Impact Force Analysis Of Valve Train Clearance Mechanism

Impact Force Analysis Of Valve Train Clearance Mechanism

Impact Force Analysis Of Valve Train Clearance Mechanism

P. Srinivasa Rao Associate Professor.

Dept. of Mechanical Engineering Mahatma Gandhi Institute of Technology Gandipet, Hyderabad-500075. A.P.,India

Impact force analysis of a valve-train clearance mechanism is done using finite element technique. The physical system which is selected for the analysis is a cam-driven-valve train mechanism with valve seat, available with present day automobiles with variety of components made of different materials. This method is sufficiently general to permit applications to other machine elements, such as gears, bearings etc or a combination thereof. Use of the method involved establishing an elastic model, specifying clearances, calculating overall stiffness matrix, overall mass matrix, overall damping matrices and determine the form of the equation of motion for each configuration for different clearances of the valve train mechanism and calculated

the impact force between pushrod and rocker arm. A computer program is developed to solve the various equations of overall stiffness matrix, overall mass matrix, overall damping matrices and few results, such as Impact forces and stresses which acting on the machine components for different speeds. Also this quantifies the maximum safe running speeds of mechanisms for different configurations depending upon the selection of materials and their mechanical properties.

Keywords: Impact force, Joint clearance, safe running speed, valve seat, finite element technique

The finite element techniques for structural analysis have been applied to the prediction of the elastic behavior of a rotating mechanism while it is being acted upon by its own inertial forces as well as arbitrary external forces[1]. In order to analyze a mechanism rather than a structure, it is necessary to consider two additional problems a mechanism will undergo rigid body motion, and it will usually experience large changes in geometry. The changes in geometry is handled easily with a relative algorithm, however accounting for the rigid body motion requires some skill, because the stiffness matrix for the mechanism is singular and the flexibility matrix, therefore, does not exist. One major assumption, which is usually applied to both mechanisms and structures, is that they are linear system and hence, are subject to methods of superposition. Even the large changes in geometry found in the mechanism are handled linearly in a piecewise fashion. Because all machinery is non-linear to some degree and it must contain

clearances if it is to move, consider the classical approach to the design of a cam- driven mechanism is to treat the mechanism as a system of rigid bodies[2]. Earlier analysis of cams briefly considered the problem of elasticity, but they dealt more with trying to optimize different cam profiles for minimum acceleration and jerk[3,4]. One early and notable contribution was that of Barken[5], who considered the valve train as an elastic system. However, he took the valve train and model it as a system of lumped masses and springs and then reduced it to an equivalent single degree of freedom spring-mass system, before solving the equation of motion. Johnson [6,7] analyzed the motion of cam mechanisms with one and two degrees of freedom and then with n-degrees of freedom. He used lumped masses and springs for his models, but discussed the desirability of using distributed parameters for the valve train model.

The problem of impact is discussed by Barkan[8] for a single degree of freedom cam

and follower system. However, he considers impact due to initial clearance only, that is, once the follower contacts the cam, it never leaves it. He also uses a coefficient of restitution, which is adequate for his system, because he is only interested in what happens after impact. As will be shown, this coefficient is of questionable value for the analysis of high speed, continuously operating machinery. Impact forces are also discussed by Johnson [9].

Dubowsky and Freudenstein[10,11] treated the problem of clearances in mechanisms, but their concern is primarily with the development of model of a joint rather than the treatment of an entire system. Briefly study of the literature shows the need for the analysis of machinery components particularly with clearances, hence, the present investigation is carried out on the impact force analysis of valve-train clearance mechanism using finite element technique. In the present analysis components of valve train mechanism are considered as bar and beam elements.

–

Figure 1 Cam-driven valve train mechanism with harmonic cam

Figure 2 System coordinates

Figure 3 Elements

Analysis of valve train mechanism with harmonic cam as shown in fig.1. The system was modeled as collection of prismatic bar and beam elements as shown in fig.2 and fig.3. The mass and stiffness properties can be derived for each of the elements as shown in fig.3. In this analysis the standard properties from automobile valve train system are taken. Those properties are listed in table 1.

Table 1: Standard dimensions

Element	Material	Modulus of Elasticity (E) N/mm2	Density kg/m3	Dimensions
				Length (mm)	Diameter (mm)
Push rod	Mild steel	202086	7720	228.6	7.62
Left portion of the rocker arm	Cast Iron	89369	7250	25.4	15.24
Right portion of the rocker arm	Cast Iron	89369	7250	38.1	15.24
Valve	Hard Chromium	199143	7530	76.2	7.62

For valve spring, the values for mass and stiffness are 0.06 kg and 42.9 N/mm respectively.

The overall stiffness matrix obtained by transferring stiffness matrices of all the elements from their local coordinate system to reference coordinate system and superposing

all stiffness matrices. The overall stiffness properties for the valve train system can be written as

5

5

i i i

i i i

[k] TT k [T]

i1

Where [T]i = coordinate transformation matrix

[k]

E A E A

1 1 1 1

L L

1 1

0 0 0 0 0

E1 A 1

L 1

12E2 I 2

3

L 2

6E 2 I 2

2

L 2

4E 2 I 2

6E 2 I 2

0 0 2 0

L 2

2E2 I 2

L

L

2 0 0 L2 0

Symm.

12E 2 I 3

3

L 3

E 3 A 4

L 4

• k s

– 6E 2 I 3 2

L 3

4E 2 I 3

L 3

– 6E 2 I 3 2

L 3

2E 2 I 3

L 3

E 3 A 4

L 4

0

4E 2 I 2

L 2

4E 2 I 3

0

L 3

E 3 A 4

L 4

Mass properties for the valve train system can be written as

[m]

5 TT m [T]

i i i i1

[m] =

1 A1L1

3

1 A1L1

6

0 0 0 0 0

AL 1 3 A L

1 1 A L2

1 3 A L22

1 1 1 2 2 2

2 2 2

2 2

3 35

2 1 0

A L32

0 0 4 20 0

A L3

2 2

2 2 2

1 0 5

0 0 140 0/p>

1 3 A L A L M

1 1 A L23

1 3 A L23

A L

2 3 3 3 4 4 s

2 3

2 3

3 4 4

35 3 3

2 1 0

A L33

4 2 0 6

A L3

2 3 2 3 3

Symm. 1 0 5 1 4 0 0

A L32 A L33

2 2 2 3

1 0 5

1 0 5 0

To accomplish this damping for the system, a viscous damping ratio[j ] for each of the

21 1 M1

0

0

1. 0

   0 0

   22 2 M 2 0

   0 23 3 M 3

   …….

   ……..

   ……..

   seven elastic mode shapes of the system, was

   selected. These mode shapes are calculated as

   to take the value of natural coordinate is unity

   ……..

   at node 1, and zero at the other node. The system damping matrix[C] was then computed. Knowing matrices [m] and [k], the eigen values[Ø], representing the elastic mode shapes of the system are computed from generalized Jacobi method, defining a set of orthogonal coordinates{} as

   { q }=[ Ø ]{ }

   Then, Damping Matrix [ C ] = ( [ Ø ]T )-1 [ D ] [ Ø ]-1

   However, in practice, the inverse of the model matrix is a large computational effort. Instead, taking advantage of orthogonally properties of the mode shapes, one can deduce the following expression for the system damping matrix, namely,

   N 2 n n T

   Diagonal stiffness and mass matrices are computed as

   C

   M

   i1 Mn

   n

   n M

   [ K ] = [ Ø ]T [ k ] [Ø] [ M ] = [ Ø ]T [ m ] [ Ø ]

   Orthogonal damping matrix is , [ D ] = [ Ø ]T [ C ] [ Ø ]

   It is now assumed that j = 0.05 for the fundamental, or lowest, frequency mode shape and j = 0.01 for all the higher frequency mode shapes. This lowest frequency mode shape is less than the next higher frequency by a factor of about thirty and as might be expected, it closely resembled the rigid body motion of the valve train as the valve spring was submerge

   softest spring in the system.

   When the valve train mechanism has clearance

   [k]

   in between pushrod and Rocker- arm, then overall stiffness matrix is

   E A

   1 1

   L

   1

   E A

   1 1

   L

   1

2. A

1 1

L

1

E A

1 1

L

1

0 0 0 0 0 0

0 0 0 0 0 0

4E 2I2

2

2

0 0 L

2E 2I2

L

L

0 0 2

6E 2I2

0 L22

12E I E A

-6E 2I3

-6E 2I3 E A

0 0 0

2 3 3 4 k 2

L

L

3 4

2 0

L

L

0 0 0

3 L

3 4

-6E 2I3 L2

3

s L

3

4E 2I3 L

3

L

3 4

2E 2I3

3

3

L 0 0

2E 2I2

2

2

0 0 L

-6E 2I3 L2

3

2E 2I3

L

3

4E 2I2

L

2

4E 2I3

0

0

L

3

6E 2I2

L22

E A E A

0 0 0

6E 2I2

3 4

L

4

0 0

6E 2I2

3 4

L

0

0

4

0

L

L

2

2

12E 2I2

0 0 L22 0 0

L22 3

When the valve train mechanism has clearance in between pushrod and Rocker- arm, then

overall mass matrix is

[m] =

3 6	0	0	0	0	0	0
1 A1 L1 1 A1 L1
6 3	0	0	0	0	0	0
	A L3			A L3 2		1 1 A L
0	0	105 0 0 1 4 0 0 210
0	0	1 3 A L A L M 1 1 A L23 1 3 A L2 3 A L 35 3 3 2 1 0 420 6
0	0	1 1 A L2 A L33 A L33 3 2 3 2 3 0 2 3 105 140 0 0 2 1 0

3 6	0	0	0	0	0	0
1 A1 L1 1 A1 L1
6 3	0	0	0	0	0	0
	A L3			A L3 2		1 1 A L
0	0	105 0 0 1 4 0 0 210
0	0	1 3 A L A L M 1 1 A L23 1 3 A L2 3 A L 35 3 3 2 1 0 420 6
0	0	1 1 A L2 A L33 A L33 3 2 3 2 3 0 2 3 105 140 0 0 2 1 0

1 A1L1

1 A1L1

2 2 2

2

2 2 2 2 2

0 2 3 3 3 4 4 s

2 3

2 3

3 4 4

0

A L3 2

1 3 A L2 3

A L33

2 3

2 3

A L3

A L3

1 3 A L2 2

0 0

0 0

2 2

2 3

2 2 2 2 3 3 0

2 2

1 4 0

420

140

105

105

420

			A L			A L
0	0	0	6	0	0	3	0

			A L			A L
0	0	0	6	0	0	3	0

3 4 4 3 4 4

1 1 A L2 2

1 3 A L2 2

1 3 A L

2 2 2 2 2 2 2

0 0 210

0 0 420

0 35

are about to impact.

.

q 2 and

.

q 8 will have

The system will have only two clearances i.e the clearance at cam and valve seat, clearance between pushrod and rocker arm, the velocity of both the pushrod and rocker arm will have the same, but unknown, velocity at the point of impact. This problem can be solved by using the principle of conservation of momentum just before and just after impact.

.

.

.

.

The equation relating the momentum at t=tn just before impact to the momentum at t=tn+1, just after impact can be written in the matrix form as shown.

different values just before impact, but will be identical just after the impact.

[m]{ q }n

+ [F] {t} = [m] { q }

n+1

Fig..4 Elements

In fig.3 assume that q2 defines the motion of upper end of the pushrod, and that a new coordinate q8 defines the linear motion of the rocker arm formerly defined by q2 as shown in

.

.

.

.

[m]{ q }n = System momentum just before impact ( is a known quantity)

fig 4. When there is contact, q8 does not exist. Now assume that the pushrod and rocker arm

Let {G}n

= [m]{ q }n

System momentum just before impact is

G1n n

1 A1L1

3

A L

1 A1L1

6

A L

0 0 0 0 0 0 dq1/dt n

1 1 1

G2 6

1 1 1

3

0 0 0 0 0 0 dq2/dt

G3 0 0

A L32

2 2

2 2

105

A L3 2

2 2

2 2

0 0 1 4 0

1 1 A L2 2

2 2

2 2

0

0

210

dq3/dt

1 3 A L

A L M

1 1 A L2

1 3 A L2 3

A L

G = 0 0 0

2 3 3 3 4 4 s

2 3 3

2 3

3 4 4

0 dq /dt

4 35

3 3 2 1 0

420 6 4

1 1 A L

A L33

A L33

2 3 3

2 3 2 3

G5 0 0 0 2 1 0

105

140

1. 0 dq5/dt

   A L3

   1 3 A L2 3

   A L33

   A L3 2

   A L33

   1 3 A L2 2

   G 0 0

   2 2 2

   2 3

   2 3

   2 2 2 3

   2 2

   dq /dt

   6 1 4 0

   420

   3

   3

   0

   0

   4 4

   4 4

   A L

   140

   105

   105

   A L

   A L

   3 4 4

   420 6

   G7 0 0 0 6

   1 1 A L2

   0 0 3

   0

   0

   1 3 A L2 2

   0 dq7/dt

2. 3 A L

   0 0

   0 0

   G8 0 0

   2 2 2

   210

   2 2

   420

   2 2 2

   35

   dq8/dt

   Since F2 = -F8

   0 n

   t 0

   {F}t = 0 F2 0

   0

   0 -t

   and, since dq2/dt = dq8/dt. The system momentum just after impact is

   0	0	0	0	0	dq1/dt
   0	0	0	0	0	dq2/dt

   0	0	0	0	0	dq1/dt
   0	0	0	0	0	dq2/dt

   [m] {dq/dt}n+1 =

   1 A1 L1

   3

   1 A1 L1

   6

   1 A1 L1

   6

   1 A1 L1

   3

   n+1

   11

   A L2 2

   A L3 2

   • A L3

0 2 2

210

2 2

105

0 0

1 1 A L23

2 2 2

140

1 3 A L2

0 dq3/dt

A L

0 0 0 2 3

2 3 3

3 4 4

dq /dt

132 A3 L3 3 A4 L4 Ms

2 1 0

420 6 4

35 3

1 1 2

3

A L33

A L33

2 3

2 3

0 0 0

2 A3 L 3

2 1 0

2 3

105

2 3

2 3

140

0 dq5/dt

1 3 A L2

A L3 2

1 3 A L2 3

A L33

A L3 2 A L33

0 2 2 2

2 2

2 3

2 3

2 2 2 3

0 dq /dt

420 1 4 0

420

3 4 4

3 4 4

A L

140

105

105

6

A L

3 4 4

0 0 0 6

0 0 3

dq7/dt

1 32 A2 L2

0 35

1 1 A L2 2

2 2

2 2

210

1 3 A L2 2

2 2

2 2

0 0 420 0

dq1/dt dq2/dt dq3/dt

Here dq4/dt and F2 are unknown quantities dq5/dt

dq6/dt dq7/dt

{G}n = [m] {dq/dt}n+1 [F] {t}

G1 n

G

1 A1L1

3

1 A1 L1

AL

1 1 1

1 1 1

6

1 A1 L1

0 0 0 0 0 0 dq1/dt n+1

G3 G	2 A L32 0 210 105 = 0 0 0
G	0	0	0
		1 3 A L2 2	A L3 2
G	0	2 2 420	2 2 1 4 0
G7	0	0	0
		1 3 A L	1 1 A L2 2

G3 G	2 A L32 0 210 105 = 0 0 0
G	0	0	0
		1 3 A L2 2	A L3 2
G	0	2 2 420	2 2 1 4 0
G7	0	0	0
		1 3 A L	1 1 A L2 2

0 0 0 0 0 -t dq /dt

2 6 3

2 2

2 2

1 1 A L 2

2 2

A L3 2

2 2

2 2

0 0 1 4 0

2

0 0 dq3/dt

1 3 A L A L M

1 1 A L23

1 3 A L2 3

A L

2 3 3 3 4 4 s

2 3

2 3

3 4 4

0 dq /dt

4 35

3 3 2 1 0

2 A L33

420 6 4

A L3

1 1 A L 3

2 3

2 3 3

2 3

0 0 dq /dt

5 2 1 0

1 3 A L2

105

A L33

2 3

2 3

A L3 2

140 5

A L33

2 3 3

2 2 2 3

0 0 dq /dt

6 420

A L

140

105

105

6

A L

A L

3 4 4

2 2 2

3 4 4

6

0 0 3

1 3 A L2 2

2 2

2 2

0 dq7/dt

G8 0

2 2

35 210

0 0 420

0 t F(2)

{G}n = [A] {S}n+1

Where {G}n = System momentum just before impact.(is a known quantity)

{S}n+1 = [A]-1 {G}n

thus impact force F(2) between pushrod and rocker arm can be determined.

The system considered in the present investigation is first shown with contact occurring only at the cam. The valve seat has been moved up so that the valve cant come in contact with it. Fig 5 shown the pushrod motion superimposed with respect to cam

rotation. As indicated fig 5, the pushrod leaves the cam at about 2430 and impacts at about 3420. Fig.6 shows that F(1), the force on the pushrod, has become zero, and hence separation has occurred. As it is expected, the impact creates vibrations in the valve train, but these are damped out in about one-half cycle, has a result the impact is very mild. It is known that the yield stress of mild steel is

225.6 N/mm2 and it is considered that the factor of safety for this material is three. So, working stress of mild steel is 75.2 N/mm2. The maximum stress on pushrod, when cam is run at 10192 rpm, is 75.06 N/mm2. So, for the valve train mechanism without valve seat, the

maximum possible running speed of cam must be equal to or less than 10192 rpm, otherwise the pushrod may behave inelasticity or it may fail. Consider valve train mechanism with valve seat, the valve seat was moved into position so that the valve would impact it. Fig.7 shows that the valve seat snatches the pushrod from the cam when the valve impacts the seat. The forces on the valve and pushrod are shown in fig.8 and fig. 9. It is known that the yield stress of hard chromium is 637.6 N/mm2 and it is considered that the factor of

safety for this material is 15, So working stress of hard chromium is 42.5 N/mm2. The maximum stress on pushrod, as well as on valve, when cam is running at 1224 rpm is

63.4 N/mm2 and 41.1 N/mm2, respectively. It

is known that the ultimate stress of cast iron is

99.55 N/mm2 and it is considered that the factor of safety for this material is 20. So, working stress of cast iron becomes 5 N/mm2. If a clearance is allowed to occur at a point with in the valve train, such as between the pushrod and rocker arm, then the maximum stress which act on the rocker arm is 1.25 N/mm2. So, for the valve train mechanism, with valve seat the maximum possible running speed is limited to 1224 rpm so as to avoid

plastic behaviour or failure of the pushrod, the rocker arm or the valve.

30

Q(1) (mm)

Q(1) (mm)

25

20

15

10

5

0

0 400 800 1200 1600

cam (degrees)

10192 rpm, no valve seat

Figure 5

F(1) (Newtons)

F(1) (Newtons)

4000

3000

2000

1000

0

0 400 800 1200 1600

cam (degrees)

10192 rpm, no valve seat

Figure 6

30

Q(1) (mm)

Q(1) (mm)

25

20

15

10

5

0

0 100 200 300 400 500

cam(degrees)

1224 rpm, with valve seat

Figure7

F(1) (Newtons)

F(1) (Newtons)

3000

2500

2000

1500

1000

500

0

F(7) (Newtons)

F(7) (Newtns)

2000

1500

1000

500

0

0 150 300 450 600

cam(degrees)

1224 rpm, with valve seat

Figure 8

0 100 200 300 400 500

cam (degrees)

1224 rpm, with valve seat

Figure 9

exclusively for this purpose to solve various equations of the valve-train mechanism, for different configurations. However, further work is required to obtain more precise model, so as to increase the precision, the two dimensional or three dimensional elements may be used for the finite element modeling and also the effect of buckling and the effect of temperatures may be included.

[1].Winfrey, R.C., "Elastic link Mechanisms Dynamics", Journal of Engineering for

Industry Trans. ASME, series B, Vol. 93, No.l, Feb 1971, pp. 268-272.

[2].Ham, C.W., Crane, E.J., and Rogers, W.L., "Mechanisms of Machinery", 4th ed., McGraw – Hill, New York, 1958. [3].Rothbart, H.A., Cams, 1st ed., wiley, New York, 1956. [4].Dudley, W.M., "New Approach to cam Design", Machine Design, vol.19, No.7,July 1947, pp. 143-148, 184.

[5].Barkan, P., "Calculation of High-speed valve motion with Flexible overhead linkage", SAE transactions, vol.61, 1953, pp.687-700.

[6].Johnson, R.C., "The Dynamic Analysis and Design of Relatively Flexible Cam Mechanisms Having more than one Degree of Freedom", Journal of Engineering for industry, Trans. ASME, series B, vol.81, No.4, Nov. 1959, pp.323-331. [7].Johnson, A.R., "Motion control for a series system of 'N' degrees of Freedom using Numerically derived and evaluated systems", Journal of Engineering for Industry, Trans. ASME,

Impact force and Impact Stress analysis of valve-train clearance mechanism is done using finite element technique. This analysis is carried out on a cam-driven-valve train mechanism with valve seat, available with present day automobiles with variety of components made of different materials. This method is sufficiently general to permit application on other machine elements, such as gears, bearings etc., or a combination thereof. Use of this method is involved, establishing an elastic model, specifying the clearances and determining the form of equations of motion for each configuration. Also this quantifies the maximum safe running speeds of mechanisms for different configurations depending upon the materials used. For the present cam driven valve train mechanism with valve seat, the maximum allowable speed is found to be 1224 rpm. A computer program is written

.

series B, vol.87, No.2, May 1965, pp. 191-204.

[8].Barkan, P., and McGamty, R.V., "A spring – Actuated, Cam

– Follower ssystem : Design Theory and Experimental Results", Journal of Engineering for industry, Trans. ASME, Series B, vol.87, No.3., Aug. 1965, pp. 279-286.

[9].Johnson, R.C., "Impact Forces in Mechanisms", Machine Design, vol.30, No. 12; June 12, 1958, pp. 138-146. [10].Dubowsky, S., and Freudenstein, F., "Dynamic Analysis of Mechanical systems with clearances, part 1 : Formation of Dynamic Model", Journal of Engineering for Industry, Trans. ASME, Series B, vol.93, No.l, Feb. 1971, pp. 305-309. [11].Dubowsky, S., and Freudenstein, F., "Dynamic Analysis of Mechanical Systems with clearances, part 2 : Dynamic Response", Journal of Engineering for Industry, Trans. ASME, series B, vol.93, No.l, Feb. 1971, pp. 310-316. [12].Chandrupatla, T.R., and Belegundu, A.D., "Introduction to Finite Elements in Engineering", Prentice – Hall of India Pvt. Ltd., New Delhi, India, 1991.

[13].Ramamurti, V., "Computer Aided Design in Mechanical Engineering", Tata McGraw-Hill Publishing Company Ltd., New Delhi, India, 1987.

[14].Mario Paz., "Structural Dynamics", C.B.S. Publishers, New Delhi, India, 1987.

[15].Winfrey, R.C., Anderson, R.V., Gnilka, C.W., "Analysis of Elastic Machinery with clearances", Journal of Engineering for industry, Trans. ASME, series B, Aug. 1973, pp.694-703