2.007 –Design and Manufacturing I Mechanisms: Kinematic Analysis and SynthesisToday’s Agenda2D Mechanisms2D Mechanisms2D Mechanisms2D MechanismsTwo Goals in our Study of MechanismsDegrees of FreedomTypes of “Joints”Mobility Criteria in 2DMobility Criteria in 2DPractice TogetherPractice TogetherThe “4 Bar” MechanismParallelogram LinkageParallelogram LinkageSlider-Crank MechanismMore Practice TogetherMore Practice TogetherNow You Do ItNow You Do ItNow You Do It AgainNow You Do It AgainOne More TimeOne More TimeRedundancyDegrees of Freedom in 3DVelocity AnalysisVelocity AnalysisVelocity Analysis3 Position SynthesisA Proper PourAutomating the TaskComputer Aided 3 Position SynthesisHomogeneous Transformation MatricesComputer Aided Mechanism Synthesis: More PossibilitiesNext StepsMIT OpenCourseWare http://ocw.mit.edu 2.007 Design and Manufacturing ISpring 2009For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.2.007 –Design and Manufacturing IMechanisms: Kinematic Analysis and SynthesisPresented by Dan Frey on 24 FEB 2009Image removed due to copyright restrictions. Please see http://www.automotive-illustrations.com/img/suspensions/accord-rear-sus.jpgSource: Fig. 4-31 in Suh, C. H. Computer Aided Design of Mechanisms Part A. Self-published book, 1989. Courtesy of Dr. C. H. Suh. Used with permission.Today’s Agenda• Collect homework #1• Distribute homework #2• Begin mechanisms–Mobility / degrees of freedom–Some basic mechanisms–Kinematic analysis–Mechanism synthesis–Solving systems of equations2D Mechanisms• Where are they found?• What do they do?2D Mechanisms• Where are they found?• What do they do?http://www.mossmotors.com/Image removed due to copyright restrictions. Please see http://mossmotors.com/Graphics/Products/Schematics/SPM-032.gif2D Mechanisms• Where are they found?• What do they do?Image from Wikimedia Commons, http://commons.wikimedia.org2D Mechanisms• Where are they found?• What do they do?Two Goals in our Study of Mechanisms• Analysis – Given a mechanism, predict its behavior– Degrees of freedom– Path of motion– Dynamics -- velocity, acceleration– Force transmission• Synthesis– Given a desired behavior, develop a mechanismDegrees of Freedom• DOF = the number of independent parameters required to completely define the entity’s location and orientation• If a body is constrained to a plane, how many DOF does it possess?• If a pin joint is added?• If a sliding joint is added?• If it rolls without slipping?Types of “Joints”• How do these attachments affect the degrees of freedom?• What about welding / clamping?Source: Figure 4 in Suh, C. H. Computer Aided Design of Mechanisms Part B. Self-published book, 1992. Courtesy of Dr. C. H. Suh. Used with permission.Norton, R. L., Design of MachineryMobility Criteria in 2D• Kutzbach criterion (to find the DOF)• Grübler criterion (to have a single DOF)F=3(n-1)-2jDOF# of bodies# of full pinned joints2j-3n+4=0Mobility Criteria in 2D• Count the rigid bodies• Every rigid body has 3 DOF• For the “joints”– Consider how many degrees of freedom each joint removes– Sum up the degrees of freedom removed• Consider the possibility of redundancy of certain “joints”• Do the arithmeticPractice TogetherBody (assume it is fixed)Pin jointsHow many degrees of freedom?Practice TogetherBody (assume it is fixed)Pin jointsHow many degrees of freedom?The answer is one degree of freedom. Three bodies capable of motion in the plane each with 3 DOF. Four full pin joints each remove 2 DOF. 3*3-4*2=1The “4 Bar” Mechanism4th“ground” link4 Pin joints3 moving “bars”Parallelogram Linkage• Where is the parallelogram?• What can we say about the resulting motion of the coupler?Parallelogram Linkage• Where is the parallelogram?• What can we say about the resulting motion of the coupler?The resulting motion of the coupler exhibits translation in x and y, but no rotationSlider-Crank Mechanism• Kinematically the same (in the limit) as a 4 bar mechanism with a very long output crankCourtesy of Design Simulation Technologies, Inc. Used with permission.More Practice TogetherBody (assume it is fixed)Pin jointsWe add the servo, horn, and linkNOW how many degrees of freedom?More Practice TogetherBody (assume it is fixed)Pin jointsWe add the servo, horn, and linkNOW how many degrees of freedom?The answer is one degree of freedom. Two more bodies capable of motion in the plane each with 3 DOF. Three more full pin joints each remove 2 DOF. 2*3-3*2=0 so no net changeNow You Do It• How many degrees of freedom does this mechanism possess?1)1 DOF2)2 DOF3)3 DOF4)4 DOF5)0 DOFBody (assume it is fixed)Pin jointsPin jointsAssume the wheel is fixed to its shaft and suspended over the table for the purpose of this questionNow You Do It• How many degrees of freedom does this mechanism possess?1)1 DOF2)2 DOF3)3 DOF4)4 DOF5)0 DOFBody (assume it is fixed)Pin jointsPin jointsAssume the wheel is fixed to its shaft and suspended over the table for the purpose of this questionThe answer is two degrees of freedom. Four bodies capable of motion in the plane each with 3 DOF. Five full pin joints each remove 2 DOF. 4*3-5*2=2.One DOF is the desired motion of the link on which the wheel is attached. One DOF is the link in the middle that is free to spin for now until the pneumatic is attached.Now You Do It Again• How many degrees of freedom does this mechanism possess?1)1 DOF2)2 DOF3)3 DOF4)4 DOF5)0 DOFBody (assume it is fixed)pneumaticExtra Pin jointsAssume the wheel is fixed to its shaft and suspended over the table for the purpose of this questionNow You Do It Again• How many degrees of freedom does this mechanism possess?1)1 DOF2)2 DOF3)3 DOF4)4 DOF5)0 DOFBody (assume it is fixed)pneumaticExtra Pin jointsAssume the wheel is fixed to its shaft and suspended over the table for the purpose of this questionThe answer is one degree of freedom. Two more bodies were added, the piston body and the piston rod. Three attachments exist. Sliding motion constrains the piston rod within the piston body (-2DOF). The piston body is attached with a full pin joint to the frame (-2DOF). Finally, the piston rod is fixedly attached to the pivot so that they move together (-3DOF) 2*3-2-2-3=-1.One DOF is removed from a system that previously had 2 DOF.One More Time• How many degrees of freedom does this mechanism