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KINEMATICS AND DYNAMICS OF PLANE MECHANISMS
JEREMY HIRSCHHORN ~
A ssociafe Professor of Mechanical Engineering The University of New South Wales Sydney, N.S. W., Australia
McGRAW-HILL BOOK COMPANY, INC. New York
San Francisco
Toronto
London
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UNIVERSITY OF MICHIGAN
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CONTENTS Preface
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Introduction .
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1-1. 1-2. 1-3. 1-4. 1-5. 1-6. 1-7. 1-8.
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Fundamentals of Vector Analysis
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Kinematics of the Plane Motion of a Particle .
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Kinematics of the Plane Motion of a Rigid Body .
Definitions and Basic Concepts . • • • Types of Plane Motion. . . . • The Velocity Pole, or Instant Center of Rotation . • Determination of Velocities by Means of the Velocity Pole . . . • Determination of Velocities by Means of Orthogonal Velocity Vectors . Polodes . . . . . . . . . . Relative Motion of Physically Connected Particles . General Motion as Superposition of Translation and Rotation • The Velocity Image. . . . . . . • • The Acceleration Image . Graphical Solution of the Velocity and· Acceleration Equations • • The Acceleration Pole, or Instantaneous Center of Accelerations .
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Definitions . . . Cartesian Reference Frame Polar Reference Frame. . Moving Reference Frame . • Summary. The Radius of Curvature Graphic Differentiation and Integration . . Method of Finite Differences . . . . . Absolute and Relative Motion; Relative Motion of Separate Particles Problems. . . . .
Chapter 3. 3-1. 3-2. 3-3. 3-4. 3-5. 3-6. 3-7. 3-8. 3-9. 3.:10. 3-11. 3-12.
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Scalars and Vectors . . . . . Vector Notation; Unit Vector; Representation of Vectors. Composition, Subtraction, and Resolution of Vectors . . Multiplication of a Vector by a Pure Number or a Scalar . The Vectorial, or " Cross," Product of Two Vectors . . The Scalar, or " Dot," Product of Two Vectors . . . Differentiation of a Vector with Respect to Time . . . Problems . . . . . .
Chapter 2. 2-1. 2-2. 2-3. 2-4. 2-5. 2-6. 2-7. 2-8. 2-9. 2-10.
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Engineering System of Units
Chapter 1.
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15 15 16 18 19 20 23 28 29 30
32 32 34 35 38 38
40 43 44 44
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CONTENTS
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3-13. Acceleration of the Velocity Pole Ps . . 3-14. Acceleration of the Center of Curvature of the Moving Polode 3-15. Problems . . . . . .
Chapter 4.
Kinematics of Simple Mechanisms .
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52 55
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56
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4-1. Definitions . • • 4-2. Degree of Freedom of a Mechanism . • • • . . . 4-3. Inversions of a Linkage . 4-4. Relative Motion of Two Planes; the Relative-velocity Pole . . . . 4-5. Relative Motion of Three Planes; Kennedy's Theorem . . . . . . . . . 4-6. Velocity Poles in Mechanisms . 4-7. Velocity Analysis of Mechanisms by Means of Velocity Poles . . . 4-8 . Velocity Analysis of Mechanisms by Means of Orthogonal Velocities . 4-9. Velocity Analysis of Mechanisms by Means of Relative Velocities . . . 4-10. Comparison of the Three Methods of Velocity Analysis . . 4-11. Acceleration Analysis of Mechanisms by Means of Relative Accelerations . . 4-12. Analysis of Mechanisms with Rolling Pairs . . . 4-13. Analysis of Mechanisms with Sliding and Slip-rolling Pairs in Motion; Coriolis Component. . . . . 4-14. Problems . . .
Chapter 5.
Kinematics of the Slider-crank Mechanism
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5-4. 5-5. 5-6. 5-7. 5-8.
5-9. 5-10.
. . • Conventional Velocity and Acceleration Diagrams . Simplified Construction of the Velocity Diagram . • • • Klein's Construction of the Acceleration Polygon . Ritterhaus's Construction of the Acceleration Polygon . . Harmonic Analysis of the Slider Acceleration of In-line Mechanisms . Approximate Expressions for the Slider Displacement, Velocity, and Acceleration in In-line Mechanisms . . . . . . . Graphical Determination of the Slider Displacement . . The Acceleration-Displacement Curve . . . . . • Problems. . •
Chapter 6. Kinematics of Complex Mechanisms 6-1. 6-2. 6-3. 6-4. 6-5. 6-6. 6-7.
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Complex Mechanisms; Low and High Degree of Complexity . Goodman's Indirect Acceleration Analysis • • • • Method of Normal Accelerations . • • • • Hall's and Ault's Auxiliary-point Method • • • Carter's Method . • • • • Comparison of Methods • • • • • • Problems . • • • • • • • • • • • •
Chapter 7. 7-1. 7-2. 7-3. 7-4. 7-5. 7-6. 7-7.
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106 107 108 109 112 113 115 116 117 119
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Fundamental Principles of Dynamics and Statics.
Dynamics of a Particle; Laws of Motion . • • • • Dynamics of the Plane Motion of a Rigid Body • • Equivalent Mass Systems . • • • • • • Work, Power, Energy; Conservation of Energy; Virtual Work D' Alembert's Principle; Inertia Force • • • • Some Simple Problems of Statics. Friction . • • • • • • •
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80 82 82
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5-1. Nomenclature and Conventions . 5-2. 5-3.
59 59 61 63 68 69 70 76
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145 146 152 156 160 162 165
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CONTENTS
Chapter 8. 8-1. 8-2. 8-3. 8-4. 8-5. 8-6. 8-7. 8-8.
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Dynamic Motion Analysis
Nomenclature and Conventions • The Energy-distribution Method . The Equivalent-mass-and-force Method . The Rate-of-change-of-energy Method Effects of Friction • • • Summary and Conclusions . Problems. • • • • •
Chapter 10.
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Introduction . • • Free-body Diagrams • lllustrative Examples Friction in Link Connections • Forces in Nonsymmetrical Linkages . Stress Determination in Moving Members Gyroscopic Effects . Problems. • •
Chapter 9. 9-1. 9-2. 9-3. 9-4. 9-5. 9-6. 9-7.
Forces in Mechanisms .
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Advanced Kinematics of the Plane Motion .
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167 167 168 181 184 188 190 193
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199 200 205 207 209 213 214
217
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10-1. 10-2. 10-3. 10-4. -5. 10-6. 10-7. 10-8. 10-9. 10-10.
The Inflection Circle; Euler-Savary Equation . • • • Analytical and Graphical Determination of d, Bobillier's Construction; Collineation Axis . • Hartmann's Construction . . The Inflection Circle for the Relative Motion of Two Moving Planes . Application of the Inflection Circle to Kinematic Analysis . Polode Curvature (General Case). Polode Curvature (Special Case); Hall's Equation . Polode Curvature in the Four-bar Mechanism; Coupler Motion . Polode Curvature in the Four-bar Mechanism; Relative Motion of the Output and Input Links; Determination of the Output Angular Acceleration and Its Rate of Change . . 10-11. Freudenstein's Collineation-axis Theorem; Carter-Hall Circle . • 10-12. The Circling-point Curve (General Case) . • 10-13. The Circling-point Curve for the Coupler of a Four-bar Mechanism .
Chapter 11. 11-1. 11-2. 11-3. 11-4. 11-5. 11-6. 11-7. 11-8. 11-9. 11-10. 11-11. 11-12.
Introduction to Synthesis; Graphical Methods
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217 222 224 225 232 233 235 239 241
243 245 248 251
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254 The Four-bar Linkage . • • 262 Guiding a Body through Two Distinct Positions . • 266 Guiding a Body through Three Distinct Positions . 267 The Rotocenter Triangle . . 277 Guiding a. Body through Four Distinct Positions; Burmester's Curve 281 Function Generation; General Discussion . 284 Function Generation; Relative-rotocenter Method. 289 Function Generation; Reduction of Point Positions 295 Function Generation; Overlay Method . . 300 Function Generation; Velocity-pole Method. 304 Path Generation; Hrones's and Nelson's Motion Atlas Path Generation; Reduction of Point Positions (Fixed Pivot Coincident 305 with Rotocenter) . . . . . . . .
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11-13. Path Generation; Reduction of Point Positions (Moving Hinge Coincident with Rotocenter) . . . . . . . . . . . . . . 309 11-14. Path Generation; Roberts's Theorem. . . . . . . . . . 314
Chapter 12.
Introduction to Synthesis; Analytical Methods .
319
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12-1. 12-2. 12-3. 12-4. 12-5.
Function Generation; Freudenstein's Equation. . . . . . . . Function Generation; Precision-point Approximation . . . . Function Generation; Precision-derivative Approximation . . . . Path Generation. . . . . . . . . . . . . . . . . Synthesis of Four-bar Mechanisms for Specified Instantaneous Conditions; Method of Components. . . . . . . . . . . . . 12-6. Synthesis of Four-bar Mechanisms for Prescribed Extreme Values of the Angular Velocity of the Driven Link; Method of Components . . .
Chapter 13.
Introduction to Synthesis; Grapho-analytical Methods . . . . . . . . . . •
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Function Generation; Precision-derivative Approximation . . . . 336 Function Generation; Precision-point and Precision-derivative Method 344. Path Generation; Coupler Curves with Approximately Circular Elements 346 Path Generation; Symmetrical Coupler Curves with Approximately Straight Elements . . . . . . . . . . . . . . . . 349 13-5. Path Generation; Coupler Curves with Extended Rectilinear Portions; Ball's Point . . . . . . . . . . . . . . . . . . 356 Bibliography
Index.
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INTRODUCTION The design of a new machine or device for the performance of an operation, or sequence of operations, associated with some particular industrial process, usually involves the following steps: 1. An assessment of the problem 2. A conception of a suitable mechanism in its skeletal form 3. A kinematic analysis, or examination of the mechanism's motion characteristics from a purely geometrical point of view, which may reveal the need for a modification of the layout 4. A static analysis, or determination of the nature and magnitude of the forces associated with the primary function of the device 5. A choice of suitable materials of construction, based on technological and economic considerations, and a tentative proportioning of the members 6. A dynamic analysis, or determination of the inertia forces and their effects on safety and operational requirements, which may disclose the need for redesign The chief purpose of this book is to provide the student with the proper tools for carrying out steps 3, 4, and 6 and to give him a basis for a rational approach to some problems of synthesis. It is also hoped that the book will prove a useful source of reference to the practicing • engmeer. Before proceeding to the detailed investigation of the kinematic and dynamic behavior of mechanisms, it will be necessary to select a suitable and consistent system of units and it will be advisable to review some fundamental notions, usually discussed in basic courses in mathematics, physics, or general mechanics. Because the concept of force is of more immediate interest to the engineer than that of mass, force is chosen as one of the three fundamental quantities in the engineering or gravitational system of units, the other two being displacement and time. The fundamental units of measure adopted in this book are, respectively, the pound (lb), the inch (in.), and the second (sec). The reasons for selecting the inch, rather than the foot, as unit of displacement are threefold: 1. Relative displacements of machine parts are generally of the order of a few inches, and sometimes amount to only fractions of an inch. 1
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KINEMATICS AND DYNAMICS OF PLANE MECHANISMS
2. Dimensions of machine elements are usually given in inches. 3. Quantities used in the analysis of stress and strain are based on the inch, e.g., modulus of elasticity (lb/ in. 2) and moment of inertia of cross section (in. 4). The accompanying table lists the most important quantities and their units of measure. ENGINEERING SYSTEM OF UNITS
Quantity
Symbol
Unit
Displacement. . . . . . . . . . . . . . . . . . . Time. . . . . . . . . . . . . . . . . . . . . . . . . . Force . . . . . . . . . . . . . . . . . . . . . . . . . .
s, x .,. F, R
• m. sec lb
Velocity, speed . . ....... .. .. .. .. . Acceleration . . . . . . . . . . . . . . . . . . . . Angular displacement . . . ...... .. . Angular velocity ......... .. . ... . Angular acceleration ... ...... . . . Mass . . . . . . . . . . . . . . . . . . . . . . . . . . Linear momentum .. .. . . ...... .. . Torque ... .. .. .... . . .. . . ... . .. . Moment of inertia. . . . . . . . . . . . . . Angular momentum . .... .. ... . . . Work . .............. .. .. ... .... . Energy ..... . ... .. ..... .. .. .. . . Power . . . . . . . . . . . . . . . . . . . . . . . .
• • v, s, :z; a, s, .i 8, cp
in./sec in./sec 1 rad rad/ sec rad/sec1 lb-sec 2 /in. lb-sec in.-Ibt lb-in.-sec 2 in.-lb-sec lb-in. t lb-in. lb-in./sec
~
w a
M mt T I g
w
s , it follows that the velocity vector is tangent to the path. y
y
0
FIG. 2-1
Acceleration. The acceleration components ax and a11 are obtained by differentiating the corresponding velocity components with respect to time: dv11 •• dvx .. (2-2) and au = = y ax= =x dr dr
Alternatively, by combining Eqs. (2-1) and (2-2), (2-3)
and The magnitude and direction of the acceleration vector follow from a
tan cf>a
=
(ax 2
+ ai)l
all
=-
az
2-3. Polar Reference Frame
This reference frame consists of a fixed point, or pole, and a fixed axis · originating at the pole (Fig. 2-2). Position. The position of the par·ticle is defined by the radius vector r =
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KINEMATICS OF THE PLANE MOTION OF A PARTICLE
17
i.e., by its distance r from the pole, and the angle 8 which the radius vector makes with the reference axis. 8 is measured from the axis in a counterclockwise sense. Displacement. Since displacement is the change in position, it may be expressed, in accordance with Eq. (1-11), as
ds
=
dr
=
+ r d8 iz
dr ir
Its magnitude and direction are given by
+ (r d8)2]l r d8 8 + A = 8 + arctan dr
ds = [(dr) 2 and
cp
=
where cp = angle between reference axis and displacement A = angle between position and displacement vectors
0
X
FIG. 2-2
Velocity.
Since velocity is the time rate of displacement,
ds dr dr • d8 • v = - = - = - lr + r - lz dT dT dT dT = iir + r8iz = Vr + Vz
(2-4)
where f = Vr is the magnitude of the radial, and rO = vz that of the lateral, velocity component. The magnitude and direction of the velocity are given by V
and
= (vr 2
cl>v = 8
+ vz2 )t
+ Av =
8
Vz
+ arctan Vr-
Acceleration. The acceleration is obtained by differentiating Eq. (2-4) with respect to time:
• a = -dr lr dT
Digitized by
+ r. -dir + -dr Ull + A!
dT
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dT
r -dO lz. dT
+ r 8. -diz dT
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KINEMATICS AND DYNAMICS OF PLANE MECHANISMS
By observing the rules for differentiating unit vectors, derived in Sec. 1-6, the above expression is reduced to a = (r - r8 2 )ir
+ (r8 + 2r8)i, =
&r
+ ar
(2-5)
where r - r8 2 = ar is the magnitude of the radial, and r8 + 2r8 = a 1, that of the lateral, acceleration component. The term 2r8 is known as the Coriolis acceleration. It will be dealt with more fully in Sec. 4-13. The magnitude and direction of the acceleration are given by
a = (ar2 and
tPa = 8
+ a,2)l
+ Xa
= 8
+
a, arctan ar
2-4. Moving Reference Frame This very important coordinate system consists of two mutually perpendicular axes which move with the particle in such a manner that one remains tangent to the path, pointing in the direction of motion, while the other points away from the center of curvature. (The center of curvature is located at the intersection of two normals to the curve, through points an elemental distance ds apart and straddling the given point. It is the center of the so-called osculation circle, which, since it has three infinitely close points in common with the curve element, approximates more closely to the curve, in the vicinity of the given point, than other tangent circles. The radius of the osculation circle is called radius of curvature; Fig. 2-3.)
c
i,. •
FIG. 2-3
Displacement.
The elemental displacement may be expressed as ds = ds i, = p dq, i,
where p = radius of curvature of path at point considered dq, = change in direction of motion Velocity. Since velocity is the time rate of displacement, V =
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S1c
=
• pWplt
(2-6)
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KINEMATICS OF THE PLANE MOTION OF A PARTICLE
19
Acceleration. By differentiating Eq. (2-6) with respect to time, the following expressions for the acceleration are obtained:
... + s. di,- =
a =
Sic
...
. .
swp1,.
Sic -
dT
whieh, combined with Eq. (2-6), and s = v dvjds, gives
a
...
= 81t -
2 v.
-
p
ln
dv. = v d 1t 8
-
2 v.
-
p
ln =
+ a, a,.
(2-7)
Alternatively,
a=
di, . + pw,. )"1t + fJWp dT (pwp
(2-8) In the foregoing equations wp and ap are, respectively, the angular speed and angular acceleration of the radius of curvature. The component along the t axis is called tangential acceleration. It represents the rate of change of speed. It should be noted that the • expressiOn
a,=
pa
familiar from elementary mechanics, is valid only if p = 0, that is, if p is either constant (circular motion) or has, at the point considered, an extreme value. The component along then axis is called normal, or centripetal, acceleration. The second name, meaning "center-seeking," is due to the fact that this component is always directed toward the center of curvature, as evidenced by the minus sign in Eqs. (2-7) and (2-8). It is important to note that the normal acceleration is independent of the rate of change of the radius of curvature. This fact permits the use of the concept of equivalent mechanisms in kinematic analysis, an application of which is illustrated in Sec. 4-13. The magnitude of the acceleration and its direction relative to the velocity vector are determined from
a and
=
(a,2 a,.
+ an )l 2
tan '1 = -
a,
2-5. Summary The important relationshipB derived in the preceding paragraphs are summarized in Table 2-1.
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KINEMATICS AND DYNAMICS OF PLANE MECHANISMS TABLE
2-1.
S UMMARY OF KI NEMATIC RE L ATIONSHIPS
Cartesian
Coordinate
Polar
v., =X•
Velocity (speed )
VII
Acceleration
a.,= x
= y•
dv., = v., dx av = ti dv 11 = VII
v,. = v, ==
t r(J
a,. =
r -
a1 = rB
Moving
v=
•
8
=pwp r8 2
a,
= v= 8 dv = vds = pwp +pap v2
+ 2t(J
a.. = -
dy
p
2-6. The Radius of Curvature Figure 2-4 shows the velocity and acceleration vectors of a particle, resolved into their cartesian and moving components. It can be seen that •
an = a Sln 11 • van = va Sin 11
Hence y
a,.
t v At
n FIG. 2-4
However, by Eq. (1-6),
va sin 11
=
mag (v X a)
= v:za11 - v11a:z
Hence and
p = -----
v;zall - v11a:z
(v:z2 + v112)f v:za11 - v11a:z
(2-9)
A positive result indicates that the center of curvature lies in the direction obtained by turning· the velocity vector through 90° in the countetclock• w1se sense.
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In Eq. (2-9), the radius of curvature is expressed in terms of the cartesian velocity and acceleration components. However, since p is a geometrical property of the path, and thus independent of the other motion characteristics of the particle, it may be obtained directly from the path equation, y = f(x) orr = f(O). The following expressions are known from the study of analytical geometry: Cartesian coordinates, with y = f(x), y' = [1
p=
. . h f( ) , Po1ar coordmates, w1t r = 8 , r
P =
Illustrative Example 1.
r2
y" -
~~:
+
(y')2]f y"
(2-10)
2 dr , d r = do' r = d() 2 :
rr"
-
*'
+ 2(r')
(2-11)
2
A particle moves as follows:
direction, az = 2 in./sec 2, Vz,O = 0, Xo = 0 y direction, a11 = -3 in./sec 2 , v11 ,o = 21 in./sec, Yo X
=0
Determine: (a) The path equation, y = f(x) (b) The acceleration of the particle (c) The velocity at T = 4 sec (d) The radius of curvature of the path at the point occupied by the particle at T = 4 sec Solution. (a) The displacement components x andy are first expressed as functions of time. By eliminating T from the resulting equations, the path equation is obtained: Vz
=
f az dT
+ C1
With the boundary condition T = 0, vz = 0, Vz
= 2r
X =
fvz dr
(a)
+ C2
With the boundary condition T = 0, x = 0, X = T2
(b) (c)
v11 = - 3T + 21 y = -jr 2 + 21r
Similarly and
(d)
Elimination ofT from Eqs. (b) and (d) yields the path equation: y
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KINEMATICS AND DYNAMICS OF PLANE MECHANISMS
(b) Since both direction:
a~
and
a = (a~2
tl>a
=
~
are constant, a has constant magnitude and
+~} 2
arctan~ as
=-= 3.61
in.jsec 2
= arctan ( -1.5} = 303°39'
(c) From Eqs. (a) and (c), vz.• = 8 in.jsec and v11 .• = 9 in.jsec.
Hence
v. = 12.04 in./ sec q,,,. = arctan t = 48°22'
and
(d) Substitution of the known values into Eq. (2-9) yields p, =
-41.5 in.
The minus sign shows that the velocity vector v, is to be turned in the clockwise sense to indicate the location of the center of curvature. It is left to the reader to verify this result by means of Eq. (2-4) and the path equation established in part a. ruustrative Example 2. The rigid bar of Fig. 2-5 revolves about its pivot at a constant speed of 1.2 rad/ sec in the counterclockwise sense.
- Guiding groove
0
~---"---+---
Xu Rod
FIG. 2-5
The bead, free to slide on the bar, is guided by a spiral groove, defined by the equation r = 4.25
where fJ is in radians. from the pivot:
(}
+-
11'
Determine, for the instant when the bead is 6 in.
(a) The velocity of the bead
(b) The acceleration of the bead Solution. In order to determine the velocity &nd acceleration components it will be necessary to express r and fJ as functions of time.
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23
In the present case,
6=w
•
=
1.2 rad/sec, const
Differentiation with respect to time yields
(a) and integration leads to 8 = 8o
+ 1.2T
(b)
Substitution of Eq. (b) into the path equation yields 4.25
T =
.
.
r•
leading to
(c)
const
(d)
7r
= 1.2 ,
11" r = 0
and
+ 8o + 1.2T
(a) In polar coordinates, the general expressions for the velocity
components are Vr
=
•
and
T
Hence the following values, corresponding to r the present case: Vr
=
1.2.In. I sec
vz = 7.2 in./sec
11"
