Fixed-Axis Rotation

SYSTEM MODELS

4.2 MECHANICAL SYSTEMS

4.2.2 Fixed-Axis Rotation

v_in k₁

k₂ v₁

v₂ v₃

m₁

a b

c (a)

0 1

I : m₁

R : b₁ C : 1/k₁

TF

C : 1/k₂

TF v_inS_f

b₁

b/a

c/a

(b)

FIGURE 4.14. Modified pulley–lever mechanism.

in the schematic diagram of Figure 4.13a turns out to be important. The modified schematic is shown in Figure 4.14a. No additional velocities need be labeled, since v1 was needed in the previous version of the model. Looking at Figure 4.13c where v₁ is exposed, the new mass element, m₁, simply becomes an —I element attached to the v₁ 1-junction. Nothing else changes, and the final result is shown in Figure 4.14b.

A straightforward extension of translational mechanical systems is fixed-axis rotation, presented next.

Construction Procedure for Fixed-Axis Rotation

1. On a schematic diagram of the physical system, use arrows and sym- bols to indicate the positive direction of absolute angular velocity for components.

These components include all individual rotational inertial elements, all prescribed input angular velocities, and the angular velocity of any other physical locations that may prove useful in establishing useful relative angular velocities. Decide on positive directions of twist for elements such as rotational springs and dampers. (The positive direction of relative rate of twist is hard to show in a schematic diagram but will be easy to see in a bond graph.)

2. Use 1-junctions to represent each distinct angular velocity from step 1.

Label the 1-junction with the angular velocity symbol from the schematic diagram. This will help to remind you which junction is associated with which angular velocity component. You might use a 1-junction to represent the reference of zero angular velocity. This will later be eliminated, but it might be helpful for establishing relative angular velocities.

3. Attach to each 1-junction any element that has that angular velocity.

In general, distinct rotational inertias are —I elements associated with distinct angular velocities represented by some of the 1-junctions. Remem- ber, positive power isalways directed into an —I, —R, or —C element.

4. Use 0-junctions to establish proper relative angular velocities.

Use the sign conventions on the 0-junctions to make the relative veloci- ties for the connecting elements conform with the twist directions assumed in the schematic diagram.

5. Eliminate the bonds with zero angular velocity and reduce to the final model.

As the first example, consider a model of a grinding wheel at the end of a torsionally flexible shaft, shown in Figure 4.15a. An electric motor is assumed to prescribe an input angular velocity,ωin(t), at the left end of the shaft. The shaft is flexible and characterized by its torsional stiffness,k_τ, with units N-m/rad. The right end of the shaft has a disk of rotational inertia,J, with units kg-m². There is also rotational damping with a coefficient having b_τ the units N-m/(rad/s), attached between the disk and ground.

Figure 4.15b is a schematic diagram of the physical system with angular velocities labeled and positive directions given. The twist in the shaft is assumed to be positive in the clockwise direction as viewed looking into the ends of the shaft, and the rotational damper torque is positive for clockwise rotation as viewed looking into end of the damper. You will notice that it is harder to show positive angular velocities clearly in a two dimensional sketch than it is to show positive linear velocities.

Motor

Flexible shaft

disk

Rotational damper

(a)

ωin

ω1

ωref = 0

k_τ b_τ

J

(b)

1 1 1

ωin ω1 ωref = 0

(c)

1 1 1

ωin ω1 ω =_ref 0

S_f

I ωin

J

Sf:0

:

:

(d)

1 0 1 1

ωin ω1 ωref = 0

S_f

I ωin

J

S_f : 0

C 1/k_τ

0

R b_τ

: :

:

:

S_f ωin:

(e)

0 1

ω1

C1/k_τ I J

R b_τ

:

:

: (f)

FIGURE 4.15. A grinding wheel model to demonstrate fixed-axis rotation. Example 1.

Figure 4.15cshows 1-junctions representing the distinct angular velocity com- ponents, including the reference of zero angular velocity. In Figure 4.15d, the flow source that prescribes ωin(t)is attached to the appropriate 1-junction, and the rotational inertia is attached to its appropriate 1-junction. In Figure 4.15e, 0-junctions are used to establish relative angular velocities across the torsional spring and torsional damper. Note that the rate of twist of the torsion spring is ωin−ω1, which is hard to indicate on the schematic diagram without using equations. In Figure 4.15f, the inertial reference is removed and some simplifi- cations are carried out to reduce to the final model.

A more complex example of fixed-axis rotation is shown in Figure 4.16. Disks of rotational inertiasJ1andJ2are rigidly attached to each end of a flexible shaft

4

τin

J₁ J₃ J₂

J

J₅

k_τ1 k_τ2

k_τ3 b_τ

ω1

ω3 ω2

ω4

ω′ ω5

R₃ R₄

ω1 ω2

τ τ

(a)

1

1

1

1 1 1

I

I I

I I

ω1

ω3

ω2

J1

J3

J2

J5

J4

ω′

ω4 ω5

Se

τin

:

:

:

:

:

: (b)

FIGURE 4.16. A system of shafts and gears. Example 2.

1

1

1

1 1 1

I

I I

I I

ω1

ω3

ω2

J1

J3

J2

J5

J4 4 ω′

ω ω5

Se

τin

0 C

0

C : 1/k_τ2

0 0

C : 1/k_τ3

R 1/k_τ1

b_τ

:

:

:

:

:

: : :

(c)

1

1

1

1 I

I

I

I

ω1 I

ω3

ω2

J₁

J3

J2

J5

J4

ω4

ω5

S_e τin

0 C : 1/k_τ1

0

C : 1/k_τ2

0

C : 1/k_τ3 R b_τ TF

3/R4

R _:

: :

:

:

:

:

:

(d)

FIGURE 4.16. (Continued)

with torsional stiffness, kτ1. Disk 2 is attached at one end to a second torsional spring with constantkτ2, which is attached to a third disk,J₃, that is free to rotate on the shaft. This third disk is of radiusR3and forms a gear set with the fourth disk with moment of inertiaJ4and radiusR4. Finally, disk 4 is attached through a torsional spring and damper, parametersk_τ3 andb_τ, to a fifth disk of rotational inertia, J5. Note that an intermediate angular velocity, ω, has been labeled in between the rotational spring and damper. This was done for convenience in anticipation of the need to establish the relative angular velocities across these two elements. We would ultimately like to predict the motion–time behavior of this system for a specified input torque,τin(t).

On the physical schematic the positive angular velocity directions are indicated with arrows and labels, and the positive torque directions for the springs are indicated by the lines with full arrows using the right-hand-rule. The isolated shaft accompanying the schematic shows this notation.

In Figure 4.16b, 1-junctions have been used to represent all the distinct angular velocities from the schematic. Moreover, the bond graph elements that have these specific angular velocities have been attached directly to these 1-junctions. These include the rotational inertias and the input torque source that is moving at the angular velocity ω1. In Figure 4.16c, 0-junctions have been used to establish relative angular velocities across the rotational spring and damper elements. The power convention for the 0-junctions was chosen to establish the positive torque directions from the schematic. Notice how useful the 1-junction for ω is for establishing the relative angular velocities across the spring and damper with parametersk_τ3 and b_τ. Also notice that there is no connection between ω3 and ω4. The two disks of rotational inertias J3 and J4

are a gear set with the kinematic relationship R3

R4

ω3=ω4. (4.4)

This relationship is enforced by a transformer as shown in Figure 4.16d. In this figure, some final simplifications were done, including the removal of the unnecessary 1-junction representing ω. After this 1-junction was removed, the two remaining 0-junctions were connected by a common bond enforcing that all connected bonds had the same torque (effort). All the bonds with a common effort can then be connected to a common 4-port 0-junction as was done in the final step, Figure 4.16d. This is a nice example of introducing 1-junctions for flows that might be convenient for model construction, and then having the modeling procedure dictate whether these 1-junctions remain in the final model.

The modeler is free to define and introduce as many flows on 1-junctions (or efforts on 0-juntions) as desired. If these extra junctions are actually unnecessary, simplification can remove them.

We now go on to model systems containing rigid bodies that can move in translation as well rotation. Here we treat the important case of plane motion.