Gelöste Aufgaben/JUMP/E-Motor and Drive-Train

Scope

The Drive-Train consists of a DC/DC-converter, a DC Motor and a gear-box.

DC/DC-converter: is supplied with the battery voltage U_B, the output voltage is controlled by the driver via setpoint “p“.
motor: is a standard DC brushed motor, the manufacturer provides only few information on its characteristics - we’ll need to improvise.
gearbox: has a gear ratio of ratio of n_G=100, its shaft rotates at speed ω_W and delivers a torque M_W to the front wheels.

The task is: provide a mathematical model for the drive train that accounts for load-alterations imposed by the driver. And we assume losses in the two converters - DC/DC and gearbox - to be negligible.

Structure

The drive train receives a "gas"-pedal position "p" from the driver and a battery-voltage U_B.

It delivers a torque M_W on the wheel and creates an electric current I_M through the motor.

The sub-model consists of DC/DC-converter, Motor and gear-box:

DC/DC Converter

Losses in the DC/DC converter shall be small - so for input port “1“ and output port “2“ we obtain

U_{B} \cdot I_{B} = U_{M} \cdot I_{M}

.

Let the “gas”-pedal-indicator “p“ control

U_{M} = p \cdot U_{B}; 0 \leq p \leq 1 .

with

0 \leq p \leq 1 and U_{1} = U_{B}

Motor

We use a common electric circuit representation for a series wound motor, the field coils are connected electrically in series with the armature coils, resistance R sums up all electrical losses in the motor.

Gearbox

Losses in the gearbox shall be small - so for input (ω_M, M_M) and output (ω_W, M_W) we obtain the fixed relation

ω_{M} \cdot M_{M} = ω_{W} \cdot M_{W}

.

Fortunately, this relationship is already "hardwired" in the Principle of Virtual Work. And we have only one differential equation for the electrical components:

\frac{d I_{B}}{d t} = \frac{U_{L}}{L}

,

the remaining equations are algebraic.

Model

Electrical Components

For the motor, we find with Kirchhoff's law that

U_{M} = U_{R} + U_{L} + e

with U_R, U_L being the differential voltage over resistance R and inductance L respectively. “e” is the back electromagnetic force with

e = k_{e} \cdot {\tilde{ω}}_{M}

and the electromotive force constant k_e. Note the ${\tilde{ω}}_{M}$ is the differential rotational velocity between rotor and stator, i.e.

{\tilde{ω}}_{M} = {\dot{ψ}}_{M} (t) - \dot{ϕ} (t) .

Employing

U_{R} = R \cdot I_{M}, U_{L} = L \cdot \frac{d I_{M}}{d t}

and using

M_{M} = k_{T} \cdot I_{M}

with the armature constant k_T, we have the complete set of equations.

From the above, we find

L \cdot \frac{d}{d t} I_{M} (t) = U_{B} (t) \cdot p (t) - R I_{M} (t) - e

and additionally the algebraic equations

\begin{array}{ll} I_{B} & = I_{M} \cdot p (t), \\ U_{R} & = R \cdot I_{M} (t), \\ U_{L} & = U_{B} \cdot p (t) - R \cdot I_{M} (t) - e, \\ U_{M} & = U_{B} \cdot p (t) \end{array}

.

/*******************************************************/
/* MAXIMA script                                       */
/* version: wxMaxima 16.04.2                           */
/* author: Andreas Baumgart                            */
/* last updated: 2021-02-08                            */
/* ref: Modelling and Simulation (TUAS)                */
/* description: virtual work of drive train electrics  */
/*******************************************************/

/*******************************************************/
/* declarations                                        */
/*******************************************************/
declare("φ", alphabetic);
declare("ψ", alphabetic);
declare("ω", alphabetic);

/*******************************************************/
/* kinematics                                          */
/*******************************************************/
kirchhoff : [U[M] = p(t)*U[B],
	     U[B]*I[B] = U[M]*I[M],
	     U[M] = U[R] + U[L] + e,
             e = k[e]*omega[M],
	     ω[M] = diff(ψ[M](t),t)+diff(φ(t),t),
	     U[R] = R*I[M](t),
	     U[L] = L * diff(I[M](t),t),
	     M[M] = k[t]*I[M](t)];

solve(kirchhoff[7], [diff(I[M](t),t)]);

Mechanical Components

The Virtual Work of d'Alembert forces, motor torque M_M and wheel torque M_W is

\begin{array}{ll} δ W & = - J_{M} \cdot {\ddot{ψ}}_{M} (t) \cdot δ ψ_{M} + M_{M} \cdot (ψ_{M} + ϕ (t)) - M_{W} \cdot δ ψ_{W} \\ = 0 \end{array}

.

Since the gearbox is built into the car, the wheel-side relative gear-box-angle is

{\tilde{ψ}}_{W} = ψ_{W} (t) + ϕ (t)

and on the motor-side

{\tilde{ψ}}_{M} = ψ_{M} (t) - ϕ (t)

.

With the gear transmission ratio

n_{G} = \frac{{\tilde{ψ}}_{M}}{{\tilde{ψ}}_{W}}

and

{\underline{q}}_{E} = (\begin{array}{c} ϕ (t) \\ ψ_{W} (t) \end{array})

we find

δ W = δ {\underline{q}}_{E} \cdot (- {\underline{\underline{M}}}_{E} \cdot {\underline{\ddot{q}}}_{E} + (\begin{array}{c} - n_{G} \cdot M_{M} \\ + n_{G} \cdot M_{M} - M_{W} \end{array}))

with

{\underline{\underline{M}}}_{E} = (\begin{matrix} (n_{G}^{2} + 2 n_{G} + 1) J_{M} & - (n_{G}^{2} + n_{G}) J_{M} \\ - (n_{G}^{2} + n_{G}) J_{M} & n_{G}^{2} J_{M} \end{matrix})

.

We have thus "returned" all state variables to the Car-Body-submodel.

/*******************************************************/
/* MAXIMA script                                       */
/* version: wxMaxima 16.04.2                           */
/* author: Andreas Baumgart                            */
/* last updated: 2021-02-08                            */
/* ref: Modelling and Simulation (TUAS)                */
/* description: virtual work of drive train mechanics  */
/*******************************************************/

/*******************************************************/
/* declarations                                        */
/*******************************************************/
declare("δ", alphabetic);
declare("φ", alphabetic);
declare("ψ", alphabetic);

/*******************************************************/
/* kinematics                                          */
/*******************************************************/
kin : [ψ[M,r] = n[G]*ψ[W,r],
       ψ[W,r] = ψ[W](t) - φ(t),
       ψ[M,r] = ψ[M](t) + φ(t)];
Q : [ψ[M](t),ψ[M,r],ψ[W,r]];
sol: ratsimp(solve(kin,Q))[1];


/* Principle of Virtual Work */
PVW: δW = - J[M]*diff(ψ[M](t),t,2)*δψ[M] + M[M]*(δψ[M]+δφ) - M[W]*δψ[W];

/* minimal coordinates */
q : [ φ(t), ψ[W](t)];
δq: [δφ   ,δψ[W]   ];

varia: [ψ[M](t)=δψ[M],ψ[M,r]=δψ[M,r],ψ[W,r]=δψ[W,r],
        ψ[W](t)=δψ[W], φ(t) =δφ];
sol : expand(append(sol, subst(varia,sol)));

PVW : expand(subst(sol,PVW));
PVW : ev(PVW,nouns);

eom : makelist(coeff(rhs(PVW),δq[i]),i,1,2);
eom : ratsimp(eom);

MM : funmake('matrix,makelist(makelist(ratsimp(coeff(-eom[i],'diff(q[j],t,2))),j,1,2),i,1,2));

rest : expand(eom + makelist(sum(MM[i,j]*diff(q[j],t,2),j,1,2),i,1,2));

print(δq, "*(-",MM,"*",transpose(diff(q,t,2)),"+",transpose(rest),") = 0")$

Consequently, the drive train has only one state-variable: I_M:

{\underline{q}}_{E} (t) = (I_{M} (t))

where

\frac{d}{d t} I_{M} (t) = \frac{U_{B} p (t) - R I_{M} (t) - e}{L}

and with this algebraic equation

e = k_{e} \cdot \underset{= {\tilde{ω}}_{M} = n_{G} \cdot {\tilde{ω}}_{W}}{\underset{⏟}{({\dot{ψ}}_{M} (t) + \dot{ϕ} (t))}}

.

Keep in mind that ${\tilde{ω}}_{M}$ and ${\tilde{ω}}_{W}$ are the rotational velocities of the gearbox-shafts relative to the car-body, thus

\begin{array}{ll} {\tilde{ω}}_{W} & = \frac{d \tilde{ψ}}{d t} \\ = (\underset{= ω}{\underset{⏟}{\frac{d ψ}{d t}}} - \frac{d ϕ}{d t}) \end{array}

.

Variables

name	symbol	unit
motor current	I_M	A

Parameter

System parameters for the motor are k_T, k_e and R_M . For the stationary ( $\frac{d I_{M}}{d t} = 0$ ) - condition, we derive

M_{M} = \frac{U_{M} k_{T} - k_{T} k_{e} {\tilde{ω}}_{M}}{R_{M}}

.

For our application, it will be easier to replace k_e, k_R and R by mechanical properties as

under a "no-load"-condition M_M=0, we have
$k_{e} = \frac{U_{M}}{{\tilde{ω}}_{N L}}$
under the "stalled motor"-condition ${\tilde{ω}}_{S t} = 0$ , we have
$k_{T} = \frac{M_{S t}}{I_{S t}}$ and $R = \frac{U_{M}}{I_{S t}}$ .

Thus for U_M = 12 V we can import measured performance parameters and then replace k_T, k_e and R_M by ${\tilde{ω}}_{N L}$ I_St and M_St.

name	symbol	value	unit
nominal voltage	U_ref	12	V
no-load rotational speed	${\tilde{ω}}_{N L}$	14800	rpm
stall-current	I_St	8.6	A
stall-torque	M_St	0.042	N m
inductance	L	0.05	H
gear tramission ratio	n_G	25	1

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References

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Gelöste Aufgaben/JUMP/E-Motor and Drive-Train

Inhaltsverzeichnis

Scope

Structure

DC/DC Converter

Motor

Gearbox

Model

Electrical Components

Mechanical Components

Variables

Parameter

Navigationsmenü

Gelöste Aufgaben/JUMP/E-Motor and Drive-Train

Scope

Structure

DC/DC Converter

Motor

Gearbox

Model

Electrical Components

Mechanical Components

Variables

Parameter

Navigationsmenü

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