Model of a crank-connecting rod system
• Let us consider the following elastic crank-connecting rod system:
θ, τ R
L
x xk
d F Pk K
Jm bm
mp bp
where
Jm shaft moment of inertia bm shaft friction coefficient K stiffness of the fictitious spring mp mass of the piston
bp piston friction coefficient θ shaft angular position ω shaft angular velocity (ω = ˙θ) τ torque acting on the shaft
˙x piston velocity F external force
• The POG dynamic model of the considered system is:
τ -
?
1 s
?
1 Jm
?
ω
-
bm
6
6
- - - H(θ) -
˙xk
H(θ) -
6
1 s
6
K
6
Fk
-
-
?
1 s
?
1 mp
?
˙x
-
bp
6
6
- -
F
where H(θ) is the function which links the angular velocity ω of the shaft to the translational velocity ˙xk of the point Pk.
• The translational position xk of point Pk can be expressed as follows:
xk(θ) = R cos θ + p
L2 −(R sin θ − d)2. Function H(θ) can be determined as follows:
H(θ) = ∂xk(θ)
∂θ = R
"
−sin θ − (R sin θ − d) cos θ pL2 − (R sin θ − d)2
#
= R
"
−sin θ − (sin θ − β) cos θ pα2 − (sin θ − β)2
#
where α = L/R > 1 and β = d/R < 1.
• From the POG blocks scheme one obtains the following POG state space model (i.e. L ˙x = Ax + Bu):
Jm
1 K
mp
˙ω F˙k
¨ x
=
−bm −H(θ) 0
H(θ) 0 −1
0 1 −bp
ω Fk
˙x
+
1 0 0 0 0 −1
τ F
• The fictitious stiffness K has been added to the system to “decouple” the physical elements and simplify the modeling procedure.
• When K → ∞, the variables ˙x and ω are constrained as follows:
˙x = H(θ) ω.
Using the following “congruent” state space transformation x = T(θ) ω, ⇔
ω Fk
˙x
| {z }
x
=
1 0 H(θ)
| {z }
T(θ)
ω
one obtains the following first order transformed and reduced system:
J(θ) ˙ω − N (θ)ω = −b(θ) ω + τ − H(θ)F where:
J(θ) = TTLT = Jm + mpH2(θ)
N (θ) = TTL ˙T = mpH(θ) ˙H(θ) = ˙J(θ)/2 b(θ) = TTAT = bm + H2(θ) bp
B(θ)u = TTBu =
1 −H(θ)
u = τ − H(θ)F
• The POG scheme of the reduced system is the following:
τ
ω | {z }
Time-varying inertia
(KEEP TOGETHER)
-
1 J(θ)
1 s
?
?
?
-
N (θ)
6
6
- -
b(θ)
6
6
- - - H(θ) -
H(θ) F
˙x
• The reduced system can also be rewritten as follows:
d
dt [J(θ)ω] + N (θ)ω = −b(θ) ω + τ − H(θ)F
The corresponding POG scheme is quite similar to the previous one:
τ
ω | {z }
(KEEP TOGETHER)
-
?
1 s
?
1 J(θ)
?
-
N (θ)
6
6
- -
b(θ)
6
6
- - - H(θ) -
H(θ) F
˙x
• The two above POG schemes are two equivalent ways of implementing this dynamic system in Simulink.
• The same result could have been obtained using the Lagrange Equations:
d dt
∂T
∂ ˙qi
− ∂T
∂qi
+ ∂U
∂qi
= Qi i = 1, . . . , N where
N degrees of freedom of the mechanical system;
qi generalized Lagrangian coordinates;
Qi generalized forces;
T kinetic energy of the system (masses and inertias);
U potential energy of the system (springs and gravitational forces);
• The kinetic and potential energies of the crank-connecting rod system are:
T = 1
2Jm ˙θ2 + 1
2mpH2(θ) ˙θ2
| {z }
˙x2
, U = 0.
Intermediate terms of the Lagrangian Equations:
∂T
∂ ˙θ = Jm ˙θ + mpH2(θ) ˙θ d
dt
∂T
∂ ˙θ
= Jmθ + m¨ pH2(θ) ¨θ + 2 mpH(θ) ˙H(θ) ˙θ
∂T
∂θ = mpH(θ) ˙θ ∂H(θ)
∂θ ˙θ = mpH(θ) ˙H(θ) ˙θ The external generalized force Q is:
Q = τ − bm ˙θ − bpH2(θ) ˙θ − H(θ) F.
• The Lagrangian equation of the system is:
d dt
Jm ˙θ + mpH2(θ) ˙θ
−mpH(θ) ˙H(θ) ˙θ = τ −bm ˙θ−bpH2(θ) ˙θ−H(θ) F.
which be rewritten as follows:
Jm + mpH2(θ)
| {z }
J(θ)
θ +m¨ pH(θ) ˙H(θ)
| {z }
N (θ)
˙θ = τ − bm + bpH2(θ)
| {z }
b(θ)
˙θ−H(θ) F.
• The obtained equations are equal to the equations obtained using POG modelling approach.
• The POG approach can be used also for non-mechanical systems.
Simulation in Simulink
• System parameters:
R 10 cm L 40 cm d 5 cm
Jm 12.5 kg cm2 bm 0 Nm/(rad/s) mp 0.8 kg
bp 0 N/(m/s)
F 0 N
τ 0 Nm
with Initial condition: ω0 = 200 rpm.
• Shaft angular velocity ω and piston speed ˙x.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
50 100 150 200 250
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−1
−0.5 0 0.5 1
ω[rpm]˙x[m/s]
Time [s]
• Energy Em stored in the system.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0.3018 0.3019 0.302 0.3021 0.3022 0.3023
Em[J]
Time [s]
• The stored energy Em is constant because F = τ = 0 and bm = bp = 0.
