4a Mechanical Models 1

MCHA2000

Modelling of Mechanical Systems Part I

A/Prof Tristan PerezLeader Mechatronics Program & Robotics Research

School of Engineering

Review

2011MCHA2000 - Mechatronics Systems | www.newcastle.edu.au

2

Power variables

Energy variables

Review


3


4

Rotational Systems

Translational Systems


5

Study of Mechanics

Statics

DynamicsKinematics

Kinetics

Kinematics – Inertial Frames

Law of Inertia (Galilei-Newton): a body that is sufficiently removed from interaction with other bodies will either continue in its state of rest or uniform motion (rectilinear).


6

Inertial Frame: a reference frame in which the Law of Inertia holds

Kinematics

Invariance in Newtonian Mechanics: Time and relative positions are the same in different frames.


7

Velocity:

Acceleration:

The time-derivative of a vector is not invariant!

Position:

(in an inertial frame)

(in an inertial frame)

In this course, we only use inertial frames.

Power & Energy Variables - Translation


8

Inertia CCR (Mass)

Particle:


9

Inertial frame

Inertia CCR (Mass)

System of Particles


10

Inertial frame

(Euler’s 1st Axiom)

Resistors (Friction)


11

Coulomb Linear Viscous Nonlinear Viscous

!!!m

vr el

FR FR

v1v2

FR

Capacitors (Springs)


12

Spring force is positive when the spring is in tension.

Elastic potential energy:

x0

es

Fs

x

∆x

PQm

FsFs

Modelling Translational Systems

1. Set the coordinate system and the positive convention for the displacements, velocities and forces.

2. Choose as states the momentum of the masses (or the velocities) and the displacement of the springs. If you require absolute positions you may need to add the integral of the velocities.

3. Use the free-body diagrams of the masses to obtain the SSR (balance of forces).

4. Look at the system configuration to determine kinematic SSR.

5. Combine the SSR and CCR to write the state-space equations.


13

Example


14

c

m1m2

b1b2b12

FT (t)

x(t)v1(t)

v2(t)

• b1 - linear friction coefficient of the truck,

• b2 - linear friction coefficient of the trailer,

• b12 - linear friction coefficient of the linkage,

• c - compliance of the linkage,

• m1 - mass of the truck,

• m2 - mass of the trailer.


15

Rotational Systems

Kinematics of Single Axis Rotation


16

v

ω

O

P

Q

rP / OrQ/ O

rP / Q

θ

Moment of a Vector


17

o

u

rP / O

Mu/ O

P

Angular Momentum of a particle (moment of momentum):

Moment of force:

Depends on the point O but is independent of P

Couples and Torque

Couple: A pair of forces that do not have the same line of action and have null resultant.


18

Torque: A torque is the resultant moment of a couple, and it is independent of the point about which the moments are taken.

Power & Energy Variables - Rotation


19

Variable General Mechanical SI unitE�ort e(t) ~T (t), Torque [Nm]Flow f (t) ~! (t), angular velocity [rad/ s]Power e(t)f (t) ~T (t) · ~! (t) [W]Momentum p(t) ~L (t), angular momentum [kg m2rad/ s]Displacement q(t) ✓(t), angular displacement [rad]Energy E (p) =

Rf (p) dp E (~L ) =

R~! (~L ) · d~L , kinet ic J

E (q) =Re(q) dq E (✓) =

R ~T (✓) · d~✓, elast ic J

Inertia CCR (Inertia about single axis)


20

Moment of inertia

Euler 2nd Axiom: ω

ri mi

Kinetic Energy:

Resistors (Friction)


21

Coulomb Linear Viscous Nonlinear Viscous

ωr el

TR TR

ω1 ω2

TR

Capacitor (Spring)


22

kT =1cT

[Nm/ rad]

Potential Energy:

TsTs

θ1 θ2

cTTs

Modelling Rotational Systems

1. Set the coordinate system and the positive convention for the angular displacements, angular velocities and torques.

2. Choose as states the angular momentum of masses (or the angular velocities) and the angular displacement of the springs. If you require absolute angles you may need to add the integral of the angular velocities.

3. Use the free-body diagrams of the masses to obtain the SSR (balance of torques).

4. Look at the system configuration to determine kinematic SSR.

5. Combine the SSR and CCR to write the state-space equations.


23

Example


24

θ1 θ2

J1J2

ω1

ω2

bT 2bT 1

cT

Te1 Te2

Structural Relations in Mechanical SystemsThere are only two ways of connecting components:

1. Share the flow (velocity)

2. Share the effort (force or torque)


25

b

m F(t)

cSharing Velocity

bm

cx1(t)

x2(t)x3(t)

Sharing Force

x(t)

vin(t)

θ1ω1

θ1ω1 ω2

θ2

Sharing angular velocity

Sharing torque

Documents

4a Mechanical Models 1