Mechanics of Translation

SYSTEM MODELS

4.2 MECHANICAL SYSTEMS

4.2.1 Mechanics of Translation

k m

mg

v₁ v_ref = 0

v_ref = 0

(a)

1 1

S_e : mg S_f : 0

I: m v₁

(b)

1 I: m

C 1/k

R: b

S_e : mg v₁

(d) 1

1

I: m

S_e : mg S_f : 0

v₁ v_ref = 0

(c)

0 0 R: b

C 1/k

+T

: :

b

FIGURE 4.9. Mechanical translation. Example 1.

with respect to inertial space, we will not need to define an inertial velocity 1-junction to be subsequently be removed.)

In Figure 4.9b, a 1-junction is shown such that any bonds that are connected to this junction will have the velocityv1, and a second 1-junction is shown such that any bonds connected to it will have the reference velocity of zero. The mass is a 1-port inertia that has the absolute velocity, v1, so the —I element is attached to that 1-junction. The gravity force,mg, is modeled as an effort source in the bond graph (any force that is a known input to a system, whether time varying or constant, is modeled as an effort source in a bond graph). Since this force is moving at the velocity v1, the effort source is attached to the 1-junction representing this velocity. The power out convention on the effort source comes from the fact that if the velocity is in the positive, downward direction and the gravity force is acting downward, then positive power is coming from the source

into the system. A flow source equal to zero is attached to the reference 1-junction to enforce that the velocity is zero.

The spring and damper in general react to the relative velocity across them.

The choice of how to properly add the velocity components at each end of these elements is a topic that causes much student frustration. For this first example, we simply note that for the elements with forces positive in tension, with the positive velocity directions indicated, the relative velocity across both the spring and damper should be v1−vref. That is, if vrel=v1−vref is positive, then the spring and the damper are extending. This is accomplished using 0-junctions with the power convention indicated in Figure 4.9c.

The reader should verify that the velocities on the C-element and the R-element bonds really do represent the rate of extension of these elements. If this is so, when the forces are positive in tension, power will flow toward theC- and R-elements as indicated by their sign half-arrows. One should review the velocity summations in Eqs. (3.18) and (3.18a) to check the sign conventions used in Figure. 4.9c.

Finally, in Figure 4.9d, the bonds with zero power are removed and the final bond graph emerges after some bond graph reductions are used. It makes sense that, since all the elements have the same velocity, they all end up on the same 1-junction. We will be able to do many mechanical system models by inspection after we gain some experience.

To formalize the bond graph construction procedure for mechanical translation, here are the steps to follow:

Construction Procedure for Mechanical Translation

1. On a schematic diagram of the physical system, use arrows and symbols to indicate the positive direction of absolute velocity components.

These elements include all individual mass elements, all prescribed input velocities, and the velocities of any other physical locations that may prove useful in establishing useful relative velocities. State whether force-generating elements, springs and dampers, are positive in tension or compression using symbols such as+T or+C.

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

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

3. Attach to each 1-junction any element that relates to the absolute veloc- ity represented by the junction.

In general, mass elements are inertias associated with absolute veloci- ties represented by some of the 1-junctions. Remember, positive power is always directed into an —I, —R, or —C element.

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

Use sign-convention half-arrows so that the connecting elements are pos- itive in compression or tension as was assumed in the schematic diagram.

This involves the sign-convention rules discussed in Section 3.3.

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

As an example of the use of the construction procedure, consider the so-called quarter-car model of Figure 4.10. This represents one corner of an automobile

(a)

1

1

1 v_s

v_us

v_in

(b)

(c) (d)

1

1

1 v_s

v_us

v_in S_e I

S_f : v_in e I

S mus

ms m_sg

m_usg

: :

::

0 C

1

1

1 vs

vus

vin

I : m_s S_e

S_f : v_in

I : m_us Se

m_sg

m_usg

0 0 R : b_s

C 1/ks

1/k_t

:

: : :

m_sg

m_usg

v_in

v_us v_s m_s

m_us k_s

k_t

b_s +C

+C

FIGURE 4.10. Quarter-car model. Example 2.

wherem_s is the sprung mass (one-quarter of the body mass);m_usis the unsprung mass, which includes the tire, wheel, and some part of the brakes and suspension;

ks,bs are the suspension spring and damper constants; andkt is the tire stiffness.

Quite some liberty has been taken in this model in representing the suspension and tire as linear elements. The system is constrained to move only vertically, and the velocity input,vin(t), at the base is representing the roadway unevenness experienced as the vehicle moves forward. The vehicle is also under the influence of gravity acting vertically downward.

Construction rule 1 is used in the schematic diagram of Figure 4.10a. The distinct velocities are labeled with arrows indicating positive directions, and, as indicated, the springs and dampers are all assumed positive in compression. In Figure 4.10b, step 2 is performed, where 1-junctions are used to represent each distinct velocity, and the junctions are labeled to remind us which velocity they represent. In this case, all the velocities are inertial velocities and we have not included a reference velocity as was done in the first example.

Step 3 is accomplished in Figure 4.10c, where the inertial —I elements, with inward power convention, are attached to the appropriate 1-junctions and labeled to indicate which mass they represent. Notice that the velocity input is represented by a flow source,Sf, attached to the 1-junction forvin. The power arrow is out from the flow source because if the tire spring is in compression (positive by assumption) and the input velocity is up (positive by assumption), then power is flowing from the source into the system.

The weight of each mass element, msg and musg, is modeled as an effort source attached to the 1-junctions having the velocity of the associated mass.

The power arrow is directed away from the 1-juntion and into the source because if the respective mass element is moving upward (positive by assumption) and the gravity force is acting downward (as it always must), then power is flowing from the system and into the source representing the gravity force. (The weight force source in the first example had a different sign convention because there, the absolute velocity was defined to be positive downward whereas here the velocities are positive upward.)

For step 4, shown in Figure 4.10d, 0-junctions are used to establish rela- tive velocities across the remaining elements. For the positive velocity directions defined, and the positive in compression force directions defined, the proper rel- ative velocity across the tire spring is vin−vus and the proper relative velocity across the suspension spring and damper isvus−v_s. These are the proper relative velocities because, if they are positive, then the respective elements will, in fact, be compressing. Notice that the relative velocities across the suspension spring and suspension damper are independently constructed, as dictated by the con- struction procedure. This is perfectly correct and can always be done; however, a simpler realization for these relative velocities will be shown below. There is one simplification that could be done for this model. The 2-port 1-junction at the bottom of Figure 4.10d could be removed because it has the sign half- arrows passing “through” it, and the flow source could be attached directly to the 0-junction. (See again Section 3.3.)

It should be recognized that we paid attention only to establishing the proper velocity components for this construction procedure, and we paid no attention to the forces. The beauty of using the power conservation properties of bond graph junctions is that we need only to constrain the velocities and the forces will automatically be balanced. For mechanical systems, if we establish proper velocities, the forces are guaranteed to be correct.

To see this, consider the free-body diagram for the quarter-car example, shown in Figure 4.11. The masses are isolated and the forces from the springs, dampers, and gravity are exposed. The arrows for the forces are directed in the assumed positive directions, namely, in compression for this example, and the positive velocities are directed upward as was assumed positive previously. Notice that we must show all the forces as having equal but opposite effects, as required by Newton’s laws. If we were to sum the forces on the sprung mass, positive upward, the resultant force,F_s, would be

F_s =F_ks+F_bs−m_sg, (4.1) and for the unsprung mass, the resultant force,F_us, would be

Fus=F_k1−F_ks −F_bs−musg. (4.2) On the bond graph of Figure 4.10d, the forces on the respective mass elements are the efforts associated with the —I elements emanating from the 1-junctions labeledvsandvus. The 1-junctions, in addition to being common flow or velocity, add efforts, or forces, according to the power convention. Amaze yourself by

F_ks

F_ks

kt

F

kt

F

F_bs

F_bs m_sg

m_usg

v_in v_us v_s

FIGURE 4.11. Free-body diagram of the quarter-car system.

adding the forces on the —I element bonds and seeing that the forces add exactly as required by Eqs. (4.1) and (4.2). The forces were constrained without further effort (no pun intended) after enforcing the velocity constraints. And we never had to show equal and opposite reaction forces.

The construction procedure used in the example for the suspension spring and damper calls for the establishment of the relative velocity across each element using 0-junctions to appropriately add velocity components. Such adherence to the construction rules will always produce a correct result. The resulting bond graph structure, referred to as a loop with a through power convention or a reducible loop, comes up a lot in bond graph modeling. (We saw a similar loop in the electrical system of Figure 4.2.)

This loop is isolated in Figure 4.12a, where it has been generalized using efforts and flows rather than forces and velocities. The relative velocities across the spring and damper are identical, and each is equal tof1−f2.

In Figure 4.12b, the relative velocity is established using a single 0-junction, and then a 1-junction is used to ensure that any bond attached to that 1-junction will have the relative velocity, f1−f2. Since both the —C and —R elements have this relative velocity, they both get attached to the 1-junction as shown in Figure 4.12b. The reader can check to see that the forces still add up properly in this alternative representation. The result is a slightly simpler bond graph emphasizing that several elements have the exactly the same relative velocity.

(It is possible to make a mistake with a loop by defining one side as positive in extension and the other in compression, which doesn’t square with the schematic diagram. The pattern in Figure 4.12b prevents that type of mistake, since in that version, both elements clearly have the same relative velocity.)

In addition to mechanical systems having one-directional translational motion, it requires only a modest extension of our procedure to deal with translational systems containing levers, pulleys, and other simple motion–force transforming devices. Consider the system shown in Figure 4.13a. A velocity input is pre- scribed in the horizontal direction on the end of a spring with constantk₁. The

0

1

C R

1

0

f₁

f₁ f₁

f₁ f₂

f₂

f₂ f₂

f₁ − f₂ f₁ − f₂ f₁ − f₂

(a)

0 1

R

C

(b) FIGURE 4.12. Loop with through power convention that can be reduced.

k1

k2

v1

v2 v³

a b

c

b1 ref 0 v = vin

p

+T +T

(a)

1ⁱⁿ 1 1 1

v v₁

v2 v_ref=0

1v³ (b)

1ⁱⁿ 1 1 1

v v₁

v2 v_ref=0

1 v3

S_f 0 0

0

Sf 0 C

R

C 1/k₁ 1/k₂

b1

vin

: :

:

: (c)

1¹ S_f 0 v

C

R C 1/k₁

1/k2

b1

vin TF

TF b/a

c/a v2

v3

(d)

: :

:

FIGURE 4.13. A pulley–lever mechanism.

other end of this spring goes around a pulley and causes vertical motion at one end of the massless lever. A spring with constantk₂and a damper with constant b₁ are located on the right side of the lever with one end of each attached to inertial ground. The pivot for the lever is located a distancea from the left end, while the spring and damper are located at distancesb andc to the right of the pivot. Figure 4.13a in addition shows positive velocity directions and indicates that both springs and the damper are assumed positive in tension. Notice that v1 is labeled only on the end of the lever, but it is the same as the horizontal velocity at the pointP.

Following the construction procedure, we first use 1-junctions to represent the distinct velocities. A good question is why these particular velocities were chosen as distinct and worthy of representation with their own 1-junctions. As mentioned in step 1 of the construction procedure, velocities are always assigned to distinct masses, as well as to other physical locations that might prove useful in establishing relative velocities across elements. You can never overspecify velocity components. If some 1-junctions turn out to not be needed or redundant, they can be eliminated during bond graph simplification.

Looking carefully at Figure 4.13a, we see that v1 (which is the velocity at pointp) will be useful for establishing the relative velocity across the horizontal spring, and velocities v2and v3 are needed for the relative velocities across the vertical spring and damper. Figure 4.13bshows the 1-junctions with the important velocities labeled. The only elements that have any of these specific absolute velocities are the source elements that establish the input velocity,vin(t), and the reference velocity,vref=0. These are shown in Figure 4.13c, where the relative velocities across theR- andC-elements have been established using 0-junctions.

Convince yourself that the half-arrows are correct for having all elements positive in tension. For example, with the velocities defined positive as they are, the spring with constant k1 will be put into tension by the relative velocity, vin−v1, and this is how the flows add on the corresponding 0-junction. The other elements follow a similar argument. The rules of construction have been followed but the model is obviously not complete. There is no connection between the 1-junctions forv1,v2, and v3.

From Chapter 3 we know that a massless lever is a transformer relating veloc- ities and forces across it according to the constitutive relationships,

b

a v1=v2 and c

a v1=v3. (4.3)

(We are assuming that the lever moves through a small enough angle such that the motion of the points at which other elements are attached is nearly in the vertical direction.)

Figure 4.13d shows the final model with the transformers installed, the refer- ence velocity eliminated, and some simplifications performed.

One of the many valuable features of bond graph modeling is the relative ease with which models can be modified. Let’s say that a mass that we neglected

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.