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Modeling and Analysis of Dynamic Systems
Dr. Guillaume Ducard
Fall 2017
Institute for Dynamic Systems and Control
ETH Zurich, Switzerland
G. Ducard c© 1 / 21
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 2 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 3 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Lagrange: 1736 -1813
G. Ducard c© 4 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Lagrange Formalism: Recipe
1 Define inputs and outputs2 Define the generalized coordinates:
q(t) = [q1(t), q2(t), . . . , qn(t)] andq(t) = [q1(t), q2(t), . . . , qn(t)]
3 Build the Lagrange function:
L(q, q) = T (q, q)− U(q)
4 System dynamics equations:
d
dt
{
∂L
∂qk
}
−
∂L
∂qk= Qk, k = 1, . . . , n
Notes:
Qk represents the k-th “generalized force or torque” acting onthe k−th generalized coordinate variable qkn: number of degrees of freedom in the systemalways n generalized variables G. Ducard c© 5 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 6 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Lagrange Equations for Constrained Systems
d
dt
{
∂L
∂qk
}
−
∂L
∂qk−
ν∑
j=1
µjαj,k = Qk, k = 1, . . . , n, (1)
Remarks:
the constraints are included using “Lagrange multipliers”: µj
j = 1 . . . ν
Number of constraints: ν with (ν < n)
n may be seen as the number of DOF
In the end, we obtain: n+ ν coupled equations to be solvedfor qk and µj (usually requires computing the time derivativeof the constraints, i.e., µ).
G. Ducard c© 7 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 8 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Nonlinear Pendulum on a Cart
y(t)
u(t)M = 1kg
ϕ(t)
l = 1m
mg
m = 1kg
Figure: Pendulum on a cart, u(t) is the force acting on the cart(“input”), y(t) the distance of the cart to an arbitrary but constantorigin, and ϕ(t) the angle of the pendulum.
G. Ducard c© 9 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Step 1: Inputs & Outputs
Input: force acting on the cart: u(t)
Output: angle of the pendulum: ϕ(t)
Step 2: System’s coordinate variables
q1 = y, q1 = y
q2 = ϕ, q2 = ϕ
Step 3: Lagrange functions
L1(t) = T1(t)− U1(t)
L2(t) = T2(t)− U2(t)
L(t) = L1(t) + L2(t)G. Ducard c© 10 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Lagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
Step 4: System’s dynamics equations
d
dt
{
∂L
∂q1
}
−
∂L
∂q1= Q1
d
dt
{
∂L
∂q2
}
−
∂L
∂q2= Q2
We are looking for dynamic equations of the form:
y(t) = f(ϕ(t), ϕ(t), u(t))
ϕ(t) = g(ϕ(t), ϕ(t), u(t))
G. Ducard c© 11 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 12 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Introduction
In general
they are described by Navier-Stokes equations.
For control purposes
simpler formulations are necessary, to build networks with buildingblocks:
ducts,
compressible nodes,
valves, etc.
G. Ducard c© 13 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Water duct in gravitational field
l
A
v(t)
v(t)
p1
p2
h
Objective
d
dtv(t) = f (p1(t), p2(t), v(t), h, ρ,A, l)
G. Ducard c© 14 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Change in momentum: Newton’s law
d~p
dt= m
d~v
dt= ~Fpressure + ~Fgravity + ~Ffriction
= [P1A− P2A] ~x+
∫
tube
~g · dm+ ~Ffriction
The mass m of the fluid in the element of tube of length l is givenby
m = ρ ·A · l → dm = ρ ·A · dl
G. Ducard c© 15 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Angle of the duct
sinα =dh
dl
Gravity force
∫
tube~g dm = g
∫
tube(− cosα ~y + sinα ~x) ρ · A · dl
= ρ · g ·A[
∫ h
0−
cosαsinα
dh ~y +∫ h
0
sinαsinα
dh ~x]
= −ρ · g · A (tanα)−1 h ~y + ρ g Ah ~x
G. Ducard c© 16 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Water duct in gravitational field
Dynamics along the ~x axis (because ~v = v~x)
ρ A ldv(t)
dt= A (P1 − P2) + ρ g A h− Ffriction,x(t)
with
Ffriction,x(t) =1
2ρ v2(t) sign [v(t)] · λ (v(t)) ·
A l
d
Remark: shape factor: ld
G. Ducard c© 17 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Outline
1 Lecture 4: Modeling Tools for Mechanical SystemsLagrange FormalismLagrange Method with Kinematic ConstraintsPendulum on a Cart
2 Lecture 4: Hydraulic SystemsWater DuctCompressible Duct Element
G. Ducard c© 18 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
∗
V in (t)∗
V out (t)p(t)
∆V = 0
∆V (t)
k = 1/(σ0V0)
Definitions of compressability
Property of a body (solid, liquid, gas, etc.) to deform (to changeits volume) under the effect of applied pressure.Defined as:
σ0 =1
V0
dV
dP
V0: nominal volume [m3], P : pressure [Pa]σ0: compressibility [Pa−1]
k0 =1
σ0is called elasticity constant
[
Pa ·m−3]
.G. Ducard c© 19 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Compressibility effects
∗
V in (t)∗
V out (t)p(t)
∆V = 0
∆V (t)
k = 1/(σ0V0)
Modeling
ddtV (t) =
∗
V in(t)−∗
V out(t) = Ainvin(t)−Aoutvout(t)
∆P (t) = k∆V (t) = 1
σ0V0∆V (t)
∆V (t) = V (t)− V0
G. Ducard c© 20 / 21
Lecture 4: Modeling Tools for Mechanical SystemsLecture 4: Hydraulic Systems
Water DuctCompressible Duct Element
Next lecture + Upcoming Exercise
Next lecture
Pelton Turbine
Electromagnetic systems
Next exercise: Online next Friday
Hydro-electric Power plant, Part I
G. Ducard c© 21 / 21