modeling and analysis of dynamic systems · lecture 2: modeling tools for mechanical systems...

Modeling and Analysis of Dynamic Systems

by Dr. Guillaume Ducard c©

Fall 2017

Institute for Dynamic Systems and Control

ETH Zurich, Switzerland

G. Ducard c© 1 / 21

Outline

1 Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum


Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum

List of Basic Modeling Elements1 Mechanical Systems

2 Hydraulic Systems

3 Electromagnetic Systems

4 Electromechanical Systems

5 Thermodynamic Systems

6 Fluiddynamic Systems

7 Chemical Systems

Case Study

Water-propelled RocketG. Ducard c© 3 / 21


Basic Modeling Methods

1 Reservoir-based approach

2 Energy Methods: often simpler when constrained orconnected systems are to be analyzed

3 Newton or Euler equations4 Lagrange formulation: appropriate for systems with:

multiple bodies that have more than 1 degree of freedomand may have some cinematic constraints



Outline




Kinetic energy: translation

Tt(t) =1

2m

(

v2x,cg + v2y,cg)

(2D-case) (1)

Kinetic energy: rotation

Tr(t) =1

2Jω2(t) =

1

2Θω2(t)a (2)

a

J or Θ is the moment of inertia [m2kg], ω is the angular speed in [rad/s]

Potential energy

U(t) = U(x(t), y(t)) (3)

can always be expressed in terms of the system body’s coordinates

Remark:location dependance for angular velocity vs. linear velocity (see scriptp.16).



Total energy

E(t) = T (t) + U(t) (4)

Mechanical power balance

The differential equation (step 3) is obtained with:

dE(t)

dt= Σk

i=1Pi(t) (5)

where Pi are the mechanical powers acting on the body

Power of a force

P = F · v (6)

Power of a torque

P = T · ω (7)G. Ducard c© 7 / 21


Outline




Simplified Vehicle Modeling

Objective: derive a model that can be used to design a robustcruise-speed controller.




γgear-box

engine

m/2

m/2 ωw

ωe

rwv(t)

Fr + Fa(t) + Fd(t)

Θe

Θw

Te

ωe: engine turn rate [rad/s], m: total weight (incl. wheel masses)[kg], rw: wheel radius [m], ωw: wheel turn rate [rad/s], γgear−box ratio, Fr rolling friction [N]




Assumptions

1 the clutch is engaged such that the gear ratio γ is piecewise constant;

2 no drivetrain elasticities and no wheel slip effects need to be considered,i.e.,: ωw(t) = γωe(t) and v(t) = rwωw(t) ;

3 the vehicle has to overcome:

rolling friction: Fr = crmgaerodynamic drag: Fa(t) =

12ρcwAv

2(t); (A is the apparent vehiclesurface)

4 all other forces are packed into an unknown disturbance Fd(t);

5 the kinetic energy of a moving part:

pure translation: 12mv2

pure rotation: 12Jω

2 ( or 12Θω2)

6 No potential energy effects need to be considered (even road).

7 The vehicle mass m includes the mass of the engine flywheel and thewheels. G. Ducard c© 11 / 21



Step 1: Inputs & Outputs

The system’s input: is the engine torque Te [Nm].It is assumed to be arbitrarily controllable (the time delay caused bythe engine dynamics is an “algebraic” variable (= no dynamics)).

The system’s output: is the car’s speed v(t) in [m/s].

Step 2: Reservoirs and level variables

Reservoirs: are the kinetic energies stored in the vehicle’s translationaland rotational moving elements

Etot =?

The “level variable” is the vehicle speed v(t) [m/s].




Step 3: Formulate a differential equation for the reservoir content

We look for dE(t)dt

= ?

The “flows” acting on the system are the mechanical powersaffecting the system, i.e.

P+(t) =? and P−(t) =?

The differential equation is therefore found by

d

dtEtot(t) = P+(t)− P

−(t).




Step 4: Formulate the algebraic relations of the flows as a functionof level variables

we look for P+(t)− P−(t) = f (v(t)).



replacements

ωe(t)

ωe(t)

Te,des(t)

Te(t)

αt(t)

Inverse Map

Engine Torque

Total Inertia

Vehicle

v(t)

Fd(t)

Fr + Fa(t)

+

+

γ




0100

200300

400

2040

6080

100

200

NmTe

αtdegrees ωe

rad/s



Outline




Example 2: Nonlinear Pendulum

x

y

z

ω

mgvϕ,cg

ϕl,m

c



Example 2: Nonlinear Pendulum

Assumptions

Pendulum assumed as thin cylinder, uniform density,

Spring slides without friction,

No friction in the pendulum’s bearing,

No external forces (total energy constant).

Modeling method used

The inverted pendulum has only 1 DOF → its model will bederived using a Total Energy approach.

Kinetic energies + potential energies are present.



x

y

z

ω

mg vϕ,cg

ϕl,m

c Step 1: Inputs/Outputs

No input (no external force actingon the system)

Output: the rod angle ϕ(t)

Step 2: Reservoirs and level variables

Reservoir: total energy of thesystem E(t) =?

Level variables: the rod angle ϕ(t)and angular speed ϕ̇(t)



x

y

z

ω

mg vϕ,cg

ϕl,m

c Step 3: Dynamics of the reservoirquantity

No energy loss:

dE(t)

dt= 0

Step 4: Algebraic relations with thelevel variables

We want to find:

ϕ̈(t) = f (ϕ̇(t), ϕ(t),m, l, c) ?


modeling and analysis of dynamic systems · lecture 2: modeling tools for mechanical systems...

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