Mechanism Analysis & Synthesis - Mechatronics

Mechanism Analysis & Synthesis K. Craig 1

Mechanism Analysis & Synthesis

Leonardo da Vinci created theSlider Crank Mechanism

Over 500 Years Ago

Forces on a machine come from applied forces and accelerations

of masses.

Dynamics of a System Stress Analysis of a System


Topics

• Terminology

• Mechanisms and Skeleton Diagrams

• Analyzing Linkage Motion; Vector-Loop Method

• Kinematic and Kinetic Simulink Simulations

• Trajectory Planning with Electronic Cams

• Slider-Crank Case Study

• Four-Bar Mechanism Case Study

• Kinematic Synthesis: Four-Bar & Crank-Slider Mechanisms

• Supplement: Rigid-Body Plane-Motion Kinetics


Terminology

• Kinematics– This is the study of the geometry of motion. It describes the

motion of bodies without reference to the forces which either cause the motion or are generated as a result of the motion. It is used to relate position, velocity, acceleration, and time without reference to the cause of the motion. Kinematics is divided into Analysis (given mechanism, find motion, one answer) and Synthesis(given motion, find mechanism, many answers).

• Kinetics– This is the study of the relation existing between the forces acting

on a body, the mass distribution of the body, and the motion of the body. It is used to predict the motion caused by given forces (Forward Kinetics – given force/torque, find motion) or to determine the forces required to produce a given motion (Inverse Kinetics - given motion, find force/torque).


• Force– Force is the vector action of one body on another. There are two

types of forces in Newtonian mechanics:

• Direct contact forces between two bodies (which are electromagnetic in nature).

• Forces which act at a distance without physical contact, of which there are only two: gravitational and electromagnetic.

• Rigid Body– A rigid body is a body whose changes in shape are negligible

compared with the overall dimensions of the body or with the changes in position of the body as a whole.

• Degrees of Freedom– This is the number of independent coordinates needed to

completely describe the motion of a mechanical system. This is a characteristic of the system itself and does not depend upon the set of coordinates chosen. A system’s DOF also defines the number of inputs (motors or other actuators) needed to control its motion. (DOF ≤ 0 → structure; DOF ≥ 1 → mechanism)


• Constraint– A constraint is a limitation to motion. If the number of

coordinates is greater than the number of degrees of freedom, there must be enough equations of constraint to make up the difference.

• Generalized Coordinates– These are a set of coordinates which describe general

motion and recognize constraint.

– A set of coordinates is called independent when all but one of the coordinates are fixed, there remains a range of values for that one coordinate which corresponds to a range of admissible configurations. A set of coordinates is called complete if their values corresponding to an arbitrary admissible configuration of the system are enough to locate all parts of the system. Hence, generalized coordinates are complete and independent.


Mechanisms

• A mechanism is a device that transforms motion to some desirable pattern and typically develops very low forces and transmits little power. It is a means of transmitting, controlling, or constraining relative motion.

• A machine typically contains mechanisms that are designed to provide significant forces / torques and transmit significant power.

• Useful Working Definitions– A mechanism is a system of elements arranged to transmit

motion in a predetermined fashion.

– A machine is a system of elements arranged to transmit motion and energy in a predetermined fashion.


• Mechanisms, if lightly loaded and run at low speeds, can sometimes be treated strictly as kinematic devices.

• Machines (and mechanisms running at higher speeds) must first be treated as mechanisms, a kinematic analysis of their positions, velocities, and accelerations must be done, and then they must be subsequently analyzed as dynamic systems in which static and dynamic forces due to those accelerations are analyzed using the principles of kinetics.

• Mechanisms come in many types, but all mechanisms are variants of a linkage.

• A linkage is made up of a collection of links and joints, one of which is grounded, and all are interconnected in a way to provide a controlled output in response to one or more inputs.


• A link is a rigid body (although flexible bodies have advantages – flexure mechanisms) of any shape that has some number of attachment points called nodes (often holes) that allow multiple links to be connected by joints.

• Links can be of any shape. The number of nodes determines the character of the link.

compliant platform


• Joints are characterized by their geometry, by the number of degrees of freedom they allow between the links they join, and by whether they are held together (closed) by a force or by their form (geometry).


• Calculating Degree of Freedom F or Mobility M– The degree of freedom of an assembly of links and joints can be

calculated with a simple equation, the Kutzbach Equation, although in some cases it may give a misleading result.

– The two-dimensional version of this equation, which applies to any planar linkage (i.e., links that move in parallel planes) is:

• F or M = degree of freedom or mobility

• N or L = number of links

• P1 or J1 = number of one-DOF joints (pin, sliding, rolling)

• P2 or J2 = number of two-DOF joints (slipping, e.g., pin-in-a-slot)

– This equation does not account for link lengths and so can predict an incorrect DOF when the link geometries have particular values.

( )

( )

1 2

1 2

M 3 L 1 2J J

F 3 N 1 2P P

= − − −

= − − −

References havedifferentidentifiers

rolling

slipping


– Shown below are two linkages that each have 5 links, six 1-DOF (J1) joints and no 2-DOF (J2) joints.

– The equation says that they both have zero DOF, i.e., they are predicted to be structures, unable to move. This is true for linkage (a), but linkage (b) can obviously move since the three binary links are the same length and are also parallel because the two ternary links have the same spacing between their nodes. This is a paradox to the Kutzbach Equation.


Pair of Externally-Rolling Gears: 1 DOF

Incorrect: DOF = 1

Correct: DOF = 0If Slipping Joint: DOF = 1

Special Geometry: Circular links are pinned to ground at

their centers

rollingjoint


statically indeterminate

statically determinate

DOF Examples

car hood mechanism

robotarm


Four-Bar Mechanisms

Car-Hood Mechanism

Door-Damper

Mechanism


FrontLoader


Kinematic Chains


Examples of Mechanisms: Open, Closed, Mixed

Link 2 is solid across the pin joint


• Skeleton Diagrams➢A skeleton diagram is a simplified drawing of a

mechanism or machine that shows only the dimensions that affect its kinematics.

Connecting Rod


Automobile Hood Mechanism: Guides Hood as it Opens & Closes

Loops: 1-2-6-5, 1-2-3-4-5, 2-3-4-5-6Only two are independent


Automobile Hood Mechanism:Four-Bar Mechanism

Single Loop: 1-2-3-4

As hood approaches the closed position, links 2 and 4 are nearly parallel, and link 3 stops rotating, thus generating uniform pressure

along the seal between the hood and car body.


Door-Damper Mechanism(Four-Bar Mechanism)

This mechanism develops a desired relationship between the rotation of the door and the rotation of link 4,

resisted by the damper. This is known as function generation.

Single Loop: 1-2-3-4

When the door is closing, the

rotational rate of link 4 relative

to link 1 increases

significantly, which makes the

damper more effective.


SmallFront Loader

3 Independent Closed Loops:1-2-3-44-5-6-84-8-7-9

2 Hydraulic Actuators(Sliding Joints)


SmallFront Loader


Folding ChairLying on its Side

(Single Loop)

Fixed

Pin-in-a-Slot JointA P2 joint


Top of an Entry Doorto a

Racquetball Court(Single Loop)


PrecorEFX 576iExerciseMachine

(Trajectory of points on the feet

is Elliptical.)

Machine consists of two kinematically

identical, 180º out-of-phase

mechanisms.


PrecorEFX 576iExerciseMachine

rolling joint


PrecorAMT 100iExerciseMachine

(only pin joints)

Machine consists of two kinematically

identical, 180º out-of-phase

mechanisms.


PrecorAMT 100iExerciseMachine


PlanetaryGearTrain

4-ring gear

2-sun gear

5-arm3-planet gears (there are 3

planet gears, 2 of which are redundant)

Gears are represented by pitch circles.Links 2, 5, and 1 are joined by a pin joint at the

center and count as two pin joints.)

F 3(5 1) 2(5) 0 2= − − − =


Rear Windshield-Wiper DriveMechanism

(Four-Bar Mechanism1-2-3-4)

motor4-rocker

F 3(4 1) 2(4) 0 1= − − − =

2-rotates continuously4-oscillates nearly 180º

wiper rotateswith gear 5gear portion of 3 drives gear 5


KinematicInversions

Kinematic InversionMechanisms that are derived from the

same kinematic chain but have a different link fixed to ground.


Crushing MachineA machine is a mechanism that transmits power.

This is a machine because it transfers power from the motor driving the crank to the boulder being crushed.


• Four-bar Linkage and the Grashof Condition

– The simplest 1-DOF mechanism is the four-bar linkage: four binary links connected by four pin joints. It will behave differently depending on the relative lengths of its four links. Its behavior can be predicted by the Grashof Criterion: S+L ≤ P+Q

– S = length of the shortest link

– L = length of the longest link

– P and Q = lengths of the remaining two links

– It matters not the order in which the links are assembled.

– If the Grashof Inequality is false, then no link can make a full revolution. The linkage is called a Non-Grashof Linkage.


– If true, then at least one link can revolve fully with respect to the others and it is called a Grashof Linkage.

– If the expression evaluates to equal, then it is a Special-Case Grashof Linkage which has “change-point” positions at which successive positions of the linkage are indeterminate. Such a linkage must be constrained to avoid these change points, or additional links added to carry it through the change points.

– The next slide shows four inversions of a Grashof Linkage. An inversion results from the grounding of a different link in the chain, thus any linkage will have as many inversions as it has links.

– The ground link is always numbered 1, the input link is always 2, the coupler is 3, and the output link is 4.

– A crank can fully revolve, a rocker cannot.


Inversions of the Grashof Four-Bar Linkage

Either of the links adjacent to the shortest is grounded.

Link opposite the shortest is grounded. Shortest link is grounded.


Special-Case Grashof Linkages

The parallelogram linkage is very useful if the links are not

allowed to become collinear, which is a change point. Over its useful range of motion, the coupler remains parallel to the

ground plane.

The parallelogram linkage can be carried through the change points by adding a second stage that is out of phase with the first. Both stages

have change points, but each occurs at a different crank angle, so the

other stage carries it through. This 5-bar linkage is a Kutzbach paradox.


Non-Grashof Four-Bar Linkage Used For Automotive Suspension

All four inversions of a Non-Grashof Four-Bar Linkage are triple rockers and give similar motions. All three moving links just oscillate until they reach extreme positions called toggle positions. They are nevertheless useful as many applications do not require that any link

make a full revolution. The automotive suspension linkage below controls the up and down motion of the wheel assemblies over bumps.

Here the wheel motion only requires the links to move through relatively small angles and never through a full revolution.


• Four-bar Crank-Slider and Slider-Crank

– If the rocker of a four-bar crank rocker linkage is increased in length indefinitely, it effectively becomes infinite in length and the linkage is transformed into a four-bar crank-slider shown. The rocker link becomes a slider block that oscillates in a straight line rather than along an arc. A straight line can be thought of as an arc of infinite radius. This is a very common linkage.


– At the heart of every internal combustion engine, is the slider-crank linkage; here the slider drives the crank. In this mode, it has two change points at top and bottom dead center and its crank must be spun to start it moving continuously, which is why you must pull the starter rope on a lawn mower. The momentum of the rotating crank carries it through the change points once it is put in motion.

Slider-crank:slider drives crank


• Cam and Follower

– A cam and follower is another variant of the four-bar linkage in which the crank takes on a contoured shape. It is then called a cam and drives a rocker or slider directly. There is no coupler link. Below are two cam-and-follower arrangements.


– Dotted lines show the effective links of an equivalent four-bar crank-slider or four-bar crank-rocker that would give the same motion as the cam for the instantaneous position shown.

– One major advantage of a cam-follower mechanism over a pure linkage is the variety of follower motion functions possible. You can think of the cam-follower mechanism as a four-bar linkage or crank-slider in which the crank and coupler are able to change their lengths as the cam rotates. The effective links are shown superposed on the mechanism in the figures.

– The effective coupler (link 3) runs from the instantaneous center of curvature of the cam contour (which changes for each cam position) to the center of curvature of the follower.


– The effective crank runs from the cam pivot to the instantaneous center of curvature of the cam contour.

– Thus, the lengths of the effective crank and effective coupler change with cam rotational position. The cam contour must be defined by appropriate mathematical functions.


Analyzing Linkage Motion

• Before we determine the forces and torques in a linkage, which are needed for motor selection and stress analysis, we first need to know the accelerations to use in Newton’s Second Law, F = ma.

• To find accelerations, we must first find position and velocity of the links because acceleration is the derivative of velocity and velocity is the derivative of position.

• Here we limit our discussion to two dimensional or planar mechanisms, which constitute most practical mechanisms used in machinery.


• Types of Motion– General plane motion: 3 DOF – x, y, ϴ

– Fixed-axis rotation: 1 DOF

– Translation: 1 DOF (straight line), 2 DOF (x and y)

• See Supplement: Rigid-Body Plane-Motion Kinetics


Kinematic Analysis: Vector-Loop Method

• Kinematic Analysis assumes that a mechanism and its dimensions are given.

• For a 1-DOF mechanism, the problem statement is:

– Given the state of motion (position, velocity, acceleration) of the input link, determine the state of motion (position, velocity, acceleration) of all other links and points of interest, e.g., mass centers.

• The Vector-Loop Method leads to a set of position equations for the mechanism that describes the changing configuration of the mechanism. The equations are nonlinear with no closed-form solution. We solve them using MatLab / Simulink.


• Vector-Loop Method Steps

– Step 1: Attach a vector to each link of the mechanism forming a closed loop or closed loops. Each vector should correspond to a link dimension or relate to a joint variable within the mechanism.

– Step 2: Write the vector-loop equation(s). The sum of the vectors in a loop is zero.

– Step 3: Choose a fixed X-Y coordinate system whose origin is at a fixed point in the vector loop.

– Step 4: Decompose the vector-loop equation(s) into scalar X and Y components. These are the position equations. List the scalar unknowns (U).

• If U > # equations, mechanism is movable

• If U = # equations, statically determinate structure

• If U < # equations, statically indeterminate structure


– Step 6: Solve the position equations for the position unknowns.

– Step 7: Differentiate the position equations with respect to time. These are the velocity equations. Solve the velocity equations for the velocity unknowns.

– Step 8: Differentiate the velocity equations with respect to time. These are the acceleration equations. Solve the acceleration equations for the acceleration unknowns.


Important

Straight Sliding Joint

1. A known magnitude or direction of the vector corresponds to a dimension of the mechanism.

2. An unknown magnitude or direction of the vector corresponds to a joint variable of the mechanism or defines the orientation of a link within the mechanism.


Circular Sliding Joint


Straight-Pin-in-a-Slot Joint


Externally-Contacting Circular BodiesWhen circular bodies are in contact, there will always be a

vector connecting their centers.


Internally-Contacting Circular Bodies


Three-Link Mechanism with 2 DOF2 5 3 1 4r r r r r+ + = +

Scalar Knowns: r1, r2, r3, r4, r5

Scalar Unknowns: θ2, θ3, θ4, θ5


Two-Loop Mechanism

2 3 1 4r r r r+ = +

8 6 1 4 5 7r r r r r r+ = + + +

8 6 2 3 5 7r r r r r r+ = + + +

Only 2 loopsare

independent

Scalar Knowns: r1, r2, r3, r4, r5, r6, r7, r8

Scalar Unknowns: θ2, θ3, θ4, θ5, θ6, θ7, r5


Two-Loop Mechanism1-2-3-41-4-5

1-2-3-4-5

Loops

Only 2 loopsare

independent


1 9 2 3 4 5

8 7 6 5

r r r r r r

r r r r

+ + + = +

= + +


Four-BarMechanism

1 DOF

+ CCW

+ CCW

+ CCW

2 3 1 4r r r r+ = +

Scalar Knowns: r1, r2, r3, r4

Scalar Unknowns: θ2, θ3, θ4


Offset Slider-CrankMechanism 1 DOF 2 3 1 4r r r r+ = +

Scalar Knowns: r1, r2, r3

Scalar Unknowns: θ2, θ3, r4


Kinematic & Dynamic Simulations

Kinematic SimulationsForward and Inverse

Dynamic SimulationsForward and Inverse


2 3 1 4R R R R+ = +

Four-Bar Mechanism


Links of the Mechanism are Constrained. This equation must be satisfied if the

mechanism remains assembled.

One Degree of Freedom

2 3 1 4R R R R+ = +

Vector-LoopEquation


Derivative of theVector Loop

Equation

ω2

is the inputMatrixForm

Second Derivative to Determine Accelerations


Vector Equations of Four Common Mechanisms



Trajectory Planning with Electronic Cams

• What do Leonardo da Vinci and the Nautilus exercise machine have in common?

• Leonardo da Vinci invented the cam hammer (see picture) around 1497 and the Nautilus exercise machine, invented around 1970, uses a cam to modulate resistance.

– The cam – an irregularly-shaped member on a rotating shaft that transfers motion – has been around for centuries. Up until recently, the study of cam design and application was foundational in a mechanical engineering curriculum. But today, it seems its study is nowhere to be found.


Leonardo da Vinci’sCam Hammer

1497

Today!


• In mechatronic design, integration is the key as complexity has been transferred from the mechanical domain to the electronic and computer software domains.

• Cams are a prime example of that mechatronic principle as mechanical cams are gradually being replaced by electronic cams.

• But transfer implies that we first understand the fundamental principles in the mechanical domain. Since MEs aren’t learning cam fundamentals anymore and it was never part of an EE’s training, motion systems today most often use crude motion trajectories that stress the machine and motor, produce unwanted vibrations, and result in poor performance.


• Trajectory planning is the computation of motion profiles for the actuation system of automatic machines, e.g., packaging machines, machine tools, assembly machines, industrial robots.

– Kinematic (direct and inverse) and dynamic models of the machine and its actuation system are required.

– Desired motion is usually specified in the operational space, while the motion is executed in the actuation space, and often these are different. The trajectoryis usually expressed as a parametric function of the time, which provides at each instant the corresponding desired position.

– Once the trajectory is defined, implementation issuesinclude time discretization, saturation of the actuation system, and vibrations induced on the load.


• In past decades, mechanical cams have been widely used for transferring, coordinating, and changing the type of motion from a master device to one or more slave systems.

• Replacing them are electronic cams, with the goal to obtain more flexible machines, with improved performances, ease of re-programming, and lower costs.

• With electronic cams, the motion is directly obtained by means of simpler mechanisms with electromechanical actuators, properly programmed and controlled to generate the desired motion profiles, which also allows synchronization of actuators on a position or time basis.


• Once the displacement and its duration have been defined, the choice of the manner of motion from the initial to the final point has important implications with respect to the sizing of the actuators, the efforts generated on the structure, and the tracking error.

• The engineer must carefully consider the different types of point-to-point trajectories which could be employed with a specific system. Both time-domain and frequency-domain analyses must be performed on the complete system, i.e., actuator, mechanism, and load, along with the motion profile, to achieve optimal performance.

• Input shaping and feedforward control are two techniques used to improve tracking performance.

• Knowledge from the past combined with new technologies results in innovation. Engineers must never forget this fact!


maxv

maxa 2

maxj 3

qC coefficient of velocity

h / T

qC coefficient of acceleration

h / T

qC coefficient of jerk

h / T

= =

= =

= =

In the following figures, h = 1 and T =1, and three profiles are shown – velocity V, acceleration A,

and jerk J – for a selection of main trajectories.


V

A

J


V

A

J


V

A

J


Slider-Crank Mechanism Kinematics and Kinetics






Initial Physical Model Simplifying Assumptions

• Rigid Homogeneous Bodies – Neglect Compliance• Disk: I = ½ m1r

2 Rod: I = (1/12) m2ℓ2

• Friction: Viscous Damping at Crank Pivot• Frictionless Revolute Joints and Prismatic Joint• Neglect External Force Fe

Slider-Crank Mechanism













Open-Loop


Parameter M-File







Segment 1

Segment 2



Inverse Kinetics




Forward Kinetics



Four-Bar Mechanism Kinematics and Kinetics



velocityequation








Parameter M-File


Dynamics of Four-Bar Mechanism

Ar2












Inverse Kinetics







Forward Kinetics



Kinematic Synthesis

• How does one design four-bar and slider-crank mechanisms for function generation?

• Freudenstein’s Equation

• This is a design (synthesis) problem where the two link lengths of a four-bar mechanism must be determined so that the rotations of the two levers within a mechanism, φ and ψ, are functionally related.

f(φ, ψ) = 0


2 3 1 4r r r r+ = +

Square both sides, add, and divide both sides by 2r2r4:

3 2 2 2

2 3 4 11 11 2 3

4 2 2 4

r r r rr rR R R

r r 2r r

− + += = =Define:

Freudenstein’s Equation

rewrite


• Freudenstein’s Equation gives a relationship between input rotation θ2 and output rotation θ4 as determined by link lengths r1, r2, r3, and r4.

• Determine a set of link lengths that will result in a (θ2-θ4) relationship that matches a desired function.

• Two four-bar mechanisms that are scaled versions of one another will have the same (θ2-θ4) relationship since the scaling factor will be eliminated in the ratios R1, R2, & R3.

• Function-generating four-bar mechanisms are scale invariant. They are also rotation invariant. This means that they can be rotated into any orientation without affecting the function generation.

• There are two basic methods for function generation:

– Point-Matching Method Focus on This

– Derivative-Matching Method (not covered)


2 2i

4 4i

= −

= −

constant constant

Differentiate2

4

d d

d d

=

=

changes in the anglesare equal

Point-Matching Method: Free to choose both θ2i and θ4i.


• Point-Matching Method

– The four-bar mechanism is designed to match three desired pairs of {φ,ψ} values that satisfy the desired function generation f(φ, ψ) = 0.

– The three pairs of values are known as precision points: {(φ)j,(ψ)j} where j = 1, 2, 3.

– The precision points are free choices of the designer.

– The designer chooses values for θ2i and θ4i. These are also free choices.

– Convert {(φ)j,(ψ)j} into {(θ2)j,(θ4)j} where j = 1, 2, 3.

– Substitute each precision point pair into Freudenstein’s Equation, one pair at a time, to generate three linear equations in the three unknowns R1, R2, and R3.


– Solve for R1, R2, and R3.

– Assume a value for one of the link lengths and then solve for the remaining three link lengths.

– The assumed link length affects the scale of the mechanism.

– The choice of precision points is a design variable that can be varied to optimize matching the desired function. Precision points at the extreme ends of the range generally are not good. This results in large errors in the middle of the range and no errors at the ends. The idea is to minimize the overall error, although sometimes it is necessary to exactly match end conditions, e.g., opening and closing a valve.

Result:


In applications where high levels of torque must be transmitted from the input to the output and the rotations are oscillatory (not continuous), a four-bar mechanism is more appropriate than a pair of gears. Mechanisms in general have several orders of magnitude higher torque density (ratio of torque capacity of a mechanism to its weight) than gears. Gears are used when a constant speed ratio is required for continuous rotations of the input and output as in a transmission. In that case, only gears will work. The following example illustrates the use of Freudenstein’s Equation to design a four-bar mechanism that generates a linear function that can replace a pair of gears with limited rotations.

Example Problem• Raise an urban bridge to allow for the passing of a ship where the

bridge needs to be rotated (lifted) a finite amount and the torque required to do this is large even if partially counterbalanced.

• Design a four-bar mechanism to replace a pair of gears with a 2:1 gear ratio (R = 2). Consider φ as the input rotation and ψ as the output rotation.

• In such a gear pair, the function generation would be linear and given by:


1 2 3

1 2 4 3

R 0.5880 R 1.4475 R 0.5790

r 10 r 6.866 r 16.886 r 17.278

= = =

= = = =

2i 4i20 40 = = initial choice

Solve

Solution


Desired Motion vs. Actual Motion

Desired


To Improve Design• Spread precision points farther apart• Choose different values of θ2i and θ4i


Point-Matching Method applied to the Crank-Slider Mechanism

– Determine the link lengths of a crank-slider mechanism so that the translation r and the rotation φ are functionally related, i.e., f(φ, r) = 0.

2 3 1 4

2 2 3 3 4

2 2 3 3 1

r r r r

r cos r cos r 0

r sin r sin r 0

+ = +

+ − =

+ − =

2 2i

4 4ir r r

= −

= −


– Rewriting the equations:

– Square both sides, add the equations together, and rearrange:

– Unlike the four-bar mechanism, function-generating crank-slider mechanisms are not scale invariant because S1, S2, and S3 are not dimensionless ratios.

– Also

3 3 2 2 4

3 3 2 2 1

r cos r cos r

r sin r sin r

= − +

= − +

2

1 4 2 2 2 3 4

2 2 2

1 2 2 1 2 3 3 2 1

S r cos S sin S r

S 2r S 2r r S r r r

+ + =

= = = − −

2 2i 2

4 4i 4

d d

r r r dr dr

= − =

= − =


– The point-matching method with Freudenstein’s Equation is a method by which a crank-slider mechanism is designed to match three desired pairs of {φ, r} values that satisfy the desired function relation f(φ, r) = 0. These pairs of values, known as the precision points, are denoted as: {(φ)j, (r)j} where j = 1, 2, 3

– These three precision points are free choices of the designer. The designer then chooses values for θ2i and r4i. These are also free choices. There are 5 free choices for optimization.

– Convert the three {φ, r} precision points into the three {θ2, r4} precision points.

– Substitute into Freudenstein’s Equation for the crank-slider mechanism {(θ2)j, (r4)j} with j = 1, 2, 3

– Result is three linear equations with three unknowns.

Solve for Sj

Then solve for rj

j = 1, 2, 3


Example:Shown is mechanical flowmeter. The flow of an incompressible fluid in a pipe passes through an orifice. The orifice causes a pressure drop so that the fluid pressure before the orifice is greater than the fluid pressure after the orifice. The pressure drop ΔP = Phigh – Plow is related to the volumetric flowrate q by the equation: ΔP = C1q

2 where C1 is determined experimentally.

q


– Pressure taps on opposite sides of the orifice feed some of the fluid into a cylinder, where each pressure exerts a force on one face of a slider. The pressure drop ΔP causes the slider to move to the right, until it is statically equilibrated by a compression spring.

– ΔP is linearly related to slider displacement ΔP = C2r where C2 is the ratio of spring stiffness to cross-sectional area of the piston (slider), and r is referenced from the position of the pin joint on the slider, which corresponds to the unstretched spring length.

– The relationship between flow velocity and the slider displacement then is:

2 21

2

2

C inr q 0.00240 q

C gpm

= =

nonlinear


• C3 is the sensitivity for a linear scale. Let C3 = 0.5 gpm/degree. Therefore q = C3φº and r = C4(φº)2 where C4 = 0.00060 in/deg2.

• Use arbitrary choices r4i = 2 inches and θ2i = 30º.

S1 = -8.6474 S2 = -0.6965 S3 = 18.0021r2 = -4.3237 in r1 = 0.0805 in r3 = 6.0583 in

Solve


Desired

Result

Desired Motion vs. Actual Motion


SupplementPlane-Motion Kinetics








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