DYNAMICS OF RIGID BODIES

DYNAMICS OF RIGID BODIES. LECTURE 4: PLANAR KINEMATICS OF RIGID BODIES (CONT.).

REVIEW FROM PREVIOUS LECTURE. Important formulas needed to remember:.

SAMPLE PROBLEMS. In the position shown, rod OABC is rotating about the y-axis with the angular velocity ω=2.4 rad/s and angular acceleration α =7.2 rad/s^2 in the directions shown. For this position, compute the velocity and acceleration vectors of point C using (a) vector equations; and (b) scalar equations..

SAMPLE PROBLEMS. In the position shown, rod OABC is rotating about the y-axis with the angular velocity ω=2.4 rad/s and angular acceleration α =7.2 rad/s^2 in the directions shown. For this position, compute the velocity and acceleration vectors of point C using (a) vector equations; and (b) scalar equations..

SAMPLE PROBLEMS. The bent rod is rotating about the axis AC. In the position shown, the angular speed of the rod is ω=2 rad/s, and it is increasing at the rate of 7 rad/s^2. For this position, determine the velocity and acceleration vectors of point B..

METHOD OF RELATIVE VELOCITY. 6.

SAMPLE PROBLEM. When the linkage is in the position shown, bar AB is rotating counterclockwise at 16 rad/s. Determine the velocity of the sliding collar C in this position..

SAMPLE PROBLEM. Bar DE is rotating counterclockwise with the constant angular velocity ω 0 = 5 rad/s. Find the angular velocities of bars AB and BD in the position shown..

INSTANT CENTER FOR VELOCITY. 9.

SAMPLE PROBLEM. Slider C of the mechanism has a constant downward velocity of 30 in/s. Determine the angular velocity of crank AB when it is in the position shown..

SAMPLE PROBLEM. Bar BC of the linkage slides in the collar D. If bar AB is rotating clockwise with the constant angular velocity of 12 rad/s, determine the angular velocity of BC when it is in the horizontal position shown..

METHOD OF RELATIVE ACCELERATION. 12.

SAMPLE PROBLEM. In the position shown, the angular velocity and angular acceleration of bar CD are 6 rad/s and 20 rad/s^2, respectively, both counterclockwise. Compute the angular accelerations of bars AB and BC in this position..

SAMPLE PROBLEM. Bar AB of the mechanism rotates with the constant angular velocity of 3 rad/s counterclockwise. For the position shown, calculate the angular accelerations of bars BD and DE..

MOTION RELATIVE TO A ROTATING REFERENCE FRAME. 15.

[Audio] Up to this point, our kinematic analysis of rigid bodies used the formulas for relative motion between points in the same body. The coordinate system used was allowed to translate but not rotate. However, there is a class of problems associated with sliding connections in which the point of interest does not lie in a body, but its path relative to a body is known. For problems of this type, it is convenient to describe the motion of the point in a reference frame that is embedded in the body. Such a coordinate system may rotate as well as translate. Thus, these equation must be used. The velocity of point P (the unknown point) is equal to the velocity of the origin of the x-y-z axes, which are in the rigid body, B, plus the relative velocity of P' of A plus the relative velocity of P with respect to the rigid body. Note that this is quite the same as the discussion with the method of relative velocity. The body is just subdivided in to parts..

[Audio] Conversely, the acceleration for rotating bodies can also be break down into parts as shown. The Coriolis acceleration aC is perpendicular to both vP/B and ω. When the scalar representation is used, the magnitude of aC is 2ωvP/B, and its direction can be determined by fixing the tail of vP/B and rotating this vector through 90◦ in the direction of ω. To understand this , let us have some sample problems..

SAMPLE PROBLEM. Crank AD rotates with the constant clockwise angular velocity of 8 rad/s. For the position shown, determine the angular speed of rod BE and the velocity of slider D relative to BE..

[Audio] Using point A, when it rotates 8 rad/s, point D tends to move as shown..

[Audio] Since, we are done analyzing point A. We should focus on point B. Point B will move after the motion of point A. Afterwards, analyze the remain point D which is the slider..

[Audio] To identify the motion needed,. SAMPLE PROBLEM.

5 in, 09t9L 1.4 in 6,40 in. SAMPLE PROBLEM. 22.

[Audio] 16.92. SAMPLE PROBLEM. The pin P, attached to the sliding rod PD, engages a slot in the rotating arm AB. Rod PD is sliding to the left with the constant velocity of 1.2m/s. Determine the angular velocity and angular acceleration of AB when θ = 60◦..

[Audio] Same as what we have been doing from the previous lectures. Let us analyze the motion of the figures. Note that the motion of point P is the result of line AP and the velocity of point P with respect to line AB. Velocity at A is zero since no velocity was given..

[Audio] Using the figures drawn in the previous slide, we can create x and y component..

[Audio] For the second part of the problem, since the problem states that velocity is constant at point P thus the acceleration at P is assumed to be zero. In addition to this, the acceleration at point A is not given. Note that the magnitude of aB is 2ωvP/B,. The encircled figure is the Coriolis acceleration. It is perpendicular to both vP/B and ω..

METHOD OF CONSTRAINTS. 27.

METHOD OF CONSTRAINTS. Kinematic constraints : geometric restrictions imposed on the motion of points in bodies. Equations of constraint : mathematical expressions that describe the kinematic constraints in terms of position coordinates. Kinematically independent coordinates : position coordinates that are not subject to kinematic constraints. Number of degrees of freedom : the number of kinematically independent coordinates that are required to completely describe the configuration of a body or a system of bodies..

METHOD OF CONSTRAINTS. The term “position coordinate” may refer to the coordinate of a point or to the angular position coordinate of a line..

METHOD OF CONSTRAINTS. As an example, consider the system shown. The figure displays three position coordinates: 1. The angle θ is the angular position coordinate of the line OA on the disk. 2. The angle φ is the angular position coordinate of the connecting rod AB. 3. The distance x B is the rectilinear position coordinate of the slider B..

[Audio] This means that we have to differentiate an equation to identify the velocity and acceleration. To understand this , let us have some sample problems..

SAMPLE PROBLEM. Rod BC slides in the pivoted sleeve D as bar AB is rotating at the constant angular velocity θ 1 = 12 rad/s. Determine the angular velocity of rod BC in the two positions where θ 2 = 30◦..

SAMPLE PROBLEM. 33. UJUJ 081.

SAMPLE PROBLEM. Differentiation this equation with respect to time it becomes:.

SAMPLE PROBLEM. 35. UJUJ 081.

SAMPLE PROBLEM. 36. UJUJ 081.

SAMPLE PROBLEM. 37. UJUJ 081.

[Audio] 16.108. SAMPLE PROBLEM. The free end of the rope attached to bar AB is being pulled down at the rate of 2 ft/s. Find the angular velocity of AB when θ = 20◦..

SAMPLE PROBLEM. Using your knowledge in Geometry and differentiating with respect to time:.

[Audio] Substituting the given in equation 1 and 2. You may solve of L and ϕ..

[Audio] Substituting the given in equation 3 and 4. You may solve of ωAB..

PRACTICE PROBLEMS. Bar AB of the mechanism rotates with constant angular velocity of 4.5 rad/s. Determine the angular acceleration of bar BC and the acceleration of slider C in the position shown. (Answers: angular acceleration BC = 31.6 rad/s^2, CCW ; acceleration C = 228 in/s^2 to the left) Bar AB of the linkage rotates with the constant angular velocity of 10 rad/s. For the position shown, determine (a) the angular velocities of bars BC and CD; and (b) the angular accelerations of bars BC and CD. (Answers: a. 5rad/s CCW; b. angular acceleration CD = 18.75 rad/s^2, CCW ; angular acceleration BC = 28.41 rad/s^2 CCW).

LAST SLIDE OF THE PRESENTATION. 43.