MOTION RELATIVE TO ROTATING AXES KINEMATICS OF RIGID BODIES MOTION RELATIVE TO ROTATING AXES

Translating+Rotating reference axes In the previous article, we have used nonrotating reference axes to describe relative velocity and relative acceleration. Rotating as well as translating reference axes are used if the body lies in the rotating system or is observed from a rotating system. Let’s consider the plane motion of two particles A and B in the fixed X – Y plane. x y X Y w,a B(x,y) A(x,y) Translating+Rotating reference axes Fixed reference axes O

Translating+Rotating reference axes We will consider A and B to be moving independently of one another. We observe the motion of A from a moving reference frame x-y which has its origin attached to B and which rotates with an angular velocity ;the vector notation will be . x y X Y w,a B(x,y) A(x,y) Translating+Rotating reference axes Fixed reference axes O

The absolute position vector of A is Time Derivatives of Unit Vectors Since the unit vectors and are now rotating with the x-y axes, their time derivatives will not be zero.

By using the cross product,

Relative Velocity We are now going to take the time derivative of the position vector

is the absolute velocity of B due to the translation of axes x-y is the absolute velocity of B due to the translation of axes x-y. If the x-y axes are not translating but only rotating , = 0.

The term appears due to the rotation of the x-y axes. Its magnitude is or and a direction normal to .

will be tangent to the path fixed in the x-y plane and its magnitude will be equal to where s is measured along the path. Its sense will be in the direction of increasing s.

Relative Acceleration The relative acceleration equation may be obtained by differentiating the relative velocity equation, * ** * **

vrel

The magnitude of is and its direction is tangent to the circle The magnitude of is and its direction is tangent to the circle. The magnitude of is rw2 and its direction is from P to B along the normal to the circle.

may be expressed in rectangular, normal and tangential or polar coordinates in the rotating system. Frequently, n and t components are used as shown in the figure. The tangential component has the magnitude where s is the distance measured along the path to A. The normal component has the magnitude . The sense of this vector is always toward the center of curvature.

Coriolis Acceleration The term seen in the figure is called the Coriolis acceleration. It represents the difference between the acceleration of A relative to P as measured from nonrotating axes and from rotating axes. The direction is always normal to the vector or and the sense is established by the right hand rule for the cross product.

The Coriolis acceleration appears when a particle or body translates in addition to its rotation relative to a system which itself is rotating. This translation can be rectilinear or curvilinear.

PROBLEMS 1. The disk rolls without slipping on the horizontal surface, and at the instant represented, the center O has the velocity and acceleration shown in the figure. For this instant, the particle A has the indicated speed u and the time rate-of-change of speed , both relative to the disk. Determine the absolute velocity and acceleration of particle A.

PROBLEMS 2. For the instant represented, link CB is rotating ccw at a constant rate N = 4 rad/s and its pin A causes a cw rotation of the slotted member ODE. Determine the angular velocity w and angular acceleration a of ODE for this instant.

PROBLEMS 3. Link OA has a constant clockwise angular velocity of 3 rad/s for a brief interval of its rotation. Determine the angular acceleration aBC of BC for the instant when q=60o.

PROBLEMS 4. Determine the angular acceleration of link EC in the position shown, where and when q=b=60o. Pin A is fixed to link EC. The circular slot in link DO has a radius of curvature of 150 mm. In the position shown, the tangent to the slot at the point of contact is parallel to AO. 150 mm

PROBLEMS 5. For the instant shown, particle A has a velocity of 12.5 m/s towards point C relative to the disk and this velocity is decreasing at the rate of 7.5 m/s each second. The disk rotates about B with angular velocity w=9 rad/s and angular acceleration a=60 rad/s2 in the directions shown in the figure. The angle b remains constant during the motion. Telescopic link has a velocity of 5 m/s and an acceleration of -2.5 m/s. Determine the absolute velocity and acceleration of point A for the position shown.

C O B A w=2 rad/s 2 m a=4 rad/s2 PROBLEMS 6. The gear has the angular motion shown. Determine the angular velocity and angular acceleration of the slotted link BC at this instant. The pin at A is fixed to the gear. w=2 rad/s a=4 rad/s2 0.5 m 0.7 m 2 m O B A C

PROBLEMS 7. The pin A in the bell crank AOD is guided by the flanges of the collar B, which slides with a constant velocity vB of 0.9 m/s along the fixed shaft for an interval of motion. For the position q=30o determine the acceleration of the plunger CE, whose upper end is positioned by the radial slot in the bell crank. .

PROBLEMS 8. Link 1, of the plane mechanism shown, rotates about the fixed point O with a constant angular speed of 5 rad/s in the cw direction while slider A, at the end of link 2, moves in the circular slot of link 1. Determine the angular velocity and the angular acceleration of link 2 at the instant represented where BO is perpendicular to OA. The radius of the slot is 10 cm. Take sin 37=06, cos 37=0.8 37o 10 cm 20 cm A O 2 1 w1=5 rad/s C 16 cm B