##### Document Text Contents

Page 1

Stress Analysis and Optimization of Crankshafts

Subject to Dynamic Loading

Farzin H. Montazersadgh and Ali Fatemi

Graduate Research Assistant and Professor, Respectively

A Final Project Report Submitted to the

Forging Industry Educational Research Foundation (FIERF)

and

American Iron and Steel Institute (AISI)

The University of Toledo

August 2007

Page 2

ii

FORWARD

The overall objective of this study was to evaluate and compare the fatigue

performance of two competing manufacturing technologies for automotive crankshafts,

namely forged steel and ductile cast iron. In addition, weight and cost reduction

opportunities for optimization of the forged steel crankshaft were also investigated. The

detailed results are presented in two reports. The first report deals with the fatigue

performance and comparison of forged steel and ductile cast iron crankshafts. This

second report deals with analyses of weight and cost reduction for optimization of the

forged steel crankshaft.

Page 104

81

that is located on the journal bearing can have limited rotation in the direction

perpendicular to the plane which central axis and load vector lay in (i.e. Fx in Figure

3.19).

The distribution of load over the connecting rod bearing is uniform pressure on

120° of contact area, shown in Figures 4.14 and 4.15. This load distribution is based on

experimental results from Webster et al. (1983). The explanation of load distribution in

the Webster et al. study is for connecting rods, but since the crankshaft is in interaction

with the connecting rod, the same loading distribution will be transmitted to the

crankshaft. For pressure P0 on the contact surface, the total resultant load is given by:

( ) 3

3

3

00 trPdtrCosPF ∫

−

==

π

π

ϕϕ (4.1)

where r is the crankpin radius and t is the crankpin length. As a result, the pressure

constant is given by:

3

0

tr

F

P = (4.2)

Force F, which is the magnitude of the total force applied to the crankshaft, can

be obtained from dynamics analysis at different angles. According to the geometry of the

forged steel crankshaft a unit load of 1 kN will result in the pressure of 1.142 MPa, as

follows:

MPa 142.1

337.2748.18

1000

0 =

××

=P

The same boundary conditions and loading were used for the cast iron crankshaft.

Since some of the dimensions are different in the two crankshafts, the applied pressure

resulting from a unit load of 1 kN is calculated to be 1.018 MPa,

Page 105

82

MPa 018.1

335.3451.16

1000

0 =

××

=P

4.1.3.2 Test Assembly FEA

Figure 4.16 shows a schematic drawing of the fixture of test assembly. As can be

seen one crankshaft end is gripped in the fixed column and the other end is gripped in a

44 cm arm to apply bending load. According to this assembly one side of the crankshaft

which is in the column will be constrained in all degrees of freedom in the FE model. A

load is then applied to the other side. Since the material behavior is fully elastic, stresses

for any load magnitude could be obtained by scaling each stress component resulting

from a unit load. These boundary conditions applied to the FE model of the test assembly

are shown in Figure 4.17.

4.2 Finite Element Analysis Results and Discussion

In Section 3.4 it was pointed out that the analysis conducted was based on

superposition of four basic loadings in the FE analysis. The unit load applied on the

connecting rod bearing was a pressure of magnitude 1.142 MPa and 1.018 MPa for

forged steel and cast iron crankshafts, respectively. Note that the resultant load F was 1

kN and because of differences in dimensions of the two crankshafts, the pressure is

somewhat different.

Section changes in the crankshaft geometry result in stress concentrations at

intersections where different sections connect together. Although edges of these sections

are filleted in order to decrease the stress level, these fillet areas are highly stresses

Page 207

184

Appendix B

MATLAB program used in dynamic analysis of the slider

crank mechanism developed using equations from Appendix A.

clc

clear

% measured weight of components

% piston 330.87gr

% piston+pin 417.63gr

% connecting-rod+bolts 283.35gr

% connecting-rod 244.89

% bolts 38.50gr

% pin 86.79gr

l1 = 36.98494e-3;

l2 = 120.777e-3;

mcrank = 3.7191;

mrod = 283.35e-3;

I2 = 662523.4802e-9;

lg = 28.5827e-3;

mp = 417.63e-3;

load = xlsread( 'load.xls' );

for theta_t = 1:145

theta = (theta_t-1)*5*pi/180;

theta_d = 2000*2*pi/60;

theta_dd = 0;

beta(theta_t) = asin(l1*sin(theta)/l2);

beta_d(theta_t) = theta_d*l1*cos(theta)/l2/sqrt(1-

(l1*sin(theta)/l2)^2);

beta_dd(theta_t) = l1/l2*(theta_dd*cos(theta)-

theta_d^2*sin(theta))/sqrt(1-(l1*sin(theta)/l2)^2) +

theta_d^2*l1^2/l2^2*(cos(theta))^2*l1/l2*sin(theta)/((1-

(l1*sin(theta)/l2)^2)^1.5);

v_pis(theta_t) = -l1*theta_d*sin(theta) -

l1^2/l2*theta_d*sin(theta)*cos(theta)/sqrt(1-

(l1/l2*sin(theta)^2));

a_rod_x(theta_t) = -l1*theta_dd*sin(theta) -

l1*theta_d^2*cos(theta) -

theta_dd*lg*l1^2*sin(2*theta)/(l2^2*2*sqrt(1-

(l1*sin(theta)/l2)^2)) -

theta_d^2*lg*l1^2/l2^2*(2*cos(2*theta)*sqrt(1-

(l1*sin(theta)/l2)^2) +

Page 208

185

sin(2*theta)*l1^2/l2^2*sin(theta)*cos(theta)/sqrt(1-

(l1*sin(theta)/l2)^2)) / (2*(1-(l1*sin(theta)/l2)^2));

a_rod_y(theta_t) = l1*theta_dd*cos(theta) - l1*theta_d^2*sin(theta)

- lg*l1/l2*theta_dd*cos(theta) + lg*l1/l2*theta_d^2*sin(theta);

a_pis_x(theta_t) = -l1*theta_dd*sin(theta) -

l1*theta_d^2*cos(theta) -

l1^2/l2*theta_dd*sin(theta)*cos(theta)/sqrt(1-

(l1*sin(theta)/l2)^2) -

theta_d^2*l1^2/l2*(2*cos(2*theta)*sqrt(1-

(l1*sin(theta)/l2)^2)+sin(2*theta)*l1^2/l2^2*sin(theta)*cos(thet

a)/sqrt(1-(l1*sin(theta)/l2)^2))/(2*(1-(l1*sin(theta)/l2)^2));

f_pis_x(theta_t) = mp*a_pis_x(theta_t) +

load(theta_t,2)*1e5*pi*.089^2/4;

f_a_x(theta_t) = mrod*a_rod_x(theta_t) + f_pis_x(theta_t);

f_a_y(theta_t) = 1/l2*((I2*beta_dd(theta_t)-(f_a_x(theta_t)*lg +

f_pis_x(theta_t)*(l2-

lg))*sin(beta(theta_t)))/cos(beta(theta_t)))+1/l2*mrod*a_rod_y(t

heta_t)*(l2-lg);

f_local_x(theta_t) = f_a_x(theta_t)*cos(theta) +

f_a_y(theta_t)*sin(theta);

f_local_y(theta_t) = f_a_y(theta_t)*cos(theta) -

f_a_x(theta_t)*sin(theta);

end

figure(2)

hold off

plot(load(:,1),f_local_x/1000, 'g--' )

hold on

plot(load(:,1),f_local_y/1000, '-.' )

plot(load(:,1),sqrt(f_local_y.^2+f_local_x.^2)/1000, 'r-' )

grid

title( 'Force Between Piston and Connecting rod @ 2000 rpm' )

xlabel( 'Crankshaft Angle (Degree)' )

ylabel( 'Force (kN)' )

legend( 'Axial' , 'Normal' , 'Magnitude' )

Stress Analysis and Optimization of Crankshafts

Subject to Dynamic Loading

Farzin H. Montazersadgh and Ali Fatemi

Graduate Research Assistant and Professor, Respectively

A Final Project Report Submitted to the

Forging Industry Educational Research Foundation (FIERF)

and

American Iron and Steel Institute (AISI)

The University of Toledo

August 2007

Page 2

ii

FORWARD

The overall objective of this study was to evaluate and compare the fatigue

performance of two competing manufacturing technologies for automotive crankshafts,

namely forged steel and ductile cast iron. In addition, weight and cost reduction

opportunities for optimization of the forged steel crankshaft were also investigated. The

detailed results are presented in two reports. The first report deals with the fatigue

performance and comparison of forged steel and ductile cast iron crankshafts. This

second report deals with analyses of weight and cost reduction for optimization of the

forged steel crankshaft.

Page 104

81

that is located on the journal bearing can have limited rotation in the direction

perpendicular to the plane which central axis and load vector lay in (i.e. Fx in Figure

3.19).

The distribution of load over the connecting rod bearing is uniform pressure on

120° of contact area, shown in Figures 4.14 and 4.15. This load distribution is based on

experimental results from Webster et al. (1983). The explanation of load distribution in

the Webster et al. study is for connecting rods, but since the crankshaft is in interaction

with the connecting rod, the same loading distribution will be transmitted to the

crankshaft. For pressure P0 on the contact surface, the total resultant load is given by:

( ) 3

3

3

00 trPdtrCosPF ∫

−

==

π

π

ϕϕ (4.1)

where r is the crankpin radius and t is the crankpin length. As a result, the pressure

constant is given by:

3

0

tr

F

P = (4.2)

Force F, which is the magnitude of the total force applied to the crankshaft, can

be obtained from dynamics analysis at different angles. According to the geometry of the

forged steel crankshaft a unit load of 1 kN will result in the pressure of 1.142 MPa, as

follows:

MPa 142.1

337.2748.18

1000

0 =

××

=P

The same boundary conditions and loading were used for the cast iron crankshaft.

Since some of the dimensions are different in the two crankshafts, the applied pressure

resulting from a unit load of 1 kN is calculated to be 1.018 MPa,

Page 105

82

MPa 018.1

335.3451.16

1000

0 =

××

=P

4.1.3.2 Test Assembly FEA

Figure 4.16 shows a schematic drawing of the fixture of test assembly. As can be

seen one crankshaft end is gripped in the fixed column and the other end is gripped in a

44 cm arm to apply bending load. According to this assembly one side of the crankshaft

which is in the column will be constrained in all degrees of freedom in the FE model. A

load is then applied to the other side. Since the material behavior is fully elastic, stresses

for any load magnitude could be obtained by scaling each stress component resulting

from a unit load. These boundary conditions applied to the FE model of the test assembly

are shown in Figure 4.17.

4.2 Finite Element Analysis Results and Discussion

In Section 3.4 it was pointed out that the analysis conducted was based on

superposition of four basic loadings in the FE analysis. The unit load applied on the

connecting rod bearing was a pressure of magnitude 1.142 MPa and 1.018 MPa for

forged steel and cast iron crankshafts, respectively. Note that the resultant load F was 1

kN and because of differences in dimensions of the two crankshafts, the pressure is

somewhat different.

Section changes in the crankshaft geometry result in stress concentrations at

intersections where different sections connect together. Although edges of these sections

are filleted in order to decrease the stress level, these fillet areas are highly stresses

Page 207

184

Appendix B

MATLAB program used in dynamic analysis of the slider

crank mechanism developed using equations from Appendix A.

clc

clear

% measured weight of components

% piston 330.87gr

% piston+pin 417.63gr

% connecting-rod+bolts 283.35gr

% connecting-rod 244.89

% bolts 38.50gr

% pin 86.79gr

l1 = 36.98494e-3;

l2 = 120.777e-3;

mcrank = 3.7191;

mrod = 283.35e-3;

I2 = 662523.4802e-9;

lg = 28.5827e-3;

mp = 417.63e-3;

load = xlsread( 'load.xls' );

for theta_t = 1:145

theta = (theta_t-1)*5*pi/180;

theta_d = 2000*2*pi/60;

theta_dd = 0;

beta(theta_t) = asin(l1*sin(theta)/l2);

beta_d(theta_t) = theta_d*l1*cos(theta)/l2/sqrt(1-

(l1*sin(theta)/l2)^2);

beta_dd(theta_t) = l1/l2*(theta_dd*cos(theta)-

theta_d^2*sin(theta))/sqrt(1-(l1*sin(theta)/l2)^2) +

theta_d^2*l1^2/l2^2*(cos(theta))^2*l1/l2*sin(theta)/((1-

(l1*sin(theta)/l2)^2)^1.5);

v_pis(theta_t) = -l1*theta_d*sin(theta) -

l1^2/l2*theta_d*sin(theta)*cos(theta)/sqrt(1-

(l1/l2*sin(theta)^2));

a_rod_x(theta_t) = -l1*theta_dd*sin(theta) -

l1*theta_d^2*cos(theta) -

theta_dd*lg*l1^2*sin(2*theta)/(l2^2*2*sqrt(1-

(l1*sin(theta)/l2)^2)) -

theta_d^2*lg*l1^2/l2^2*(2*cos(2*theta)*sqrt(1-

(l1*sin(theta)/l2)^2) +

Page 208

185

sin(2*theta)*l1^2/l2^2*sin(theta)*cos(theta)/sqrt(1-

(l1*sin(theta)/l2)^2)) / (2*(1-(l1*sin(theta)/l2)^2));

a_rod_y(theta_t) = l1*theta_dd*cos(theta) - l1*theta_d^2*sin(theta)

- lg*l1/l2*theta_dd*cos(theta) + lg*l1/l2*theta_d^2*sin(theta);

a_pis_x(theta_t) = -l1*theta_dd*sin(theta) -

l1*theta_d^2*cos(theta) -

l1^2/l2*theta_dd*sin(theta)*cos(theta)/sqrt(1-

(l1*sin(theta)/l2)^2) -

theta_d^2*l1^2/l2*(2*cos(2*theta)*sqrt(1-

(l1*sin(theta)/l2)^2)+sin(2*theta)*l1^2/l2^2*sin(theta)*cos(thet

a)/sqrt(1-(l1*sin(theta)/l2)^2))/(2*(1-(l1*sin(theta)/l2)^2));

f_pis_x(theta_t) = mp*a_pis_x(theta_t) +

load(theta_t,2)*1e5*pi*.089^2/4;

f_a_x(theta_t) = mrod*a_rod_x(theta_t) + f_pis_x(theta_t);

f_a_y(theta_t) = 1/l2*((I2*beta_dd(theta_t)-(f_a_x(theta_t)*lg +

f_pis_x(theta_t)*(l2-

lg))*sin(beta(theta_t)))/cos(beta(theta_t)))+1/l2*mrod*a_rod_y(t

heta_t)*(l2-lg);

f_local_x(theta_t) = f_a_x(theta_t)*cos(theta) +

f_a_y(theta_t)*sin(theta);

f_local_y(theta_t) = f_a_y(theta_t)*cos(theta) -

f_a_x(theta_t)*sin(theta);

end

figure(2)

hold off

plot(load(:,1),f_local_x/1000, 'g--' )

hold on

plot(load(:,1),f_local_y/1000, '-.' )

plot(load(:,1),sqrt(f_local_y.^2+f_local_x.^2)/1000, 'r-' )

grid

title( 'Force Between Piston and Connecting rod @ 2000 rpm' )

xlabel( 'Crankshaft Angle (Degree)' )

ylabel( 'Force (kN)' )

legend( 'Axial' , 'Normal' , 'Magnitude' )