FOUNDATION of MECHANICS 1 MECHANICS of MACHINES MECHANISM & MACHINE SCIENCE
The Mechanics of Machines deals with problems associated with the movement and equilibrium of mechanical systems. The M.o.M. main topics are:
• Composition of the machines: degrees of freedom of a mechanism, mechanical efficiency... ; • Tribology: contact between the organs of the machines during their relative motion, power transmission, friction, energy dissipation, wear, rolling friction, lubrication...; • Theory of mechanisms: machine behavior from the functional viewpoint; linkages, couplings , cams, gears and gearing, brakes... • Dynamics: calculation and balancing of inertia actions, coupling between actuator and oppgerating machine, functionin g of machines and plants in steady states or for transients… MACHINE
• Mechanical system that transmits and transforms force and motion (i.e. energy) t o perf orm a specifi c t ask :
– MECHANISM (speed reducers and variators, belt/chain transmissions, cam mechanisms , coupler ... ) – MOTOR (IC engines, electric motors, hydraulic motors...) – GENERATOR (pumps and compressors, dynamos, turbines...) – OPERATING (or WORKING) MACHINE (machine tools, automatic m., packaging m., farming m., textile m., lifting and transport m., vehicles, robots...) pmppump speed reducers
IC engine industrial robot MODELLING
• Starting point: defining a proper model of the examined mechilhanical sys tem, cons idiidering a llthll the aspec ts we are interested in (and only those) and assuming reasonable hypotheses and approximations.
REAL PHYSICAL MATHEMATICAL SYSTEM MODEL MODEL
. SOLUTION: analytical numerical graphical MODELLING
Slider-crank mechanism
piston
conrod
accessories ANALYSES
• Three kinds of analysis in Mechanics of Machines:
• Kinematic analysis (+ Kinematic synthesis)
• Static analysis – Kinetostatic • Dynamic analysis (Inverse dynamic) – Dynamic (Direct dynamic) COMPOSITION OF MECHANISMS basic definitions Rigygid body: solid body in which the distance between any two given points is constant.
A basic concept in Kinematics is the Degree of Freedom (DOF)
Number of independent kinematic variables required to completely define the configuration of a system at any instant of time .
• DOFs of a free rigid body in a 2D Space (plane): 3
• DOFs of a free riidbdigid body in a 3D Space: 6 COMPOSITION OF MECHANISMS basic definitions
• Link • Kinematic element & Kinematic joint (or pair) • Mechanism: system of bodies designed to convert motion of or one or several bodies into forces on
constrained motion of or other bodies. forces on COMPOSITION OF MECHANISMS COMPOSITION OF MECHANISMS basic definitions
• Link • Kinematic element & Kinematic joint (or pair) • rigid vs. flexible joints • contact surface vs. contact point/segment joints - Lower ppgair: rigid AND contact surface - Higher pair: non rigid OR point/segment contact •planar vs. sphiherica l vs. generic jitjoints • linkage: planar mechanism with lower joints only KINEMATIC JOINTS
REVOLUTE
PRISMATIC
HELICAL KINEMATIC JOINTS
CYLINDRICAL Revolute PLANAR HIGHER PAIR
PLANE on PLANE
SPHERICAL KINEMATIC JOINTS
Relative motion 2-1 V M 0 (21)t Rolling
V M (21)n 0 V M 0 (21)t Sliding • Projections of the kinematic V M (21)n 0 (Rolling possible as well) elements in the plane: common tangent line in M
V M (21)t • Contact kept during the Impact relative motion V M (21)n 0 or Separation CONJUGATE PROFILES KINEMATIC JOINTS
DOF Class Conventional name Rotation Translation Helical motion R (Revolute) 1
1 C1 P (Prismatic) 1 H (Helical) 1
RT 2 C (Cylindrical) 1 1 2 C2 CS (Planar higher pair) 1 1 R 1 1 S (Spherical) 3
SA 2 1 3 C3 SL 2 1 PP (Plane on plane) 1 2
SC 3 1 4 C4 SE 3 1 CC 22
5 C5 S5 32 Bold: most used joints Red: planar joints Highlighted: lower pair COMPOSITION OF MECHANISMS basic definitions No link is Kinematic chain vs. Mechanism One link is fixed a priori chosen as the frame Kinematic chains (examples)
STEPHENSON WATT kinematic chain kinematic chain
Mechanisms (from previous kin. ch.) DOFs of MECHANISMS
Calculating the DOFs of a mechanism: the Grübler’s formula (DOF := Number of independent kinematic variables required to completely define the configuration of a system at any instant of time) m = number of links [1 link is the frame]
ci = number of joints [(6 – i) DOFs are constrained] leaving i DOFs free
3D) lm6( 1) 5 ccccc12345 4 3 2
2D) lm 3( 1) 2 cc12 DOFs of MECHANISMS
Calculating the DOFs of a mechanism: the Grübler’s formula
l = mechanism DOFs NJ = number of joints
m = numbflikber of links li = DOFs le ft f ree b y th e i-th jjitoint
NJ 3D) lm 6( 1) (6 li ) i1
NJ 2D) lm 3( 1) (3 li ) i1 l number of configuration variables to be actuated bfby means of motors or oth hdi/hier devices/mechanisms DOFs of MECHANISMS
Examples l = 1 l = 1
Crank–slider mechanism Four-bar linkage
l = 1 l = 1
Belt & Pulleys transmission Cam system CAM SYSTEMS: overview DOFs of MECHANISMS
Examples l = 3 l = 3
l =6= 6 l =6= 6 DOFs of MECHANISMS
Examples
l = 2 DOFs of MECHANISMS
Caution in using Grubler’s formula! 1 A AM = MB • 3D vs. 2D Spaces R R M R 5 2 R 3 4 B RR
• Redundant constraints
• Ineffective/meaningless constraints 2
3 1 lm3( 1) 2 cc12
4 lm6( 1) 5 ccccc12345 4 3 2 DOFs of MECHANISMS
Ineffective/meaningless constraints
McPherson Suspension