The mass properties of a vehicle affect its motion in all directions, translational and rotational. Previously this author has dealt with how mass properties affect automotive longitudinal acceleration and automotive lateral acceleration . Now a consideration is in order of how mass properties affect automotive vertical acceleration. Of course, lateral or longitudinal inputs can lead to vertical responses; every aspect of a vehicle’s dynamics is interconnected with every other aspect, but it is convenient to divide up automotive dynamics as if the subject were truly a matter of independent motions in the longitudinal, lateral, and vertical directions. Initially, this paper will investigate the significance of mass properties with regard to automotive ride (transmission of road shock & vibration) and road-holding (maintaining contact at the tire/road interface) through the use of simple, undamped, 1-DOF models. Later, the full story of how mass properties influence the bounce and pitch motions of the sprung mass will necessitate recourse to more complex 2-DOF models. The mass properties of greatest relevance to this investigation will prove to be the “sprung mass”, the “unsprung masses”, the “sprung mass distribution” (longitudinal, lateral, and vertical c.g.), the rotational inertias of the rotating portions of the “unsprung masses”, and the “sprung mass” longitudinal and lateral mass moments of inertia.

1. Brian Paul Wiegand, PE
70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011

2.  PERSONAL INTEREST
• SAWE: WROTE PAPERS & JOURNAL ARTICLES
• SAE MEMBER: SEMINAR, PUBLICATIONS
• PERSONAL LIBRARY: BOOKS, TECH PAPERS
 SIGNIFICANCE
• ACCELERATION / BRAKING, MANEUVER, RIDE
• FUEL ECONOMY, EMISSIONS, SAFETY
 EXPAND SAWE SCOPE
• AEROSPACE MASS PROPERTIES
• MARITIME MASS PROPERTIES
70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 2

3. 1. ALL BASIC AUTOMOTIVE PERFORMANCE CAN BE DEVIDED INTO
ONE OF THREE CATEGORIES:
1. LONGITUDINAL: ACCELERATION & BRAKING
2. LATERAL: TURNING, ROLLOVER, DIRECTIONAL STABILITY
3. VERTICAL: SHOCK, VIBRATION, PITCH & ROLL MOTION
2. ALL OTHER AUTOMOTIVE ISSUES CAN BE RELATED TO THE
ABOVE:
 FUEL ECONOMY / EMMISSIONS
 SAFETY: PASSIVE & ACTIVE
 NVH: NOISE, VIBRATION, & HARSHNESS
3. THEREFORE, THREE SAWE PAPERS:
1. “MASS PROPERTIES & AUTOMOTIVE LONGITUDINAL
ACCELERATION”, SAWE PAPER #1634, ATLANTA, GA, 21-23 MAY
1984.
2. “MASS PROPERTIES & AUTOMOTIVE LATERAL ACCELERATION”,
SAWE PAPER #3528, HOUSTON, TX, 14-19 MAY 2011.
3. “MASS PROPERTIES & AUTOMOTIVE VERTICAL ACCELERATION”,
SAWE PAPER #3521, HOUSTON, TX, 14-19 MAY 2011.
4. AND A FOURTH SUPPORTING PAPER:
4. “AUTOMOTIVE MASS PROPERTIES ESTIMATION”, SAWE PAPER
#3490, VIRGINIA BEACH, VA, 22-26 MAY 2010.
70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 3

4.  SHOCK
 SHOCK TRANSFER: SPRING RATE, SPRUNG MASS, UNSPRUNG MASS,
ROLLING RADIUS
 ROAD CONTACT
 ROAD CONTACT PARAMETERS: SPRING RATE, MASS RATIO
 VIBRATION
 RESPONSE GAIN: SPRING RATE, DAMPING, MASS RATIO
 GYROSCOPIC REACTION
 GYROSCOPIC REACTION: ROLLING MASS, ANGULAR VELOCITY
 RIDE MOTIONS
 BEAT FREQUENCY
 PRINCIPAL MODES: BOUNCE & PITCH
 CONJUGATE CENTERS
 “FLAT RIDE”: FRONT-REAR SUSPENSION INTERACTION
70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 4

5. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 5
A VERY SIMPLE MODEL IS ALL THAT
IS NEEDED TO INVESTIGATE SHOCK
TRANSMISSION TO SPRUNG MASS:

6. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 6
TO INVESTIGATE THE EFFECT OF THE
UNSPRUNG MASS A SOMEWHAT MORE
COMPLICATED MODEL IS NEEDED:
EARLY AUTOMOTIVE
DESIGNERS WERE WARY OF
REDUCING THE UNSPRUNG
MASS “TOO MUCH”:

7. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 7
PROBABLY THE OLDEST KNOWN TECHNIQUE FOR REDUCING ROAD
SHOCK TRANSMISSION IS TO INCREASE THE WHEEL RADIUS:

8. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 8
FROM THE SPRING-MASS SYSTEM
EQUATIONS THE FOLLOWING CAN
BE DERIVED:
NOTE THE APPEARANCE OF THE UNSPRUNG-TO-SPRUNG MASS RATIO:
“mus/ms”. THIS RATIO CAN BE REDUCED BY INCREASING THE SPRUNG
MASS, BUT AS THAT ADVERSELY EFFECTS SO MANY PERFORMANCE
CRITERIA (ACCELERATION, FUEL ECONOMY, ETC.) THAT IS NOT A
GOOD IDEA. SO…….

9. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 9
WE’LL INVESTIGATE WHAT HAPPENS AS
THE UNSPRUNG MASS IS VARIED:
Road Contact Parameters as Unsprung Weight Varies Range Range
Quarter Model, General SHM Equations, at 30 mph: 4.3 1.13
1 2 4 3 5 6 7 8
Ws Wus chg k ds fn T min l max d
lb lb %Wus lb/in in cpm sec (ft) (in)
920 73 -50% 193.3 4.8 86.04 0.697 4.3 5.13
920 80 -45% 193.3 4.8 86.04 0.697 4.5 5.17
920 93 -36% 193.3 4.8 86.04 0.697 4.9 5.24
920 106 -27% 193.3 4.8 86.04 0.697 5.2 5.31
920 117 -19% 193.3 4.8 86.04 0.697 5.5 5.37
920 145 0% 193.3 4.8 86.04 0.697 6.1 5.51
920 173 19% 193.3 4.8 86.04 0.697 6.6 5.65
920 184 27% 193.3 4.8 86.04 0.697 6.9 5.71
920 197 36% 193.3 4.8 86.04 0.697 7.1 5.78
920 210 45% 193.3 4.8 86.04 0.697 7.3 5.85
920 290 100% 193.3 4.8 86.04 0.697 8.6 6.26

10. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 10
WHAT’S HAPPENING MAY BE BETTER
UNDERSTOOD GRAPHICALLY:
Road Contact Parameters vs Unsprung Weight
0.0
2.0
4.0
6.0
8.0
10.0
0 100 200 300 400
Unsprung Weight (lb)
MinLength(ft),Max
Depth(in)
min l (ft)
max d (in)
CONCLUSION: FOR PAVED ROADS MINIMUM “l” MORE
IMPORTANT (TEND TOWARD DECREASING “mus”)

11. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 11
NOW WE’LL INVESTIGATE WHAT HAPPENS
AS THE SPRING CONSTANT IS VARIED:

12. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 12
AGAIN, WHAT’S HAPPENING MAY BE
BETTER UNDERSTOOD GRAPHICALLY:
CONCLUSION: AS DECREASING “mus” TENDS TO WORK
AGAINST MAX “d” THE SPRINGS MAY BE SOFTENED
TO COMPENSATE.
Road Contact Parameters vs Spring Stiffness
0.0
2.0
4.0
6.0
8.0
10.0
12.0
0.0 100.0 200.0 300.0 400.0
Spring Stiffness @ Wheel (lb/in)
UndulationMinL(ft),
Maxd(in)
min l (ft)
max d (in)

13. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 13
ROAD SURFACE
INPUT TO A
VEHICLE
SUSPENSION
SYSTEM IS
CHARACTERIZED
BY ROAD’S “PSD”
(POWER SPECTRAL
DENSITY). PSD’s
FOR VARIOUS
ROAD TYPES ARE
EMPIRICALLY
DETERMINED, BUT
MAY BE REDUCED
TO TYPE
REPRESENTATIVE
MATHEMATICAL
MODELS.
ROAD PSD’s ARE IN
THE SPATIAL
DOMAIN, BUT FOR
STUDY OF AN
AUTOMOBILE
MOVING OVER THE
ROAD AT SOME
VELOCITY “V”
(ft/sec) THE
CONVERSION TO
TIME DOMAIN IS:
FREQUENCY:
ft/sec x cycles/ft =
cycles/sec
POWER:
ft/sec x in2/cycles/ft
= in2/cycles/sec

14. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 14
THE RESULTING
TIME DOMAIN PSD’s
FOR VARIOUS
VELOCITIES
THROUGHOUT THE
VEHICLE
OPERATIONAL
ENVELOPE MAY BE
DOUBLE
DIFFERENTIATED
(d2/dt2) TO OBTAIN
PLOTS OF ROAD
VERTICAL
ACCELERATION /
FREQUENCY vs.
FREQUENCY.
FOR EACH VEHICLE
WEIGHT
CONDITION FROM
“1-UP” TO GVWR
THERE IS A
TRANSMISSIBILITY
FACTOR BY WHICH
THE ROAD
ACCELERATION
INPUT PLOTS MAY
BE MULTIPLIED TO
OBTAIN VEHICLE
RESPONSE
ACCELERATION
PLOTS.

15. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 15
TRANSMISSIBILITY FACTORS
SPRUNG MASS FREE BODY: UNSPRUNG MASS FREE BODY:
SPRUNG MASS DYNAMIC
EQUILIBRIUM:
UNSPRUNG MASS DYNAMIC
EQUILIBRIUM:
THE RESULTING SET OF SECOND ORDER DIFFERENTIAL EQUATIONS……

16. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 16
…SOLVED FOR THE TRANSMISSIBILITY FACTORS:
SPRUNG MASS ACCELERATION RESPONSE TO ROAD ACCELERATION INPUT:
SPRUNG MASS ACCELERATION RESPONSE TO AXLE FORCE INPUT:
SPRUNG MASS ACCELERATION RESPONSE TO BODY FORCE INPUT:

17. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 17
THE TRANSMISSIBILITY FACTORS PLOTTED:

18. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 18
UNSPRUNG MASS EFFECT ON ROAD INPUT-TO-
SPRUNG MASS TRANSMISSIBILITY FACTOR:
DECREASING UNSPRUNG MASS
REDUCES TRANSMISSION FOR
FREQUENCIES GREATER THAN
SPRUNG MASS RESONANCE.
ALTHOUGH MASS RATIO IS SHOWN
ONLY UNSPRUNG MASS WAS VARIED >

19. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 19
UNSPRUNG MASS EFFECT ON SPRUNG MASS
RESPONSE -TO-SPRUNG MASS INPUT
TRANSMISSIBILITY FACTOR:
AGAIN, SINCE THE SPRUNG
MASS RESONANCE
FREQUENCY DOESN’T
CHANGE, ONLY THE
UNSPRUNG MASS IS BEING
VARIED. SINCE SPRUNG MASS
(“BODY”) INPUT CONSISTS OF
SUCH THINGS AS ENGINE
VIBRATION, EXHAUST
PULSATIONS, ETC., THE
UNSPRUNG MASS VARIATION HAS VIRTUALLY NO EFFECT.

20. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 20
SPRUNG MASS RESPONSE-TO-UNSPRUNG MASS
INPUT:
“AXLE” (UNSPRUNG MASS)
LEVEL INPUT FORCE DUE TO
IMBALANCE, MISALIGNMENT,
OUT-OF-ROUND, & STIFFNESS
VARIATIONS.
FORCE / SPRUNG MASS x GAIN =
RESPONSE

21. 70th Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 21
NOTE THAT
HALVING THE
WHEEL RADIUS
RESULTS IN
1/16th THE
INERTIA BUT
ONLY 1/8th THE
GYRO
REACTION
TORQUE
TORQUE REACTION DIRECTION – FLEMMING’S
RIGHT HAND RULE:

22. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
22
WHEN BOUNCE & PITCH INTERACT:
“BEAT” FREQUENCY (BARF!)
BOUNCE & PITCH CAN’T INTERACT WHEN EQUAL:
WHICH REDUCES TO WHEN THE “DYNAMIC INDEX” IS ONE:

23. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
23
NODE 1 WOULD BE REGARDED
AS A “PITCH” NODE AND NODE
2 WOULD BE REGARDED AS A
“BOUNCE” NODE.
THE EQUATIONS OF DYNAMIC
EQUILIBRIUM ARE WRITTEN
FOR SPRUNG MASS FREE BODY
AND SOLVED. NOTE THESE
TWO EQUATIONS HAVE ONE
TERM IN COMMON:
“(krlr-kflf)/Ms”

24. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
24
GENERAL
EQUATIONS OF
MOTION >>>>>>>>
UNCOUPLED PURE
BOUNCE & PURE
SPRING MOTIONS >>
BECOME:
WHEN:
I.E.,
LOCATION OF NODES >>
COEFFICIENTS >>>>
FWD FROM C.G. IS POS., AFT
FROM C.G. IS NEG.

25. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
25
THE S.C. IS A SPECIAL
POINT AS A FORCE
PRESSING DOWN AT
THE S.C. WILL CAUSE
ONLY VERTICAL
MOVEMENT, NO
ROTATION.
HOWEVER, WHEN RELEASED A BOUNCE AND A ROTATION
MOVEMENT WILL RESULT. THIS IS CALLED “STATICALLY
UNCOUPLED” AND/OR “DYNAMICALLY COUPLED” MOTION.
TO LOCATE THE SPRING CENTER >>>>>>>

26. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
26
FIND COEFFICIENTS:
FIND NODE POINT LOCATIONS:
FIND FREQUENCIES @ NODES:
A FORCE AT “J” WILL ONLY
CAUSE A ROTATION AT “H”,
AND VICE VERSA.

27. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011
27
TIME DIFFERENTIAL BETWEEN FRONT AND REAR
WHEELS ON BUMP:
MOTION PER TIME (“t”) EQUATION:
PITCH ANGLE RESULTING FROM FRONT & REAR
HEIGHTS:

28. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 28
JUST AS IT IS IMPORTANT TO GET RIDE MOTIONS AS SIMPLE AS
POSSIBLE AND AS CLOSE TO TOLERABLE FREQUENCY AS POSSIBLE,
IT IS ALSO IMPORTANT HOW THE NODE POINTS ARE LOCATED
WITH RESPECT TO THE PASSENGERS.
1931 CADILLAC V12 LIMOUSINE

29. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 29
THE ESSENCE OF AUTOMOTIVE DESIGN WITH REGARD
TO VERTICAL ACCELERATIONS IS TO MINIMIZE SHOCK,
VIBRATION, & ADVERSE GYROSCOPIC REACTIONS; AND
TO GET AUTOMOTIVE RIDE MOTIONS AS SIMPLE AND
SMOOTH AND COMFORTABLE FOR HUMAN
SENSITIVITY AS POSSIBLE.
TO THIS END CERTAIN MASS PROPERTIES PLAY
DETERMINING ROLES: SPRUNG MASS, UNSPRUNG
MASS, SPRUNG MASS C.G. LOCATION, SPRUNG MASS
MOMENT OF INERTIA IN PITCH, SPRUNG MASS PITCH
RADIUS OF GYRATION, UNSPRUNG MASS (ROLLING)
MOMENT OF INERTIA; ALL AS DEMONSTRATED.

30. FIVE MINUTES ARE ALLOCATED FOR
ASKING QUESTIONS OF THE AUTHOR
70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 30

31. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 31

32. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 32
1958 JAGUAR XK150S

33. 70TH Annual International Conference of the Society of Allied Weight Engineers, Inc.
Houston, TX, 14-19 May 2011 33
1980 FORD FIESTA S