M. Bhavani

Department of Mechanical Engineering Lakireddy Balireddy College of Engineering Mylavaram, India

J. Venkata Somi Reddy

Department of Mechanical Engineering Lakireddy Balireddy College of Engineering Mylavaram, India

Abstract Leaf springs are used in suspension systems. The past literature survey shows that leaf springs are designed as generalized force elements where the position, velocity and orientation of the axle mounting gives the reaction forces in the chassis attachment positions. The present work attempts to find the maximum pay load of the vehicle by performing static analysis using ANSYS software. The obtained results are compared analytically and found good agreement. The optimality conditions such as maximum bending stress and the corresponding pay load are designed with proper consideration of the factor of safety. To assess the behavior of the different parametric combinations of the leaf spring, the modal analysis using ANSYS software for the natural frequencies is carried out and the corresponding mode shapes are obtained. The natural frequencies are compared with the excitation frequencies at different speeds of the vehicle with the various widths of the road irregularity. These excitation frequencies are validated with analytical results.

Keywords Leaf Spring, ANSYS, pay load, natural frequency, camber

  1. INTRODUCTION

    A spring is an elastic body, whose function is to distort when loaded and to recovers its original shape when the load is removed. Semi-elliptic leaf springs are almost unanimously used for suspension in light and heavy industrial and commercial vehicles such as TATA- 407, LPT-1109, LPT- 1613, utility vehicles like Tata Sumo, Tata safary, Scorpio, Quallise etc. The spring consists of a number of leaves called blades as shown in Fig 1.. The blades are varying in length. The blades are usually given an initial curvature or cambered so that they will tend to straighten under the load. The leaf spring is based upon the theory of a beam of uniform strength. The lengthiest blade has eyes on its ends. This blade is called main or master leaf, the remaining blades are called graduated leafs. All the blades are bound together by means of steel straps. The front eye of the leaf spring is constrained in all the directions, where as rear eye is not constrained in X-direction. This rear eye is connected to the shackle. During loading the spring deflects and moves in the direction perpendicular to the load applied. The springs are initially cambered. More

    cambered leaf springs are having high stiffness, so that provides hard suspension. Use of longer springs gives a soft suspension, because when length increases the softness increases. Generally rear springs are kept longer than the front springs. Sometimes the leaf springs are provided with metallic or fabric covers to exclude dirt. The covers also serve to contain the lubricant used in between the spring leaves. In case of metal covers, the design has to be of telescopic type to accommodate the length of cover after the change of spring length.

    The suspension system consists of a spring and a damper. The energy of road shock causes the spring to oscillate. These oscillations are restricted to a reasonable level by the damper, which is more commonly called a shock absorber. When the rear wheel comes across a bump or pit on the road, it is subjected to vertical forces, tensile or compressive depending upon the nature of the road irregularity. These are absorbed by the elastic compression, shear, bending or twisting of the spring. The mode of spring resistance depends upon the type and material of the spring used.

    Fig 1. Leaf Spring and its parts

  2. LITERATURE REVIEW

Zliahu Zahavi et all. [1] discussed the behavior of structures of leaf springs under practical conditions. Practically, a leaf spring is subjected to millions of load cycles leading to fatigue

failure. They performed free vibration analysis which determines the frequencies and mode shapes of leaf spring.

A.strzat and T.Paszek [2] performed a three dimensional contact analysis of the car leaf spring. They considered static three dimensional contact problem of the leaf car spring. The solution was obtained by finite element method performed in ADINA 7.5 professional system. Numerical results were verified with experimental investigations. The static characteristics of the car spring was obtained for different models and compared with experimental investigations.

Fu-Cheng Wang [3] performed a detailed study on leaf springs. Classical network theory is applied to analyze the behavior of a leaf spring in active and passive suspensions. Typically, these situations involved in specifying a soft response from road disturbances.

Shahriar Tavakkoli, Farhang Aslani, and David S. Rohweder

[4] performed analytical prediction of leaf spring bushing loads using MSC/NASTRAN and MDI/ADAMS. Two models of leaf spring in MSC/NASTRAN and MDI/ADAMS were created to compare the bushing loads predicted by each model. Geometric non-linear capability of MSC/NASTRAN (SOL

106) was used to predict the bushing loads in MSC/NASTRAN model. The quasi-static simulation capability of MDI / ADAMS was used to predict the bushing loads in MDI/ADAMS model.

  1. Rajendran and S.Vijayarangan [5] performed a finite element analysis on a typical leaf spring of a passenger car. Finite element analysis has been carried out to determine natural frequencies and mode shapes of the leaf spring. A simple road surface model was considered.

    C.Madan Mohan Reddy, D.RavindraNaik, Dr M.Lakshmi Kantha Reddy [1] conducted study on analysis and testing of two wheeler suspension helical compression spring. The study on suspension system springs modelling, analysis and testing was carried out.

    The present work discusses the behavior of the different parametric combinations of the leaf spring, the modal analysis is carried out using ANSYS software.

    1. ANALYSIS OF LEAF SPRING

      The material selected for analysis is Manganese Silicon Steel and variables such as thickness, camber, span and no. of leaves are considered as variables. Bending stress is computed under different loading conditions.

      1. Material Properties of leaf spring

        Material = Manganese Silicon Steel Youngs Modulus E = 2.1E5 N/mm2 Density = 7.86E-6 kg/mm3 Poissons ratio = 0.3

        Yield stress = 1680 N/mm2

      2. Geometric Properties of leaf spring

        1. Variation of thickness

          The thickness of leaves changes from 7 mm to 10 mm with the interval of 1mm, the lengths, radius of curvatures, rotation angles are computed. The Sample readings for thickness 7 mm are shown in table 1.

          Thickness of leaves = 7 mm to 10 mm Other parameters are taken as: Camber = 80 mm

          Span = 1220 mm Number of leaves = 10

          Number of full length leaves nF = 2 Number of graduated length leaves nG = 8 Width = 70mm

          Ineffective length = 60 mm Eye diameter = 20 mm Bolt diameter = 10 mm

          Table1. Length of leaves when thickness is 7 mm

          Leaf number

          Full leaf Length (mm)

          Half leaf Length (mm)

          Radius of Curvature (mm)

          Half Rotation angle

          (degrees)

          1

          1240

          620

          2372.625

          p>14.972

          2

          1240

          620

          2379.625

          14.928

          3

          1108

          554

          2386.625

          13.299

          4

          978

          489

          2393.625

          11.708

          5

          846

          423

          2400.625

          10.096

          6

          716

          358

          2407.625

          8.519

          7

          584

          292

          2414.625

          6.929

          8

          454

          227

          2421.625

          5.371

          9

          322

          161

          2428.625

          3.798

          10

          190

          95

          2436.625

          2.235

        2. Variation of camber

          The camber varies from 80mm to 110 mm with the interval of 10mm, the lengths, radius of curvatures, rotation angles are computed. The Sample readings for Camber 80 mm are shown in table 2.

          Camber = 80 mm to 110 mm Span = 1220 mm

          Thickness of leaves = 7 mm Number of leaves = 10

          Number of full length leaves = 2 Number of graduated length leaves = 8 Width = 70mm

          Ineffective length = 60 mm Eye diameter = 20 mm Bolt diameter = 10 mm

          Table2. Length of leaves when camber 80 mm

          Leaf number

          Full leaf Length (mm)

          Half leaf

          Length (mm)

          Radius of Curvature (mm)

          Half Rotation angle (degrees)

          1

          1240

          620

          2372.625

          14.972

          2

          1240

          620

          2379.625

          14.928

          3

          1108

          554

          2386.625

          13.299

          4

          978

          489

          2493.625

          11.705

          5

          846

          423

          2400.625

          10.096

          6

          716

          358

          2407.625

          8.519

          7

          584

          292

          2414.625

          6.929

          8

          454

          227

          2421.625

          5.371

          9

          322

          161

          2428.625

          3.798

          10

          190

          95

          2436.625

          2.235

        3. Variation of span

          The span varies from 1120 mm to 1420 mm with the interval of 100 mm, the lengths, radius of curvatures, rotation angles are computed. The Sample readings for span of leaf spring 1120 mm are shown in table 3.

          Span of leaf spring = 1120 mm to 1420 mm Span = 1220 mm

          Thickness of leaves = 7 mm Number of leaves = 10

          Number of full length leaves = 2 Number of graduated length leaves = 8 Width = 70mm

          Ineffective length = 60 mm Eye diameter = 20 mm Bolt diameter = 10 mm

          Table 3. Length of leaves when Span 1120 mm

          Leaf number

          Full leaf Length

          (mm)

          Half leaf Length (mm)

          Radius of Curvature

          (mm)

          Half Rotation angle (degrees)

          1

          1140

          570

          2007

          16.272

          2

          1140

          570

          2014

          16.216

          3

          1020

          510

          2021

          14.459

          4

          900

          450

          2028

          12.714

          5

          780

          390

          2035

          10.981

          6

          660

          330

          2042

          9.259

          7

          540

          270

          2049

          7.55

          8

          420

          210

          2056

          5.852

          9

          300

          150

          2063

          4.166

          10

          180

          90

          2070

          2.489

        4. Variation of no. of leaves

          The number of leaves varies from 9 to 12 with the interval 1, the lengths, radius of curvatures, rotation angles are computed. The Sample readings for no. of leaves 9 are shown in table 4. number of leaves = 9 to 12

          Span = 1220 mm

          Thickness of leaves = 7 mm Number of leaves = 10

          Number of full length leaves = 2 Number of graduated length leaves = 8 Width = 70mm

          Ineffective length = 60 mm Eye diameter = 20 mm Bolt diameter = 10 mm

          Table 4. Length of leaves when Number of leaves 9

          Leaf number

          Full leaf

          Length (mm)

          Half leaf Length (mm)

          Radius of

          Curvature (mm)

          Half Rotation angle (degrees)

          1

          1240

          620

          1917.5

          18.526

          2

          1240

          620

          1924.5

          18.459

          3

          1092

          546

          1931.5

          16.196

          4

          946

          473

          1938.5

          13.980

          5

          798

          399

          1945.5

          11.751

          6

          650

          325

          1952.5

          9.537

          7

          502

          251

          1959.5

          7.339

          8

          356

          178

          1966.5

          5.186

          9

          208

          104

          1973.5

          3.019

      3. Bending Stress of leaf spring

        The bending stress of a leaf spring may be obtained from the formula:

        Static Bending stress = 6

        2 (2+3)

        Where W = Static Load in Newton L = Half span of leave spring in mm. b = width of the leaf spring in mm.

        t = thickness of the leaf spring in mm. nG = number of graduated leaves.

        nF = number of full length leaves.

        1. Geometric Properties of leaf spring

          The geometrical properties are mentioned below: Camber = 80 mm

          Span = 1220 mm Thickness = 7 mm

          Width = 70 mm

          Number of full length leaves nF = 2 Number of graduated leaves nG = 8 Total Number of leaves n = 10

        2. Material Properties of leaf spring

          The material properties of the leaf spring material are presented:

          Material = Manganese Silicon Steel Youngs Modulus E = 2.1E5 N/mm2 Density = 7.86E-6 kg/mm3 Poissons ratio = 0.3

          Yield stress = 1680 N/mm2

          The bending stress for different loading conditions from 1000 N to 15000 N is given in the table 5.

          Table 5 variation of Bending Stress with load

          Load (N)

          Bending Stress (N/mm2)

          1000

          48.502

          2000

          97.005

          3000

          145.075

          4000

          194.010

          5000

          242.512

          6000

          291.015

          7000

          339.517

          8000

          88.020

          9000

          436.52

          10000

          485.025

          11000

          533.527

          12000

          582.03

          13000

          630.532

          14000

          679.035

          15000

          727.537

      4. Modeling of Road Irregularity

        An automobile assumed as a single degree of freedom system traveling on a sine wave road having wavelength of L as shown in Fig.2. The contour of the road acts as a support excitation on the suspension system of an automobile .The period is related to by t=2/ and L is the distance traveled as the sine wave goes through one period.

        L = v.t = 2 v/.

        so, Excitation frequency = 2 v/L

        L = width of the road irregularity (WRI)

        V = speed of the vehicle

        The variation of road irregularities is highly random. However a range of values is assumed for the present analysis i.e. 1m to 5m for the width of the road irregularity (L).

        Fig 2. An automobile traveling on a sine wave road

        1. Variation of Exciting Frequency with Vehicle Speed The variation of exciting frequency with vehicle speed for assumed width of road irregularity. At low speeds the wheel of the vehicle passes over road irregularities and moves up and down to the same extent as the dimensions of the road irregularity. So, the frequency induced is less. If the speed

          increases and the change in the profile of the road irregularity is sudden, then the movement of the body and the rise of the axles which are attached to the leaf spring are opposed by the value of their own inertia. Hence, the frequency induced also increases. The exciting frequency is very high for the lower value of road irregularity width, because of sudden width. The following table 6 shows the variation of exciting frequency with vehicle speed.

          ten nodes having three degrees of freedom at each node: translations in the nodal x, y, and z directions. The element also has plasticity, creep, swelling, stress stiffening, large deflection, and large strain capabilities.

          Fig 5. element Solid 92: 3D- 10 Node Tetrahedral Structural solid with Rotations

          Table 6. Variation of Exciting Frequency with Vehicle Speed

          Speed

          Frequency

          Frequency

          Frequency

          Frequency

          Frequency

          (Kmph)

          Hz

          Hz

          Hz

          Hz

          Hz

          (at WRI

          (at WRI =

          (at WRI

          (at WRI

          (at WRI

          =1m)

          2m)

          =3m)

          =4m)

          =5m)

          20

          5.5500

          2.77

          1.8518

          1.3888

          1.11111

          40

          11.1111

          5.54

          3.7037

          2.7777

          2.22222

          60

          16.6666

          8.31

          5.5555

          4.1664

          3.33333

          80

          22.2222

          11.08

          7.4074

          5.5552

          4.44444

          100

          27.7777

          13.85

          9.2593

          6.9440

          5.55555

          120

          33.3333

          16.66

          11.1111

          8.3333

          6.66666

          140

          38.8888

          19.44

          12.9630

          9.7222

          7.77777

      5. Geometric Modeling of Leaf Spring

        The solid model of the leaf spring is modeled in CATIA software under part design module are shown in figures 3 and 4.

        Fig 3. Full model of Leaf Spring

        Fig 4. Front eye of the leaf spring

        1. Analysis of leaf spring by ANSYS

          The stress, strain analysis is carried out using ANSYS software under static loading conditions for a given specifications. The natural frequencies and mode shapes are computed performing the modal analysis to assess the behavior of the leaf spring with various parametric combinations. The element Solid 92: 3D- 10 Node Tetrahedral Structural solid with Rotations is considered for analysis. Solid92 has a quadratic displacement behavior and is well suited to model irregular meshes. The element is defined by

        2. Static Analysis

      Static analysis is to be performed to find the allowable stresses. The meshing and boundary conditions are given in ANSYS and are shown in Fig 6 and Fig7.

      Fig. 6. Meshing, Boundary conditions and loading of leaf spring

      Fig.7. Deformed and undeformed shape of leaf spring

      The Von-Mises stresses with different loading conditions are tabulated and presented in Table 7.

      Table 7 variation of Von-Mises stress with load

      Load (N)

      Von-Mises Stress (N/mm2)

      1000

      50.095

      2000

      101.856

      3000

      150.428

      4000

      200.712

      5000

      254.640

      6000

      305.150

      7000

      356.535

      8000

      407.469

      9000

      458.124

      10000

      509.928

      11000

      560.270

      12000

      611.204

      13000

      662.064

      14000

      713.071

      15000

      727.537

    2. RESULTS AND DISCUSSIONS

      1. Static Analysis

        Static analysis is performed to find the Von-Mises stress by using ANSYS software and these results are compared with bending stresses calculated in mathematical analysis at various loads and are tabulated in Table 8.

        Load (N)

        Von-Mises stress N/mm2

        Theoretical

        ANSYS

        1000

        48.502

        50.095

        2000

        97.005

        101.856

        3000

        145.075

        150.428

        4000

        194.010

        200.712

        5000

        242.512

        254.640

        6000

        291.015

        305.585

        7000

        339.517

        356.565

        8000

        388.020

        407.469

        9000

        436.520

        458.124

        10000

        485.025

        509.928

        11000

        533.527

        560.270

        12000

        582.030

        611.204

        13000

        630.532

        662.264

        14000

        679.035

        703.071

        15000

        727.537

        764.005

        Table 8 variation of Von-Mises stress with load comparison between theoretical and ANSYS

        1

        NODAL SOLUTION

        STEP=1 SUB =1 TIME=1

        SEQV (AVG) DMX =1.786

        SMN =.165967

        SMX =458.124

        MX

        ANSYS 10.0

        FZEB 25 2010

        Y 13:24:56

        X

        .165967 101.934

        51.05 152.819

        203.703

        254.587

        305.471

        356.356

        407.24

        458.124

        Fig. 9 Vn-mises Stress contour plot of Front eye of leaf spring

        1

        NODAL SOLUTION

        STEP=1 SUB =1 TIME=1

        SEQV (AVG) DMX =1.786

        SMN =.165967

        SMX =458.124

        MX

        FEB 25 2010

        13:31:04

        .165967

        51.05

        101.934

        152.819

        203.703

        254.587

        305.471

        356.356

        407.24

        458.124

        From Theoretical and ANSYS the allowable design stress is found between the corresponding loads 8000 to 10000 N, the near corresponding safe loads are given in Table 9.

        Table 9. Von-Mises stress under different loading conditions Comparison between Theoretical and ANSYS

        Load (N)

        Von-Mises stress N/mm2

        Theoretical

        ANSYS

        9500

        460.770

        481.916

        9700

        470.470

        494.565

        9900

        480.174

        504.243

        Fig 8 Variation of Von-mises stress with load

        Fig. 10. Von-mises Stress contour plot of Rear eye of leaf spring

      2. Modal Analysis

        From the leaf spring specification width is fixed and other parameters namely thickness, camber, span and number of leaves are taken for parametric variation. First ten modes are considered for analysis. Variations of natural frequencies with spring parameters are studied.

        1. Variation of natural frequency with span

          The variation of natural frequency is computed with different span and tabulated in Table 10 and variations of arc radius are shown in Table 11.

          Table 10. Variation of natural frequency with Span

          Span in mm

          Frequency Hz at

          1120

          1220

          1320

          1420

          Mode 1

          2.464

          2.362

          2.038

          1.741

          Mode 2

          3.700

          3.624

          3.053

          2.603

          Mode 3

          7.168

          6.874

          5.916

          5.042

          Mode 4

          14.409

          14.240

          12.192

          10.416

          Mode 5

          15.553

          15.412

          14.862

          12.995

          Mode 6

          18.224

          17.652

          15.246

          13.413

          Mode 7

          26.208

          25.596

          23.181

          20.292

          Mode 8

          31.377

          31.126

          26.798

          22.968

          Mode 9

          31.690

          31.149

          27.889

          24.651

          Mode 10

          46.567

          45.532

          39.470

          33.549

          Table 11. Variation of Arc radius with Span

          Span in mm

          Arc radius in mm

          1120

          2000.000

          1220

          2365.600

          1320

          2762.500

          1420

          3190.625

          Fig. 11. Frequency vs span for different modes

          It is observed from the Fig 11. that, when span increases the spring becomes soft and hence the natural frequency decreases. Every three modes are in one set of range. There is a considerable gap between mode3 to mode4, mode6 to mode 7 and mode 9 to mode10. It is observed from the Fig. 8.4 that the decrease of frequency value with the increase of span is very high for mode10 compared to remaining modes.

        2. Variation of natural frequency with camber

          The variation of natural frequency is computed with different span and tabulated in Table 12 and variations of arc radius are shown in Table 13.

          Table 12. Variation of natural frequency with Camber

          Camber in mm

          Frequency Hz at

          80

          90

          100

          110

          Mode 1

          2.362

          2.344

          2.344

          2.414

          Mode 2

          3.624

          3.527

          3.446

          3.571

          Mode 3

          6.874

          6.791

          6.760

          7.063

          Mode 4

          14.240

          14.107

          13.988

          14.361

          Mode 5

          15.412

          16.006

          16.642

          15.395

          Mode 6

          17.652

          17.506

          17.458

          18.122

          Mode 7

          25.596

          25.625

          25.741

          27.007

          Mode 8

          31.126

          30.895

          30.702

          30.557

          Mode 9

          31.149

          31.215

          31.391

          31.667

          Mode 10

          45.532

          45.241

          45.109

          45.915

          Table 13. Variation of Arc radius with Camber

          Camber in mm

          Arc radius in mm

          80

          2165.60

          90

          2112.20

          100

          1910.50

          110

          1746.36

          Fig. 12. Frequency vs Camber for different modes

          It is noticed from the Fig 12 that it shows the variation of natural frequency with camber. When camber increases the spring becomes stiff and hence the natural frequency increases. Every three modes are almost in one set of range. There is a considerable gap between mode 3 to mode4, mode6 to mode 7 and mode 9 to mode 10. It is observed from the Fig.12 that the increase of frequency value with the increase of camber is very high for mode 10 compared to remaining modes.

        3. Variation of natural frequency with thickness

          The variation of natural frequency is computed with different span and tabulated in Table 14.

          Table 14. Variation of natural frequency with thickness

          Thickness in mm

          Frequency Hz at

          7

          8

          9

          10

          Mode 1

          2.362

          2.744

          2.945

          3.270

          Mode 2

          3.624

          3.697

          3.650

          3.702

          Mode 3

          6.874

          7.950

          8.502

          9.403

          Mode 4

          14.240

          14.293

          14.092

          14.116

          Mode 5

          15.412

          15.710

          15.656

          15.736

          Mode 6

          17.652

          20.244

          21.716

          23.691

          Mode 7

          25.596

          26.669

          26.678

          27.275

          Mode 8

          31.126

          31.233

          30.955

          31.004

          Mode 9

          31.149

          34.630

          37.249

          40.714

          Mode 10

          45.532

          51.608

          51.091

          51.025

          Fig. 13. Frequency vs Camber for different modes

          <>It is observed from the Fig 13. that it shows the variation of natural frequency with thickness of the spring. When thickness increases the natural frequency also increases. Its natural frequency increases like variation of natural frequency with camber, but with thickness the natural frequency increasing rate is lesser than that of variation of natural frequency with camber. Every three modes are almost in one set of range. There is a considerable gap between mode3 to mode4, mode6 to mode 7 and mode 9 to mode 10.It is observed from Fig. 13 that the increase of frequency value with the increase of thickness is very high for mode9 and mode10 compared to remaining modes.

        4. Variation of natural frequency with number of leaves The variation of natural frequency is computed with different span and tabulated in Table 15.

      Table 15 variation of natural frequency with number of leaves

      ANSYS 10.0

      1

      FEB 25 2010

      17:53:51

      NODAL SOLUTION STEP=1

      SUB =1 FREQ=2.362 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      Z

      XMX Y

      MN

      DMX =.210091

      SMX =.210091

      XV =.803253

      YV =.246554

      ZV =.542214

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-97.519 Z-BUFFER

      0

      .023343

      .046687

      .07003

      .163404

      .186748

      .210091

      No. of leaves

      Frequency Hz at

      9

      10

      11

      12

      Mode 1

      2.240

      2.362

      2.527

      2.559

      Mode 2

      3.571

      3.624

      3.440

      3.285

      Mode 3

      6.506

      6.874

      7.262

      7.301

      Mode 4

      14.202

      14.240

      14.006

      13.720

      Mode 5

      16.473

      15.412

      16.410

      15.887

      Mode 6

      17.044

      17.652

      18.873

      19.360

      Mode 7

      25.060

      25.596

      26.346

      26.188

      Mode 8

      30.567

      31.126

      30.789

      30.446

      Mode 9

      30.906

      31.149

      33.002

      33.751

      Mode 10

      42.669

      45.532

      48.544

      50.096

      Fig.15 Mode 1 (Camber 80 mm, Span 1220 mm)

      1

      ANSYS 10.0

      FEB 25 2010

      17:55:21

      NODAL SOLUTION STEP=1

      SUB =3 FREQ=6.874 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      Z

      X

      DMX =.284142

      Y

      SMX =.284142

      MN

      MX

      XV =.944828

      YV =.285095

      ZV =-.161307

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-87.384 Z-BUFFER

      0

      .031571

      .063143

      .094714

      .220999

      .252571

      .284142

      Fig. 16 Mode 3 (Camber 80 mm, Span 1220 mm)

      Fig. 14. Frequency vs No. of leaves for different modes

      Figure 14 shows the variation of natural frequency with number of leaves of the spring. Even though the number of leaves increases there is no considerable increase in natural frequency, it is almost constant. It is observed from the Fig.14 every three modes are in gradual increment, there is considerable increase in natural frequency from mode3 to mode4, there is much increase in natural frequency from mode6 to mode7 and there is very much in increase in natural frequency from mode 9 to mode 10.

      The mode shapes for modes 1, 3 & 10 and for different parameters like Camber, Span, Thickness of leaves and number of leaves are presented in the following Figures 15 to 23.

      ANSYS 10.0

      1

      MN

      Z

      X Y

      MX

      FEB 25 2010

      17:57:38

      NODAL SOLUTION STEP=1

      SUB =10 FREQ=45.532 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.459032

      SMX =.459032

      XV =.942515

      YV =-.32687

      ZV =-.069442

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-102.976 Z-BUFFER

      0

      .051004

      .102007

      .153011

      .357025

      .408029

      .459032

      Fig. 17. Mode 10 (Camber 80 mm, Span 1220 mm)

      1

      MN

      Z XY

      MX

      Fig. 18. Mode 1 ( Thickness 8 mm )

      1

      MN

      Z XY

      MX

      Fig. 19. Mode 3 ( Thickness 8 mm )

      1

      MN

      Z XY

      MX

      Fig. 20. Mode 7 (Thickness 8 mm)

      FEB 25 2010

      19:10:02

      NODAL SOLUTION STEP=1

      SUB =1 FREQ=2.744 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.196781

      SMX =.196781

      XV =.79749

      YV =-.582004

      ZV =-.159

      *DIST=701.825

      *XF =.086462

      *YF =15.074

      *ZF =-43.349 A-ZS=-106.601 Z-BUFFER

      0

      .021865

      .043729

      .065594

      .131187

      .153052

      .174916

      .196781

      FEB 25 2010

      19:10:53

      NODAL SOLUTION STEP=1

      SUB =3 FREQ=7.951 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.265981

      SMX =.265981

      XV =.79749

      YV =-.582004

      ZV =-.159

      *DIST=701.825

      *XF =.086462

      *YF =15.074

      *ZF =-43.349 A-ZS=-106.601 Z-BUFFER

      0

      .029553

      .059107

      .08866

      .177321

      .206874

      .236427

      .265981

      FEB 25 2010

      19:11:24

      NODAL SOLUTION STEP=1

      SUB =7 FREQ=26.669 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.242804

      SMX =.242804

      XV =.79749

      YV =-.582004

      ZV =-.159

      *DIST=701.825

      *XF =.086462

      *YF =15.074

      *ZF =-43.349 A-ZS=-106.601 Z-BUFFER

      0

      .026978

      .053956

      .080935

      .161869

      .188847

      .215825

      .242804

      1

      MN Z

      XMX Y

      Fig. 21. Mode 1 ( umber of leaves 10)

      1

      MN

      Z

      X

      Y

      MX

      Fig. 22. Mode 3 (Number of leaves 10)

      1

      MN

      Z

      X Y

      MX

      Fig. 23. Mode 10 (Number of leaves 10)

      ANSYS 10.0

      FEB 25 2010

      17:53:51

      NODAL SOLUTION STEP=1

      SUB =1 FREQ=2.362 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.210091

      SMX =.210091

      XV =.803253

      YV =.246554

      ZV =.542214

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-97.519 Z-BUFFER

      0

      .023343

      .046687

      .07003

      .163404

      .186748

      .210091

      ANSYS 10.0

      FEB 25 2010

      17:55:21

      NODAL SOLUTION STEP=1

      SUB =3 FREQ=6.874 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.284142

      SMX =.284142

      XV =.944828

      YV =.285095

      ZV =-.161307

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-87.384 Z-BUFFER

      0

      .031571

      .063143

      .094714

      .220999

      .252571

      .284142

      ANSYS 10.0

      FEB 25 2010

      17:57:38

      NODAL SOLUTION STEP=1

      SUB =10 FREQ=45.532 USUM (AVG) RSYS=0

      PowerGraphics EFACET=1 AVRES=Mat

      DMX =.459032

      SMX =.459032

      XV =.942515

      YV =-.32687

      ZV =-.069442

      *DIST=700.36

      *XF =-.144703

      *YF =14.731

      *ZF =-38.884 A-ZS=-102.976 Z-BUFFER

      0

      .051004

      .102007

      .153011

      .357025

      .408029

      .459032

      The following table shows the variation of exciting frequency with vehicle speed.

      Table 16 variation of natural frequency with vehicle speed

      Spe ed in Km ph

      Freque ncy at WRI =

      1 m

      Freque ncy at WRI =

      2 m

      Freque ncy at WRI =

      3 m

      Freque ncy at WRI =

      4 m

      Freque ncy at WRI =

      5 m

      20

      5.5500

      2.77

      1.8518

      1.3888

      1.11111

      40

      11.1111

      5.54

      3.7037

      2.7777

      2.22222

      60

      16.6666

      8.31

      5.5555

      4.1664

      3.33333

      80

      22.2222

      11.08

      7.4074

      5.5552

      4.44444

      100

      27.7777

      13.85

      9.2593

      6.9440

      5.55555

      120

      33.3333

      16.66

      11.1111

      8.3333

      6.66666

      140

      38.8890

      19.44

      12.9630

      9.7222

      7.77777

      Fig 24. Variation of Excitation frequency with vehicle speed (wri = width of road irregularity in meters)

      The variation of exciting frequency is computed with vehicle speed as parameter under dynamic conditions, for assumed width of road Irregularity. At low speeds the wheel of the vehicle passes over road irregularities and moves up and down to the same extent as the dimensions of the road irregularity.

      So, the frequency induced is less. If the speed increases and the change in the profile of the road irregularity are sudden, then the movement of the body and the rise of the axles which are attached to the leaf spring are opposed by the value of their own inertia. Hence, the frequency induced also increases. The exciting frequency is very high for the lower value of road irregularity width, because of sudden width.

      It is noted that the some of the excitation frequencies are very close to natural frequencies of the leaf spring, but they are not exactly matched, hence no resonance will take place.

    3. CONCLUSIONS AND FUTURE SCOPE OF WORK

The leaf spring is considered for analysis under static and dynamic loading conditions. The spring is modeled using CATIA software and analysis is carried out in ANSYS for different loading conditions. The following conclusions are made.

  • The Leaf spring has been modeled using solid tetrahedron 10 node element.

  • By performing static analysis it is concluded that the maximum safe load is 9900 N for the given specification of the leaf spring from the analysis.

  • In model analysis, the leaf spring width is kept as constant and variation of natural frequency with leaf thickness, span, camber and numbers of leafs are studied.

  • It is observed from the present work that the natural frequency increases with the increase of thickness of leaves as well as camber, and decreases with decrease of thickness of leaves as well as camber.

  • The natural frequency decreases with increase of span, and increases with decrease of span.

  • The natural frequency almost constant with number of leaves.

  • The natural frequencies of various parametric combinations are compared with the excitation frequency for different road irregularities.

  • This study concludes that it is advisable to operate the vehicle such that its excitation frequency does not match the above determined natural frequencies i.e. the excitation frequency should fall between any two natural frequencies of the leaf spring.

  • An extended study of this nature can be used along with appropriate sensors and microprocessors to enable achievement of optimum speed at which one can drive the vehicle with maximum comfort.

  • In this work no contact elements are considered only nodal coupling has taken, instead of nodal coupling contact elements can be considered.

  • Also, instead of steel leaf springs the composite material can be considered to optimize the cost and weight of the vehicle through experimental setup.

REFERENCES

  1. G Zliahu Zahavi(1991) The finite element method in machine design, A soloman press book Prentice Hall Englewood cliffs.

  2. A Skrtz, T.Paszek,(1992) Three dimensional contact analysis of the car leaf spring, Numerical methods in continuum mechanics 2003, Zilina, Skrtz republic.

  3. In-Cheng Wang,(1999) Design and Synthesis of Active and Passive vehicle Suspensions.

  4. Shahriar Tavakkoli, Farhang, Daved S.Lohweder,(2001) Practical prediction of leaf spring , Loads using MSC/NASTRAN & MDI/ADAMS.

  5. I.Rajendran & S. Vijayarangan(2002) A parametric study on free vibration of leaf spring.

  6. Lupkin P. Gasparyants G. & Rodionov V. (1989) Automobile Chassis- Design and Calculations, MIR Publishers, Moscow.

  7. Manual on Design & Application of Leaf Springs (1982),SAE HS-788. Canara Springs Catalogue-1992.Workshop Manual of. TATA 407 ( LCV )

  8. Jiang Xu, Wenjin, Liu, Wear characteristic of in situ synthetic TiB2 particulate reinforced Al matrix composite formed by laser cladding, Wear 260 (2006) 486492.

  9. Christophe, G.E. Mangin, J.A. Isaacs, J.P. Clark, MMCs for automotive engine applications, JOM (1996) 4951.

  10. D. Hull and T. W. Clyne, An Introduction to Composite Materials, Second edition, Cambridge Solid State Science Series.

  11. AjitabhPateriya, Mudassir Khan, Structural and thermal analysis of spring loaded safety valve using FEM, International journal of mechanical engineering and robotics research, 4(1), 430-434. 3.

  12. C. Madan Mohan Reddy, D. RavindraNaik, Dr M. Lakshmi Kantha, Study on analysis and testing of two wheeler suspension helical spring.

  13. Pozhilarasu V. and T ParameshwaranPillai, Performance analysis of steel leaf spring with composite leaf spring and fabrication of composite leaf spring, International journal of engineering research and Science & Technology, 2(3), 102-109. 4.

  14. Aishwarya A.L., A. Eswarakumar, V.Balakrishna Murthy, Free vibration analysis of composite leaf springs, International journal of research in mechanical engineering &technology, 4(1), 95-97. 5.

  15. T. N. V. Ashok Kumar, E. VenkateswaraRao, S. V. Gopal Krishna, Design and Material Optimization of Heavy Vehicle Leaf Spring, International Journal of Research in Mechanical Engineering & Technology, 4(1), 80-88. 6.

  16. K.A. SaiAnuraag and BitraguntaVenkataSivaram, Comparison of Static, Dynamic & Shock Analysis for Two & Five Layered Composite Leaf Spring, Journal of Engineering Research and Applications, 2(5), 692- 697.

  17. E. Mahdi a, O.M.S. Alkoles a, A.M.S. Hamouda b, B.B. Sahari b, R. Yonus c, G. Goudah, Light composite elliptic springs for vehicle suspension, Composite Structures, 75, 2428.

  18. PankajSaini, AshishGoel, Dushyant Kumar, Design and analysis of composite leaf spring for light vehicles, International Journal of Innovative Research in Science, Engineering and Technology 2(5), May 2013.

  19. ManasPatnaik, NarendraYadav, RiteshDewangan, Study of a Parabolic Leaf Spring by Finite Element Method & Design of Experiments, International Journal of Modern Engineering Research, 2(4), 1920-1922.

  20. H.A.AI-Qureshi, Automobile leaf spring from composite materials,Journal of materials processing technology, 118(2001).

  21. Ashish V. Amrute, Edward Nikhil karlus, R.K.Rathore, Design and assessment of multi leaf spring, International journl of research in aeronautical and mechanical engineering.

  22. Rupesh N Kalwaghe, K. R. Sontakke, Design and Anslysis of composite leaf spring by using FEA and ANSYS, International journal of scientific engineering and research, 3(5), 74-77.

  23. Mahmood M. Shokrieh , DavoodRezaei, Analysis and optimization of a composite leaf spring, Composite Structures, 60 (2003) 317325.

  24. U. S. Ramakant&K. Sowjanya, Design and analysis of automotive multi leaf springs using composite material, IJMPERD 2249-6890Vol. 3,Issue 1,pp.155-162,March 2013

  25. Mr. V. K. Aher *, Mr. P. M. Sonawane , Static And Fatigue Analysis Of Multi Leaf Spring Used In The Suspension System Of LCV, (IJERA) 2248-9622Vol. 2, Issue 4,pp.1786-1791,July-August 2012

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