Plane Kinematics of Rigid Bodies q Kinematics study

Plane Kinematics of Rigid Bodies q Kinematics Østudy of body motion without reference to force. q Rigid Body Ø It has dimensions. (particle doesn’t have it). Ø distance between 2 points in the body remains unchanged. Ø assumption validity? i. e. there is no real rigid body. q Plane Motion Ø Definition: All parts of the body move in parallel planes. 1

Plane Motion Definition: All parts (points) of the body move in parallel planes. Motion Plane Movement of one cutting face (corresponding to its motion plane) describes movement of the whole body. Treat the body as thin slab object. Any slab object (cutting face) is okay, But we usually use the plane where the object’s C. G. is in. C. G. All corresponding points in other motion plane have the same velocity and acceleration 2

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Rigid-Body Plane Motion I (Pure) Translation II (Pure) Rotation III General Motion 4

Type of Plane motion C. G. q (Pure) Translation Definition: A line between two points in the body remains parallel through out the motion. Motion of one point can be used to describe motion of the whole body. Rectilinear Translation Curvilinear Translation C. G. Treat it as a particle 5

Types of Plane Motions (cont. ) q Fixed-axis (Pure) Rotation: All , move along circular path with center at Not having the same velocity and acceleration (depends on circle radius r) Use techniques for particle (circular) motion (n-t, r- ) Important Property of Fixed-Axis Rotation axis motion plane Rotation Axis Motion Plane

Types of Plane Motions (cont. ) q General Plane motion: Ø need new techniques in this chapter 7

5/2 Rigid Body’s Rotation same w, a? b anay il ne Angle between any two lines on “rigid” object does not change during the time “Rotating” concept is basically a concept on rotation of “line”. a concept on (whole) rigid body. Define angular position (+/-) angular velocity + any Reference 0 for rigid body angular acceleration Any lines on a rigid body in its plane of motion have the same angular displacement, velocity and acceleration 8

(a) Angular Motion Relations Define l y an angular position (+/-) angular velocity angular acceleration + any Reference Similar to rectilinear motion Sign convention of all variables must be consistent! 9

Angular Motion Relations q Observed the similarity with the linear motion 2) Graphical meanings 1) Integrals Calculation 10

a=4 t CCW: + w = - 60 p a=4 t w=0 a=4 t w= + 203. 50 A flywheel rotating freely at 1800 rev/min clockwise is subjected to a variable counterclockwise torque which is first applied at time t = 0. The torque produces a counterclockwise angular acceleration = 4 t rad/ss, Determine the total number of revolutions, clockwise plus counterclockwise, turned by the flywheel during the first 14 seconds of torque application. + 11

Fixed-Axis (Pure) Rotation (scalar notation) Can we find? rw Rigid body P rww whole body (rigid body) points ra O w a Point P n-t coord: Fixed-Axis (Pure) Rotation Only ! 13

The pinion A of the hoist motor drives gear B, which is attached to the hoisting drum. The load L is lifted from its rest position and acquires an upward velocity of 2 m/s in a vertical rise of 0. 8 m with constant acceleration. As the load passes this position, compute (a) the acceleration of point C on the cable in contact with the drum (b) the angular velocity and angular acceleration of the pinion A. w, a CCW 14

n n t t floor- fixed Non-slipping share the same Non-slipping Why? share the same =0 =0 Without gear teeth, the ball is not guaranteed to roll without slipping. Two possible motion: - roll without slipping - roll with slipping. If it do really roll without slipping, It should have some “motion constraint”. [ fixed relation between w and v ] (Next session: we will find out this) In case of No gear-teeth 15

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Fixed-Axis (Pure) Rotation (vector notation) The body rotating about the O axis q Represent “angular velocity” w as Vector q Ø q Direction: “Right-hand rule” From the last section, the magnitude of the velocity is Think in 3 D Vector We consider only Plane motion of rigid body q The cross product can be used to establish the direction: 18

Fixed Axis Rotation (vector notation) q Differentiating the velocity with respect to time: Direction depends on We consider only “Plane motion” of rigid body Direction: 19

Fixed-Axis (Pure) Rotation (vector notation) 20

The rectangular plate rotates clockwise. If edge BC has a constant angular velocity of 6 rad/s. Determine the vector expression of the velocity and acceleration of point A, using coordinate as given. Rigid body 21

Equations Review: (Pure) Translation Movement of one point describes movement of the whole body. (Pure) Rotation rotation of one line describes rotation of the whole body. Whole body shares the same “angular” quantities. an ine l y + Reference General Plane Motion Absolute motion Relative motion 23

5/3 Absolute Motion (of General Plane Motion) q Use geometric constraint (which define the configuration of the body to obtain the velocity and acceleration (in general motion) q Idea: write a (position) constraint equation which always applies regardless of the system’s configuration, then differentiate the equation to get velocity and acceleration. w? for complex constraint method of relative motion may be easier. 24

w? Define the displacement and its positive direction. The variable must be measured from the fixed reference point or line. Find the equation of constraint motion. Equation must be true all during the motion. Differentiate it to find (angular) velocity and acceleration. 25

5/56. Express the angular velocity and angular acceleration of the connecting rod AB in terms of the crank angle for a given constant. b 0 26

motion constraint SP 5/4 A wheel of radius r rolls on a flat surface without slipping. Determine angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the points comes into contact with the surface on which the wheel rolls w Point O: rectilinear motion Point C’s trajectory 27

SP 5/4 A wheel of radius r rolls on a flat surface without slipping. Determine angular motion of the wheel in terms of the linear motion of its center O. Also determine the acceleration of a point on the rim of the wheel as the points comes into contact with the surface on which the wheel rolls Not depend on a(t), w(t) are! velocity =0 Acceleration in the direction of axis x = 0 O D C’’ When Point D comes to contact the surface, It also has a velocity (=0), and 28 acc. as above.

Acceleration in the direction of axis x = 0 velocity =0 O O C’’ Analogy No slipping Rel. vel. =0 Rel acc. = 0 29

Non-slipping Condition O t floor- fixed C’’

“no slipping” implies: 1) Contact point { C , C’ } on two body has no relative velocity. 2) Contact point { C, C’ } on two body has same tangential component of acceleration 31

Each cables do not slip. Load-supporting pulleys are rigid body. A O B O and L has same vertical velocity & acceleartion 32

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General Plane Motion of Rigid Body นยามการเคลอนทของวตถเกรง -Introduction -Rotationวธอธบายการเคลอนทแบบหมน - Calculation methods usually for any t -Absolute motion ใชคำนวณ V -Relative velocity ใชคำนวณ V Instantaneous Center of Zero Velocity (ICZV) ใชคำนวณ V -Relative acceleration ใชคำนวณ V A Motion relative to rotating axes ใชคำนวณ V A A using constraint equation Observer is at the point of rigid body where its velocity = 0 Translati ng-only observer using its geometric shape at that instant usually for Translating some t and rotating (instant) Observer 35

5/4 Relative Velocity q relative velocity concept Different viewpoint General Plane Motion = Translation + Rotation Motion of point (observer) B, detected by O = Motion of plate moving “translationally” “simultaneous” Since the distant between the two points on a rigid body is constant, an observer at one point will see the other point move in a circular motion around it! Wait! B really sees A moving circularly? O B sees A has no movement !? !? !? 36

Applying the relative concept q Observer B is on the plate Rotating observer (attached to B) Rotating frame Which one? Only this case q Observer B is sitting on the magic carpet. non-rotating observer (attached to B) non-rotating frame A A A B B B see A no moving at all B see A having a velocity perpendicular with its distance. 37

Relative Velocity q (non-rotating observer) We use: non-rotating observer (frame) attached to B Absolute world: Relative world: same? Only when non-rotating observer (see next page) Observer O detects: Observer at B see A moving in a circle around it Observer B detects: 38

Translating observer see same w, a as absolute Observer The rotating of rigid body = The rotating of line compared with “fixed” reference axis same w same a B B’s reference line B O O’s reference line 39

Translating-Rotating observer sees different w, a with absolute Observer The rotating of rigid body = The rotating of line compared with “fixed” reference axis different w different a B B’s reference line B O O’s reference line The rotating B’s “reference line”, observed by absolute observer. 40

Understanding the equation absolute Hint on solving problems q Any 2 points the and unknown Identify the on known same rigid body A q Above equation (2 D: “ 3 D-fake”) can be solved when there at most 2 unknown scalar quantities B q Above equations usually contains 5 scalar quantities (not including position vector r) Non-rotating, Moving with B Important key: always perpendicular to line AB. Its direction can be deduced from Also works with A as the observer 41

Relative Vector Analysis on General Plane Motion Fixed-Axis (Pure) Rotation w P G w w of observer at G = w (of rigid body) of observer at O G P 42

Velocity at A is key point to find w A, B on the same rigid body (bar AB) Solved by Vector Analysis 44

From i k A y O x + j C B + 3 D vector calculation (i, j, k): Sign indicates angular direction (right hand rule) 45

A, B on the same rigid body (bar AB) D M ? ? Solved by Graphical Method CW CW need to find the angular direction from the figure 46

middle link 47

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Note: relative velocity technique Direction is simply found: q Ø graphical solution: simplest, be careful about sign / Ø 3 D (i, j, k) vector: complicated, automatic sign indication : same in absolute and relative world q Non-rotating To know q B A of the rigid body B A 49

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D w. ABC Graphical solution M ? ? CCW 51

v. B D M ? ? v. A Vector solution Direction? CW 52

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Relative Velocity (Part 2) Another usage of the relative velocity equation q A on OD D B on screw B A 2 points need not be in the same rigid body parerell Ø For constrained sliding contact between two links in a mechanism. Ø Pick points A and B as coincident points, one on each link (the points may be imaginary). some reason later! The observer on B no longer see A moving around it in a circle. c. c. w. 54

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B 0. 175 P f O 0. 175 cos 0. 4 -0. 175 cos M D ? ? CW Ans 56

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w=? Q on slot C V=1. 5 =30 Non-slipping condition 0 D y C b x v. Q v. P v. O O 100 200 v. Q/P ? ? Plus CCW 58

5/6 relative Relative acceleration Acceleration concept q Different viewpoint General Plane Motion = Translation + Rotation Motion of point (observer) B, detected by O = Motion of plate moving “translationally” “simultaneous” Since the distant between the two points on a rigid body is constant, an observer at one point will see the other point move in a circular motion around it! Wait! B really sees A moving circularly? O B sees A has no movement !? !? !? 61

Relative Acceleration q (non-rotating observer) We use: non-rotating observer (frame) attached to B Absolute world: Relative world: same? Only when non-rotating observer (see the proof at relative velocity part) Observer O detects: Observer at B see A moving in a circle around it Observer B detects: 62

Understanding Equations Non-rotating : the same both in absolute and relative (translation-only) world Hint on solving problems q Identify the known and unknown q Above equation (2 D: “ 3 D-fake”) can be solved when there at most 2 unknown scalar quantities q Above equations usually contains 6 scalar quantities (not including position vector r) 63

= Given : + (constant) Find : 64

Given : Find : (constant) (at this instant) ANS 65

Find : (at this instant) Find velocity first, Before acceleration 66

5/124 The center O of the disk has the velocity and acceleration shown in the figure. If the disk rolls without slipping on the horizontal surface, determine the velocity of A and the acceleration of B for the instant represented. Non-slipping condition a w D You can calculate using point O and D. I. C. Z. V 69

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w. OA v. E O v. A B w. BD v. D 71

a. OA w. OA=20 O 72

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General Plane Motion (non-rotating observer) any point: A, B (on same rigid body moving in GPM) Graphical Approach Cross-Vector Approach New technique I. C. Z. V B is special point : I. C. Z. V 75

Note: relative velocity technique Direction is simply found: q Ø graphical solution: simplest, be careful about sign / Ø 3 D (i, j, k) vector: complicated, automatic sign indication : same in absolute and relative world q Non-rotating To know q B A of the rigid body B A 76

Checkpoint: circular motion Fixed-point rotation (Rotation) Don’t know it is fixed-point rotation or not (General Motion) ? valid method? from rotation point Yes! but show your reason! I. C. Z. V concept 78

General Plane motion 5/5 Instantaneous Center of Zero Velocity q Extension theory using relative velocity. B A P Z w q w of observer at C = w of rigid body (in Absolute Observer’s perception) Z is called I. C. Z. V (the point where its velocity at that instant is zero) - can find w easily by geometry - can find velocity and its direction of any points easily by geometry each point on the body can be though of as rotating around point Z. For calculating v and w only 79

Finding an I. C. Z. V. not a rigid body z B General Plane motion “instantaneous” Translational motion A I. C. Z. V A A B z B z A B A z w of observer at C = w of rigid body (in your perception) A D B at Inf. I. C. Z. V for calculating instantaneous velocity only az usually 0 (Even vz = 0)

Arm OB of the linkage has a clockwise angular velocity of 10 rad/s in the position shown where = 45°. Determine the velocity of A, the velocity of D, and the angular velocity of link AB for the instant shown. w of what? OA, AB, BO solved by relative velocity I. C. Z. V D 350 M 381 w You have to find Direction yourself ? ? Direction of (absolute) velocity of two point in the same rigid body Thus, we can locate the instantaneous center of velocity, which is point C 82

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v. C y x a r. C v. B ICZV v. A Vector Diagram r. B CCW 84

Non-slipping motion ICZV of P 1 CW ICZV of sun ICZV of A CW ICZV of P 2 Sun Gear: Fixed-Axis (Pure) rotation CCW Planet Gear: general Plane motion 85

ICZV 100 mm ICZV 86

I. C. Z. V Non-slipping condition Absolute motion rectilinear I. C. Z. V n circle t 5/140 I. C. Z. V 87

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VD 1 v. C 1 I 8 rad/s v. C 1 =8 r 4 rad/s VD 0 ICZV VD 2 v. H 3 CW II III v. H 2 = 8 r v. C 2 d ICZV v. H 3 VD 3 CCW

Practice Before Rotating Observer 91

5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position. CCW ICZV of AB CW 92

5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position. CCW ICZV of AB =0 CW 93

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ICZV of DAB is at infinite CW 3 D vector solution 0 Graphical solution CW 95

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middle link 97

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5/134 The sliding collar moves up and down the shaft, causing an oscillation of crank OB. If the velocity of A is not changing as it passes the null position where AB is horizontal and OB is vertical, determine the angular acceleration of OB in that position. CCW ICZV of AB =0 CW 100

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ICZV at inf. AB translational. CW CW 102

CCW 103

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