Automobile Vibration Analysis

8/17/2019 Automobile Vibration Analysis

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University of Missouri-ColumbiaMechanical and Aerospace Engineering Department

Automobile Vibration Analysis

MAE 3600

System DynamicsProject

Fall 2009

Eric Booth (12907660) (25%) – Equations of motion, write-up

Evan Kontras (12157638) (25%) – State-space model, plots

Will Linders (13942861) (25%) – Write up, optimization

Brad Pyle (11869469) (25%) – Natural frequencies, modes, FBD, write-up


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2

Table of Contents

1. Introduction p. 3

2. Modeling p. 4

3. Analysis and Results p. 7

4. Parameter Selection and Optimization p. 13

5. Design Considerations p. 16

7. Conclusion p. 17

6. References p. 18


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3

Introduction

The purpose of this report is to study the vibration of an automobile when running from a

smooth section to a bumpy section of a road. A four-degree-of-freedom model, shown in Fig. 1,

was developed for the study. The car was approximated as a flat plate with mass equal to the car,

the suspension was represented by four spring-and-damper systems attached to the four corners

of the plate, and the driver was approximated as a block mass supported by another spring-and-

damper system. The forces resulting from the unbalanced inertia force of the engine were taken

into consideration as well. The car was assumed to have three degrees of freedom; one for rolling

( x1), one for pitching ( x2, assumed to be positive opposite the sense of Fig. 1), and one for

heaving ( x3). In addition, the driver was assumed free to move vertically ( x4), giving the model

its fourth and final degree of freedom.

The analysis required derivation of equations of motion, calculation of natural

frequencies and mode shapes, state space analysis, and graphical depictions of system responses

over different input and constraint conditions. The system was then optimized to minimize the

vertical steady state vibration of the driver, keeping realistic constraints in mind. Finally,

recommendations were made for improving the system design.

Fig. 1. Working model of the automobile with the suspension system and the driver [1].


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4

Modeling

Several assumptions were made while developing the mathematical model of the system

shown in Fig. 1. First, it was assumed that, due the large mass of the engine, the center of gravity

of the car would be located a distance e1 from the front of the plate rather than at its center. The

driver was also assumed to be located a distance a1 by a2 from the C.G. Second, the bumpy

surface of the road was assumed to cause four separate displacement inputs, one to each spring-

damper system on the plate, given by z 1, z 2, z 3, and z 4, where

z 1(ξ) = 1.2 z 3(ξ) = Asin(2πξ/λ)1(ξ) (1)

z 2(ξ) = 1.2 z 4(ξ) = Asin(2π /λ)1( ) (2)

with = (ξ – e1 – e2), ξ = Vt , and 1(ξ) is a unit step function. In these equations, ξ is the

horizontal distance the car travels with a velocity equal to V over a wavy surface of amplitude A

and wavelength λ. Lastly, the road was assumed to be smooth and the car assumed to be at

equilibrium before hitting the bumpy surface, making all initial conditions zero.

The values and symbols for system parameters used for calculations are given in Table 1.

Table 1. Values and symbols for system parameters used for calculations.

Parameter Symbol Value Parameter Symbol Value

Mass of Driver m 70 kg Car Velocity V Variable

Mass of Car M 3500 kg Road profile z( ξ ) Variable

Radius of gyration (rolling) r 1 0.43 m Unit step function 1(t) N/A

Radius of gyration

(pitching)

r 2 0.5 m Road Profile Amplitude A 0.04 m

Displacement co-ordinates

of driver from CG

a1, a2 0.2 m,

0.25 m

Displacement coordinates

of edges from CG

e1, e2,

e3

1.1 m, 1.4

m, 0.6 m

Spring Constant (Wheel 1) k 1 10000

N/m

Spring Constant (Wheel 3) k 3 8000 N/m

Spring Constant (Wheel 2) k 2 10000 N/m Spring Constant (Wheel 4) k 4 8000 N/m

Spring Constant (Driver) k 5 110000

N/m

Damping Coeff. (Wheel 1) b1 800 Ns/m




Damping Coeff. (Driver ) b5 20 Ns/m


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5

The equations of motion were derived using Newton's Second Law applied to the free-

body diagram shown below in Fig. 2. The force on the plate that causes heaving is given by,

65432133

F F F F F F x M F x

, (3)

where3

x is the vertical acceleration of the center of gravity of the plate. The force on the

driver’s seat is given by,

2112345211234544 a xa x x xba xa x x xk xm F x , (4)where

4 x and

4 x are the vertical velocity and acceleration of the driver,

3 x is the velocity of the

center of gravity of the plate, and1

x and2

x are rolling and pitching angular velocities

respectively. The moments on the plate about the x and y axes are given by,

253432333111 a F e F e F e F e F x J M x (5)

1152423121122 sin et F a F e F e F e F e F x J M y (6)

where,1

x and2

x are rolling and pitching angular accelerations and J 1 = Mr 12and J 2 = Mr 2

2 are

moments of inertia about x and y axes. Note that x M and y M are moments that cause rolling

and pitching respectively.

Fig. 2. Free body diagram of the automobile

x2

x

e1

x4

2213344

22133444

xe xe x z b

xe xe x z k F

2113311

21133111

xe xe x z b

xe xe x z k F

2213333

22133333

xe xe x z b

xe xe x z k F

t F t f F sin)(6

2213322

21133222

xe xe x z b

xe xe x z k F

2112345

21123455

xa xa x xb

xa xa x z k F

x1

e2

e3

x3

a1a2

y


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6

The forces F1 through F5 of the spring-and-damper systems acting on the plate were found to be,

211331121133111 xe xe x z b xe xe x z k F (7)




211234521123455 xa xa x xb xa xa x z k F (11)

where,1

z ,2

z ,3

z , and4

z are vertical velocities of the wheels. The force due to the engine, F6, is

a function of time and is given by,

t F t f F sin)(6 (12)

where, F is constant that depends on the engine type and is the angular velocity of engine

vibrations.

The force equations (Eq. 7-12) were inserted into the equations of motion (Eq. 3-6) and

combined to create the final matrix formulation of the governing equations shown below.

4

3

2

1

551525

554321514322112543213

51514322115

2

143

2

221

2

152143321231

525243213521433212315

2

24321

2

3

4

3

2

1

2

2

2

1

)()()()()(

)()()()(

000

000

000

000

x

x

x

x

bbabab

bbbbbbbabbebbeabbbbbe

bababbebbebabbebbebaabbeebbee

bababbbbebaabbeebbeebabbbbe

x

x

x

x

m

M

Mr

Mr

…

4

3

2

1

551525

554321514322115231423

51514322115

2

143

2

221

2

152143321231

525231423521433212315

2

24321

2

3

x

x

x

x

k k ak ak

k k k k k k k ak k ek k ek ak k k k e

k ak ak k ek k ek ak k ek k ek aak k eek k ee

k ak ak k k k ek aak k eek k eek ak k k k e

…

t f e

z

z z

z

k k k k k ek ek ek e

k ek ek ek e

z

z z

z

bbbbebebebeb

ebebebeb

0

1

0

00000000

1

4

3

2

1

4321

42322111

43332313

4

3

2

1

4321

24231211

34333231

(13)


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7

Analysis and Results

3.1 Natural Frequencies and Mode Shapes

The first step in the analysis was to compute both the undamped natural frequencies of

the system and the corresponding mode shapes. A MATLAB program was created that used the

spring and mass matrices from Eq. 13 to compute the natural frequencies and mode shapes

(shown in Fig. 4) based on the Eigenvalues and Eigenvectors of the system. The natural

frequencies obtained were,

Hz 505.04 , Hz 710.03 , Hz 266.12 , Hz 404.61 (14)

and the corresponding mode shape values were,

ω1 ω2 ω3 ω4

0000.12520.01991.00000.1

9842.00110.00069.00201.0

0125.00489.00000.10166.0

0274.00000.10079.00274.0

Modes (15)

The mode shapes are shown in Fig. 3 below.

Fig. 3. Mode shape plots for the system.


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8

The natural frequencies of the system are significant because when they are within the

range of a human being’s natural frequency (4-8 Hz) [2] resonance will occur causing the motion

experienced by the driver to be both exaggerated and uncomfortable. When optimizing the

system, the natural frequencies should be made to lie outside this range.

The mode shape values shown in Eq. 15 represent how the system responds to the natural

frequencies shown above each column. Each row gives the system’s response in a particular

degree of freedom. Because the highest value for the first mode occurs in the row corresponding

to DOF x4, the first mode results primarily in vertical displacement of the driver. Likewise, the

second mode results primarily in pitching motion, the third in rolling motion, and the fourth in

heaving motion.

3.2 State-Space Formulation

In order to further analyze the model, the equations of motion were converted to state-

space. The state-space model consists of the following pair of equations,

u D xC y

u B x A x

(16)

where x and x are state space variables and their derivatives, u is the input matrix, y is the output

matrix, and A, B, C, and D matrices are constants. Eq. 16 can be expanded into Eq. [17-21].

8

7

6

5

4

3

2

1

x

x

x

x

x

x

x x

= A

8

7

6

5

4

3

2

1

x

x

x

x

x

x

x x

+BU B =

m

M

Mr

Mr

1000

01

00

001

0

0001

0000

0000

0000

0000

2

2

2

1

(17,18)

U =

0

)(

)()()(

)(

4433221144332211

1443344332221122111

44332211443322113

t f z k z k z k z k z b z b z b z b

t f e z k z k z b z be z k z k z b z be

z k z k z k z k z b z b z b z be

(19)


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9

A =

m

k

m

k

m

k a

m

k a M

k

M

k k k k k

M

k ak k ek k e

M

k ak k k k e

Mr

k a

Mr

k ak k ek k e

Mr

k ak k ek k e

Mr

k aak k ek k ee

Mr

k a

Mr

k ak k k k e

Mr

k aak k ek k ee

Mr

k ak k k k e

555152

554321512114325243213

21

51

21

51211432

21

5

2

143

2

221

2

1

21

5213422113

2

1

52

2

1

5243213

2

1

5212113423

2

1

5

2

24321

2

3

)()()()(

)()()()())()((

)())()(()(

0000

0000

0000

0000

…

…

m

b

m

b

m

ba

m

ba M

b

M

bbbbb

M

babbebbe

M

babbbbe

Mr

ba

Mr

babbebbe

Mr

babbebbe

Mr

baabbebbee Mr

ba

Mr

babbbbe

Mr

baabbebbee

Mr

babbbbe

555152

554321512114325243213

2

1

51

2

1

51211432

2

1

5

2

143

2

221

2

1

2

1

5213422113

21

52

21

5243213

21

5212113423

21

5

2

24321

2

3

)()()()(

)()()()())()((

)())()(()(

1000

0100

0010

0001

(20)


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10

8

7

6

5

4

3

2

1

3

2

1

00001000

00000010

00000001

x

x x

x

x

x

x

x

y

y

y +0U (21)

The transfer function,

4

4

43

3

32

2

21

1

11 )( U

U

Y U

U

Y U

U

Y U

U

Y sY (22)

was used to solve the equations in MATLAB where U1-4 are the Laplace transforms of Eq. 19.

3.3 Frequency Response due to Engine

Once the state-space model was developed, the response of the system to enginevibrations (Ω) was analyzed. This was done by making all z inputs zero, thus simulating idling

conditions. The amplitude of the engine vibrations was assumed to be a non-zero constant. The

frequency response functions (FRFs) were plotted using a MATLAB program entitled Auto1.m

and are shown in Fig. 4 below. The vibration frequencies ranged from 0 to 10 Hz, which covers

all the natural frequencies.

References

1. Human Vibration. Bruel & Kjaer, Choayang University of Technology.http://www.cyut.edu.tw/~hcchen/downdata/human%20vibration.doc

Fig. 4. Frequency Response Functions due to engine vibrations.


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11

The peaks observed in Fig. 4 correspond to the natural frequencies of the system. This

means that as the engine vibration frequencies approach a natural frequency, resonance occurs

and the system response increases. The resonance response observed for the roll angle at 0.71

Hz, for the pitch angle at 1.2 Hz, and for the driver displacement at 6.4 Hz are easily explained

when Fig. 3 is reexamined. The roll angle response at 6.4 Hz is observed because the driver is

offset from the rolling axis and, when excited, produces a moment causing the vehicle to roll.

The driver response observed at 0.5 Hz results from the fact that the driver is anchored to the car.

If the car heaves, the driver is displaced, and 0.5 Hz is the frequency at which heaving is

observed.

3.4 Response to Single Bump

The next step of the analysis examined the automobile’s response when traveling over asingle bump. The engine excitation was assumed to be zero and the bump was modeled as an

impulse input. Because there is only a single bump in the road the inputs will be,

)()(1 t V

At z

, )()( 13 t t

V

At z

, 0)()( 42 t z t z . (22)

where, t 1 is the time delay given by, V

eet 211

. Assuming A=0.04m , λ=0.2m, and V=48 km/hr ,

the outputs were plotted using MATLAB and are depicted in Fig. 5.

Fig. 5. Response of automobile to single bump input

0 1 2 3 4 5 6 7 8-5

-4

-3

-2

-1

0

1

2

3

4

5x 10

-4

Time (s)

R e s p o n s e

Roll Angle (rad)

Pitch Angle (rad)

Driver Displacement (m)


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12

The initial jump observed in all the outputs of Fig. 5 corresponds to the time delay (t 1) of

0.1875 seconds before the second tire hits the bump. Secondly, the plot shows that the

suspension system effectively damps all responses, allowing the system to return to its

equilibrium state in about 8 seconds.

3.5 Response to Continuous Road Profile

The next step in the analysis was to evaluate the steady-state displacement of the driver at

when the car travels over a continuously bumpy road surface described by Eq. 1 and 2. The

engine vibrations were assumed to be zero and b5 was assumed to be 300 Ns/m. The velocity of

the car and the wavelength of the road were varied and related according to,

/2 V (23)

where describes the excitation frequency due to the road profile. The response of each output

was computed using MATLAB and the results are shown in Fig. 6 below.

Fig. 6. Steady-state responses of automobile traveling over continuous bumpy surface

10 20 30 40 50 60 70 80 90 100 110 1200

0.005

0.01

0.015

0.02

=0.5

Velocity (km/h)

R o l l A n g l e S t e a d y - S

t a t e A m p l i t u d e ( r a d )

=1

=1.5

=2

=2.5

=3

=5

=10

=15=20 =25 =30

20 40 60 80 100 1200

0.01

0.02

0.03

0.04

0.05

=0.5

Velocity (km/h)

P i t c h A n g l e S t e a d y -

S t a t e A m p l i t u d e ( r a d )

=1

=1.5

=2

=2.5

=3

=5

=10

=15

=20=25

=30

20 40 60 80 100 1200

0.05

0.1

=0.5

Velocity (km/h)

D r i v e r P o s i t i o n S t e a d

y S t a t e A m p l i t u d e ( m )

=1 =1.5=2 =2.5 =3 =5

=10

=15=20

=25 =30


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13

In addition, the steady-state response of each output was plotted as a function of the

ground excitation frequency for different road wavelengths and is shown in Fig. 7 below. As can

be seen, the largest displacements occur at the natural frequencies. This was expected from the

results observed in Fig. 4. Although the excitation source differed, the resulting responses

exhibited similar trends.

Fig. 7. System Responses as a function of excitation frequency.

Parameter Selection and Optimization

Parameter selection and optimization began by first identifying all relevant design

constraints. The most important of these constraints is the health and safety of the driver.Frequencies in the range of 4-8 Hz [1] cause resonance in organs of the human body which could

be painful to the driver, while frequencies in the range of 16 Hz to 20 kHz are audible and may

be uncomfortable. Hence, the natural frequencies of the system must be below 4 Hz or between 8

and 16 Hz. Thus, the fourth mode natural frequency of the current design, 6.4 Hz, must be

changed. The first way to change the natural frequency would be to install a vibration absorber.

0 1 2 3 4 5 6 7 80

0.005

0.01

0.015

0.02

Frequency Hz

R o l l

A n g l e S t e a d y - S t a t e A m p l i t u d e ( m )

0 1 2 3 4 5 6 7 80

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Frequency Hz

P i t c h A n g l e

S t e a d y - S t a t e A m p l i t u d e ( m )

0 1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

Frequency Hz

D r i v e r P o s i t i o n S t e a d y S t a t e A m p l i t u d e ( m )

=0.5

=1

=1.5

=2

=2.5

=3

=5

=10

=15

=20

=25

=30


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A vibration absorber would consist of a mass (ma) suspended from the driver seat by a spring

(k a). The spring and the mass could be adjusted until,

Hz m

k

a

a 404.6 (24)

causing the displacement of the driver at this frequency to become negligible. This would also

cause two displacement peaks to occur at frequencies on either side of 6.404 Hz. If one or both

of the peaks occur within the 4-8 Hz range, then the values selected for the vibration absorber

and the driver seat spring and damper would need to be adjusted until the peaks lied outside the

4-8 Hz range.

The second method to alter the natural frequencies so that they will occur outside the

4-8 Hz range would be to change the spring constants of the system. To accomplish this, the

fourth mode natural frequency was plotted against the spring constant k 5 to obtain the plot shown

in Fig. 8. Based on the results, a new spring constant of 2,500 N/m was selected for k 5. This

changes the natural frequencies of the system to,

Hz 5034.01 , Hz 7072.02 , Hz 9695.03 , Hz 270.14 (25)

which are all in the acceptable range.

Fig. 8. Plot of first mode natural frequency versus driver seat spring constant (k 5 )

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6

x 105

0

2

4

6

8

10

12

14

16

Spring Constant (k5)

N a t u r a l F r e q u e n c y

( H z )


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15

Manufacturing concerns dictate that the spring constant for k 5 be reduced, rather than

increased. If the spring constant were increased it would have to be increased beyond

approximately 175 000 N/m to force the natural frequency above 8 Hz. With common

manufacturing process and parts, this would prove to be an impractically high stiffness constant.

In addition, manufacturing constraints prevent any changes in the mass of the automobile

because it would require a complete retooling of the manufacturing plant.

Because any changes in the suspension system’s spring constants would alter the natural

frequencies of the system and since the mass of the system cannot be changed, the only

parameters available for optimization are the damping constants. The damping constant for the

driver seat was varied while the suspension system’s damping constants were held constant. This

was done because examining one damper is more cost-effective than examining four and because

any changes in the suspension dampers may cause the car to bottom out. Optimization was

performed by examining the Transmissibility (TR) of the system. Transmissibility is the ratio of

the output displacement to the input displacement and is given by [3],

222

2

)2()1(

)2(1

TR (26)

wherenm

b

2

5 ,

,

m

k 5 , and

v

2 .

A MATLAB program was developed to calculate the transmissibility of the driver’s seat

with respect to the car as a function of β. The results were plotted and are depicted in Fig. 9.

Fig. 9. Transmissibility as a function of β for different damping ratios

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

1

2

3

4

5

6

7

=0.01

=0.1

=0.2

=0.4=0.6

=0.8=1

T R

= / n


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16

Based on the results shown in Fig. 9, a damping coefficient of 1.0 was selected. From the

relation,

nm

b

2

5 (27)

using m = 70 kg, and ωn = 1.270 Hz, b5 was calculated to be 177.8 Ns/m. Figure 9 also shows

that at higher driving speeds, lower damping ratios are desirable. The displacement at the natural

frequency as well as those at higher speeds may be minimized by using a dynamic damping

system which changes the damping ratio as a function of vehicle velocity.

Design Considerations

The mathematical model used for the analysis is simplified and its accuracy could be

improved by including several other considerations. The first of these is that the analysis

considered only a single bump hitting two tires and a continuous sine wave curve. Further

analysis could be performed for a variety of other road surfaces such as potholes, random bumps,

and non-sinusoidal road profiles. Secondly, the model did not consider the vibration absorption

characteristics of the tires. Another set of spring-and-dampers could be included underneath the

suspension system to account for the tires. Thirdly, the car was modeled as a rigid plate, but in

reality, the shape is both more complex and is not rigid. The car itself would deform under

external forces which would affect all the system responses. Fourth, the center of gravity of the plate was offset in only along the length of the plate. The center of gravity is most likely offset

along the width of the plate as well as vertically. Fifth, the location of the engine input would

most likely be both closer to the center of gravity and offset from the major axis of the car, as

opposed to being at the edge of the car and on the major axis as assumed in the model. Sixth,

because the engine is actually mounted to the car at several locations, it should not be modeled as

single input acting at a single point, but rather multiple inputs acting at multiple points. Seventh,

when the car was actually in motion, the engine forces were assumed to be zero, which does not

reflect reality. Lastly, only a single occupant was considered. Consideration of passengers would

make the system more complex but provide a more accurate estimate of real system responses.


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17

Conclusion

The above analysis, with some limitations, examined the vibrations of an automobile

suspension system, and the following conclusions were established. The natural frequencies for

the system were found to be 6.404 Hz, 1.266 Hz, 0.710 Hz and 0.505 Hz for the first, second,

third and fourth modes respectively. The resulting mode shapes were plotted as well. The

pitching, rolling and heaving responses for a single bump were plotted and analyzed. Thereafter,

responses were plotted for a continuous road profile with different wavelengths ranging from 0.5

to 30 m and velocities ranging from 3 to 120 km/hr. The maximum displacements were observed

at about 55 km/hr. These results may be used to simulate real suspension system responses

within acceptable error limits, and are good testing tools.

It was realized that the fourth mode, which causes maximum displacement of the driver’s

seat, has a natural frequency of 6.4 Hz, which is within the uncomfortable frequency range of the

human body (4-8 Hz). For this reason, the spring constant for the driver’s seat was reduced in

order to force the fourth mode natural frequency into the comfortable range. A spring constant of

2500 N/m was chosen which resulted in a new natural of 1.27 Hz. This also changed the other

mode natural frequencies, however they all remained in the acceptable range.

In order to minimize the driver’s displacement, fur ther optimization was performed. This

was accomplished by changing the driver seat damping coefficient using the concept of

transmissibility. It was observed that higher damping coefficients resulted in lowerdisplacements near the natural frequency. Based on this information, an optimal driver seat

damping coefficient of 177.8 Ns/m was selected. In addition, a dynamic damping system was

recommend to decrease the response at higher speeds.

Based on the results, a number of recommendations were made to improve the

mathematical model that may be used if more accurate analysis is desired.


18/18

Documents

Automobile Vibration Analysis