Topic 1 - Planar Linkage Kinematics - Faculty of ...grideout/7945_W10/Topic1_KinBackground_Edits3.… · Topic 1 - Planar Linkage Kinematics Goals Review kinematics and relative motion

ENGI 7945 Machine Dynamics Topic 1, Planar Mechanism Kinematics Date:

Topic 1 - Planar Linkage Kinematics

GoalsReview kinematics and relative motion from Mechanics IIEvaluate vector cross products

Analytical (using cross products of i-j-k)Using right-hand rule

Define and discuss "vector loops"Apply vector loops to generate equations for general motion analysis

1-1. Why We're Doing This

A mechanism is a chain of constrained bodies. Because they're constrained, their motions arerelated. If we have a 1 degree of freedom (DOF) mechanism, then if we know the motion of theinput link we should be able to calculate the motion of any other link or point on the mechanism.In other words, one actuator (e.g., motor or hydraulic piston) is required to drive the mechanism.If we have a 2 DOF mechanism, then two actuators would be required to drive the system.

Actuator(motor)

Mechanism(constrained links)

Output motionKnown input,

e.g.

2 2 2θ ,θ ,θ e.g., slider speed, rocker angle

Actuator(motor)

Mechanism(constrained links)

Output motionKnown input,

e.g.

2 2 2θ ,θ ,θ e.g., slider speed, rocker angle

The motion of any mechanism link can be categorized as shown below.

Translation Rotation General

orientation does not changeany two points have sameposition, velocity and accel'ncurvilinear or rectilinear

one point is fixedvelocity of point is proportionalto distance from pivottangential and normal accel.

anything goesrelative motionequations arecritical

Hibbeler Fig. 16-2 - Compound Mechanism (Four-Bar and Slider Crank)

Application to dynamics

If you know the motion (position, velocity and acceleration) of every link in a mechanism, then youcan find the angular acceleration of each link, and the acceleration of each centre of gravity.Then, using Newton's Law, you can find the motor torques or piston forces required to create thatmotion. That's "inverse dynamics", and is relevant in robotics. Knowing the desired path of therobot arm, what motor torques are required?

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Suppose the wheel is the input link (motor driven), and we know its angular motion. In MechanicsII, you learned how to calculate the instantaneous velocity and acceleration of the slider, and theangular velocity and acceleration of the rocker and coupler links. This was done by writing avector equation for relative motion between two points.

Slider-Crank A-B-E Four-Bar A-B-C-D

Relative position:

B/EBE rrr B/CBD/CD rrrr

Relative velocity:

B/EBE vvv B/CBC vvv

Relative acceleration:

B/EBE aaa B/CBC aaa

Notes:We must differentiate the relative position equation twice, in order to get the relative velocityand acceleration equations.The preceding equations are vector equationsEach vector term has an i and j componentCollect the terms in i to get one equation, and the terms in j to get a second equationSolve two equations in two unknowns

Motion of points on a link rotating about a fixed axis is comparatively easy. Confusion often arisesover relative motion between two points on a link in general plane motion.

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1-2. 2D Fixed Axis Rotation - Tangential Velocity

Consider body rotating with angular velocity about a fixed axis through O:

ωrv

rωv P

Hibbeler Eq's (16.8-9)

r = rP = position vector from origin O to point P

= rP/O = position vector of P w.r.t. O

If O fixed, v = vP = vP/O ("velocity of P w.r.t. O")

magnitude = r = rP = rP/O = OPperpendicular to OPdirection determined by direction of

Hibbeler Fig. 16-4d

ˆ ˆω ωk θk

ˆ ˆ ˆα αk ωk θk

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1-3. 2D Fixed Axis Rotation - Tangential and Normal Acceleration

Hibbeler Eq's (16.11-13)

2ωra

αra

rωωa

rαa

rωωrαa

n

t

Pn

Pt

PP

Hibbeler Fig. 16-4f

If O fixed, a = aP = aPO ("acceleration of P w.r.t. O")

aP/O = (aP/O)t + (aP/O)n

(aP/O)t magnitude = rP/O = OPperpendicular to OPdirection determined by direction of

(aP/O)n magnitude = rPO2 = OP2

directed from P to O

ˆ ˆω ωk θk

ˆ ˆ ˆα αk ωk θk

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1-4. 2D General Plane Motion - Relative Motion

The following equations apply to any two points in the universe. The velocity and accelerationequations are simply the first and second derivatives of the relative position equation.

A/BAB

A/BAB

A/BAB

aaa

vvv

rrr

A/Br

is position vector of B w.r.t. A= position vector from A to B

B/AA/B

B/AA/B

B/AA/B

aa

vv

rr

2

2

dt

rd

dt

vda

rdt

dv

generalIn

Hibbeler Fig. 16-10

Relative motion between two points on a rigid body

Given a rigid body undergoing general plane motion with angular velocity , pick two points, Aand B. Suppose you were moving with point A, but always facing the same way so that the bodywas rotating beneath you.

How would point B appear to be moving?rigid body, therefore fixed distance between you (point A) and point BB doesn't get closer or further away - it appears to be moving perpendicular to youif your location, point A, stopped moving and became a fixed rotation axis, but didn'tchange - you would notice no difference in the movement of point B

General plane motion:

ωrv

rωv

A/BA/B

A/BA/B

Hibbeler Fig. 16-10 (d)

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2ωra

αra

rωωrαa

A/BnA/B

A/BtA/B

A/BA/BA/B

Hibbeler Fig. 16-23 (c)

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Comments

While the motion of a link rotating about a fixed axis may appear fundamentally different thanthat of a link in general plane motion, the velocity and acceleration vectors have the sameform.In the Hibbeler Figs. 16-4 on a previous page, velocity and acceleration of P were really"relative" velocity and acceleration with respect to the fixed point O. Therefore theexpressions for velocity and acceleration of P (with respect to O) shouldn't look different fromthose for B with respect to A above.In Mechanics II, to find instantaneous motion of a mechanism given an input, you would writethe relative velocity and acceleration equations, express each vector in terms of i and jcomponents, and work out the cross products. Gathering i terms gives one equation, andgathering j terms gives another equation. See Hibbeler Examples 16.8, 16.17, 16.18

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1-5. Vector Cross Products

Given two vectors A and B, the cross product is written

C A B "C equals A cross B"

Magnitude

If the angle between A and B is , then cross product magnitude C = AB sin.

Direction

If C = A X B, then C will be perdendicular to A and B.

There are two possible vectors perpendicular to both A and B.The correct one is determined by the right hand rule as shown.

point fingers of right hand towards A1.curl fingers towards B2.right thumb points in direction of C3.

Hibbeler Fig. 4-7

Note:A X B = -(B X A)(A X B) X C does not equal A X (B X C)

Hibbeler Fig. 4-8

Notes on velocity and acceleration cross-products

the cross-product of two vectors is always perpendicular to the two vectors and vectors, in 2D, will always be into or out of the page the cross-product of or and a position vector r must be perpendicular to r.the exact direction of the cross-product is determined by the right hand rule

Right-hand rule

To find direction of cross-product of and r:point middle finger of right hand in direction of (into or out of page)curl middle finger towards ryour extended right thumb points in the direction of x r (in the plane of the page)

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Example Systems

After reviewing some fundamentals, we'll analyze linkages such as the following:

Hibbeler P16-111 Hibbeler P16-128

In the first two examples (four-bar and slider-crank),each link has fixed length. In the example on the left,the length of "link" CB changes. Link CD hasconstant length, but the location of point B (and thedistance CB) are what's important. You could movepoint D farther away by lengthening the slot, but thatwouldn't affect the motion (assuming the block at Bdoesn't hit the end of the slot).

The "rotating and sliding" nature of the block at Ballows "link" CB to take on whatever length isrequired as the crank rotates.

This introduces additional terms to the relativevelocities and accelerations, including a Coriolisacceleration. Coriolis acceleration is oftenmemorized and is poorly understood. As you'll seesoon, Coriolis acceleration is something thatappears if you differentiate your velocity vectorcorrectly, and there's no need to worry about it. Ifyou have a mechanism with a rotating sliding joint,Coriolis acceleration should appear and you'llrecognize it when it does.

Hibbeler P16-141

We will use the "loop closure method" to write general equations for mechanism motion (position,velocity and acceleration). These equations will allow instantaneous analysis as in Hibbeler, oreasy coding into Matlab for continuous simulation of mechanism motion. That's far more usefulthan finding motion at a single instant. Once you identify vector loops and define the vectors, thenchain and product rule will allow you to mindlessly generate motion equations.

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1-6. Identifying Inputs and Outputs in Mechanism Analysis

For planar mechanisms, each link has a length and an angle.

Each link length or angle may beconstanta known inputan output to be solved for

For a mechanism with n DOF, if you specify n link motions as inputs, then you can calculate themotion of any other link.

Mechanism motion can be described by relative motion vector equations.

A 2D vector equation can be written as two scalar equations (e.g., equations relating i and jcomponents). Therefore, when we do position, velocity or acceleration anaysis, there can only betwo unknown outputs for every relative motion equation that we need to write.

If you look at a mechanism and there are only two unknown outputs, then you can analyze themechanism using one relative motion equation (and its derivatives). As we will see, these aremechanisms with one kinematic loop.

If there are four unknowns, then you require two relative motion equations for each of position,velocity and acceleration analysis. Such a mechanism will have two kinematic loops.

For the five mechanisms below, given the input motion, write the outputs.

Four-bar

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Slider-crank

Scotch yoke

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Quick-return mechanism

2DOF mechanism

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1-7. Vectors

In dynamics and robotics, there are two typical ways to describe a vector. Both have the followingin common:

vector is a sum of (components x unit vectors) ... in other words, a sum of productsto differentiate a vector, you need to apply the product rule from calculus

dt

dBAB

dt

dABA

dt

d

Cartesian Radial-Transverse (Polar)

The unit vectors of the Cartesian vector are fixed. To differentiate the Cartesian vector, simplydifferentiate the components.

To differentiate a vector expressed in terms of rotating unit vectors, both the components and theunit vectors must be differentiated.

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It's common in advanced kinematics to attach unit vectors to each link. For mechanisms withslender links, the radial direction is along the link, from one joint to the next. Each link will have itsown

unit vectors, making a different angle n with respect to horizontal, and having a different angular

velocity n. To add vectors that are expressed in different body fixed frames, you must convert

them so that their unit vectors are all in the same perpendicular directions. Often we convertthem all to the standard i-j inertial frame with fixed horizontal and vertical unit vectors. Beingfamiliar with the conversions between rotating and fixed i-j unit vectors will also help you reconcilewhat we do next with what you learned in Mechanics II.

nn θr iandi

jθcosiθsini

jθsiniθcosi

nnθ

nnr

n

n

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1-8 - Loop Analysis - General Expressions for Mechanism Motion

ReadWaldron and Kinzel Chapter 5 handout

Goals

Determine how linkage positions, velocities, and accelerations vary as a function of time - not justat an instant

more useful - going beyond instantaneous analysisderive equations useful for computer implementation (Matlab, Excel, etc.)write "loop closure equations" and express position, velocity and acceleration in absolutecoordinates (i-j unit vectors fixed to base, instead of radial-transverse rotating unit vectors)

Position Analysis and Choosing Loop Vectors

As we have discussed, any mechanism can be thought of as containing one or more positionvector loops (kinematic loops).

Loop vectors are generally drawn from one joint to another. Often they originate at the end of alink that's attached to the base.

Each loop vector is a relative position vectorvector from a point A to a point B is rB/A = position of B with respect to Aderivatives of rB/A are vB/A (velocity of B with respect to A) and aB/A (acceleration of B withrespect to A)

Each loop vector's magnitude (length) and/or direction (angle) may be fixed or varying.

Coriolis Alert when both length and angle vary.

Each loop should present at most two unknown vector quantities (magnitude or direction), just likeeach relative motion equation gave at most two unknowns.

Every loop (position) vector has the same form:

In i-j (x-y) components:

In column vector form (better for computer implementation):

All angles are measuredcounterclockwise from + xaxis. Clockwise = negative.

ˆ ˆ ˆ ˆ(cos sin ) cos sinr r i j r i r j

cos

sin

rr

r

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Choosing bad loop vectors will give you too many unknowns, or give you useless information. Eachvector should tell you something about a link, for example:

the angular velocity (acceleration) of that vector should tell you the angular velocity(acceleration) of a corresponding linkthe magnitude of the vector should tell you the length of a telescoping link, or the location of aslider relative to some reference point.

The analytical expressions for velocity and acceleration have been derived using body-fixed rotatingunit vectors. If we express every position vector in the Cartesian form

cos

sin

rr

r

then we can differentiate the components, and get the same vectors in the end.

Example 1 - Slider-Crank with Coupler Point

Link O2A has angular velocity of 10

rad/s CCW, and angular acceleration of

50 rad/s2 CW at the instant shown.

Using loop closure analysis, writegeneral expressions for unknownpositions, velocities and accelerations.

These expressions can be coded into acomputer to solve for the motion atsuccessive instants in time.

Substitute instantaneous values to find the instantaneous motion.In the next Problem Set you will have to simulate the continuous motion of this mechanism.

Key words: loop closure, coupler point, assembly modes

Example 2 - Raphson Slide

If angular velocity of link 2is 10 rad/s CCW (constant),write general expressions tofind velocity andacceleration of link 4.

Find v4 and a4 when 2 =

60o.

Handwritten Solutions Generated in Lecture

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Form of Cartesian Vector for Relative Position, Velocity and Acceleration

For a general link, which may or may not have one end fixed

22

22/2 sin

cos

r

rrr AB

222

222

22

22/2

cos

sin

sin

cos

r

r

r

rvv AB

222

222

22

22

222

222

22

22

22

22/2

cos2

sin2

sin

cos

cos

sin

sin

cos

r

r

r

r

r

r

r

raa AB

If your loop vector doesn't change length, then cross off the r-dot and r-double-dot terms.Likewise, cross off -dot and -double-dot terms if the vector doesn't rotate (for example, positionvector to a slider-crank slider).

Example 3 - Telescoping Link Given 4 = 1 rad/s constant, find

velocity and acceleration of point B.

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Examples - Choosing Loop VectorsFigures on this page from Waldron and Kinzel.

Offset slider-crank

Four-bar with telescoping link

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Multiloop mechanism - pump from Cleghorn Appendix A

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Multiloop mechanism - level-luffing crane from old Mechanisms exam

A level-luffing crane is designed to keep the load moving horizontally as the jib is rotated. Askeletal representation of a level-luffing crane is shown below.

The jib is rotated by extending the actuator at a constant velocity of 1 ft/sec. At the instantshown, the goal is to find the velocity vector of the load, and to confirm that it is movinghorizontally. Draw loop vectors, clearly indicating vector numbers, angles, and directions to allowcalculation of vector vG.

Write your loop equations, differentiate them, and use the given position information andvelocity input to calculate vG (x and y components, and overall magnitude and direction).

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Topic 1 - Planar Linkage Kinematics - Faculty of ...grideout/7945_W10/Topic1_KinBackground_Edits3.… · Topic 1 - Planar Linkage Kinematics Goals Review kinematics and relative motion