modeling and analysis of dynamic systems · lecture 2: modeling tools for mechanical systems...
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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
G. Ducard c© 2 / 21
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
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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
G. Ducard c© 4 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Outline
1 Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
G. Ducard c© 5 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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).
G. Ducard c© 6 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Outline
1 Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
G. Ducard c© 8 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
Objective: derive a model that can be used to design a robustcruise-speed controller.
G. Ducard c© 9 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
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]
G. Ducard c© 10 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
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
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
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].
G. Ducard c© 12 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
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).
G. Ducard c© 13 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
Step 4: Formulate the algebraic relations of the flows as a functionof level variables
we look for P+(t)− P−(t) = f (v(t)).
G. Ducard c© 14 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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
G. Ducard c© 15 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Simplified Vehicle Modeling
0100
200300
400
2040
6080
100
200
NmTe
αtdegrees ωe
rad/s
G. Ducard c© 16 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Outline
1 Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
G. Ducard c© 17 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
Example 2: Nonlinear Pendulum
x
y
z
ω
mgvϕ,cg
ϕl,m
c
G. Ducard c© 18 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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.
G. Ducard c© 19 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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)
G. Ducard c© 20 / 21
Lecture 2: Modeling Tools for Mechanical SystemsMechanical Systems: Energy & PowerExample 1: Simplified Vehicle ModelingExample 2: Nonlinear Pendulum
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) ?
G. Ducard c© 21 / 21