Page 360 - Cam Design Handbook

THB11 9/19/03 7:34 PM Page 348

348 CAM DESIGN HANDBOOK

2
J Ê R pinion ˆ
J = load Á ˜
eq
h Ë R ¯
gear
2
b Ê R pinion ˆ
b req = b + rout Á ˜ .
rin
h Ë R ¯
gear
The equivalent damping and inertia both involve the mechanical advantage provided by
the gear ratio as before. The mechanical efficiency, however, alters not only the equiva-
lent damping of the system, but also its inertia.
11.6 EXAMPLE: MODELING AN AUTOMOTIVE
VALVE-GEAR SYSTEM

As an example application of the ideas presented in this chapter, consider the process of
producing a simple model of an automotive valve-gear system, the mechanism that con-
nects the engine valves to the camshaft. This system not only illustrates a large number
of the concepts discussed in this chapter, but has also been the focus of a number of pre-
vious modeling efforts (Barkan 1953, Rothbart 1956, Chen 1982, Pisano and Freudenstein
1983; Pisano, 1984; Hanachi and Freudenstein, 1986). The variety of approaches and
levels of detail involved in these models provide a valuable reminder that engineering
models are not unique descriptions of the system and that no single model can satisfy all
needs. The models by Barkan (1953) and Chen (1982), in particular, include many more
degrees of freedom than the one presented here, though they can ultimately be reduced.
Hanachi and Freudenstein (1986) produce some interesting analytical models for specific
types of damping in this system so the model can be used as a design tool without the
need for experimental determination of damping. They also explicitly consider the oper-
ation of the hydraulic valve lifter, assumed here to be a solid mass.
Figure 11.22 illustrates the physical components of the valve-gear system used in
the LS-1 engine on a Chevrolet Corvette and Fig. 11.23 shows a schematic of the assem-
bly. Beginning the model after the cam, five moving masses can be identified: the lifter,
the pushrod, the rocker arm, the valve and the spring. The mass of each component
can be obtained by weighing each element separately and the moment of inertia of the
rocker arm about its axis (a fixed point of rotation) can be obtained from experiment or
analysis of a solid model. Values for each of these are listed in Table 11.2 and represent
roughly the values for the real system. To shorten the modeling process a bit, the rocker
arm and lifter are assumed to be rigid (Pisano and Freudenstein, 1983). Given the complex
geometry of these shapes, it should of course be verified from finite element models or
from experiment that these components do have the highest stiffness. For the remaining
elements, some estimates of the stiffness can be obtained from basic mechanics of mate-
rials. The coil spring has 5.5 active coils, a wire diameter of 0.175in, a spring diameter
6
of 1.0in and an assumed shear modulus of 11 10 psi for a steel spring. The stiffness is
¥
therefore
psi
)( .
6
Gd 4 (11 ¥100 175in ) 4
K = = = 230 lb in
cs 3 3
8 DN ( 81in ) (5 5 . )
The valve stem and the pushrod resemble long, slender rods of circular cross-section.
2
Both are made of steel with a cross-sectional area of 0.075in . The valve stem has a
length of 4.75in while the pushrod has a length of 7.4in. The stiffnesses of these compo-
nents are