The propeller models presented in Section 2.1 are all quasi-static. Dynamic effects are introduced by considering the propeller motor and shaft, and also by flow dynamics.
2.2.1
Shaft dynamics
The torque balance for the propeller shaft is written:
Isω = Q˙ mp− Qa− Qf(ω), (2.50)
where Qmpis the motor torque inflicted on the propeller shaft, Isis the moment
of inertia for the shaft, propeller, and motor, ω is the shaft angular velocity, and Qf(ω) is the shaft friction. The friction may for most applications be viewed as
a sum of a static friction — or starting torque — Qsand a linear component:
Qf(ω) = sign(ω)Qs+ Kωω, (2.51)
where Kω is a linear friction coefficient. If desired, more sophisticated friction
models may be used, including e.g. nonlinear elements, Stribeck friction, and other hysteresis effects. This has not been considered in the current work, and the model in (2.51) is considered sufficient for control design purposes.
In general, friction is assumed to be more significant on small thrusters typically used on underwater vehicles and in experimental setups, than on large thrusters used on surface vessels.
2.2 Dynamic effects 25
Remark 2.4 The propeller torque Qa in (2.50) should in general also include
an added mass term due to hydrodynamic forces in phase with ˙ω. This is dis- cussed in the context of propeller vibrations in e.g. Wereldsma (1965) and Par- sons and Vorus (1981). However, the added mass will depend on propeller shaft speed, advance velocity, and propeller submergence, and extensive model knowl- edge is required in order to include such terms. Neglecting the added mass will give a reduced rotational inertia, and hence faster dynamics. If the necessary model knowledge is available, the added mass could be included in Is.
2.2.2
Motor dynamics
Electric motor
The inner torque control loop — or inner current loop — is inherent in the design of most applied control schemes for variable speed drive systems. The torque is controlled by means of motor currents and motor fluxes with high accuracy and bandwidth. By the cascaded control principle, an outer control loop — e.g. a speed controller — sets the commanded torque setpoint of the inner loop. For this principle to work, the inner current loop must have a much higher bandwidth than the outer loop. In the design of the outer control loop, the closed loop of an electric motor and its current controller may be assumed to be equivalent with a first order system (Leonhard, 1996):
˙ Qm=
1 Tm
(Qcm− Qm), (2.52)
where Qm is the motor torque, Tm is the time constant, and Qcm is the com-
manded motor torque from the thruster controller. This model is applicable to both AC and DC motors; once the current loop is closed, there is little difference between the two types of machines. From a practical point of view, the current loop is usually embedded in the motor drive, and is not at the disposition of the user. The main importance is therefore to account for its dynamics in the design of the higher-level controllers (Leonhard, 1996). According to Nilsen (2001), the dynamics of the current loop are, in the context of shaft speed controller design, dominated by the filter on the shaft speed measurement. The motor power Pm
is given by:
Pm= Qm2πnm, (2.53)
where nm is the motor speed. The rated (nominal) torque and power for con-
tinuous operation of the motor are denoted QN and PN. The corresponding
rated motor shaft speed nN is given from:
PN = QN2πnN. (2.54)
The maximum torque Qmax and power Pmaxfor the motor are usually set to:
Qmax= kmQN, Pmax= kmPN, (2.55)
Diesel engine
A diesel engine is modelled in a similar way as the electric motor. According to Blanke (1981), the diesel engine dynamics may for control design and propulsion performance evaluation be approximated by a time constant Tmand a time delay
τm. The diesel engine transfer function becomes:
Qm(s) = e−sτm
Ky
1 + sTm
Y (s), (2.56)
where s is the Laplace operator, Ky is the motor torque constant, and Y is
the fuel index (governor setting). This means that Qcm in (2.52) is given by
Qcm= KyY , where Y is the control signal from the diesel controller. The diesel
engine power is proportional to the fuel flow (Blanke, 1994). From Blanke (1981), the time constant is empirically found to be:
Tm≈
0.9 2πnm
, (2.57)
and the time delay can be approximated by half the period between consecutive cylinder firings. A diesel engine with N cylinders rotating at speed nmrps then
has the time delay:
τm≈
1 2nmN
. (2.58)
2.2.3
Accounting for gears
Many electrically driven propulsion units are equipped with a gearbox between the motor and the propeller shaft. In this work, Qm is the motor torque as
seen from the motor, and n is the shaft speed as seen from the propeller. The relations between the motor torque Qm, the motor shaft speed nm, the motor
torque Qmpseen from the propeller, and the propeller shaft speed n, are given
by the gear ratio kg:
Qmp = kgQm, (2.59)
n = nm/kg. (2.60)
For thruster controller design and analysis, all torques and shaft speeds are as seen from the propeller, with the exception of Qm and Qcm. For example, the
commanded torque Qc from the controller is referred to the propeller, and Qcm
calculated from Qcm= Qc/kg. The various terms are illustrated in Figure 2.7.
By the principle of conservation of energy, the rotational inertia Isseen by the
propeller can be transformed to the rotational inertia Imseen by the motor by:
Isωω˙ = Imkgωk˙ gω,
⇓
2.2 Dynamic effects 27 Qm Ta Qc Tr n Controller 1/kg kg Propeller and shaft Motor Qcm Qmp
Figure 2.7: Block diagram of the controller, motor, propeller/shaft, and gears.
2.2.4
Bollard pull relationships
When the vessel is stationary such that Va = 0, and the thruster motor is
operated at its max continuous rating, the thruster is said to be in the bollard pull condition. The bollard pull thrust, torque, power, and shaft speed (seen from the propeller) are denoted Tbp, Qbp, Pbp, and nbp, respectively. From
(2.60), nbp is given from nN by:
nbp= nN/kg, (2.62)
and Tbp and Qbp are given from KT 0, KQ0, and nbp by:
Tbp= ρD4KT 0n2bp, Qbp= ρD5KQ0n2bp. (2.63)
At the bollard pull condition (with Qm = QN), the steady-state rotational
dynamics (2.50) must satisfy:
kgQN= Qs+ Kω2πnbp+ Qbp, (2.64)
and PN, Pbp, and the mechanical efficiency ηmmust satisfy:
Pbp= Qbp2πnbp = ηmPN. (2.65)
That is, ηmis given from the power lost in static and linear friction:
ηm= Qbp QNkg
=kgQN− Qs− Kω2πnbp QNkg
. (2.66)
If higher-order friction terms were to be included, similar relationships would apply.
Remark 2.5 In industrial applications, a constant mechanical efficiency is usu- ally assumed. This implies that Pa = ηmPm for any shaft speed, where ηm is
found from (2.65). This is equivalent to using a purely quadratic friction model, as opposed to the static plus linear model in (2.51). If the true friction contains significant static and/or linear terms, the assumption of a purely quadratic model will under-estimate the friction losses for n < nbp.
2.2.5
Propeller flow dynamics
Experimental results have shown that overshoots in thrust and torque during transient operation cannot be explained by steady state theory. An investiga- tion into the effects of flow dynamics on the performance of low-level thruster controllers and underwater vehicle positioning was initiated by Yoerger et al. (1991). In this work, a lumped parameter one-state model for the shaft dynam- ics and flow dynamics was proposed:
Isω˙ = Qmp− K1ω |ω| ,
Ta = K2ω |ω| , (2.67)
where an instantaneous relationship between the propeller shaft speed and the flow rate is included in the parameters K1 and K2. The shaft friction is ne-
glected.
To describe overshoots in thrust found in experimental data, Healey et al. (1995) extended (2.67) to include the flow velocity Vp through the propeller as
a state. From momentum theory, see e.g. Durand (1963), Vpis connected with
the advance velocity Vaand the induced velocity in the wake UA through:
Vp= Va+
1
2UA. (2.68) In the work of Healey et al. (1995), a control volume of water around the pro- peller was modelled as a mass-damper system, leading to the following two-state model:
Isω˙ = Qmp− Kωω − Qa,
mfV˙p = Ta− dfVp|Vp| ,
Ta = Ta(ω, Vp),
Qa = Qa(ω, Vp), (2.69)
where mf can be viewed as the mass of water in the control volume, and df is
a quadratic damping coefficient. The mappings T (ω, Vp) and Qa(ω, Vp) employ
foil theory with sinusoidal lift and drag curves and the local angle of attack at radius 0.7R to model the propeller blade lift and drag. Healey et al. (1995) also proposed using a time constant on the thrust and torque for open propellers to model lift and drag build-up. A summary of the theory and assumptions behind these models, as well as experimental verification, can be found in Whitcomb and Yoerger (1999a).
Discrepancies between (2.69) and experimental results were attempted ex- plained by Bachmayer et al. (2000). Here it was concluded that the inclusion of rotational flow dynamics did not improve model performance, whereas the use of non-sinusoidal lift and drag curves specific to the tested propeller gave significant performance improvements. Note that these models all assume that Va= 0, and hence only are applicable for station-keeping operations with small