FOUNDATION of MECHANICS 1MECHANICS of MACHINES

MECHANISM & MACHINE SCIENCE

The Mechanics of Machines deals with problems associated with themovement and equilibrium of mechanical systems. The M.o.M. main topics are: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 ;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 brakeslinkages, couplings, cams, gears and gearing, brakes...

• Dynamics: calculation and balancing of inertia actions, coupling between actuator and operating machine, functioning of machines and plants in p g g psteady states or for transients…

MACHINE

• Mechanical system that transmits and transforms force and motion (i ) t f ifi t k(i.e. energy) to perform a specific task:

– MECHANISM (speed reducers and variators, belt/chain transmissions, cam mechanisms coupler )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...)

p mp

speed reducers

pump

IC engine industrial robot

MODELLING

• Starting point: defining a proper model of the examined h i l t id i ll th tmechanical system, considering all the aspects we are

interested in (and only those) and assuming reasonable hypotheses and approximationshypotheses and approximations.

REAL SYSTEM

PHYSICAL MODEL

MATHEMATICAL MODEL

SOLUTION:

analytical

numerical

graphical

MODELLING

crankshaft

Slider-crank mechanism

piston

conrod

accessories

ANALYSES

• Three kinds of analysis in Mechanics of Machines:

• Kinematic analysis (+ Kinematic synthesis)

• Static analysis

• Dynamic analysis– Kinetostatic

(Inverse dynamic)

– Dynamic(Direct dynamic)

COMPOSITION OF MECHANISMSbasic definitions

Rigid body: solid body in which the distance between any two g y y ygiven points is constant.

A basic concept in Kinematics is the Degree of Freedom (DOF)

Number of independent kinematic variablesrequired to completely define the configurationof a system at any instant of timeof a system at any instant of time.

• DOFs of a free rigid body in a 2D Space (plane):

DOF f f i id b d i 3D S

3

6• DOFs of a free rigid body in a 3D Space: 6

COMPOSITION OF MECHANISMSbasic definitions

• Link

• Kinematic element & Kinematic joint (or pair)• Kinematic element & Kinematic joint (or pair)

• Mechanism: system of bodies designed to convert

motion of

for one or several bodies intoforces on

constrained motion of

forces onor other bodies.forces on

COMPOSITION OF MECHANISMS

COMPOSITION OF MECHANISMSbasic definitions

• Link

• Kinematic element & Kinematic joint (or pair)• Kinematic element & Kinematic joint (or pair)• rigid vs. flexible joints

• contact surface vs. contact point/segment joints

- Lower pair: rigid AND contact surfacep g

- Higher pair: non rigid OR point/segment contact

l h i l i j i t• planar vs. spherical vs. generic joints

• linkage: planar mechanism with lower joints only

KINEMATIC JOINTS

REVOLUTE

PRISMATIC

HELICAL

KINEMATIC JOINTS

CYLINDRICAL RevolutePLANAR HIGHER PAIRPLANAR HIGHER PAIR

PLANE on PLANEPLANE on PLANE

SPHERICAL

KINEMATIC JOINTS

Relative motion 2-1

Rolling ( 21) 0tMV

g( 21) 0nMV

Projections of the kinematic Sliding(Rolling possible as well)( 21) 0MV

( 21) 0tMV

• Projections of the kinematic elements in the plane: common tangent line in M

(Rolling possible as well)( 21) 0nMV

V

• Contact kept during the relative motion CONJUGATE PROFILES

Impact or Separation( 21) 0nMV

( 21) tMV

CONJUGATE PROFILES

KINEMATIC JOINTS

DOF Class Conventional name Rotation Translation Helical motion

1 C1

R (Revolute)P (Prismatic)H (Helical)

11

1

2 C2

RTC (Cylindrical)CS (Planar higher pair)R

2111

11

1R 1 1

3 C3

S (Spherical)SASL

322

11SL

PP (Plane on plane) 1 2

4 C4

SCSE

33

114 E

CC 2 2

5 C5 S5 3 2

Bold: most used joints Red: planar joints Highlighted: lower pair

COMPOSITION OF MECHANISMS

Kinematic chain vs. MechanismNo link is fixed a priori

One link is chosen as

basic definitions

fixed a priori chosen as the frame

Kinematic chains (examples)

STEPHENSON WATTSTEPHENSON kinematic chain

WATT kinematic chain

Mechanisms (from previous kin. ch.)

DOFs of MECHANISMS

Calculating the DOFs of a mechanism: the Grübler’s formula g(DOF := Number of independent kinematic variables required to

completely define the configuration of a system at any instant of time)

m = number of linksc = number of joints

[1 link is the frame][(6 i) DOFs are constrained]ci = number of joints

leaving i DOFs free[(6 – i) DOFs are constrained]

1 2 3 4 56 ( 1) 5 4 3 2l m c c c c c 3D)

3( 1) 2l m c c2D) 1 23( 1) 2l m c c 2D)

DOFs of MECHANISMS

Calculating the DOFs of a mechanism: the Grübler’s formula g

b f li kNJ = number of jointsl = mechanism DOFsl DOF l ft f b th i th j i t

JN

m = number of links li = DOFs left free by the i-th joint

16 ( 1) (6 )i

il m l

3D)

3( 1) (3 )JN

il m l 2D)1i

l number of configuration variables to be actuatedb f h d i / h iby means of motors or other devices/mechanisms

DOFs of MECHANISMS

Examplesp

Crank–slider mechanism Four-bar linkage

l = 1 l = 1

Four bar linkage

l = 1 l = 1

Cam systemBelt & Pulleys transmission

CAM SYSTEMS: overview

DOFs of MECHANISMS

Examplesp

l = 3 l = 3

l = 6 l = 6l = 6 l = 6

DOFs of MECHANISMS

Examplesp

l = 2

DOFs of MECHANISMS

Caution in using Grubler’s formula!g

• 3D vs. 2D SpacesR

A1

AM = MB

5R

RB

23

4

MR

R

• Redundant constraints

B4R R

• Ineffective/meaningless constraints 2

4

3 1

6( 1) 5 4 3 2l m c c c c c1 23( 1) 2l m c c

41 2 3 4 56 ( 1) 5 4 3 2l m c c c c c

DOFs of MECHANISMS

Ineffective/meaningless constraintsIneffective/meaningless constraints

McPherson SuspensionMcPherson Suspension