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Cojocaru V., Micloşină C.-O.
STRESSES AND DEFORMATIONS ESTIMATION ON A FORK SUBMITTED TO CYCLICAL ASYMMETRICAL LOADS Assist. Eng. Vasile Cojocaru, Lect. Dr. Eng. Călin-Octavian Micloşină ”Eftimie Murgu” University of Resita, Faculty of Mechanics and Materials Engineering Traian Vuia Square, No. 1-4, 320085 Resita, Romania e-mail:
[email protected],
[email protected] Abstract: The paper presents the finite element method analysis of a mechanical system with two asymmetrical loads moved on Viviani’s trajectories. Using a CAD model a motion study was performed and the loads and restraints were exported to a static analysis. The variation of Von Mises stresses and maximum deformations were determined at four values of angular velocity of the motor. Keywords: Viviani’s curve, asymmetrical part, Von Misses stresses, deformation, finite element method. 1. Introduction The use of inertial forces in the propulsion of mobile robots was studied by Varga [1,2]. The mathematical model developed [1] is based on the design of a system where eight asymmetric parts are moved on Viviani’s trajectories in order to obtain a linear motion. A Viviani’s curve (figure 1) is generated by the intersection of a cylinder of radius r and center C1 (r,0,0) with a sphere of radius 2r and center C2 (0,0,0). This three-dimensional curve can be described by the parametric equations [3]:
x = r (1 + cos t ) y = r sin t 1 z = 2r sin t 2
A 3D model with two asymmetrical loads [1], [2] moved on Viviani’s trajectories was build in order to study the dynamics and the stability of the system. This two loads have an initially offset of 180°, one relative to the other. The two rotations have the same axis and are rotated in opposite direction using a double planetary bevel gear (ratio i=1). The CAD model was made using SolidWorks 2011 [4]. The components of the system were individually modeled and a material was assigned to each part. In the assembly file [figure 2] the outer casing O.C. was fixed to the ground and the others components were linked by geometrical constraints to the outer casing. The work was focused on the study of the stresses and deformations of the fork F which supports the shaft with the two asymmetrical parts E1 and E2. The analysis was made in the fork file using Design Scenario option from Simulation module. The restraints and the loads were imported from a motion study defined in the assembly file.
Fig. 1. The Viviani’s curve a) front view, b) right view, c) shaded isometric view and d) top view Fig. 2. The assembly (section view) Robotica & Management, 16-2 / 2011
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Stresses and Deformations Estimation on a Fork Submitted to Cyclical Asymmetrical Loads 2. The Motion Study
The motion study was performed using the next calculus assumptions: The motor is defined by constant angular velocity n applied on the cylindrical surface of the fork top (figure 3); The gravity is applied on Z axis, for all the assembly parts (axis X is the fork symmetry axis); The time of simulation is 5 seconds, and the animation is made at 25 frames per second.
The material used for the fork model was the unalloyed steel 1.0503 (DIN C45) with the mechanical properties listed in table 1. Table 1. Material properties Yield strength: 2.75e+008 N/m^2 Tensile strength: 4.2e+008 N/m^2 Elastic modulus: 2.1e+011 N/m^2 Poisson's ratio: 0.28 Mass density: 7800 kg/m^3 Shear modulus: 7.9e+010 N/m^2 Thermal expansion coefficient: 1.1e-005 /Kelvin High quality mesh was used with 15097 elements and 24484 nodes (figure 5).
n
Fig. 3. The motor position with the constant angular velocity “n” 3. The Static Analysis
Fig. 5. The mesh details
From the motion analysis the loads and the restraints were exported to static analysis (figure 4). The frame rate and the time of simulation gave the total number of static analysis cases in Design scenario. For each case of analysis the amplitudes and directions of the forces varied due to the variation of fork position related to the gravitation axis and due to the rotation of the asymmetrical parts on Viviani’s trajectories.
4. The Influence of the Angular Velocity The study was run at four different values of angular velocity: 500 rpm, 1000 rpm, 1500 rpm and 2000 rpm. For each velocity were analyzed the most unfavorable results from Design Scenario cases. The von Mises stresses distribution shows an increase linked to the angular velocity.
Fig. 6. The Von Mises stresses for n=1500 rpm Fig. 4. Exported loads and restraints Robotica & Management, 16-2 / 2011
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Cojocaru V., Micloşină C.-O. For an angular velocity of 1500 rpm the maximum value of von Mises stresses is 50.1 MPa. At 2000 rpm the maximum reach at 139.9 MPa.
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the rotation of fork position related to the gravitation axis; The deformations in the fork reach a maximum values of 0.141 mm at n=2000 rpm. This deformation is dangerous for the fits between the fork and the spindle; The deformations of the fork arms are not equals. The causes are the same as in the case of unsymmetrical distribution of the stresses.
Fig. 7. The von Mises stresses for n=2000 rpm The gradient of variation (figure 8) of von Mises stresses tends to an exponential evolution near to the angular velocity of 1500 rpm.
Fig. 9. The displacement distribution for n=500 rpm
Von Mises Stress [MPa] 160 140 120 100 80 60 40 20 n [rot/min] 0 0
500
1000
1500
2000
Fig. 8. The von Mises stresses variation related to angular velocity The resultant displacement diagrams show a maximum of 0.020 mm at n=500 rpm and 0.141 mm at n=2000 rpm. Stands out a significant difference between the two arms of the fork, from 0.007 to 0.017 for n=500 rpm and from 0.103 to 0.122 for n=2000 rpm. 5. Conclusions The finite element method analysis performed on a fork submitted to asymmetrical loads, driven at four values of angular velocity, led to the next conclusions: The von Mises stresses reach high values for angular velocities bigger than n=1500 rpm; The von Mises stresses distribution is unsymmetrical due to the 360° cycle of rotation of asymmetrical parts and due to
Fig. 10. The displacement distribution for n=2000 rpm 6. References [1] Varga Şt.: “Forţa ortocentrală, un concept de propulsie pt. roboţi zburători” (Orthocentral Force, a Concept of Propulsion for Flying Robots), Robotica & Management, Vol. 8, No. 1, pp. 21-28, 2003. [2] Varga Şt.: “Performanţe realizabile de roboţi zburători acţionaţi cu propulsoare ortocentrale” (Achievable Performances of Flying Robots Driven by Orthocentral Propellers), Robotica & Management, Vol. 8, No. 2, pp. 25-32, 2003. [3] Gray, A. "Viviani's Curve." in Modern Differential Geometry of Curves and Surfaces with Mathematica, Ed. Boca Raton, CRC Press, pp. 201-202, 1997. [4] Nedelcu D.: “Proiectare si simulare numerica cu SolidWorks” (Digital prototyping & numerical simulation with SolidWorks), Ed. Eurostampa, Timisoara, 2011.
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