Contact Stress Distribution of a Pear Cam Profile with Roller Follower Mechanism

The problem of this paper is the high contact stress at the point of contact between the cam and the follower. A pear cam and roller follower mechanism were studied and analyzed for different position of the follower and different contact compression load. The objective of this paper is to study the effect of contact compression load on the contact stress distribution of the cam profile at the point of contact. Four different positions of the follower with the cam was considered (0°, 90°, 180°, and 270°). The theory of circular plate was applied to derive the analytic solution of the contact stress. The numerical simulation had been done using ANSYS Ver. 19.2 package to determine the contact stress, while SolidWorks software was used to investigate follower displacement, velocity, and acceleration. Four distinct values of the compression contact load, such as 3.121 N, 6.242 N, 9.364 N, and 12.485 N, were used in the numerical simulation. In the experiment setup, a photo-elastic technique was carried out in the field of polarized light to exhibit the stress distribution on the cam specimen. The annealed PSM-4 backalate material was used in the experiment setup. The experimental value of contact stress was checked and verified analytically and numerically at the point of contact. The innovation in this paper the use of spring-damper system which reduce the value of contact stress at the point of contact. The contact stress was maximum 2.136 MPa when the follower located at 270° with the cam, while the contact stress was minimum 1.802 MPa when the follower located at 180° at compression load 12.485 N.


Introduction
Cam-follower system is connected to a bionic robotic through a linkage mechanisms to keep the robot walking on a horizontal territory. The contact stress between the cam and the follower is crucial especially at high speed. In this paper, spring-damper system was used at the end of the follower stem to retain the contact stress as much as possible. Pear cam with roller follower mechanism is extensively used in paper processing machines and automatic assembly lines. Sundar et al. [1] used Hertzian contact theory to calculate the contact stiffness at the point of contact. They analyzed numerically the coefficient of restitution. The Hertzian laws were used with a damping term to accommodate the energy loss during the impact [2]. The stiffness model for the active and dwell periods of the globoidal fluctuation due to the contact. They investigated cam was used by Kuang et al. to characterize the mesh stiffness the correlation between the dynamic response of the globoidal cam system and the motor driving speed numerically and experimentally [3]. A cam profile with concave-convex surface of contact had been selected between the cam and follower to suppress the high contact forces at high speeds. Moreover, a hertz contact theory of stress analysis is important to improve the contact fatigue life cycles of cam-follower system. Hua et al. found that the smooth surface of cam profile provides a (53%) reduction in maximum subsurface von Mises stress [4]. Contact stress is reduced to (41%) experimentally by taking into consideration the use of a constant velocity in a dwell stroke [5]. Hsu et al. [6] used a cam mechanism with a translating follower of a dual concave faces to reduce the contact stress over the constant-breadth and the constant-diameter of the cam. Yousuf and Hadi Chin. J. Mech. Eng. (2021) 34:24 Fabien et al. [7] selected a cam with dwell-rise-dwell profile to reduce the peak of contact stress over a range of speeds. Angeles et al. [8] calculated the contact pressure into a series of a harmonic waves of a local deformation and sub-surface stress fields. Optimal-control-theory was used by Chew et al. to optimize the contact stress and pressure angle of D-R-D and D-R-R-D cam profiles numerically [9,10].
The problem of this paper is the high contact stress at the point of contact between the cam and the follower. The contribution of this research is the use of springdamper system (one degree of freedom system) at the end of the follower stem. The innovation in this paper the use of spring-damper system which reduce the value of contact stress at the point of contact. The aim of this paper is to study the effect of contact compression load on the contact stress distribution of a pear cam profile with roller follower mechanism.

Analytic Solution of Dynamic Response
The follower with simple harmonic motion was assumed since it is widely used and simple to design. Cam-follower mechanism was treated as a one-degree-of-freedom system as indicated in Figure 1 [11].
The system with a mass (m = 0.2759 kg), spring stiffness (k = 7 N/mm), and viscus damper coefficient (c = 0.875 N·s/mm) [12] was considered in the solution of one degree of freedom equation. The equation of motion of cam-follower mechanism is as below: where: After simplification the equation of motion will be: The homogeneous solution of Eq. (2) is: The particular solution of Eq. (2) is: where C 1 , C 2 , C 3 and C 4 are constants. The first and second derivatives of Eq. (4) are: Substitute x P ,ẋ P ,ẍ P into Eq. (2) to obtain: where The complimentary solution is as below: Substitute Eq. (3) and Eq. (4) into Eq. (5) to obtain: The constants C 1 and C 2 are calculated from the initial boundary conditions as illustrated below: (2) x + 2βẋ + ω 2 x = F sin(�t). (3) (4) x P = C 3 sin(�t) + C 4 cos(�t), 2β� ω 2 cos(�t).  After applying the boundary conditions on Eq. (6), the constants C 1 and C 2 will be: The general solution of follower response is as below:

Numerical Simulation of Contact Model of Basic Kinematic Relationships
SolidWorks software was used to draw a disc cam, roller follower, and the two guides to investigate follower linear displacement, velocity, and acceleration at the point of contact [13]. Before the cam start spinning, a spring with elastic constants such as spring constant (k 1 = 38.0611 N/ mm), length (l = 28 mm}), spring index (C 1 = 7), preload extension (Δ = 37 mm), coil diameter (d = 0.8 mm) and outside diameter (OD = 6.4 mm) was selected. In this paper, the spring force was expressed as an external force in SolidWorks software to keep the cam and follower in permanent contact. The spring force was applied at the point in which it has the coordinates (x: 0, y: 161.76 mm, z: 0) between the follower body and the installation table.
The contact expression force function was as below [14]: The spring and damper were applied at the end point of the follower stem in which it has the coordinates (x: 0, y: 227.79 mm, z: 0). The contact model of a pear cam and roller follower with its dimensions is illustrated in Figure 2.
The area of contact can be assumed to be semi-ellipsoid in SolidWorks software with a rectangular contact zone of a half-width (b) and length (L) (contact between two parallel cylinders), as below [15]: 2β� ω 2 cos(�t).
The maximum depth (z 1 ) underneath the surface of contact in terms of damping ratio was occurred when (z 1 /b = 0.786) as illustrated in the following equation [15,16]: The penetration distance (δ) is related to the field surface displacement when (ν 1 = ν 2 = ν) and ( E 1 = E 2 ) which characterized by the following equation [17]: where The impact between the follower and the cam had been occurred due to the impulse and momentum theory of the follower velocity: Hertzian contact theory predicts the value of the exponent (n) based on the flattening law of the shape of the first asperity. The exponent value (n) is ranging from 1.5 for spherical punch to 2 for wedge shaped punch [15,18].
The contact parameters, such as the contact force ( P ), contact bodies stiffness ( K b ), damping ratio ( ζ ), penetration (δ), sliding contact velocity ( V C ), and exponent (n), were entered as an inputs in SolidWorks software to calculate follower linear displacement.

Cam Profile Layout
The points of cam profile were calculated by rotating the radius of curvature of the roller follower in an opposite direction to the rotation of the cam [19]. The parameters (ρ(θ), S(ψ), ψ and θ) were used in the equation of cam profile as follows [20]:

Where
The points of cam profile were calculated using Eq. (15) and Eq. (16): Figure 3 shows the verification design of analytic and numerical cam profile. The numerical cam profile was drawn using SolidWorks software. The analytic cam profile was designed by applying Eq. (15) and Eq. (16) at the point of contact between the cam and the follower.
There was no error at the base circle of the cam because the contact between the cam and the follower when (a 1 = 85 • ) . The analytic solution of Eq. (15) and (13) Eq. (16) gives the exact value of Cartesian coordinates of the cam profile at the point of contact with no error. The tangent command in SolidWorks software was used between flank No. (1) and the nose or between flank No.
(2) and the nose which gives an error to the Cartesian coordinate during the drawing. The error between the analytic solution and SolidWorks software is around (8.823%).

Follower Motion Verification
A pear cam with return-rise-dwell-return-rise-dwellreturn profile was proposed. The cam starts spinning with a return stroke for ( 20 • ) followed by a rise stroke with ( 80 • ) and a dwell motion for the next ( 8 • ). Return stroke for the next ( 67 • ) of the cam rotation had been occurred accompanied by a rise stroke with ( 101 • ) and a dwell motion for the next ( 7 • ) followed by return stroke for the next ( 77 • ) of the cam cycle. Follower displacement was determined using Eq. (7), while the numerical simulation of follower displacement was done using SolidWorks software. Figure 4 indicates the verification of follower displacement.

Numerical Simulation of Contact Stress
The numerical simulation of bending deflection was done by using ANSYS Ver. 19.2 package [21]. Linear elastic isotropic model was used to calculate the bending deflection of the cam profile. Element (PLANE 42) was assigned to create the mesh generation of finite element analysis. This element was defined by four nodes which is having two degrees of freedom at each node: (translations in the nodal x, y directions). The translation degrees of freedom were assumed to be zero (clamped boundary conditions) at the hub radius when ( r = r 1 ) of the motor shaft. In contact analysis, element (TARGE 170) and element (CONTA 174) had been used to create the contact mesh generation between the target and the contact surfaces. The disc cam represented the source, while the roller follower was the target. In this simulation, two load steps were used. In the first load step, a small displacement was imposed on the target surface because of the small value of the radius of curvature. In the second load step, the small imposed displacement was deleted in which the force was applied at the point of contact. Four different values of compression loads, such as 3.121 N, 6.242 N, 9.364 N, and 12.485 N, were considered in the simulation. The follower was located at 0°, 90°, 180°, and 270° with the cam.

Contact Load Calculation
Both cam and roller follower were assumed to be parallel cylinders with the length of (L = 10 mm) in the calculation of contact load. A convex surface of the cam profile was considered to have positive radius of curvature, while the concave surface was negative [22]. In this paper, the contact pressure showed that the intensity of the pressure between the two contacting surfaces could be represented by the elliptical or semi-ellipsoid regions as indicated in Figure 10, [23]. Therefore, this small region exhibits a very high contact stress specifically during the active region of the cam, when the most significant loading of the followers occurs [5].
The maximum contact load is given by the volume of a small elliptical region (contact between two curved surfaces) as below [24]: The values of (m 1 ) and (n 1 ) were taken from Table 1 based on the pressure angle between the cam and the follower [24]:

Experiment Setup of Photo-Elastic Technique
The experimental setup was carried out using a photoelastic technique [25,26]. Four different positions of the follower with the cam were considered (0°, 90°, 180°, and 270°) by taking into consideration four distinct value of contact compression load, such as 3.121 N, 6.242 N, 9.364 N, and 12.485 N. A photo-elastic stress analysis is one of the most powerful techniques in which it gives an overall picture for the stress field quickly. Photo-elastic approach deals with the systems which is having complicated geometry, such as (pear cam), complicated loading conditions or both. Photo-elastic rig is constructed from a plane polariscope with a light source, a polarizer, the specimen, and an analyzer that is always crossed with respect to the polarizer, as indicated in Figure 11. Plane polariscope with a dark field was used to show the regions of stress intensification.
In this paper, the zero degree direction of polarization was assumed to be vertical in which it rotates the polarizer and analyzer together in order to determine the principal stress directions. Pear cam and roller follower were made from an annealed plastic of PSM-4 backalate material with modulus of elasticity (4 MPa.) and Poisson's ratio (0.5). Machining process was used during the manufacturing of the cam and the follower in which a

Figure 10
Contact load distribution between two curved bodies [24]  smooth surfaces were created and left no residual stress on the model. The PSM-4 specimen was placed in the trough to remove the cracks caused by machining and after machining, the specimen was placed in the polariscope to ensure that there is no residual stress occurred.
The follower axis passed longitudinally through the axis of pear cam when the follower located at 0° and 180° degrees and transversely through the axis of pear cam when the follower located at 90° and 270°. In this paper, photo-elasticity technique had been applied over two stages. In the first stage, a material calibration was performed using four-point bending test in order to calculate the stress fringe constant. In the second stage, a sequence of colored brands was implemented based on a birefringence phenomenon of the pear cam geometry. Four-point bending test was used effectively to extract the value of stress fringe constant. The straight beam was supported by two ends, while a transverse downward loading had been applied at the two points as illustrated in Figure 12a. A sample of straight beam of four-point bending test was manufactured using ASTM (D6272) dimensions, as indicated in Figure 12b. The value of stress fringe constant (f σ ) can be determined experimentally by inducing a difference between σ 1 and σ 2 in a model that is made from PSM-4 material, as in below: By substituting Eq. (18) into Eq. (17) and after simplification we can obtain the value of quantity (Z), as in below: where Hence, ( f σ ) can be determined by plotting the quantity (Z) versus the fringe order (N), as shown in Figure 13 in which the stress fringe constant represents the slope.
From Figure 13, it can be found that the value of stress fringe constant is (f σ = 178.323 N m. fringe ) . The mass with the value (m 1 = 1.272 kg) was used in the experiment  since the fringe order (N) was more obvious and clear. The photo-elasticity experiment of four-point bending test is shown in Figure 14. Figures 15, 16, 17, 18 reflect the stress distribution for different positions of the follower (0°, 90°, 180°, and 270°) with the cam using photo-elastic technique.
In this paper, photo-elastic technique represents the three dimensional of experimental stress analysis in two dimensional form using the state of plane stress. The pear cam with the hub diameter (25 mm) was mounted on the aluminum shaft, while the aluminum shaft with the hub diameter (6 mm) was installed on the square grooving key. The sample test of pear cam and roller follower were cut off from a 6.35 mm thick plate of PSM-4 material. The pear cam and the aluminum shaft were suspended on the bottom ruler, while the roller follower with the diameter (5 mm) was installed on the upper ruler to make the contact between the cam and the follower. The upper and bottom rulers were suspended on the photoelastic apparatus. The weights was hanged on a hook load pan to exhibit the stress analysis in the field of polarized light in order to represent the birefringence phenomenon of the pear cam geometry.

Analytic Solution of Contact Stress Analysis
The general equation of circular plate in terms of (r, Ɵ, t) is [27]: where Substitute M r , M θ , and M rθ into Eq. (21), it can be obtained [27]: The homogeneous solution of Eq. (22) will be [28]: where For non-symmetric cam profile, the deflection, slope, and the moment must be infinite at the center of the plate. Then the homogeneous solution of Eq. (22) will be [28]: By applying the infinite series theorem on the homogeneous solution of Eq. (24), as in below: Cam profile was unsymmetrical (odd function), the symmetric terms (even function) was ignored, as in follows: Also the same procedure was processed on the second term: As stated above: The homogeneous solution of Eq. (22) will be [28]: where A and B are constants.

The constants A and B is as follows:
The general solution of the bending deflection of the cam profile at the point of contact will become: 3,5 [Cr 4 sin(mθ)]sin(�t), r 4 sin(mθ) = rmθ.

�
The bending deflection of Eq. (30) can be derived once and twice to obtain the contact stress at the point of contact, as in below [29]: After integration, (σ r , σ θ , and σ rθ ) will be: [(r 1 θ P sin(r 1 θ P ) + cos(r 1 θ P ))sin(rθ) + (θ P cos(r 1 θ P ) − sin(r 1 θ P )) cos(rθ)]sin(ω n t) + [ The principal stresses are: The general von Mises stress equation is illustrated in the following equation: The single trigonometric (m) has been taken as a n infinite series (m = 1, 2, 3,..., ∞) in which there will be a compatible between the analytic and numerical solutions. The percentage of error will be under (10%).  (33a)

Results and Discussions
deflection of the points on flanks No. (1) and flank No.
(2). Figures 21,22,23,24 show the Von Mises contact stress contour when the follower located at 0°, 90°, 180°, and 270° with the cam at compression load 12.485 N. The contours of contact stresses were determine by using ANSYS software. All values of the contact stress are in MPa. Maximum contact stress is 0.8758 MPa when the follower located at 270° with the cam, while the minimum contact stress is 0.2373 MPa when the follower located at 180°. The contact stress has a minimum value for all compression loads when the follower located at 180° with the base circle of the cam profile since the acceleration fluctuation was equal to zero. Figure 25 shows the verification of contact stress versus the compression load at different followers' positions with the cam. The analytic results was done using Eq. (34) of Von Mises stress, while the simulation results were done using ANSYS software. In general, the analytic results were very close to ANSYS results within the percentage of error. The percentage of error between the two sets of results are acceptable since the results are under 10% at 0°, 90°, 180°, and 270°, respectively. Figure 26 shows the variation of contact stress with the compression load at different followers' positions 0°, 90°, 180°, and 270° with the cam. The contact stress is increased linearly with the increasing of compression load. The contact stress is maximum when the follower located at 270° with the cam, while the contact stress is minimum when the follower located at 180° with the cam at compression load 12.485 N.

Conclusions and Future Work
This study analyzed and discussed the calculation of contact stress distribution at the point of contact for path No. (1) and path No. (2). The numerical procedure was done using ANSYS package and SolidWorks software. SolidWorks software was used to calculate follower displacement, velocity, and acceleration, while ANSYS package was carried out to determine the contact stress. The theory of circular plate was applied to derive the general equation of the bending deflection and contact stress distributions. In the experiment setup, four points bending test and birefringence analysis were conducted using photo-elastic technique to determine the contact stress. The contact stress was maximum 2.136 MPa when the follower located at 270° with the cam, while the contact stress was minimum 1.802 MPa when the follower located at 180° at compression load 12.485 N. The contact stress has a minimum value for all compression loads when the follower located at 180° with the base circle of the cam profile.
The recommendation of the future work is as below: (1) Contact stress reduction using genetic algorithm technique. (2) Study the effect of Hertzian contact stress on the bending deflection of cam profile. σ r , σ θ , σ rθ : Circular bending and twisting stresses, N/mm 2 ; σ 1 , σ 2 : Principal stresses, N/mm 2 ; σ V : Von Mises stress, N/mm 2 .

Figure 25
Verification of contact stress when the follower located at 0° and 90°, 180°, and 270° with the cam profile