Kinematics Dynamics ch-4, Lecture notes of Dynamics

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CHAPTER FOUR

PLANE KINEMATICS OF RIGID BODIES

Introduction

• In particle kinematics , we develop the relationship governing the

displacement, velocity , and acceleration of points as they moved along a straight or curved paths.

• In rigid body kinematics we use these same relationships but must also

account for the rotational motion of the body as well.

• Thus rigid body kinematics involves both linear and angular

displacements, velocities and accelerations.

PLANAR KINEMATICS OF A RIGID BODY

• There are cases where an object cannot be treated as a particle. In these cases the size or shape of the body must be considered.
• For example, in the design of gears, cams, and links in machinery or mechanisms, rotation of the body is an important aspect in the analysis of motion.

• We will now start to study rigid body motion. In this chapter, the analysis will be limited to planar motion.
• Planar motion (often called General Plane Motion) is a specific type of movement where every point in a rigid body moves within a single plane or in planes that are parallel to one another.

• When all points move along straight lines, the motion is called rectilinear translation.
• When the paths of motion are curved lines, the motion is called curvilinear translation.

Cont…

• Rotation about a fixed axis. In this case, all the particles of the body, except those on the axis of rotation, move along circular paths in planes perpendicular to the axis of rotation.

Cont…

Cont…

• The wheel and crank (A and B) undergo rotation about a fixed axis. In this case, both axes of rotation are at the location of the pins and perpendicular to the plane of the figure.
• An example of bodies undergoing the three types of motion is shown in this mechanism. C

E

A

D B

C

E

A

D B

• The piston (C) undergoes rectilinear translation since it is constrained to slide in a straight line. The connecting rod (D) undergoes curvilinear translation, since it will remain horizontal as it moves along a circular path.
• The connecting rod (E) undergoes general plane motion, as it will both translate and rotate.

Cont…

• Note, all points in a rigid body subjected to translation move with the same velocity and acceleration.
• The velocity at B is vB = vA+ drB/A/dt (^).
• Now drB/A/dt = 0 since rB/A is constant. So, vB = vA, and by following similar logic, aB = aA.

Cont…

• The rotation of the a rigid body is described by its angular motion.
• By considering the figure, the relation
• Differentiating with respect to time gives
• During a finite interval

2 1 2 1

    

  and  

B. Rotation

• Angular velocity, , is obtained by taking the time derivative of angular displacement:

  d^  dt  rad s  +

d d or d d

d rad s

d dt

d dt

d

    ^ 

2

• Similarly, angular acceleration is

ANGULAR MOTION RELATIONS

• For rotation with constant angular acceleration, the integral become

o ^ o 

o

o t t

t

    

   

  

  

  

 

2

2

1 2 2

0 2

• (^) 𝜃o and 𝜔o are the initial values of the body’s angular position and angular velocity. Note these equations are very similar to the constant acceleration relations developed for the rectilinear motion of a particle.

Cont…

• These quantity may be expressed alternatively using the cross product relationship of vector notation.
• From the definition of the vector cross product, we see that the velocity vector v is obtained by crossing  into r, and gives the correct magnitude and direction for v

v  r   r

 

Cont…

• The acceleration is obtained by differentiating the cross product expression for v, which gives

stands for the angular acceleration of the body

  v r

r r

a v r r

   

    

     

  

 

Cont…