Transcript B Mech Engg 4-Dynamics

Chapter 16
Planar Kinematics of a Rigid
Body
TEXTBOOK: ENGINEERING MECHANICSS TAT I C S A N D D Y N A M I C S
1 1 T H E D . , R . C . H I B B E L E R A N D A . G U P TA
COURSE INSTRUCTOR: MISS SAMAN SHAHID
Rigid Body Motion
 The study of planar
kinematics is important for
the design of gears, cams
and mechanisms used for
many mechanical
operations.
 When all the particles of a
rigid body move along
paths which are equidistant
from a fixed plane, the
body is said to undergo
PLANAR MOTION.
 There are three types of
rigid body planar motion:
 1) Translational (rectilinear
and curvilinear)
 2) Rotation about a fixed
axis
 3)General plane motion.
1- Translation
 Translation: this type of
motion occurs if every line
segment on the body remains
parallel to its original
direction during the motion.
 When the paths of motion for
any two particles of the body
are along equidistance
straight lines the motion is
called rectilinear translation,
and if the paths of motion are
along curved lines which are
equidistant the motion is
called curvilinear translation.
 2- Rotation About A Fixed
Axis:
 When a rigid body rotates
about a fixed axis, all the
particles of the body, except
those which lie on the axis of
rotation, move along circular
paths.
 3) General Plane Motion:
 When a body is subjected to
general plane motion, it
undergoes a combination of
translation and rotation.
The translation occurs
within a reference plane,
and the rotation occurs
about an axis perpendicular
to the reference plane.
Translation- Position
 Consider a rigid body which is
subjected to either rectilinear or
curvilinear translation in the x-y
plane.
 the location of points A and B in
the body are defined from the
fixed x,y reference frame by
using position vectors rA and rB.
 The translating x’,y’ coordinate
system is fixed in the body and
has its origin located at A,
hereafter referred to as the BASE
POINT. The position of B with
respect to A is denoted by the
RELATIVE POSITION VECTOR
r(B/A). By vector addition,
Translation: Velocity & Acceleration
 A relationship between the
instantaneous velocities of A and B
is obtained by taking the time
derivative of the position equation,
which yields: vB=vA
 Since these vectors are measured
from x, y axes and the magnitude
of rB/A is CONSTANT .
 Similarly: aB=aA (taking the time
derivative of the velocity equation)
 The above two equations indicate:
 “all points in a rigid body
subjected to either rectilinear or
curvilinear translation move with
the same velocity and
acceleration”
Angular
Dynamics
 When a body is rotating about a fixed axis, any point P
located in the body travels along a circular path.
 Angular Motion: Consider the body shown and the
angular motion of a radial line r located within the
shaded plane and directed from O on the axis of rotation
to point P.
 Angular Position: It is defined by the angle θ, measured
between a fixed reference line and r.
 Angular Displacement: The change in the angular
position, which can be measured as a differential dθ (in
degrees of radians). 1rev=2π rad.
right hand rule:
“the fingers of the right hand are curled with the sense of
rotation, so that in this case thumb, or dθ, points
upward. If both θ and dθ are directed counterclockwise,
and so the thumb points outward from the page.”
Cont.
 Angular Velocity: the time rate of change in the
angular displacement is called the angular velocity
ω= dθ/dt (in rad/s)
 Counterclockwise rotation is chosen as positive (and
vice-versa).
 Angular Acceleration: it measures the time rate of
change of the angular velocity. α=dω/dt
The line of action of α is same as ω, however, its sense of
direction depends on whether ω is increasing or
decreasing.
Eliminating ‘t’ between ω and α euqations:
Constant Angular Acceleration
 If the angular acceleration of the body is constant,
will yield a set of formulas which relate the body’s
angular velocity, angular position, and time.
2-Rotation About a Fixed Axis
 Motion of point P: As the rigid
body rotates point P travels
along a circular path of radius
r and center at O.
 Position: the position of P is
defined by the position vector
r, which extends from O to P.
 Velocity: the direction of v is
tangent to the circular path.
The fingers of the right hand are
curled from ω toward r(p) and
the thumb indicates the
correct direction of v, which is
tangent to the path in the
direction of motion.
Cont.
 Acceleration (tangential): it
represents the time rate of
change in the velocity’s
magnitude. If the speed of P
increasing, then a(t) acts in
the same direction as v; if the
speed is decreasing, a(t) acts
in the opposite direction of v;
and finally, if the speed is
constant, a(t) is zero.
 Acceleration (normal): it
represents the time rate of
change in the velocity’s
direction. The direction of
a(n) is always toward O, the
center of the circular path.
Example
 The gears used in the
operation of a crane all
rotate about fixed axes.
Engineers must be able
to relate their angular
motions in order to
properly design this gear
system.