ENGI 7945 Machine Dynamics Topic 1, Planar Mechanism Kinematics Date: Topic 1 - Planar Linkage Kinematics Goals Review kinematics and relative motion from Mechanics II Evaluate 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 are related. If we have a 1 degree of freedom (DOF) mechanism, then if we know the motion of the input 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 motion Known input, e.g. 2 2 2 θ , θ , θ e.g., slider speed, rocker angle Actuator (motor) Mechanism (constrained links) Output motion Known input, e.g. 2 2 2 θ , 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 change any two points have same position, velocity and accel'n curvilinear or rectilinear one point is fixed velocity of point is proportional to distance from pivot tangential and normal accel. anything goes relative motion equations are critical 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 you can 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 that motion. That's "inverse dynamics", and is relevant in robotics. Knowing the desired path of the robot arm, what motor torques are required? 1 Prof. G. Rideout
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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?
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.
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
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
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)
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.
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.
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.
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.
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.
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.
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
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).