Kinematics of Rigid Body

7/30/2019 Kinematics of Rigid Body

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Chapter 2Kinematics of Rigid Bodies

2.1 lntroductionRigid BodyA rigid body is a body consisting of a large number of particles, with the relative distancebetween these particles being constant. When the body moves, every particle in the bodywill move according to its own locus or path depending on the nature of the motionundergone by the rigid body.Types of Rigid Body Motion.The motion of a rigid body may be categorized into :o Rectilinear motion - or motion in a straight lineo Circular motion - or rotation about afixed axiso Curvilinear motion - or general motionFigure 2.I shows a mechanism, known asslider-uanft, which is widely used as the basisfor numerous machines. In this mechanism,piston B undergoes rectilinear motion, whilecrank OA undergoes circular motion.Connecting rod AB undergoes curvilinearmotion.These types of motion will be studied in greaterdetail in the ensuing sections..

2.2 Rectilinear MotionIn this type of motion, all the particles in a rigid bodymove in paths that are straight and parallel to oneanother as shown in Figure 2.2.In this figure, it can be seen that all the particles in thebody will have exactly the same motion, in terms ofdirection and magnitude. Consequently, the motion ofthe rigid body can be represented by the motion of asuitably chosen point or particle. Such point is usuallythe centre of gravity of the body, and its motion can beanalysed using the method for kinematics of particles.

Figure 2.1

Figure2.2


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2.3 Circular MotionIn this motion, the entire rigid body rotates about a fixed point.Strictly speaking, the body rotates about an axis that passesthrough the fixed point, perpendicular to the plane of therotation. The particles on the body move in circular paths asshown in Figure 2.3.The radii of these paths will differ fromone another, according to the distance of each particle from thecentre of rotation.Notice that all the particles will undergo the same amount ofrotation about the fixed point. In other words all the particleswill have the same angular displacement about the fixed pointO. Therefore :0r=0,where 0, is angle AOA' and 0u is angle BOB'Successive differentiation with respect to time will yield :

( or rrlT : roa )

Figure 2.3

oo: ouand

6r:6, (or a7 :crs)It can be seen that in this type of motion, the angular motion of the rigid body can beknown by studying the angular motion of a suitably chosen particle in the body, using themethod of particle kinematics studied earlier.

2.4 Gurvilinear Motion ln Two DimensionsIn this type of motion, a rigid body undergoes general motion in a plane. Hence thismotion is also called general planar motion. This motion may be regarded as being acombination of linear motion and rotation. In Figure 2.4 (a) a rigid body moves fromposition (1) to position (2).

(2)

Figure 2.4a


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This motion maybelow.

be accomplished in two stages as shown in Figures 2.4 (b), (c) and (d)

1)Figure 2.4bLinear motion from (1) torotation about A, from (1')(2).

(1'), followed by ato the final position

( 1')

Figure 2.4cInitial rotation about A from (1) to (1'),followed by linear motion from (1') to thefinal position (2).

Figure 2.4dInitial rotation about B from (1) to (1'),followed by linear motion from (1') to(2).

lf we scrutinize Figures 2.4 (b) to (d), wewill notice important characteristics ofrigid body motion, namely that :, The linear component of the motion will dffir depending on the particle chosen asour reference point.ii) The rotational component of the motion is the same in all three /igures, Regardlessof which particle is taken as a reference point, the magnitude of rotation is the same,

and the direction of rotation is also the same.ii, The relative rotation between the particles is the some os the absolute rotation of thewhole rigid body.The last observation above is in fact a very important property of rigid body motion, and

can be clarified by the figures above. Notice that in Figures 2.4b and c, particle B rotatesabout particle A through an angle ,0u. This is rotation of B relative to A, and it takesplace in a clockwise direction. In Figure 2.4d, particle A rotates about particle B through

(2)

(l)

/A\t reilAJt2\ \B/'',frGN

.A


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direction. Thus we can say that uou = Be A .If we continue this line of analysis to otherpoints in the bodY, we can see that :t}n = ul, = n9c = rOp = ...,.. etc.

Successive differentiation with respect to time will yield :etc. andetc.

l@B = B0A = A@c = 8oD = ......AdB= adA= A&C= BdD=,,...,2.5 Velocity DiagramThe characteristics of rigid body motion identifiedabove form the basis of a method for analyzing thevelocities of rigid bodies, known as the velocitydiagram method. To help understand this method let usconsider the case of a rod (or ladder) sliding against avertical wall as shown in Figure 2.5.Inthis motion, therod moves from position (1) to position (2).As in Section 2.4 above, we may regard this motion ascomprising a linear motion from (1) to (l'), followedby a rotation about B from (1') to (2), as shown inFigure 2.6. Consequently, the linear velocity of everypoint in the rod is the same as the linear velocity ofpoint B. In addition, due to the rotation about B' otherpoints on the body will also have a tangential velocityrelative to B. For example, point A will have a velocityrelative to B ( called uv o ) which acts perpendicular tothe radius BA. This relative velocity will have amagnitude of :

nlA = uauxBAwhere u co, is the angular velocity of A relative to B.Thus point A has two velocity components, oneproduced by the linear motion, and another producedby rotation about B. These two velocity componentscan be combined vectorially to give the total velocityof A as shown in Figures 2.7a and 2.7b.

AJA

B_BFigure 2.5

Figure 2.6

AvA

B-B

Figure2.Ta Figure 2.7b


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Figure 2.7a shows the velocity components of point A, i.e. vu and uv, being combinedusing the parallelogram law of vector addition, to produce the resultant vu . Notice thatthis resultant velocity of point A must act along the surface of the vertical wall. The sameresultant can also be obtained by the method of completing the polygon as shown in Figure2.7b, and this is the technique used in the vetocity diigram method. In this case, theconstruction of the velocity diagramcomplies with the u..io, equation:v.a = ln I at.t

The velocity diagram method is sometime referred to as the relative velocity method. Wewill illustrate a proper procedure for using the velocity diagram method by means of anexample.

Example 2.1A rod of length 50 cm. slides on a vertical wall as shownin Figure Ex2.la. When 0 = 60" end B of the rod has avelocity va = 2 cm/s to the right. Determine the velocity

Iof end A and the angular velocity of the rod at that Iinstant.Solution:Notice that in this problem the velocity of point B iscompletely known. In other words, its magnitude anddirection are known. Next, the velocity of A relative to Bmust be at right angles to the rod AB since this rod isrigid. And finally, the total velocity of A must be along t''NutY Lrzthe vertical wall. With these pieces of information we .* pro...d to construct the velocitydiagram for A.This diagram is drawn starting with a line o-b to represent the known velocity va = 2cm/s. This line is drawn in the same direction as vo ( i.e. running left to right ), and is of asuitable length to represent 2 cmls. The start of the line is labeled ,o, to indicate that theline represents an absolute velocity. Next, we draw a second line which runs perpendicularto rod AB. This line represents the direction of the velocity of A relative to B. This line isdrawn passing through 6. Finally, we draw a third linewhich runs vertically to represent the direction of thevelocity of A. This line is drawn passing through oo' toindicate that this velocity is also an absolute velocity.We will finally get a velocity diagram as shown inFigure Ex2.1b. The intersection between the second andthird lines is labeled 'a' to show that it represents thetip of the vectors for point A.From this diagram, it can be seen that :'u - tan6o' and 'u - sin6o"v A nv,q

l-- yBFigure Ex2.Ia

Figure Ex2.1b


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ThereforevB2-r^---,---rlvA = -- -: = -- -- = 1.155 cm/s downwards, and" tan60" tan60' :

ovo = ,'u - 2 =2.309cm/ssin60" sin60' ::But BvA = na.tx AB = uro, x 50

aa.q = u!! = ry = 0.0461 8 rad/s anticlockwise.5050:More practice work with different arrangements of rigid bodies can help make the drawingof velocity diagrams almost an intuitive process. Then, this method can become an easyand efficient way of finding the velocities of rigid bodies in planar motion.

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Kinematics of Rigid Body