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Flores and Ambrsio compared several continuous force models from elastic deformation and energy dissipation aspects. Poincar maps and journal center trajectories show that the magnifications of forces and moments are affected by the revolute clearance joints and depend on the clearance size and input crank speed. f\ For a given set of parameter values, initial condition and time scale affected the appearance of the response. Since link 2&3 are joined, link 2 will show the same motion as link 3. The simulation model presented in Figure 1 is completed for the three cases and basic conditions listed in Table 2. n{Hy0.Da#GhD J;)I.n 0&3qA\0]j-.hPg%~#9]hwcx4-vSh6nk[ :UDv"(#HZAx `Bj8G3VY//IX:etp]'5lSAGQ.F*UA1=$t[W;>m\T)3l^`]PZ#mGl#)p YFRI ^:,&$9 Wx|*#_[%Mk)#?0'= ~QM^QDt=*I/*/%R-4Ll-9vksEA, > sJ5 Swift, H. L. Swinney, and J. They describe an average rate of divergence or convergence of two neighboring trajectories in phase space, thus providing an indication of the sensitivity of a dynamic system to small local perturbations and distinguishing the chaotic and nonchaotic behaviors of the system [42, 43]. A tangential friction force is involved in the simulation model based on Coulombs friction law. In general, periodic signals give peaks at a fundamental and its harmonics; quasi periodic signals give peaks at linear combinations of two or more irrationally related frequencies; and chaotic dynamics give broad band components to the spectrum [39]. Ravn used the continuous approach and the L-N force model to describe a mechanism with a rotational joint clearance. Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. From the aspect of dynamic response, the influence resulted from the crucial factors of clearance size, input speed, rigidity and flexibility of bodies, lubrication, and friction in joint, which have been investigated previously to accurately predict the dynamic response of the mechanical system with clearance [2628]. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. When faced with the import wizard, choose delimited (as shown [36] studied the effects of clearance size and input crank speed on the dynamic response of a typical plane crank-slider mechanism with two revolute clearance joints. Activate your 30 day free trialto continue reading. fvL6 VFIjy3?OR))(ra%.E em2;%IG=d9;T-*Y#gd[a)MBcu2x rt;s|qRj6z3`5("*V\ht}Al-xt2G}U5>yP pSgUAs4/8xA$mPR++$GNdCl(._JD?9D7p&$B?P55Pd&Ir6Q ~=)i(0g1HD b!L1 Yiy DRq7l5Y+{uN>j|}:7=}*PP(+/d(ziSoW|Kkxv])zmAvHa4o `wrf,-$&i {Y .JTMxBtH"?x W24EPB1 ak The result shows that the transformation from regular to chaotic motion depends on a critical value of the input speed, slightly over the value, the clearance response characterized by nonperiodicity, and sensitive to initial conditions. <> The motion trajectories of the clearance joint elements and the relationship between the force and deformation indicated the L-N model led to a much smoother dynamic response of the system than other models [18]. Over the last decades several formulations for the damping factor have been proposed, which was compared in detail by Alves et al. The chaotic behavior of the deep impact mechanics also could be analyzed further. The two figures demonstrate that the impact-contact between the slider and link 3 affects the accuracy of link 3 motion, increases in the clearance, and decreases in the precision of the motion. Figure 3 exhibits the velocity responses of link 3 mass center . This means the impact-contact is more serious in the larger clearance joint. Tang et al. If the contact instant and point are not detected precisely, the penetration and the contact force may become abnormally large and eventually result in the termination of the integration process. At the same time, the tangential relative sliding and viscous phenomenon at contacting point should also be considered through a friction force. [19], who used this approach for modeling the impact mode equations for a basic mechanical system. mechanism. B. Dingwell and L. C. Marin, Kinematic variability and local dynamic stability of upper body motions when walking at different speeds,, M. T. Rosenstein, J. J. Collins, and C. J. It was proved from Poincar map that the phenomena of chaotic, periodic, and subharmonic response could be found in the special conditions. 4 0 obj and are the absolute velocity thresholds for the transition from dynamic friction to static friction. It can be characterized by two items in a normal contact force. The table is organized with rows indicating the time steps and columns for all The numerical integration algorithm continues in the whole solution process of equations. The simulation model was built with a nonlinear impact model. No related content is available yet for this article. Figures 5(a) and 5(b) show the comparison of the FFT of center displacement data when the clearance is and 0.1mm, respectively. The periodic motion is observed for low values of the crank speed and coefficient of restitution. Track your progress, build streaks, highlight & save important lessons and more! The clearance is defined as the difference of the two diameters: the diameter of link 3, 2, and the slider, . System exhibits chaotic behavior under specific conditions. Instant access to millions of ebooks, audiobooks, magazines, podcasts and more. In this section, we study the effect of initial conditions on the dynamic responses of the planar mechanism with two driving links and prismatic pair clearance. 7/24/2014 1 y-position, joint1's x-velocity, joint1's y-velocity, joint1's x-acceleration, So link will rotate in counter clockwise direction. See our Privacy Policy and User Agreement for details. This will is the relative tangential velocity of the contact point. State University, Tutorial It is clear that the system having clearance exhibits nonlinear behavior at different link 1 speeds. The complexity and importance of the clearance joints have attracted investigation for centuries. Free access to premium services like Tuneln, Mubi and more. A recommended value for is 0.1mm and is one percent of the stiffness coefficient of the material. Prof. @ Dr. Akhilesh Das Gupta Institute of Technology and Management, New Delhi, Static and Dynamic Balancing of Rotating Mass, All possible questions - KINEMATICS OF MACHINERY / UNIT I, STEERING GEAR MECHANISMS (kinematics of machines), GIEDEEAM SOLAR and Gajanana Publications, LIC, Velo & accel dia by relative velo & accl method, Unit 8: Torsion of circular shafts and elastic stability of columns, Unit 6: Bending and shear Stresses in beams, Unit 5 shear force and bending moment in beams, Dynamics of Machines: Question bank unitwise from vtu old question papers, Mechanics of Materials: Question Bank from old VTU Question papers, Robots: state of-the-art and future trends, Screw thread measurements and Gear measurement, Mechanical measurements and Measurement systems, Introduction to Non-Traditional Machining, Be A Great Product Leader (Amplify, Oct 2019), Trillion Dollar Coach Book (Bill Campbell). From Figure 5(b), it can be seen that the Fourier spectrum is a continuous broadband spectrum and has broader peaks as compared to Figure 5(a), which indicates that the motion of the system is not periodic or quasi-periodic. Performing this simulation has shown that all the kinematic and dynamic characteristics of the mechanism at different motion modes could be displayed as a continuous form. In this section, the dynamic behaviors of the mechanism with a series of clearance sizes are mainly exhibited and discussed. Clearance joints are found in all real mechanical systems, instead of ideal joints, which ignore the existence of tolerance and wear on joint elements. xZoHi@E!c'3UU7 qt=~U}lwnmpjk(;Gs0oB_$4f'+4]q-t kdpqj)Gnqg08bb`"$q.>r1*[INZ According to different assumptions on the impact process (continuous or instantaneous), there are two main approaches to formulating the dynamic equations of motion of mechanism with clearance, namely, continuous and discontinuous; we will discuss them separately. Periodic motion corresponds to closed trajectories and chaos motion corresponds to a group of random separated points of the never closed trajectories (strange attractor) in a certain region [40]. Copyright 2017 Lijuan Wu et al. Figure 9 shows the largest Lyapunov exponents calculated for the four groups of input speeds. When impact-contact occurs, a contact force is added simultaneously in the basic equations of motion as an internal form. Four possible states of the relative motion. Enjoy access to millions of ebooks, audiobooks, magazines, and more from Scribd. Other measures like small time step, short simulation time, simple model, and so forth are also effective for finding the contact instant exactly. When link 4 is moved upwards, it will push link 3 upwards but since link 3&4 are joined by a revolute joint, the link will have to rotate in anticlockwise direction to move upwards. of the joint and link state variables. G8Z%AXr07 Farahanchi and Shaw [33] studied the influence of clearance size, joint friction, crank speed, and impact parameters on the response of a crank-slider mechanism with slide clearance. In this section, the clearance size is set to the same series as Section 3.1. identified by its time-stamped filename. Numerical simulation has been done in ADAMS using a specific input force. A negative Lyapunov exponent indicates local stability, whereas a positive Lyapunov exponent indicates local instability. ES3323 - Project 1 Report - Modeling Watt's Linkage, Mechanical Joints in LS-Dyna for Explicit Analysis, Hareesha N Gowda, Dayananda Sagar College of Engg, Bangalore, ANKIT SAXENA Asst. <>>> Note: you will have to tell Excel to show other file types to find Hareesha N G, Dept of Aero Engg, DSCE, The normal contact force is characterized by Hertz contact theory and an energy dissipation term. is the maximum damping coefficient related to the material, which represents the loss of energy resulting from collisions. The frequency and the amplitude of fluctuation increase as the clearance size increases. In order to get a more detailed understanding of the movement, one might prefer to export the FFT (fast Fourier transform) converts a time domain signal of infinite duration into a continuous spectrum composed of an infinite number of sinusoids. Blore. Thus, the main focus is to build a force model which takes into account the material, velocity, and deformation of collision bodies. frames of reference appear on the screen. Learn faster and smarter from top experts, Download to take your learnings offline and on the go. 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First open Excel and then Excel's open Timoshenko and Gere suggested a linear force model which considered the linear elastic deformation on the contact surface and energy consumption in the collision process [13]. Then the largest Lyapunov exponent is calculated through the slope of the linear best-fit of the Euclidean distance. In this paper, the effect of clearance size and input speed on the dynamic performance is investigated on a mechanism with two driving links. Thus, the work of modeling the dynamic model correctly and figuring out the relationships between the response and parameters of a system with clearance is very critical and still in progress. This approach was first proposed by Kane [24], Marghitu and Hurmuzlu [25], and Schwab et al. A number of valuable works have focused on the methodology for modeling and simulation of mechanical systems with clearance joints, as well as prediction and quantification of the dynamic response based on the system parameters of clearance size, system inputs, material, and surface properties. In compression process, a small elastic deformation, namely, penetration , occurs at the contacting point as shown in the figure. The second discontinuous approach assumes the impact occurs instantaneously and the system configuration does not significantly change during the short time contact. Two rotation speeds are considered inputs on link 1 and link 2, respectively, to drive the mechanism. If you continue browsing the site, you agree to the use of cookies on this website. The chaotic behavior is related to the unpredictable and random motion states between the two elements of the clearance joints, due to which there is uncertainty in the contact and the friction force. Under ADAMS, the normal contact force is described by the impact function based on a typical nonlinear spring-damper model, which is composed of two items: the nonlinear pure elastic deformation force expressed by Hertz contact law and the damping force expressed by hysteretic damping item. At the prismatic pair, a clearance is introduced by the difference of slider bore and link 3 in the diameter direction. Lijuan Wu, Dan B. Marghitu, Jing Zhao, "Nonlinear Dynamics Response of a Planar Mechanism with Two Driving Links and Prismatic Pair Clearance", Mathematical Problems in Engineering, vol. Furthermore the system is extremely sensitive to initial conditions of the clearance size. A Poincar section consists of plotting the value of two components: derivative of the state vector versus the vector itself, that is, and [41]. You are welcome to change the name, but it should be uniquiely When the penetration reaches , the damping coefficient raises to the maximum and the damping force reaches maximum. B. The initial angles are deg and deg. The largest Lyapunov exponents increase with the increase in clearance whether the input speed is constant or variable. The damping force is related to relative penetration velocity and damping coefficient , which was widely expressed as a product of a hysteresis damping factor and an exponential item of penetration. Also, the velocity and position response shows the decrease in precision with the increase in input speeds. Experimental data was presented later to verify the continuous analysis method [17]. Largest Lyapunov exponents for variable input speeds and a series clearance size. [37] studied the response that resulted from the constant crank speed and the different clearance size. M. W. Hirsch, S. Smale, and R. L. Devaney, A. Wolf, J. The simulation results show that the dynamic of the mechanism system is changed dramatically. A series of clearance sizes and a group of input speeds, constant and variable, are individually set as the different operating conditions. The simulation results indicate that the largest Lyapunov exponents are dependent on the clearance size and the input speed. [46] studied the nonlinear performance of mechanisms with clearance under various conditions and parameters; chaotic phenomenon was presented in their works. 2: Creating a Crank-and-Slider. The characteristics of the bodies are given in Table 1. %PDF-1.5 Over the last years a good number of works [2023] have performed studies to modify the hysteresis damping for both large and small coefficients of restitution. This change can be seen from the increase fluctuations of motion as the clearance size and input speed increase. Activate your 30 day free trialto unlock unlimited reading. {pV|7zG:)%9pH* rI*~@EK``b The damping factor is defined as a function of the contact stiffness, initial contact velocity, and coefficient of restitution. Poincar sections are mathematical abstractions which are often useful in highlighting the behavior of systems in terms of periodic, quasi-periodic, and chaotic or nonlinear motion. The largest Lyapunov exponents calculated at the condition of variable input speeds and a series clearance size are shown in Figure 11. However, the system with clearance exhibits the chaotic nature with the dense points filling up the section in fractal structure. is the maximum deformation point. In present work, the damping factor is expressed in maximum damping coefficient form and the damping coefficient is expressed as a step function to avoid discontinuity of damping force:where is the maximum penetration, which determines when the damping force reaches maximum. The contact process is considered as a continuous compression process and a gradual restitution process. Following the order of joints in the 2017, Article ID 4295805, 12 pages, 2017. https://doi.org/10.1155/2017/4295805, 1Department of Mechanical Engineering, Auburn University, 1418 Wiggins Hall, Auburn, AL 36849-5341, USA, 2Department of Mechanical Engineering, Shanghai Dianji University, Shanghai, China. For accuracy and efficiency variable time step integral algorithms are widely utilized to gradually narrow the scope of the detection according to the calculation of the distance between the contacting bodies. In a clearance prismatic pair, there are four possible states of the relative motion which are illustrated in Figure 2: (a) no contact; (b) impact-contact on a single point on the end face of slider; (c) impact-contact on two points on the opposing sides; (d) impact-contact on two points on the same side. The authors declare that there are no conflicts of interest regarding the publication of this paper. endobj Largest Lyapunov exponent for different link 1 speeds and clearance 0.5mm. Then the stability and accuracy of the mechanism motion are quantified by comparing the dynamic responses deduced from different conditions. below), and indicate that the 'delimiters' are tabs and/or commas. This is easily done by In fact, because of the frequent transition between the different states, the contact instant detection is another important task except the solution of the dynamic equation for numerical issue in impact-contact events. Seneviratne and Earles [32] studied the periodic and chaotic performance of a four-bar mechanism through changing the crank speed and the realistic damping of link. The impact-contact duration, in addition to material deformation, is accompanied by energy dissipation. Phase plane describes the relationship between the trajectories evolution and the various state parameters, in which each possible state corresponds to one unique point (attractor). 2: Creating a Crank-and-Slider, Copyright 2014 Matthew I. Campbell, Oregon In order to design a mechanism it is useful to have some additional shapes or It can also be concluded that, as the points increase, they get more dispersed with the increase in clearance, which further indicates that the system is more unpredictable. is a constant value. The degrees of freedom of the mechanism become four from two when the clearance is set. APIdays Paris 2019 - Innovation @ scale, APIs as Digital Factories' New Machi Mammalian Brain Chemistry Explains Everything. Now customize the name of a clipboard to store your clips. endobj State University When , the mode is free fly; otherwise, which means impact or contact occurs; then the contact and the deformation point can be calculated bywhere is the initial contacting point on the slider surface. The nonlinear characteristic of the mechanism system is analyzed using FFT, phase plane, Poincar section, and largest Lyapunov exponents. However, it has been concluded that the stability and precision of system motion are considerably degraded as a result of the impact between the elements of the clearance joint. If you continue browsing the site, you agree to the use of cookies on this website.


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