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Mechanisms Linkages 1. The elliptic trammel is an instrument for drawing ellipses. Please prove that the point P describes an ellipse. 2. Please draw the equivalent linkages for following mechanisms. 3. (a) (b) For a four-bar linkage, link 2 is the input link and link 4 output link. Please prove the min and max transmission angles occurring at θ 2 = 0 o and θ 4 = 180 o , respectively. 4. For a four-bar linkage, the lengths of links are r1 = 100 mm, r2 = 60 mm, r3 = 90 mm and r4 = 90 mm. Please calculate the two limit positions of the mechanism. 5. Please design an offset slider-crank mechanism with stroke=20mm and offset= 10 3 mm and time ratio=13/11. 6. Design a quick-return mechanism with 20mm stroke; the working stroke is to take 0.3 sec and the return stroke 0.1sec. 7. This question concerns the classification of four-bar linkages. Link lengths: r2 , driver crank; r3 , coupler; r4 , driven link; r2 =100mm, r3 =200mm, r4 =300mm. Find the ranges of values for r1 if the linkage is as follows: (a) Grashof mechanism (b) Crank-rocker mechanism 1 (c) Drag-link mechanism (d) Double-rocker mechanism (e) Change-point mechanism (f) Triple-rocker mechanism 8. Please draw the skeleton sketch and explain the function of the following mechanisms. (a) Toggle mechanism (b) Universal joint (c) Oldham coupling (d) Scotch yoke (e) Parallel mechanism (f) Quick-return mechanism 9. List the number of links, the numbers of various joints and calculate the degree of freedom for the following mechanisms. 2 Kinematics 1. For a four-bar linkage, link 2 is the input link and link 4 output link. Please derive the position equation of link 4. 2. For a four-bar linkage, link 2 is the input link and link 4 output link. The position analysis is finished and the velocity of link 2 is known. Please derive the velocity equation of link 4. 3. Please state Kennedy’s theorem and prove it. 4. Please locate all the instant centers of the following four-bar linkage and calculate the output velocity of link 4 when the input velocity is ω 2 = 10 rad/sec ccw. Please use instant-center method and the link lengths measured in the figure. 3 2 4 1 5. P.4 P.5 Please locate all the instant centers of above slider-crank mechanism and calculate the output velocity of link 4 when the input velocity is ω 2 = 10 rad/sec 6. ccw. Please use instant-center method and the link lengths measured in the figure. Please calculate the output velocity of link 6 when the input velocity is ω 2 = 10 rad/sec ccw. Please use instant-center method and the link lengths measured in the figure. 3 7. Please locate all instant centers for following mechanisms. 8. For the six-bar mechanism shown below, please use graphical method to find the positions of links when θ 2 = 30 o . Use the lengths measured in the figure. 9. For the six-bar mechanism shown above. When the position analysis is finished and input velocity ω 2 is known, please derive the velocity equations of all links in matrix form. 10. For the inverted slider-crank mechanism shown below, please calculate the position, velocity and acceleration of the output link 4 with θ 2 = 120 o , ω2 =100rpm ccw and α 2 =0. O2 O4 = 3.0cm and PO2 =7.0cm. 4 Cams 1. For a given displacement diagram, how to reduce the pressure angle of the cam? 2. Please state the definition of undercutting. How to solve the problem of undercutting? 3. A cycloidal cam with a flat-faced follower has a rise of 20mm in an angle of β = 75 o . Base-circle radius Rb is 80mm. Radius R g of the cutter is 50mm. For values of θ of 30 o and 60 o calculate the cam profile. Also calculate the location of the cutter to produce the cam profile. 4. The total displacement h of the follower occurs as the cam rotates an angle β . For any angle of cam rotation θ , the displacement is s. Please derive the equation of 4-5-6-7 polynomial curve. ⎡ θ θ θ θ ⎤ s = h ⎢35( )4 − 84( )5 + 70( )6 − 20( )7 ⎥ β β β β ⎦ ⎣ 5. A disk cam is to give its follower a rise through travel h during a cam rotation α . This motion is preceded by a dwell and is also followed by a dwell. Determine the polynomial motion that will satisfy all displacement, velocity, and acceleration conditions at the beginning and end of the travel. 6. Please explain following nomenclature for cam. (a) Base circle (b) Pitch curve (c) Pressure angle 5 7. A disk cam with an offset translating roller follower is shown below. Please derive the equations of the cam profile and cutter location. 8. What is the definition of positive-drive cam or conjugate cam? Which are positive-drive cams in the following figures. 6 9. For a disk cam with a translating radial roller follower, the follower is to move outward 10mm with cycloidal motion during 180 o of cam rotation, is to dwell for the next 60 o , and is to return with constant velocity motion during the last 120 o of cam rotation. The roller radius is 5mm, the base circle 30mm and the cam speed is 18rpm. The 360 o of cam rotation is divided into twelve equal divisions. (a) Plot the displacement and velocity curves in the following figures. (b) Plot the cam profile. Displacement 10 8 6 4 2 0 0 90 180 270 360 0 90 180 270 360 10 5 Velocity 0 -5 -10 7 Gears 1. Please explain following nomenclature for gear. (a) Base circle (b) Pitch circle (c) Pressure angle (d) Theory of gearing (e) Contact ratio (f) Module (g) Backlash (h) Circular pitch 2. A spur gear having 32 teeth and a pressure angle of 20 o has a module of 6.5 and an addendum of 5.5. Determine the number of teeth on the smallest pinion that will mesh with the gear without inference. 3. A pair of spur gears has 16 and 24 teeth, a module of 6.5mm, and a 20 o pressure angle. (a) Determine the center distance (b) If the center distance is increased 3mm, find the resulting pressure. 4. Two 20 o gear have a diametral pitch of 4. The pinion has 28 teeth, while the gear has 56 teeth. Determine the center distance for an actual pressure of 20 o . What is the actual pressure if the center distance is increased by 0.2in? 5. Two 25 o full-depth spur gears have a velocity ratio of 1/3. The diametral pitch is 5 and the pinion has 20 teeth. Determine the contact ratio (number of teeth in contact) for the gears. Two standard 20 o full-depth gears have a module of 8 mm. The larger gear has 30 teeth, while the pinion has 15 teeth. Will the gear interfere with the pinion? 7. Suppose input 1 turns at 120rpm ccw and input 2 turns at 360 rpm cw, determine the speed and direction of rotation of the output shaft. 6. 8 8. For the planetary train shown below, let tooth numbers be N S =40, N P =20 and N R =80. Find planet carrier speed if the sun gear rotates ccw at 100rpm and the ring gear cw at 300 rpm. 9. Calculate the speed reduction ratio ω5 for the simple planetary train with sun ω4 gear 2 fixed. 10. Calculate the speed reduction ratio ω2 for the following mechanism. Link 2 is ω6 the input and link 6 output. (b) 9 11. A conventional three-speed automotive transmission is show below. Gear 1 is driven by the engine. Gears 4, 5, 6, and 7 rotate as a unit. Gear 8 is an idler that meshes with 7. These gears are always in motion when the shaft to the engine is motion. Gears 2 and 3 can slide axially on the splined shaft to the driving wheels. In the first speed, Gear 3 is shifted to the left to mesh with gear 6. For the second speed, Gear 2 is shifted to the right to mesh with gear 5. For the third speed, gear 2 is shifted to engage its clutch teeth with those of gear 1. For reverse, Gear 3 is shifted to the right to mesh with idler 8. Please calculate the speed reductions for all speeds. 12. A conventional automotive differential mechanism is show below. Please derive the arm speed in terms of the speeds of left axle and right axle. Also explain how does it work? 10