FUNDAMENTALS Topic 4 Gerald Rothenhofer 9/21/2009 Linkages FUNDAMENTALS Topic 4 Gerald Rothenhofer 9/21/2009 What is a linkage? Who uses linkages? Who rides a bike?

What is a Linkage? A mechanical linkage is a series of rigid links connected with joints to form a closed chain, or a series of closed chains. Each link has two or more joints, and the joints have various degrees of freedom to allow motion between the links. A linkage is called a mechanism if two or more links are movable with respect to a fixed link. Mechanical linkages are usually designed to take an input and produce a different output, altering the motion, velocity, acceleration, and applying mechanical advantage. A linkage designed to be stationary is called a structure. en.wikipedia.org/wiki/Linkage_(mechanical)

History Leonardo da Vinci (1452, 1519), Codex Madrid I. Industrial Revolution was the boom age of linkages: cloth making, power conversion, speed regulation, mechanical computation, typewriting and machining

Linkages Today In many applications (typewriting) linkages have been replaced by electronics. Still linkages can have a cost advantage over electronic solutions: Couple different outputs by a mechanism rather than using one motor per output and electronics to achieve the coupling. Current applications: Sports Equipment, Automotive (HVAC modules), Precision Machinery (Compliant Mechanisms), Medical Devices

Linkage Categorization Planar Linkages Three bar Four bar Slider Crank Five bar Six bar …be creative… Spatial Linkages

Degrees of Freedom Planar Linkages: F=3*(N-1)-2*J1-Jh F – total degrees of freedom N – number of links J1 – constraints by 1DOF joints Jh – constraints by 2DOF joints How many degrees of Freedom in MTB frame? How many degrees of freedom in MTB frame? What types of planar linkages are possible (using 1DOF joints)? How would the degree of freedom formula look like for a 3d linkage (ball joints)? F=6*(N-1)-Jx

Four Bar Grashof: The sum of the shortest (S) and longest (L) links of a planar four-bar linkage must be smaller than the sum of the remaining two links (P, Q). In this case the shortest link can rotate 360degree relative to the longest link. L + S < P + Q: crank-rocker, double-crank, rocker-crank, double-rocker L + S = P + Q: crank-rocker, double-crank, rocker-crank, double-rocker,  note: linkage can change its closure in singularity positions (all links aligned) If L + S > P + Q, double-rocker, no continuous rotation of any link Is a parallelogram a good four bar linkage?

Transmission Angles – Four Bar A – ground link B – input link C – coupler D – output link A B D C γ Practical use of zero transmission angles can be made in locking mechanisms Transmission angle in slider crank linkage? Angle between coupler and output link should be 40º≤γ≤140º  Zero torque at output link if γ=0º or γ=180º

Transmission Angles – Slider Crank α a b y x T Θ2 Θ1 Especially important in critical position such as within the main working range or high load positions Minimize α Minimize |θ2-90°| No stick condition: 1/tan(α)<μ

Four Bar Synthesis A – ground link B – input link C – coupler D – output link A B D C Function Generation (input/output relation) Line Path Generation (line on coupler) Point Path Generation (coupler point)

Four Bar Two Position Synthesis Instant center example: longest link is the input, rotates with omega, how do you determine the speed of any other point on the linkage?

Four Bar Three Position Synthesis Where are the instant centers, depending on which link is the ground link, do velocity analysis, make sure that links do not collide, if collision occurs one or more links can be replaced by circular slot Replace a link by finding coupler point that approximately moves on a circle within the required range of motion

Cognate Mechanisms Provide identical motion of a point or link Here: coupler point cognate

Four Bar Function Generation Two angular displacements Only one initial position; either primary or secondary side can be chosen freely (here 60°) E.g.: Primary side moves by 2x 20 ° Secondary side moves by 35°+30°

Crank Rocker Design Design in extreme positions Typically design for crank movement >180º depending on required transmission ratio i.e. rocker should move slowly when load is heavy, the return fast In this example rocker moves through 60º while the crank moves through 180º+10º=190º

Slider Crank Synthesis I Two point synthesis make sure that crank only rotates +/- (required angular stroke)/2  slider will move required stroke

Slider Crank Synthesis II Three point synthesis by geometrical inversion

Other Basic Four Bar Design Methods Approximate function generation Approximate coupler point path generation Uncorrelated with input Correlated with input Slider crank synthesis by approximation Sometimes it is desirable to replace a slider crank by a four bar in order to reduce guide way friction e.g. instrumentation -> approximate coupler point path generation Mechanism Design: Chapter 3

What is possible with advanced design methods? Four coupler position synthesis In some cases five coupler position synthesis is achievable Straight line motion Complex linkages (more than four bars) Spatial linkages

Five Bar How many degrees of freedom? Why does it work? In theory: two degrees of freedom, but friction in connecting link (between blades) eliminates one degree of freedom. How many degrees of freedom? Why does it work?

Six Bar Watt & Stephenson Linkages approximate dwells or better MTB suspension

Four Bar Analysis I rp ψ d ex ey a b c Θ2 Θ1 Θ2 is a complicated trigonometric function of Θ1, Θ2=f(Θ1)

Four Bar Transmission Ratio

Slider Crank Analysis x0 y0 R a b θ1 θ2 F, V T1, ω1 φ F is maximum available force (no friction or other loads taken into account)

Slider Crank Transmission Ratio x0 y0 R a b θ1 θ2 F, V T1, ω1 φ The transmission ratio determines the relation of slider (flapper) position and motor angle.

Power Budget Assumes that load forces are constant y x Tin δΘin δl Assumes that load forces are constant Average and max load forces should be used to check for safety factors δl  arc length between starting point and end point of slider movement

Slider Friction I α a b y x T Θ2 Θ1 RF F

Slider Friction II F F║ F┴ α a y x T Θ2 Θ1

Questions ?