In the previous section a pendulum-like position controlled system was investigated. Here, the pendulum is changed to a rotating disk which helps to eliminate the nonlinear effects due to large angular displacements and the speed and position dependent normal force loading the motor axis. This setup can be shown in Fig. 2.8 where the disk ensures that the only (mechanical) nonlinear effect is due to dry friction. Otherwise, the control realization and the electrical parameters are the same as those of the previously described model.
DC motor Encoder
Disk
Figure 2.8: Brushed DC motor with encoder and with the disk
It results, without considering the effect of torsional stiffness, the following equation of motion
Jϕ(t) +¨ τCsgn ( ˙ϕ(t)) =−kpϕ(tj−1), t∈[tj, tj+1). (2.21) Using the same experimental setup with another parameter settings collected in Tab. 2.3.
The parameter κwas selected to beκ= 0.5 Nm/rad, which results thatkp= 15.5 mNm/rad.
The experimental data presented in Fig. 2.9 with the solid black line, and the presented time history of vibration shows further as well the concave vibration decay. In the same figure, the solid red line corresponds to the numerical simulation of the sampled-data system. This sim-
Table 2.3: Parameters for calculations κ 0.5 Nm/rad ϕ0 2.11 rad
ts 10 ms ω0 0 rad/s
J 431 gcm2 τC 2.6155 mNm
-2 -1 0 1 2
Time [s]
Rotation[rad]
experiment
0 1 2 3 4 5
simulation
Figure 2.9: Simulation versus experimental results
Time [s]
0 -4 -2 0 2 4
Rotation[rad]
5 10 15 20
ϕ0=1.301 ϕ0=1.288 ϕ0=1.382 ϕ0=1.150 ϕ0=1.090
ts=0.01 s
kp=0.0186 Nm/rad unstable motion
stable motion set ofϕ0
Figure 2.10: Experimentally predicted stability limit (unstable limit cycle)
ulation was implemented in Mathematica by using the commandNDSolveValue utilizing event handlers for the sampling and for detecting velocity reversals. During simulation, the combined second moment of inertia of disk and the rotor is J, and the friction torque was selected to be τC in order to achieve a good (quantitative) agreement with the experimental result.
During the experiments it was realized that the concave shape becomes more pronounced at larger initial positions and it shows a certain dependence on the initial conditions. Ex- amples for this are shown in Fig. 2.10 where green and red lines correspond to the cases of stable and unstable motions, respectively. In all stable cases the vibration envelope is con- cave. Furthermore, it can be concluded that there has to be an unstable limit cycle around the desired zero reference position, because of the motion is sensitive to the initial position.
If the effective model in Eq. (2.21) is further simplified without considering the unit delay in the model of the continuous-time driving torque reconstruction, the following effective
-2 -1 0 1 2
Time [s]
Rotation[rad]
experiment
0 1 2 3 4 5
simulation with delay
simulation without delay
Figure 2.11: Simulation versus experimental results model can be considered
Jϕ(t) +¨ τCsgn ( ˙ϕ(t)) =−kpϕ(tj), t ∈[tj, tj+1), (2.22) The simulation result is shown as dashed green line in Fig. 2.11 and it is compared with the previous simulation and with the new experimental data. Based on this, it can be concluded that the simulation results, which does not include the effect of the additional unit delay, and the experimental results are initially in a very good agreement. However, a slight difference can be recognized between the two time histories in their frequencies towards the end of the motion. When simulations with and without a unit delay are compared, it can be seen that the effect of delay is negligible in case of the current parameter setting.
Based on these, the corresponded unit delay is not negligible from the model of the experimental setup, but, from the viewpoint of the identification of the concave vibration shape, the unit delay is not necessary. Although, the neglected unit delay can have an effect on the system dynamics in case of different parameters, the results clearly show that the simplest model that qualitatively reproduce the observed concave envelope vibrations is the one in Eq. (2.22). Further simplification in the control model will results in a qualitatively different behavior.
If in Eq. (2.22) the effect of sampling is neglected, then the control law reduces to a continuous-time proportional controller with zero reference position. This simplified equation of motion forms the model of a sliding-oscillator described by the following equation of motion Jϕ(t) +¨ τCsgn ( ˙ϕ(t)) =−kpϕ(t) ⇒ ϕ(t) +¨ ω2nϕ(t) +f0ωn2sgn ( ˙ϕ(t)) = 0, (2.23)
where the undamped natural angular frequency ωn = p
kp/J and f0 =τC/kp. Considering the initial conditions as follows ϕ(0) =ϕ0 and ˙ϕ(0) = 0, the consecutive piecewise segments of the solution can be combined in a special closed form
ϕ(t) = (ϕ0−f0(2n+ 1) cos (ωnt)) + (−1)nf0, with n=jωnt π
k
, (2.24)
where b·c denotes the floor function. This solution with specific system parameters is pre- sented as the solid blue curve in Fig. 2.12.
0.0
Time [s]
-1 -0.5 0.0 0.5 1
Position[rad]
1 2 3 4 5 6
±φ(t)
±Φ(t) ϕ(t)
f0 = 0.05 rad ϕ0 = 1 rad ωn= 2πrad/s
Figure 2.12: Time history of vibration of the sliding-oscillator
In Eq. (2.24), the term multiplying the cosine function gives the amplitude decay, and considering a positive offset with f0 the upper linear vibration envelope is approximated by the function
φ(t) =ϕ0−2f0jωnt π
k
. (2.25)
It is presented as the green sawtooth shape function in Eq. (2.24) which shows that every second local minimum of this function sits exactly on the vibration peaks. These local minima can be connected by a continuous function by removing the floor function. Then the approximated vibration envelope and the exact amplitude decay function is described by
±Φ(t) with
Φ(t) = ϕ0−2f0ωn
π t, (2.26)
and the corresponding lower and upper amplitude decay function are shown in red in Fig. 2.12.
If the effect of Coulomb friction is neglected in Eq. (2.23), the model further simplifies to the model of a mass-spring oscillator. It follows, that the vibration has no amplitude decay as the total mechanical energy is conserved.
Finally, if in Eq. (2.22) the effect of Coulomb friction is neglected, the equation of motion becomes
Jϕ(t) =¨ −kpϕ(tj), t ∈[tj, tj+1) (2.27) and the desired zero position is always unstable (see later in Sec. 3.1).
Based on the above analysis, if the motion is stabilized by Coulomb friction, the unstable nature of the sampled-data systems due to the temporal discretization results in concave envelope vibrations. The concave envelope vibrations is identified as the following sampled- data generalized mechanical system model where the continuous-time generalized driving force fu is realized by a zero-order-hold resulting that fu is constant over a sampling period m¨x(t) +fCsgn ( ˙x(t)) =fu(tj), t∈[tj, tj +ts), (2.28) where x(t) represents the generalized coordinate as a function of time t corresponding to the modeled degree-of-freedom, m is the generalized mass that takes its meaning based on the definition of x and fC denotes the magnitude of the generalized dry friction force. In addition, tj denotes the jth sampling instant and ts is the sampling time.