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CAD and Analysis of the Whitworth Quick Return Mechanism

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CAD and Analysis of the Whitworth Quick Return Mechanism Computer-Aided Design and Analysis of the Whitworth Quick Return Mechanism Matt Campbell Stephen S. Nestinger Computer-Aided Mechanism Design, Project Department of Mechanincal and Aeronautical Engineering University of California Davis, CA 95616 March 2004 Table of Contents Section Page 1 Introduction 2 2 Kinematic Analysis of the Whitworth Quick Return Mechanism 3 2.1 PositionAnalysis ................................ ........ 3 2.2 VelocityAnalysis ................................ ........ 5 2.3 Acceleration Analysis . .......... 5 3 Dynamic Analysis of the Whitworth Quick Return Mechanism 6 3.1 ForcesonEachMember .............................. ...... 7 4 Description of the Software Package CQuickReturn 12 4.1 Getting Started with the Software Package CQuickReturn ................... 13 4.2 Solving Complex Equations . ......... 15 5 Conclusion 16 6 Acknowledgments 16 7 Reference 16 8 Appendix A: CQuickReturn API 17 CQuickReturn — Documentation on the CQuickReturn Class 17 CQuickReturnClass .................................. ....... 17 ˜CQuickReturn ...................................... 19 animation .......................................... 19 CQuickReturn....................................... 21 displayPosition .................................... ..... 21 getAngAccel ........................................ 23 getAngPos.......................................... 24 getAngVel.......................................... 25 getForces .......................................... 26 getPointAccel ...................................... 27 getPointPos ........................................ 28 getPointVel ........................................ 29 getRequiredTorque .................................. ..... 30 plotAngAccel ....................................... 31 plotAngPos ......................................... 33 plotAngVel ......................................... 35 plotCGaccel........................................ 37 plotForce .......................................... 41 plotSliderAccel .................................... ..... 43 plotSliderPos ...................................... 45 plotSliderVel ...................................... 46 plotTorque......................................... 48 setAngVel.......................................... 49 i setForce........................................... 50 setGravityCenter ................................... ..... 50 setInertia ......................................... 51 setLinks........................................... 52 setMass ............................................ 53 setNumPoints ....................................... 53 sliderAccel ........................................ 54 sliderPos.......................................... 55 sliderRange ........................................ 56 sliderVel.......................................... 57 uscUnit ............................................ 58 9 Appendix B: Source Code 59 9.1 Source Code of Classes and Functions . ............ 59 Index 92 ii Abstract This report discusses the design and implementation of a software package for the computer-aided design and analysis of the Whitworth Quick Return Mechanism. The kinematic analysis of the Whitworth Quick Return Mechanism is discussed with details on how the position, velocity, and accerleration were calculated. The dynamic anaylsis of the Whitworth Quick Return Mechanism follows. A description of the software package is given as well as the package API. A Whitworth Quick Return Mechanism class was created that allows users to easily calculate the position, velocity, acceleration, and forces at each linkage point given an input torque. The class also allows users to find the required input torque given an input force. The class utilizes the available xlinkage software to display an animation of the Whitworth Quick Return Mechanism. 1 1 Introduction The implementation of a Whitworth Quick Return Mechanism can be useful for applications requiring slow initial force and a quick reset operation. The design of such a Whitworth Quick Return Mechanism can be tedious to do by hand. High-end commercial computer applications are available that can help in designing a Whitworth Quick Return Mechanism but these applications are usually large and come with other pack- ages that are not essential to the specified task. These commercial applications are also expensive for the general user. For students earning a degree in mechanical engineering, these black-box commercial software packages are suitable for explaining some basic principals and concepts with traditional graphic methods. In order to fully comprehend the subject matter, students must utilize numerical and analytical methods to solve complicated engineering problems. A general purpose Whitworth Quick Return Mechanism software package is required that allows the general public to be able to quickly design and implement a Whitworth Quick Return Mechanism. Such a package would require a simple user front end and easy to understand API. Users should be allowed to fully integrate the software package into their own code with the ability to either choose to specify an input torque or required output force. Students most benefit from an open soft- ware package as compared to a black-box software package. Students are able to go through and examine the available source code and modify it to solve similar problems. By learning from examples, students will better understand the principles and numerical aspects of the subject matter. Utilizing the C/C++ interpreter, Ch, a Whitworth Quick Return Mechanism software package has been cre- ated to facilitate in the design and analysis of a Whitworth Quick Return Mechanism [1]. The package contains the CQuickReturn class giving users the ability calculate the position, velocity, and acceleration of each linkage. The fundamental methods used to analyze a Whitworth Quick Return Mechanism can be found in [2]. Users can also plot the output motion of any linkage. Utilizing the xlinkage software available in the Ch Mechanism toolkit, the class provides a function to create an animation file that can be displayed by xlinkage showing the movements of the Whitworth Quick Return Mechanism over time [3]. 2 2 Kinematic Analysis of the Whitworth Quick Return Mechanism Figure 1. Vector representation of the Whitworth Quick Return Mechanism. Looking at Figure 1 the Whitworth Quick Return Mechanism can be broken up into multiple vectors and two loops. Utilizing these two loops, the following sections will go through the kinematic analysis of the Whitworth Quick Return Mechanism. 2.1 Position Analysis For the Whitworth Quick Return Mechanism shown in Figure 1, the displacement analysis can be formulated by the following loop-closer equations r1 + r2 = r3 (1a) r3 + r8 + r5 = r6 + r7 (1b) Using complex numbers, Equations 1 become iθ1 iθ2 iθ3 r1e + r2e = r3e (2a) iθ3 iθ8 iθ5 iθ6 iθ7 r3e + r8e + r5e = r6e + r7e (2b) where the link lengths r1, r2, r5, r7, and angular positions θ1, θ6, and θ7 are constants. Angular position θ2 is an independent variable; angular positions θ3, θ8, θ4, and θ5 are dependent variables. Looking at Figure 1, it can be seen that θ8 = θ3 = θ4 and r4 = r3 + r8. Substituting and rearranging Equations 2 to have all of the unknowns on the left hand side and all of the knowns on the right hand side gives iθ4 iθ1 iθ2 r3e = r1e + r2e (3a) iθ4 iθ5 iθ6 iθ7 r4e + r5e − r6e = r7e (3b) 3 Looking at Equations 3, it can be seen that Equation 3a has 2 unknowns and Equation 3b has 3 unknowns. Utilizing Euler’s equation, eiθ = cos θ + i sin θ, Equation 3a can be broken up into two equations, one comprising of the real numbers and the other comprising of the imaginary numbers. r3 cos θ4 = r1 cos θ1 + r2 cos θ2 (4a) r3 sin θ4 = r1 sin θ1 + r2 sin θ2 (4b) Squaring Equations 4, adding them together, and simplifying gives 2 2 r3 = (r1 cos θ1 + r2 cos θ2) + (r1 sin θ1 + r2 sin θ2) (5) q Dividing Equation 4b by Equation 4a and simplifying gives r1 sin θ1 + r2 sin θ2 θ4 = arctan (6) r1 cos θ1 + r2 cos θ2 Knowing θ4, Equation 3b now only has 2 unknowns and becomes iθ6 − iθ5 iθ4 − iθ7 r6e r5e = r4e r7 (7) iθ iθ4 iθ7 Since the right hand side of Equation 7 is constant, we let re = r4e − r7e and use it in the rest of the calculations. Breaking Equation 7 up into real and imaginary parts gives r6 cos θ6 − r5 cos θ5 = r cos θ (8a) r6 sin θ6 − r5 sin θ5 = r sin θ (8b) Solving Equations 8 for r6 gives r cos θ + r5 cos θ5 r6 = (9a) cos θ6 r sin θ + r5 sin θ5 r6 = (9b) sin θ6 where Equation 9a is used when cos θ6 > 0 and Equation 9b is used when cos θ6 = 0. Substituting Equation 9a into Equation 8b gives r cos θ sin θ6 − r sin θ cos θ6 sin(θ5 − θ6)= (10) r5 Solving for θ5 we find r cos θ sin θ6 − r sin θ cos θ6 θ5a = θ6 + arcsin (11a) r5 r sin θ cos θ6 − r cos θ sin θ6 θ5b = θ6 + π − arcsin (11b) r5 Knowing all of the angular positions and the length of r6, we can find the position of the output slider, link 6, using P6 = r4 + r5 (12) 4 2.2 Velocity Analysis For the Whitworth Quick Return Mechanism shown in Figure 1, the velocity analysis can be formulated by taking the time derivative of the loop-closer equations. Taking the time derivative of Equations 2 gives iθ4 iθ4 iθ1 iθ1 iθ2 iθ2 r˙3e + r3iω4e =r ˙1e + r1iω1e +r ˙2e + r2iω2e (13a) iθ6 iθ6 iθ4 iθ4 iθ5 iθ5 iθ7 iθ7 r˙6e + r6iω6e =r ˙4e + r4iω4e +r ˙5e + r5iω5e − r˙7e − r7iω7e (13b) Equations 13 can be simplified since r˙1 =r ˙2 =r ˙4 =r ˙5 = 0, the links are assumed to be rigid members that may not elongate, ω1 = 0, link 1 is a rigid link that is unable
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