# Problems in Chapter 20 (3D Kinematics) kinematics ω α

## Transcription

Problems in Chapter 20 (3D Kinematics) kinematics ω α

Problems in Chapter 20 (3D Kinematics) Ú How to analyze the kinematics problems r r 1. Determine the angular velocity ( ω ) and angular acceleration (α ) of a body at a specific instance or r r Determine the velocity ( v ) and acceleration ( a ) of a point in the body at a specific instance. Select the appropriate XYZ (fixed) and/or xyz (moving) frames - Coincident origins & axes at the instance of interest 2. In case of motion of one moving object (Fixed or Translational R.F.) r r r r - vB = v A + ω × rB / A r r r r r r r a B = a A + α × rB / A + ω × (ω × rB / A ) 3. In case of motions of several moving objects (Rotating R.F.) r& r - xyz: Attached to a moving object then, determine Ω & Ω of xyz r r r r r - vB = v A + Ω × rB / A + (vB / A ) xyz r r r r r r& r r r r aB = a A + Ω × rB / A + Ω × (Ω × rB / A ) + 2Ω × (vB / A ) xyz + (a B / A ) xyz Example 1. The conical spool rolls on the plane without slipping. If the axle has an angular velocity of ω1 = 3 rad/s and an angular acceleration of α1 = 2 rad/s2 at the instant shown, r r determine ω and α of the spool at this instant. Example 2. One end of the rigid bar CD shown in figure slides along the horizontal member AB, and the other end slides along the vertical member EF. If the collar at C is moving towards B at a speed of 3 m/s, determine the velocity of the collar at D and the angular velocity of the bar at the instant shown. The bar is connected to the collar at its end points by ball-and-socket joints. Example 3. The pendulum shown in figure consists of two rods; AB is pinsupported at A and swings only in the YZ plane, whereas a bearing at B allows the attached rod BD to spin about rod AB. At a given instant, the rod have the angular motions shown. Also a collar C, located 0.2 m from B, has a velocity of 3 m/s and an acceleration of 2 m/s2 r r along the rod. Determine v and a of the collar at this instant. Example 4. At the instant shown, rod BD is rotating about the vertical axis with an angular velocity ω BD = 7 rad/s and an angular acceleration α BD = 4 rad/s2. Also θ = 60o and link AC is rotating downward such that θ& = 2 rad/s and θ&& = 3 rad/s2. r r Determine v and a of point A on the link at this instant. Z, z, z’ X, x, x’ Y, y, y’