Chapter 7Problems7.1 Problem Set: MechanismsThe dimensions of the planar mechanisms, shown in Figs. 7.1-7.15, are given inTables 7.1-7.15, respectively. The angle of the driver link 1 with the horizontal axisis φ. The constant angular speed of the driver link 1 is n and is given in the tables.1. Determine the type of motion (rotation, translation, and complex motion) foreach link, the connectivity table, the structural diagram, the contour diagram, theindependent contours, the number of degrees of freedom, the dyads, and the typeof the dyads.2. Find the positions of the joints and the angles of the links with the horizontalaxis when the angle of the driver link 1 with the horizontal axis is φ . Write aMATLABRprogram for the positions of the mechanism for the given angle φ.3. Write a MATLAB program for the positions of the mechanism for a completerotation of the driver link 1, φ ∈[0◦,...,360◦], using different methods.4. Write a MATLAB program for the path of a point on a link with general planemotion.5. Write a MATLAB program for the animation (movie) of the mechanism for acomplete rotation of the driver link 1, φ ∈ [0◦,...,360◦].6. Find the velocities and the accelerations of the mechanism, using different meth-ods, for the given position when the driver link 1 makes an angle φ with thehorizontal axis, The constant angular speed of the driver link 1 is n. Write aMATLAB program.7. The link bars of the mechanism are homogeneous rectangular prisms with thewidth h = 0.01 m and the depth d = 0.001 m. The sliders have the width wSlider=0.050 m, the height hSlider= 0.020 m, and the same depth d = 0.001 m. The linksof the mechanism are homogeneous made of steel having a mass density ρSteel=8000 kg/m3. The gravitational acceleration is g = 9.807 m/s2. The driver link 1has a constant angular speed n. The external force or moment applied on the lastlink 5 is opposed to the motion of the link and has the value: |F5ext| = 2000 N if275276 7 Problemsthe last link 5 has a translational motion or, |M5ext| = 3000 N·m if the last link 5has a rotational motion.Determine the joint forces and the motor moment Mmrequired for the dynamicequilibrium of the considered mechanism when the the driver link 1 makes anangle φ with the horizontal axis. Write MATLAB programs for different methodsof calculating the dynamical forces.Table 7.1 Mechanism 1No AB AD BC CD φ n[m] [m] [m] [m] [◦] [rpm]1 0.08 0.19 0.21 0.12 60◦5002 0.10 0.24 0.26 0.14 120◦6003 0.18 0.43 0.47 0.27 210◦7004 0.15 0.36 0.40 0.21 330◦8001AnφCBD2300Fig. 7.1 Mechanism 17.1 Problem Set: Mechanisms 277Table 7.2 Mechanism 2No AB BC CD CE EF a b c φ n[m] [m] [m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.15 0.40 0.37 0.23 0.23 0.30 0.45 0.37 330◦5002 0.10 0.27 0.25 0.15 0.15 0.20 0.30 0.25 120◦8003 0.12 0.32 0.30 0.18 0.18 0.24 0.36 0.30 240◦9004 0.20 0.55 0.50 0.30 0.30 0.40 0.60 0.50 300◦1000abc24CE31AnφBD0F50Fig. 7.2 Mechanism 2278 7 ProblemsTable 7.3 Mechanism 3No AB BC CD DE EF a b φ n[m] [m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.02 0.03 0.03 0.03 0.06 0.03 0.01 30◦6002 0.05 0.20 0.20 0.31 0.30 0.10 0.06 150◦7003 0.09 0.11 0.07 0.11 0.12 0.035 0.025 240◦9004 0.22 0.27 0.17 0.25 0.28 0.09 0.055 330◦110024CE1AnB5F0φ0D0ba3Fig. 7.3 Mechanism 37.1 Problem Set: Mechanisms 279Table 7.4 Mechanism 4No AB BC CE CD EF a b c φ n[m] [m] [m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.09 0.40 0.25 0.12 0.21 0.22 0.35 0.40 60◦4002 0.15 0.67 0.45 0.22 0.32 0.33 0.60 0.65 135◦6003 0.22 1.00 0.65 0.35 0.60 0.55 0.90 1.20 240◦8004 0.16 0.70 0.50 0.25 0.48 0.40 0.60 0.70 300◦10002E31AnBD0F50ab0φcC4Fig. 7.4 Mechanism 4280 7 ProblemsTable 7.5 Mechanism 5No AB BC CD CE EF a b c φ n[m] [m] [m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.10 0.37 0.15 0.15 0.25 0.12 0.36 0.09 45◦10002 0.08 0.30 0.12 0.12 0.20 0.10 0.30 0.08 135◦11003 0.12 0.45 0.18 0.18 0.30 0.15 0.45 0.14 225◦12004 0.06 0.24 0.09 0.09 0.18 0.075 0.225 0.070 315◦130024CE31AnBD0bφc0F50aFig. 7.5 Mechanism 57.1 Problem Set: Mechanisms 281Table 7.6 Mechanism 6No AB AC BD DE EF a b φ n[m] [m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.06 0.18 0.27 0.12 0.08 0.02 0.01 225◦6002 0.08 0.20 0.30 0.16 0.11 0.025 0.13 45◦7503 0.05 0.12 0.18 0.10 0.065 0.017 0.085 135◦9504 0.04 0.10 0.16 0.09 0.063 0.013 0.065 315◦115024CE31nBDbF5aAφ000Fig. 7.6 Mechanism 6282 7 ProblemsTable 7.7 Mechanism 7No AB AD BC a φ n[m] [m] [m] [m] [◦] [rpm]1 0.08 0.20 0.10 0.15 300◦10002 0.06 0.15 0.08 0.50 210◦12003 0.10 0.25 0.12 0.50 150◦13004 0.12 0.30 0.15 0.50 30◦150024CE31nBD500Aφ00aFig. 7.7 Mechanism 77.1 Problem Set: Mechanisms 283Table 7.8 Mechanism 8No AB AD BC CE EF a φ n[m] [m] [m] [m] [m] [m] [◦] [rpm]1 0.22 1.00 0.65 0.18 0.70 0.60 300◦10002 0.25 1.10 0.70 0.20 0.80 0.65 120◦12003 0.20 1.00 0.55 0.15 0.60 0.52 60◦15004 0.18 0.80 0.40 0.10 0.50 0.43 225◦180024CE3BD5001n0AφFa90Fig. 7.8 Mechanism 8284 7 ProblemsTable 7.9 Mechanism 9No AB AC CD DE φ n[m] [m] [m] [m] [◦] [rpm]1 0.22 0.08 0.20 0.60 240◦5002 0.18 0.08 0.20 0.50 30◦6003 0.15 0.05 0.18 0.45 120◦7004 0.08 0.03 0.07 0.20 330◦80024E31nBD50CAφ9000Fig. 7.9 Mechanism 97.1 Problem Set: Mechanisms 285Table 7.10 Mechanism 10No AB AE BC CD a φ n[m] [m] [m] [m] [m] [◦] [rpm]1 0.03 0.07 0.05 0.08 0.05 30◦4002 0.10 0.25 0.17 0.30 0.09 120◦6003 0.12 0.30 0.20 0.35 0.10 200◦8004 0.09 0.35 0.25 0.40 0.12 300◦100024CE31nBD5Aφ00D0aFig. 7.10 Mechanism 10286 7 ProblemsTable 7.11 Mechanism 11No AB CD DE AC φ n[m] [m] [m] [m] [◦] [rpm]1 0.08 0.14 0.40 0.04 45◦3002 0.10 0.16 0.45 0.03 135◦4003 0.25 0.40 1.00 0.10 225◦5004 0.22 0.30 0.90 0.06 315◦6000A0CE1254B03φDnFig. 7.11 Mechanism 117.1 Problem Set: Mechanisms 287Table 7.12 Mechanism 12No AB AC BD a b φ n[m] [m] [m] [m] [m] [◦] [rpm]1 0.12 0.30 0.50 0.08 0.15 60◦5002 0.10 0.30 0.50 0.07 0.12 150◦7003 0.09 0.30 0.45 0.06 0.10 240◦9004 0.08 0.25 0.40 0.05 0.09 330◦11000φA1DCBE324500nbaFig. 7.12 Mechanism 12288 7 ProblemsTable 7.13 Mechanism 13No AB BC CD a b φ n[m] [m] [m] [m] [m] [◦] [rpm]1 0.20 0.21 0.39 0.30 0.25 45◦5002 0.18 0.17 0.35 0.27 0.26 135◦10003 0.10 0.25 0.15 0.225 0.30 240◦15004 0.22 0.23 0.45 0.30 0.32 315◦200034φA12BD05C0n0abEFig. 7.13 Mechanism 137.1 Problem Set: Mechanisms 289Table 7.14 Mechanism 14No AB AC CD DE a φ n[m] [m] [m] [m] [m] [◦] [rpm]1 0.10 0.20 0.12 0.08 0.13 20◦5002 0.08 0.15 0.10 0.10 0.12 110◦6003 0.15 0.30 0.20 0.14 0.25 200◦7004 0.07 0.15 0.10 0.06 0.11
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