400 5 Kinematics of Rigid Bodies

Transcription

1 4 5 Kinematics of Rigid odies The absolute angular acceleration of link 4 and the acceleration of are: alpha4=[,,88.9] (rad/sˆ) a5=[, ,] (m/sˆ) 5. Problems 5. The dimensions for the mechanism shown in Fig are given in the table. The driver link rotates with a constant angular speed of n = 5 rpm. Find the velocities and the accelerations of the joints and the angular velocities and the angular accelerations of the links for the case when the angle of the driver link with the horizontal axis is =. a n [m] [m] [m] [ ] [rpm] The dimensions for the mechanism shown in Fig are given in the table. The driver link rotates with a constant angular speed of n =, rpm. Find the velocities and the accelerations of the joints and the angular velocities and the angular accelerations of the links for the case when the angle of the driver link with the horizontal axis is = 45. a b n [m] [m] [m] [m] [m] [m] [ ] [rpm] , 5. The dimensions for the mechanism shown in Fig. 5.5 are given in the table. The driver link rotates with a constant angular speed of n = 5 rpm. Find the velocities and the accelerations of the joints and the angular velocities and Fig Problem 5. a

2 5. Problems 4 b a Fig Problem 5. Fig. 5.5 Problem 5. the angular accelerations of the links for the case when the angle of the driver link with the horizontal axis is = 45. n [m] [m] [m] [ ] [rpm]

3 4 5 Kinematics of Rigid odies Fig. 5.5 Problem The dimensions for the mechanism shown in Fig. 5.5 are given in the table. The driver link rotates with a constant angular speed of n =, rpm. Find the velocities and the accelerations of the joints and the angular velocities and the angular accelerations of the links for the case when the angle of the driver link with the horizontal axis is = 7. n [m] [m] [m] [m] [ ] [rpm] The dimensions for the mechanism shown in Fig. 5.5 are given in the table. The driver link rotates with a constant angular speed of n =,5 rpm. Find the velocities and the accelerations of the joints and the angular velocities and the angular accelerations of the links for the case when the angle of the driver link with the horizontal axis is =. n [m] [m] [m] [m] [ ] [rpm].8...,5

4 5. Problems 4 Fig. 5.5 Problem F L a L b 4 y n x Fig. 5.5 Problem The mechanism in Fig. 5.5 has the dimensions: = 5 mm, = 5 mm, = 5 mm, = mm, F = mm, L a = 55 mm, and L b = 5 mm. The constant angular speed of the driver link is n = rpm. Find the velocities and the accelerations of the mechanism for = =. 5.7 The mechanism in Fig has the dimensions: = mm, = 6 mm, =, mm, L a = 5 mm, and L b = 5 mm. The driver link rotates with a constant angular speed of n = 6 rpm. Find the velocities and the accelerations of the mechanism for = =. 5.8 The planar mechanism considered is shown in Fig The driver link is the rigid link (the link ). The following numerical data are given: =.4 m, =.6 m, =.5 m, =.5 m. The constant angular speed of the driver link is,5 rpm. Find the velocities and the accelerations of the mechanism when the angle of the driver link with the horizontal axis is =.

5 44 5 Kinematics of Rigid odies 5 L a L b y 4 n x Fig Problem The dimensions of the mechanism shown in Fig are: = mm, = mm, = 5 mm, = 5 mm, and L a = 4 mm. The constant angular speed of the driver link is n = 4 rpm. Find the velocities and the accelerations of the mechanism when the angle of the driver link with the horizontal axis is = The dimensions of the mechanism shown in Fig are: = 8 mm, = 9 mm, and = mm. The constant angular speed of the driver link isn = 88 rpm. Find the velocities and the accelerations of the mechanism for = = regular tetrahedron O in motion is shown in Fig The length of the sides of the regular tetrahedron is L, O = O = O = = = = L. t the instantaneous moment t the velocity of the point O is v O, the velocity of the point is v parallel to v O,(v v O ), and has the same sense v v O >. The velocity of the point is parallel to the plane O, v plane (O ). Find the instantaneous angular velocity of the regular tetrahedron.

6 5. Problems 45 Fig Problem 5.8 L a 5 4 n y x Fig Problem parallelipeped is rotating along a fixed axis Δ (Δ = ), as shown in Fig. 5.59, with the angular velocity ω = kt,wheret is the time. The sides of the paralelipiped are a, b,andc. Find the velocities and the accelerations of the points and as a function of time. The numerical values are k = 5rad/s, a = b = m,andc = 4m.

7 46 5 Kinematics of Rigid odies y n 4 5 x Fig Problem 5. Fig Problem Repeat Problem 5. for the regular prism shown in Fig The numerical values are = = = mand = m. 5.4 Repeat Problem 5. for the regular tetrahedron with the sides equal to m, as shown in Fig The rigid link ( = L) is moving in the vertical plane xoy, as shown in Fig The absolute value of the velocity of the point is v = t +m/s, where t is the time. t the initial moment t = the point is at the origin O. Find the angular velocity and acceleration of the link and the velocity and acceleration of the point as a function of time. 5.6 The straight rod, shown in Fig. 5.47, is moving in a vertical plane. The trajectories of the end points and are shown in Fig (a) If the velocity of the point is given, v, find the angular velocity of the rod and the velocity

8 5. Problems 47 Fig Problem 5. Fig. 5.6 Problem 5.

9 48 5 Kinematics of Rigid odies Fig. 5.6 Problem 5.4 Fig. 5.6 Problem 5.5

10 5. Problems 49 Fig. 5.6 Problem 5.6 v of the point ; (b) Repeat (a) for two particular cases: () the trajectory of the point is a straight line Δ and the trajectories of the point is a circle with center O; () the trajectories of the points and are circles (Fig. 5.6).