25.01.2017

System modeling (151-0573-00) Ducard

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Name: .....................................................................

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Exam Conditions

Exam Duration: 120 min + 15 min reading time at the beginning!

Number of Questions: 41 (57 points total)

Rating: The maximum number of achievable points is indicated for

each question.

Points are only awarded for completely correct solutions; ex-

cept for multiple-choice questions with true/false-statements,

where half of the point are awarded if all but one statement

was evaluated correctly.

Permitted aids: 20 A4-sheets (=40 pages)

No additional aids or electronic devices (no calculators, no

smartphones, etc.)

The assistants are not allowed to help.

Important: Solutions do not have to be justified. Only the given end

result is evaluated.

Write down your solutions at the indicated locations only.

Good luck!

System modeling Page 1

Topic: Excavator arm dynamics

Description: Imagine, you are an engineer working at a company for construction machines.You are responsible for developing a next-generation excavator (see Figure 1), therefore it isnecessary to model several subsystems.

Figure 1: Excavator.

As a first step, you will investigate the excavator’s arm. The excavator’s arm can be modeledas a double pendulum (see Figure 2) with two weightless, stiff rods (length l1, l2) and two pointmasses (m1, m2) at the end of the rods. The first rod is free to rotate about a frictionlesspivot in the origin O, whereas the second rod is free to rotate about a frictionless pivot in point1. In order to control the arm, the two forces F1 and F2 acting on the point masses m1 andm2, respectively, can be used. Both forces are in positive y direction. In order to describe thesystem dynamics, the two absolute angles α and β shall be used as generalized coordinates, i.e.~q = (α β)T . Both angles, α and β, are shown in positive direction in Figure 2. Gravitationalacceleration acts in −y direction.

α

β

l1 l2

m1

m2

O

~ey

~ex

F1

g

F2

Point 1

Point 2

Figure 2: Simplified model for excavator’s arm.

Page 2 System modeling

Q1 (2 points) In the following, choose true or false for each statement.

Statement true false

The Lagrange formalism can be used only for holonomic systems.

The minimum possible number of generalized coordinates equals thenumber of degrees of freedom.

A point mass features a moment of inertia equal to zero (with respectto its center of gravity).

x1 and x2, i.e. the x-coordinates of point masses m1 and m2 respec-tively, are a possible set of generalized coordinates for the excava-tor’s arm (see Figure 2).

Q2 (1 point) Please derive the potential energy U1 of point mass m1 as function of thegeneralized coordinates.

U1(α, β) =

Q3 (1 point) Please derive the potential energy U2 of point mass m2 as function of thegeneralized coordinates.

U2(α, β) =

Q4 (1 point) Please derive the kinetic energy T1 of point mass m1 as function of thegeneralized coordinates and their derivatives.

T1(α, β, α̇, β̇) =

Q5 (2 points) Please derive the kinetic energy T2 of point mass m2 as function of thegeneralized coordinates and their derivatives. (Hint: cos(α) cos(β)− sin(α) sin(β) =cos(α+ β))

T2(α, β, α̇, β̇) =

Q6 (1 point) Please derive the generalized force vector ~Q1 resulting from the force~F1 = (0 F1)

T .

~Q1 =

System modeling Page 3

Q7 (2 points) Please derive the generalized force vector ~Q2 resulting from the force~F2 = (0 F2)

T .

~Q2 =

Your colleague was so kind and derived the kinetic and potential energies for you:

U = a · (sin(α)− sin(β)) (1)

T = b · (α̇2 + β̇2) (2)

where a and b are known constants. Use U and T for the following two questions.

Q8 (1 point) Please derive the equation of motion for the generalized coordinate α byusing Lagrange II for holonomic systems. The generalized force in α direction isgiven as Q̃1.

α̈ =

Q9 (1 point) Please derive the equation of motion for the generalized coordinate β byusing Lagrange II for holonomic systems. The generalized force in β direction isgiven as Q̃2.

β̈ =

Page 4 System modeling

Topic: Electric motor

Description: You decide now to analyze, calculate and identify several aspects and componentsof a DC electric motor. A schematic of the considered system is shown in Figure 3. The elementsor variables of the electrical subsystem are the inductance L, the current I(t), the resistance R,the system’s input u(t) and the induced voltage Uind(t), while the mechanical ones are the loadtorque Tl(t), the motor torque Tm(t), the moment of inertia θ and the rotational speed ω(t).If not specified during the exercise, no rotational friction is to be considered in the mechanicalpart of the DC motor.

Figure 3: DC motor schematic

Q10 (2 points) Complete the causality diagram of the DC electric motor, by connectingthe given sub-blocks with the right variable(s) denoted in Figure 3. Please make surethat all the input(s) and output(s) of each sub-block are named.

System modeling Page 5

Q11 (1 points) Which system modeling approach(es) could be used to derive the dy-namics of this system?

System modeling approach yes no

Reservoir-based approach

Lagrange

Kirchhoff’s and Newton’s law

Q12 (1 points) The brushless DC electric motor has:

Statement true false

Permanent magnets on the rotor

Mechanical commutation of the current in the rotor coil

Permanent magnets on the stator

Electrical commutation of the stator current

The differential equations describing the mechanical and electrical dynamics of the DC electricmotor are:

L · dI(t)

dt= −R · I(t)− Uind(t) + u(t)

θ · dω(t)

dt= Tm(t)− Tl(t)

(3)

where the induced voltage is Uind(t) = κind · ω(t), the motor torque is Tm(t) = κm · I(t), κm isthe motor constant and κind is the generator constant.

Q13 (2 points) In steady-state conditions and for a constant input voltage u(t) = U0,write the motor torque as a function of the motor speed, i.e. Tm = f(ω).

Tm(ω) =

Q14 (2 points) In steady-state conditions and for a constant input voltage u(t) = U0,what happens if the circuit’s resistance Rold is replaced with a new one, where Rnew <Rold?

Statement true false

The current I decreases.

The speed ω at zero torque production, i.e. when Tm = 0, increases

The motor torque Tm at zero speed, i.e. when ω = 0, increases

The slope of Tm = f(ω) changes.

Page 6 System modeling

Q15 (2 points) Assume that at time t = 2s the input voltage u(t) and the load torqueTl(t) change as shown in Figure 4. All the components and constants of the DCelectric motor remain the same, except the inductance L.

0 2 4 6 8 10

Time [s]

600

700

800

Inpu

t vol

tage

u [V

]

0 2 4 6 8 10

Time [s]

200

300

400

500

600

Load

torq

ue T

l [Nm

]

0 2 4 6 8 10

Time [s]

4

4.5

5

5.5

Spe

ed ω

[rad

/s]

L0

L1

0 2 4 6 8 10

Time [s]

200

300

400

500

600

700

Mot

or to

rque

Tm

[Nm

]

L0

L1

Figure 4: Input, disturbance and state variables evolutions.

The state variable evolution of the motor speed and the one of the current are shownin Figure 4 for two different inductances, i.e. L0 and L1. Please choose the rightanswer:

� L0 < L1

� L0 > L1

� L0 = L1

System modeling Page 7

While modeling the mechanical part of the DC electric motor, you notice that the speed youachieve is not the expected one. For this reason you introduce a friction term Tf (t). Thedifferential equation of the mechanical part and the friction torque Tf (t) are:

θ · dω(t)

dt= Tm(t)− Tl(t)− Tf (t)

Tf (t) =√ω · (a · ω + b · ω2)

(4)

Q16 (2 points) To estimate the lumped parameters a and b, 3 steady-state measurementswere performed. During all the 3 measurements the load torque Tl(t) was keptconstant and equal to Tl(t) = Tl,0, while the motor torque Tm(t) was changed Tm =

(Tm,0, Tm,1, Tm,2)T and the speeds ω = (ω0, ω1, ω2)

T were measured. Establish thelinear system y = H · π + e, where e are the measurement errors, that can be usedto find the parameters a and b by applying the least squares method.

y = H = π =

While modeling the mechanical part of the DC motor, you decide to estimate the moment ofinertia θ. The differential equation of the DC motor mechanical part is:

θ · dω(t)

dt= Tm(t)− Tl(t) (5)

Since the measurements are not steady-state, the time derivative of the speed is approximatedusing the forward Euler approach, where t = 0, Ts, 2Ts and Ts is the sampling time at which themeasurements where taken:

θ · dω(t)

dt= θ · ω(t+ Ts)− ω(t)

Ts= Tm(t)− Tl(t) (6)

Q17 (2 points) To estimate the lumped parameter θ, during the transient operation3 points were measured. During the experiment, the load torque Tl(t) was keptconstant and equal to Tl(t) = Tl,0, while the motor torque Tm(t) was changed Tm =

(Tm,0, Tm,1, Tm,2)T and the speeds ω = (ω0, ω1, ω2)

T measured. Establish the linearsystem y = H · π + e, where e are the measurement errors, that can be used to findthe parameters θ by applying the least squares method.

y = H = π =

Page 8 System modeling

Topic: Diesel engine air-path

Description: In the following questions, we are going to look at the engine of the excavator.The compressor of the turbocharged Diesel engine compresses ambient air (p1, ϑ1), which issubsequently cooled again by the intercooler. To minimize the nitrous oxide emissions of theDiesel engine, it is equipped with exhaust gas recirculation (EGR) and a selective catalyticreduction (SCR) catalyst:

Compressor

Intercooler

EGR cooler

EGR throttle

TurbineSCR catalyst

∗mcp1,ϑ1 p2,ϑ2,V2

∗mEGR

∗mcyl

∗mt p3,ϑ3,V3p4,ϑ4,V4

∗mSCR,in

∗mSCR,out

∗mNH3

VSCR

Cylinder

Exhaust manifold

Intake manifold

Figure 5: Engine schematic.

In the following questions, only the variables introduced in this schematic drawing as well as thevariables and constants listed in the following table shall be used:

Name Symbol Unit

open area of the EGR throttle AEGR(t) m2

rotational speed of the turbocharger ωtc(t) rad/s

isentropic coefficient of air κ -specific ideal gas constant of air R J/(K kg)specific heat constants of air cp, cv J/(K kg)intake manifold volume V2 m3

exhaust volume (between turbine and SCR catalyst) V4 m3

SCR catalyst volume VSCR m3

discharge coefficient of the EGR throttle cd -turbine efficiency ηt -

The following assumptions are made:

• All the gases (i.e. fresh air and exhaust gas) are perfect gases and have the same constantproperties (i.e. cv, cp, R, κ).

• The intake and exhaust manifolds (V2, V3, V4) can be modeled as gas receivers with constantvolume.

• The pressure losses over the intercooler and over the EGR cooler can be neglected.

• The EGR throttle can be modeled as an isenthalpic throttle.

System modeling Page 9

Q18 (2 points) Complete the Simulink model of the intake air-path using the followingblock types:

1s

Integrator

?

Gain

Product

Replace the place holders ”?” with the correct gains. Each block type may be usedmultiple times.

Q19 (2 points) Determine an analytic (symbolic) description of the mass flow throughthe EGR valve (from the Exhaust manifold to the Intake manifold), given:

• The isentropic coefficient of exhaust gas is κ = 1.4.

• The pressure in the exhaust manifold is p3 = 4.8 bar.

• The pressure in the intake manifold is p2 = 1.3 bar.

∗mEGR(t) =

Q20 (1 point) Give an analytic (symbolic) description of the exhaust gas temperatureexiting the turbine:

ϑt,out(t) =

Page 10 System modeling

Q21 (1 point) Give an analytic (symbolic) description of the exhaust gas enthalpy flowinto the SCR.

∗HSCR,in(t) =

Q22 (1 point) Give an analytic (symbolic) description of the mass m4 in the gas receiverafter the turbine and before the SCR catalyst.

ddtm4(t) =

Q23 (2 point) Now, assuming that the mass m4(t) is known, determine an analytic(symbolic) description of the exhaust gas temperature ϑ4 in front of the SCR, given:

• The in- and outflowing enthalpies (∗Ht,out(t) and

∗HSCR,in(t), respectively) are

known.

• Heat losses to through the exhaust wall can be neglected.

• m4(t) is not constant.

ddtϑ4(t) =

Q24 (2 points) The power of the engine can be increased by increasing the air mass flow

into the cylinder∗mcyl(t). Which of the following changes could be made to increase

the maximum engine power in steady-state, assuming:

• the air mass flow∗mcyl(t) from the intake manifold (p2(t), ϑ2(t)) into the cylinder

is determined by an isenthalpic throttle with constant counter-pressure pcyl;

• the EGR throttle is closed at maximum engine power.

Change to the engine true false

increasing the turbine efficiency

reducing the moment of inertia of the turbocharger

reducing the intake manifold volume V2increasing the heat removed by the intercooler

insulating the exhaust manifold (V3)

System modeling Page 11

Description: In a next step, we want to derive a model for the amount of ammonia (NH3)required to reduce the NO-emissions in the exhaust gas. NH3 is injected into the SCR catalyst

at the mass flow rate∗mNH3 . First, this injected NH3 is adsorbed on the surface of the catalyst:

NH3 ↔ NH∗3

where NH∗3 is the adsorbed ammonia. Then, the adsorbed ammonia reacts with the nitrogen

oxide (NO) in the following reduction reaction:

NO +NH∗3 +

1

4O2 → N2 +

3

2H2O

Assumptions:

• It can be assumed that the adsorbed ammonia is distributed uniformly in the catalyst.

Thus, it behaves like a gas with concentration [NH∗3 ] =

nNH∗3

VSCR.

• The SCR catalyst can be modeled as a continuously stirred reactor with constant volumeVSCR.

• The concentrations of O2 and N2 are much higher than the concentration of the otherreactants. Thus, the reaction velocity is independent of [O2] and [N2].

Variables and constants to be used in the following questions:

Name Symbol Unit

mass flow of NH3 entering the catalyst∗mNH3(t) kg/s

NH3 concentration in the catalyst [NH3](t) mol/m3

NH∗3 concentration in the catalyst [NH∗

3 ](t) mol/m3

NO concentration in the catalyst [NO](t) mol/m3

H2O concentration in the catalyst [H2O](t) mol/m3

SCR catalyst volume VSCR m3

molar mass of NH3 MNH3 kg/molrate of NH3 adsorption rads 1/srate of NH3 desorption rdes 1/srate of NO reduction rred (m3/mol)/s

Q25 (2 points) Derive the differential equation describing the dynamics of the ammoniaconcentration [NH3] in the SCR catalyst.

ddt [NH3](t) =

Q26 (2 points) Derive the differential equation describing the dynamics of the adsorbedammonia concentration [NH∗

3 ] in the SCR catalyst.

ddt [NH

∗3 ](t) =