Çok amaçlı kremayer-pinyon mekanizmalarının kinematik sentez ve analizi
Kinematic synthesis and analysis of the rack and pinion multipurpose mechanism
- Tez No: 39622
- Danışmanlar: PROF.DR. AYBARS ÇAKIR
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 57
Özet
Kremayer-pinyon mekanizmasx, belirli üç noktadan geçen yörüngeyi ve bu noktalardan birinde istenen hızı sağlayan bir mekanizmadır, üç çubuk mekanizmasına çok benzer olmasına rağmen, bir çok avantaja sahiptir. îlk olarak/ kremayer-pinyona her zaman teğet olduğundan, iletim açısı daima pinyonun kavrama açısını doksan dere ceye tamamlayan açıdır. İkincisi, kremayerde hem dönme hem öteleme hareketi meydana geldiğinden, mekanizmanın tasarımı için çok sayıda çözüm mümkündür. Monoton olan ve olmayan hareketler için de fonksiyon üretimi yapar. Bu yüzden bu tek serbestlik dereceli mekanizmanın sente zi, mekanizmayı makina ve mekanizma dizaynı için çok değerli yapar. Kremayer-pinyon mekanizmasının oyuncaklar ve di ğer eğlence gereçleri üretimi ve paketleme endüstrisin de uygulamaları vardır. Bu mekanizmanın tersi olan tah- riğin pinyondan yapıldığı mekanizma da, mekanik uçak kontrol aygıtlarında, hastane ve laboratuvar cihazların da, otomotivde direksiyon bağlantılarında ve imalat mon taj hattı gibi bir çok alanda kullanılır. vı -
Özet (Çeviri)
The rack and gear mechanism presented here con- sists of a crank,- gear and rack that can be used to gene- rate a function för three specified positions. The met- hod of solution för the design and analysis utilizes Ga¬ uss Eliminative Method and some iterative methods of ki- nematic synthesis as well as basic algebra and geometry. The designer has the capability to specify pinion radi- us. The synthesis of the mechanism will yield numerous designs satisfying a number of prescribed conditions. in this way, a mechanism designer may choose öne of many suitable designs för a particular application; The general procedure for synthesizing the rack and pinion mechanism up to seven precision conditions is developed. To illustrate the medhod, the mechanism has been synthesized in closed form for three precision con¬ ditions of path generation, two positions of function generation, and a velocity condition at öne of the pre¬ cision points. This mechanism has a number of advanta- ges över conventional four-bar mechanisms. First, since the rack is always tangent to the pinion, the transmissi- on angle is always 90 degree minus the pressure angle of the rack. Second, with both translation and rotation of the rack occurring, multiple outputs are available. Ot- her advantages include the generation of monotonic func- tions for a wide variety of motion and nonmonotonic func- tions for a full range of motion as well as nonlineer amplified motions. By including the velocity specifica- tion, the designer has considerably more control of the output motion. The method of solution developed here uses Gauss Eliminative Method, Newton-Raphson Method, basic algebra and geometry. The rack and pinion mechanism is composed of an input crank, r^, whose rotation is specified by the solu¬ tion of kinematic eguations. The rack/ d, is in nonslip contact with the pinion such that it rotates by y and translates. The offset, -c^t is rigidly connected to d and allows for generality but in some cases may be omitt- ed. The pinion, whose radius vector is r3, rotates as the rack rotates and translates. Vector r g defines the tracer point with respect to the tip'of the offset and is rigidly attached to the rack. The fixed link, O^C^/ connects.the two fixed pivots. This specialized.mechanism is similar to the prismatic mechanism in that vectors r2, & and r^ ali rotate with the same angle, since d is always perpendicular to r^. The rack and pini- on mechanism, however, produces and additional output, which is the rotation of pinion. The rack and pinion mechanism has industrial applications in the packaging industry as well as toys and other leisure equipment. The inversion of the mechanism where the pinion is the driver, has been used in mechanical aircraft control de- vices, hospital and laboratory eguipment, rack and pini¬ on automative steering linkages and several tasks on ma- nufacturing assembly lines. The mechanism where the pinion is the driver has been also used in rack-and-pinion automative steering linkages. A mathematical model is developed för the design synthesis of :rack-and-pinion steering linkages. The general objective is to minimize the difference bet- ween the steering centers över the full range of steer¬ ing angle inputş while fitting into a reasonable space. Because there is a substantial amount of design art in these systems and the mathematical representation is not clear, the model, constraints and objective actually evolve to the eventual desired form. The problem has multiple optima, and practical and heuristic considera- tions are used to choose suboptimal but more realistle solutions, önce satisfactory optimal solutions are iden- tified. These involve manipulation of the objective function, constraint set, and intitial guesses. Both leading, and trailing link designs are considered, the formes are being slightly better. While there is considerable literatüre on compu- ter analysis, design, and optimization of mechanisms, there appear to be few published applications to the design synthesis of Vehicle Steering Systems. it is attempted to relate this design art to spesific erite-1 ria, mathematically expressed, and then employ computer- aided design techniques to synthesize optimal linkages. in so doing, the design synthesis problem actually evol- ves, since the problem as initially formulated was in complete-as are most real design problems. The system considered is the rack-and-pinion steering linkage, currently öne of the most widely used systems. The advantages of rack-and-pinion steering are its superior response to steering inputş, simpli-r.:. city, and relative rugged-ness. There are two practi¬ cal implementations of the rack-and-pinion steering mechanism. Both, a trailing link design and a leading link design are symmetric. And, in positioning the front wheels of a car to guide its direction, ideally the center of curvature should be the same for ali four wheels.Design Restrictions Since r3 is a free choice in the solution method, theoretically an finite number of solutions are posslb- le. However many solutions are rejected because of un- reasonable link length ratio, too large an offset för the rack, negative value for at least öne of the things which implies the sudden flipping of the rack ör if the input crank interferes with the pinion. The restricti- ons to insure that the rack remains in contact with the pinion, and that the input crank does not interfere with the pinion are; r3 < rı+r2-0l02 The condition for complete rotatability of the crank is given as; rl+r3 < °1°2 If the above reguirements are met, then complete rotation of the crank is insured without the problem of branching. Önce the mechanism has been determined it is checked for adherence to practical design restric.tions. These restrictions include elimination of extraneous roots and certain geometries which make the mechanism unusable. °1°2+ r3 < rl+r2 The rack and pinion mechanism is a versatile mec¬ hanism because it can perform path and function genera- tion simultaneously and it has good transriıission charac- teristics since the transmission angle is always egual to 90 degree minus the pressure angle of the rack. A velocity condition is also taken into consideration which makes this mechanism more useful in practical app- lications. The synthesis of this single degree of fr'ee- dom mechanism for multiple output (path and function) makes it very valuable in machine and mechanism design. Engineering is based on the fundamental sciences of mathematics, physics, and chemistry. in most cases, engineering involves the analysis of energy from some source to öne ör more outputs, using öne öre more of the basic principles of these sciences. Solid mechanics is öne of the branches of physics which, among others, con- tains three majör subbranches: Kinematics, vhich deals with the study of relative motion; statics, which is the study of forces and moments, apart from motion; and kine- tics, which deals with the action of forces on bodies.The combination of kinematics and kinetics is preferred to as dynamics. This text describes the appropriate mat- hematics f kinematics and dynamics required to accomplish mechanism design. A mechanism is a mechanical device that has the purpose of transferring motion and force from a source to an output. A linkage consists of links, generally considered rigid, which are connected by joints, such as pins, or prismatic joints, to form open or closed loops. Such kinematic chairs, with at least one link fixed, be come mechanisms if at least two other links retain mobi lity, or structures if no mobility remains. In other words, a mechanism permits relative motion between its rigid links? a structure does not. Since linkages make simple mechanism and can be designed to perform complex tasks, such as nonlinear motion and force transmission, they will receive much attention. Some of the linkage design techniques presented here are the result of a re surgence in the theory of mechanisms based on the availa bility of the computer. Many of the design methods were discovered the 1960s, but long, cumbersome calculations discouraged any further development at that time. A large mojority of mechanisms exhibit motion such that all the links move in parallel planes. Planar rigid-body motion consists of rotation about axes perpen dicular to the plane of motion and translation-where all points in the body move along parallel straight or pla nar curvilinear paths and all lines embedded in the body remain parallel to their original orientation. Combina tions of rotation around up to three non-parallel axes. and translations in up to three different directions are possible depending on the constraints imposed by the joints between links (spherical, helical or cylindirical) Mechanism are used in a great variety of machines and devices. The simplest closed-loop linkage is the four-bar, which has three moving links (plus one fixed link) and four revolute, pivoted or pin joints. The links is connected to the power source or prime mover is called the input link. The output link connects the moving pivot to ground pivot. The coapler of floating link connects the two moving pivots. All motion observed in nature is relative motion; that is, the motion of the observed body relative to the observer. For example, the seated passeng.er on a bus is moving relative to the waiting observer at the bus stop, but is at rest relative to another seated passenger. Conversely, the passenger moving along the aisle of the bus in motion relative to the seated passenger as well as relative to the waiting observer at the bus stop. - x -The study of motion, kinematics, has been refer red to as the science of relative motion. Design and analysis of machinery and mechanisms relies on the de signer's ability to visualize relative motion of machi nery components major. Degrees of Freedom The next step in the kinematic analysis of mecha nism, is to determine the number of degrees of freedom. By degrees of freedom we mean the number of independent inputs required to determine the position of all links of the mechanism with respect to ground. There are hund reds of thousands of different linkage types that one could invert. Most mechanism tasks require a single input to be transmitted to a single output. Therefore, single-deg- ree-of-freedom mechanisms, those that have constrained motion, are the types used most frequently. The process of drawing kinematic diagrams and determining degrees of freedom of more complex mechanisms are the first steps in both the kinematic analysis and synthesis process. In kinematic analysis, a particular given mechanism is investigated based on the mechanism geometry plus possibly other known characteristics (such as input angular velocity, angular accelaration etc.) Kinematic synthesis, on the other hand, is the process of designing a mechanism to accomplish a desired task. Both the type as well as the dimensions (dimensional synthesis) of the new mechanism can be part of kinematic synthesis. The fundamentals described in this chapter are most important in the initial stages of either analysis or synthesis. The ability to visualize relative motion, to reason why a mechanism is designed the way it is, and the ability to improve on a particular design are marks of a successful kinematician. Although some of this ability comes in the form will help put mechanism design into perspective: The structure or methodology of design is described, including the place of kinematic analysis and synthesis. The first decision to be made is to choose the cam and follower types. The specific application may dictate the combination of the cam and follower some factors that- should enter into the decision are geomet- 'ric considerations-type of output (rotary or translati- onal), distance between cam shaft and the center of the required oscillatory output, space allowed for cam and follower; dynamic considerations-rotational speed of cam, loading on cam and follower and masses to be moved; environmental considerations-projected environmental - xi -conditions in which the cam will be required to operates as well as environmental requirements of the cam system (e.g., noise, cleanliness); and economic matters-first and maintenance cost, number of duplicate systems, and so on. Once a cam and follower pair has been chosen, the follower motion must be synthesized. In most cases a cam-follower is required to be displaced through a specified rise or fall (return). - xix -
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