- 3 3g M3 12 Op T12 M M3 3g Y 2 A O R A 2 Tindr A B Figure A Above Shows A Pin In Slot Mechanism Working On A H 1 (237.26 KiB) Viewed 12 times
3 [3G M3 12 OP t12 m₂ m3 3G Y+ 2. A O r₁ A 2 Tindr (a) (b) Figure (a) above shows a pin-in-slot mechanism working on a horizontal plane. It comprises a crank (2), a slotted link (3), and the ground link (1). The crank has a disk pin (P) sliding inside the slotted link. The crank is to be rotated continuously, making the slotted link illustrating an oscillating motion. Notations m₂ and m3 denote the mass centres of the crank and the slotted link, while ₁, 12, and 3G represent the kinematic dimensions of the ground link and the crank as well as the location of the mass centre m3. Note that the geometry of the crank has been specifically designed so that the mass centre m₂ is perfectly coincident with the centre of the rotation joint (O). Assume that the contact between the crank and the swinging link is perfect, so there i neither a gap nor friction between the pin and the slot. To analyze the dynamics of the mechanism, a vector loop and a coordinate system are established as shown in Fig. (b). Suppose that the kinematic dimensions and the inertia parameters of the mechanism are: ₁ = 5.4 m, r₂ = 3.5 m, and /3G = 7.3 m m₂ = 3.7 kg, m3 = 6.7 kg /2 = 0.75 kg-m², /3 = 1.05 kg-m² Now, we wish that the mechanism can be driven in a way that the crank can maintain a constant angular speed w₂ = 5.9 rad/s (CCW) within the whole range of motion. Based on this motion requirement, we have completed the kinematics analysis for the mechanism, and it shows that at the instant when the crank angle 8₂ = 67°, the motion status of the slotted link (3) is: 03= 141.38°, W3 = 1.0775 rad/s, a3 = 14.4295 rad/s² By writing the force equilibrium equations and the moment equations about the mass centres, the inverse dynamics of the mechanism can be formulated as the following matrix equation: 0 0 0 01 f12x 0 1 0 00 f12y 0 A 0 0 1 f32x 10 1 01 0 00 -3.2218 00 -1 00 0 0 0 1.3349 0 0 1 0 1 0 0 £32y 0 -396.1635 с 15.151 1 0 -1 0 -1.6708 4.5567 B 0 0 0 0 f13z £13M t12 0 -0.799 where fij, and fij, are the forces of which link i applies onto link j along the x- and y-directions, respectively, and t12 is the motor torque applied onto link 2. Note that the last row equation represents a geometrical constraint between f322 and f32y, as there is no friction force between the pin and the slot (i.e., the orientation of the force vector f32 is completely defined by the position of the mechanism. What are the values of A, B, and C in the matrix equation? Provide your answers in the blanks below to a precision of 4 decimal digits neglecting the units. For example, if the answer is 1.524834 kg- m², then input "1.5248". 1) What is the value of A? 2) What is the value of B? 3) What is the value of C? O r₁ √3 →X