4- AUTOMOTIVE VERTICAL DYNAMICS (damping, shock, vibration, pitch & bounce, flat ride)

Brian Paul Wiegand, B.M.E., P.E.
1
AUTOMOTIVE DYNAMICS and DESIGN

As previously noted, 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 purely a
matter of independent motions in the
longitudinal, lateral, and vertical directions.
This vertical section of the course will
investigate 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 the bounce and
pitch motions of the sprung mass will
necessitate recourse to more complex 2-DOF
models.
AUTOMOTIVE DYNAMICS and DESIGN 2

Controlling vertical motion is the raison d’être of the
suspension system, which involves the physics of what
may be regarded as a series of interacting mass-spring
systems. The most complete model of an automotive
suspension system generally used may be illustrated as
follows:

Even this 10-DOF model has far too
many DOF (degrees of freedom) for
analysis by classical means. Modern
computer analysis can handle such a 10-
DOF model very easily and precisely, but
such easy precision is too often
obtained at the expense of human
understanding. By bringing the model
into the realm of human understanding
through appropriate simplifications the
matter becomes amenable to those most
powerful of human traits: thought and
imagination.

“…detailed models…have some disadvantages.
Engineers…may not have access to the geometric design
data…when the full set of input parameters is assembled; the
(general purpose) programs run slower than (specific purpose)
programs that are less complex…(…multibody programs might
be 50 times slower than a…program…for a specific vehicle
dynamics model).”
“…something has been lost during the evolution from the older
models to the newer. The insight and expertise that underlay
the old (specific purpose) models are often lacking in modern
multibody models. Although the modern models are…highly
detailed, their accuracy in predicting vehicle response…is
sometimes not as good as…40 years ago.”
(“A Generic Multibody Program for Simulating Handling and Braking”
Michael W. Sayers and Dungsuk Han, 1995 International Association
for Vehicle Simulation Dynamics (IAVSD) Symposium, Ann Arbor, USA)

“…building and running the finite element
model is complicated, time consuming, and
costly. In a ride quality analysis…approximate
solutions are enough to determine vehicle
parameters such as…damping & stiffness. Thus,
simplified models will satisfy requirements.”
(Dukkipatti, Rao V.; Jian Pang, Mohamad S. Qatu,
Gang Sheng, and Zuo Shuguang; Road Vehicle
Dynamics, Warrendale, PA; SAE R-366, 2008,
Page 261.)

“…problems are now routinely solved with
the use of computers, running….analysis
programs…(which) are complex…requiring
the user to place a certain amount of trust in
the program...it has become obvious to us
that the use of canned software has not
improved the general understanding…in fact
the opposite has occurred. “Blind trust” in
the computer has led to some amusing (and
time consuming) design flaws…”
(Milliken, William F.; and Douglas L. Milliken,
Chassis Design, Principles and Analysis (Based
on…Notes by Maurice Olley), Warrendale, PA;
SAE R-206, 2002, page 448.)

1-DOF models, which are referred to as “quarter-
car” models, are the simplest suspension
models possible. Three of these models are as
depicted:

Ignoring damping (coefficients “cs” and “ct”) for
now, the natural frequencies of the sprung (“fs”)
and unsprung (“fus”) mass systems can readily
be determined by some simple equations
corresponding to the models shown:

Using the values ms = 1.907 lb-sec2/in (736.35 lb,
333.95 kg), ks = 117.60 lb/in (21.00 kg/cm), mus =
0.336 lb-sec2/in (129.90 lb, 58.91 kg), and kt =
2266.5 lb/in (404.68 kg/cm), the simple equation
for the sprung mass resonance produces:
We will find the sprung mass system “bounce”
(resonance) frequency by successive equations
of increasing precision to compare the results.

Two of the three simple equations can be readily
expressed as intermediate equations of a little
more complexity and precision:
For the unsprung mass the suspension and tire
springs act in parallel:
For the sprung mass the suspension and tire springs
act in series:

For the same parameter values as before, to find
the sprung mass “bounce” (resonance) frequency
via the intermediate equation, first the combined
in series spring constant must be found:
Use this “kcs” value in the intermediate equation:

To obtain an exact model of the sprung mass
motion the interaction of the sprung and unsprung
mass motions must be recognized:

The equation for the sprung system with springs
combined in series is now:
And the unsprung system spring coefficient is now:
Where “x” is in effect dividing up the effect of the
tire spring between the sprung and unsprung
systems, producing the equations:

Equating these two linked frequencies results in:
So then the quantity “ktx/mus” can be substituted
for “((ks×kt(1-x))/(ks+kt(1-x)))/ms” under the radical
sign in the expression for “fs” and the resulting
exact equation is:
Where the interrelating factor “x” is obtained from
an iterative solution of:

For the same parameter values as before, to find
the sprung mass “bounce” (natural) frequency by
exact equation, 1st the “x” factor must be found:
The substitution of the “x” value into the exact
sprung mass frequency equation produces:

Damping is a necessary evil; it increases
the level of road shock transmission, but
without damping the automotive spring-mass
system would be “conservative”. That is, any
motion imparted to the vehicle by road
surface irregularities would continue
unabated, perhaps to be added to, or to be
made more complex by, successive
disturbances. There are a number of damping
methods, but viscous fluid damping in the
form of hydraulic “shock absorbers”
(dampers) is what is generally used on
vehicles today.

In viscous fluid damping the resistance force “F” is
proportional via the damping coefficient “c” to the
flow velocity “V”; this is typical of viscous fluid
behavior:
Throughout the previous section damping was
neglected. The damped natural frequency of the
sprung mass system is not very different from the
undamped natural frequency, at least for
conventional automotive design, as calculated
previously per the simple equation:

When the values “ms = 1.907 lb-sec2/in” and “ks =
117.60 lb/in” are input then the simple sprung
mass undamped natural frequency becomes:
To get the frequency when damping is included,
we simply have to multiply the undamped
frequency by a quantity “ ”; the “” (XI) is
called the “damping ratio” and is a dimensionless
(no units) parameter.

The “damping ratio” just is the ratio of the
damping coefficient “c” to a spring-mass system
characteristic “4k/m” which is called the “critical
damping”:
For the previous simple sprung mass example
assume a “” of 0.3 (conventional automotive
damping ratios tend to range around 0.2 to 0.4);
the damped natural frequency would be:

The effect of damping on sprung mass system
behavior is illustrated by the following diagrams:
Damping
vibration effect
Damping road
shock effect

Just as the two spring values, “ks” and “kt”, can
be combined to obtain a single spring rate, so can
the two damping values be combined to form a
single damping coefficient. For the sprung mass
both the “dampers”, suspension and tire, act in
series giving rise to the combined damping rate
“ccs”:
For the unsprung mass the dampers are
considered to be in parallel, with the combination
resulting in the combined damping coefficient
“ccp”:

The primary function of any suspension system is the
“isolation” of the sprung mass from road shock. To
consider the matter of shock attenuation note the
“quarter-car model” as it approaches a 2 inch step
at velocity “V”:

The equations of SHM state that the “period”, or
time for one complete oscillation, “t” is:
This period is the reciprocal of the sprung-mass
natural frequency, which brings us back to
simple sprung mass frequency equation:
The maximum acceleration “amax” that the sprung
mass will endure is expressed as:

An easy substitution for “fs” results in an
even simpler formula:
From this equation it is easy to see that there
are two principal ways to decrease road shock:
1) “Softer” spring.
2) Greater sprung mass.
Of course, it would be stupid to increase the
sprung mass just to get less road shock, but
even “softening” the springs has its drawbacks.

Another important function of the suspension is
to keep the tires in firm contact with the ground;
in this regard note the dip in the road surface
condition:

Using the equations of SHM for the spring-
unsprung mass system the following
relationships may be derived:
The test for the spring-unsprung mass system
is: what is the minimum length “l” and maximum
depth “d” that the system can traverse at speed
“V” without losing contact.

Unfortunately, the length and depth results move in
opposite directions; when there is an improvement in “l”
then there is a worsening in “d”, indicating a necessary area
of design compromise. A better understanding of how this
works may be obtained by a consideration of the parameters
as the unsprung mass “mus” is varied about the nominal
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

Now that we have seen how road contact parameters “l”
and “d” vary as the unsprung mass “mus” is varied about the
nominal, let’s see how those same parameters vary as the
spring stiffness “ks” is varied about the nominal

And, of course, there is the matter of what happens to
the road contact parameters “l” and “d” as the sprung mass
“ms” is varied about the nominal

The effects may be even better understood when plotted:

So far, the discussion of the suspension’s primary
function in attenuation of road disturbance input
has been limited to the relatively simple matter
of road shock. This simplicity is due to the fact
that road shock tends to consist of large scale
discrete events, but there is another far more
complex type of road disturbance: road
vibration. Road induced vibration results from
the general roughness of the road surface which
presents itself to a moving vehicle as an infinite
series of small scale events which have an
adverse effect on ride quality.

Human perception of ride is mainly tactile,
which involves vibrations in the range of 0-25
Hz. Above that range vibration is perceived as
noise; human aural sensitivity involves vibrations
in the 25–20,000 Hz range. A vehicle that does
not adequately attenuate road vibration input
may not only inflict unpleasant sensation, blur
vision, and create noise, but in extreme cases
can inflict damage, if not to the passenger then
to the vehicle and/or its cargo. Extreme cases
tend to occur on those occasions when there is a
coincidence of natural frequency and frequency
of excitation resulting in a condition known as
“resonance”.

What the human body can endure, which can
be surprisingly extreme, is very different
from what human’s consider comfortable
with regard to shock and vibration tolerance.
The subject of comfort is a matter of a
complex interplay between frequency,
amplitude (which relates to acceleration and
power), directional orientation, and the
exposure time. For a graphical presentation
of the discomfort zone in a frequency vs.
amplitude plot the following “Figure 20”
borrowed from SAE J6a “Ride and Vibration
Data Manual” sums up the situation:

HUMAN LONGITUDINAL VIBRATION TOLERANCE:

Raynaud’s Syndrome (Secondary) is associated with
high frequency vibrations in the range of 20 to 200
cps. Severe enough exposure can damage the nerves
and joints. Early vehicles often generated such
vibration through the steering system.
Motion sickness is associated with very low
frequency vibrations in the 0.1 cps to 0.8 cps range.
Fatigue is associated with exposure to vibrations in
the range of 4 to 8 cps. The Technical Committee 108
of the International Organization for Standardization
recommended limiting RMS acceleration of 0.1 g at
4.0-to-8.0 cps to 1.5 hours, and increasing that limit
to 4.0 hours at 1.0 cps.
SOME SPECIAL HUMAN VIBRATION CONDITIONS:

Lastly it should be noted that the human perception of ride
is not just tactile, but acoustic as well. Human hearing is in the
20 to 20,000 cps range. However, even “sounds” that are
outside the range of human hearing, infrasound and
ultrasound, can still have physiological effects. Considerable
“noise” below 25 cps (infrasound) is generated in automobiles,
especially with the windows open. Visual recognition times
were found to increase with sound levels in the 2 to 15 cps
range at strengths greater than 105 dB, and visual tracking
error increased in this range at only 96 dB. To further
complicate matters, prolonged exposure to such infrasound
can generate feelings of euphoria; all of which does not sound
conducive to safe driving, but should sound conducive to
developing vehicle designs that not only bring shock and
tactile vibration to within desirable levels, but sound as well.
HUMAN RESPONSE TO AUDITORY VIBRATIONS:

Although small scale road surface roughness is the
major cause of vibrations there are other possible
causes: rotational imbalances / dimensional &
stiffness variations / misalignment / engine torque
flux / aerodynamic buffeting. The main objective of
the suspension system is still to attenuate the effect
of such disturbances on the sprung mass, but in the
case of vibration this goal is principally achieved by
avoiding resonance. Road vibration input must be
modeled as an almost infinite number of continuous
sine wave “mini-shock” functions all varying in
amplitude, frequency, and phase; so the matter can
be dealt with only by statistical means.

Road surface
excitation is
treated as a
broad band
random input
to the vehicle
suspension
system best
described by
its PSD (Power
Spectral
Density):

Once a PSD has been
generated, there must
be a PSD conversion
from the spatial
domain to the more
useful time domain.
The time domain is
more useful because
the most significant
measure of ride quality
is the level of vertical
accelerations (d2z/dt2)
experienced by the
vehicle passengers.

Now that the road
vertical acceleration
input for each
frequency at a
particular vehicle
velocity (“V”) has
been obtained this
process may be
repeated at suitable
velocity intervals to
cover the entire
vehicle velocity
range.

To determine this
relationship of input
to output we return to
the 2-DOF model
wherein the sprung
and the unsprung
masses did not move
independently of each
other but were linked.
From the sprung mass
free body diagram the
sprung mass “dynamic
equilibrium” equation
is obtained…

…and then rearranged into the proper format for
solution as a second order differential equation:
From the unsprung mass free body diagram…
…the unsprung mass dynamic equilibrium equation
is obtained…

…and then also rearranged into the proper format
for solution as a second order differential equation:
The solutions to these two differential equations
are complicated but determinable by classic
methods, although customarily the matter is first
simplified by dropping the tire damping terms
(involving the “ct” parameter); this practice has
lead to a wide-spread lack of appreciation of the
significance of the tire damping.

The solutions for sprung mass response to road
input, sprung mass response to axle input, and
sprung mass response to direct sprung mass input
are:
These solutions constitute the three main vibration
“transmissibility factors” or “gains”…

These solutions involve the consolidated
symbolism:  = mus/ms C = cs/ms K1 = ks/ms
K2 = kt/ms j = -1
And the general symbolism:
ms = quarter car model sprung mass.
mus = quarter car model unsprung mass.
cs = damping coefficient for the suspension damper (“shock absorber”).
ct = damping coefficient for the tire damping (hysteresis).
ks = spring rate for the suspension spring.
kt = spring rate for the tire spring quality (at particular inflation pressure).
zs = vertical displacement of the sprung mass.
zus = vertical displacement of the unsprung mass.
zr = vertical displacement input due to road surface roughness.
Fs = force directly inflicted on the sprung mass (aero buffeting, engine
vibration, etc.).
Fus = force directly inflicted on the unsprung mass (wheel imbalance,
out of round, etc.).

When these gain functions are plotted against the
frequency in Hz (Hz = cps = /2) the result tends to
look as follows:

The road acceleration inputs of Slide #39 for a
particular velocity, when multiplied by the
appropriate gain factor from Slide #45, will
generate the sprung mass vertical acceleration
response spectrum at that velocity as per the
equation:
Where:
Gzs(f) = Sprung (“s”) mass vertical (“z”)
acceleration PSD response.
Hzs(f) = Sprung (“s”) mass vertical (“z”) gain
or transmissibility.
Gzr(f) = Road (“r”) vertical (“z”) acceleration
PSD input at a velocity

Plotted, the resulting
sprung mass vertical
acceleration response
spectrum at that
velocity (the multiple
velocities are not
shown for clarity, but
such a plot would
“move” with vehicle
velocity in a fashion
similar to that of Slide
#39) would be:

The sprung mass - road input is the transmissibility
function of the greatest interest, so let’s see how
this sprung mass gain varies with the unsprung
mass:

Now, let’s look at this variation in a more
generalized way, in terms of the familiar unsprung
to sprung mass ratio “”, with the dependent axis
made non-linear in the interest of visual clarity:

Now let’s see the effect of suspension spring rate
variation on the sprung mass response (not the
gain as previous, which is why the dependent axis
value is zero at zero input and not unity) to road
vibration input:

Another way to look at the effect of spring rates on
sprung mass gain to road input is to observe the
effect of variation in the spring rate ratio “kt/ks”:

Having investigated the spring ratio effect on the
sprung mass gain (with reservations as noted), now
let’s see the damping ratio “” effect on the sprung
mass response to road input gain:

A similar plot of sprung mass gain to road input vs.
damping ratio variation, but over a wider range of
variation, and this time with a linear dependent axis,
is presented:

All of the preceding has concentrated on the
damping of the “shock absorber”, totally neglecting
the damping of the tire. Considering again the
equation for the damping ratio:
The damping ratio “” could be made to include the
effect of the tire damping coefficient “ct”, and not
just the “shock absorber” damping coefficient “cs”,
by using the in series combo damping coefficient:

The damping ratio “” would then be:
However, remember that the tire damping
coefficient “ct” terms were dropped when the
equations for sprung mass gains were derived;
any effects due to tire damping variation are
simply not present. As the gain equations do not
include any “ct” terms, it is not certain what
including “ct” in the damping ratio determinations
would represent.

Tire damping was included in the derivation of the
equation for sprung mass gain due to road input in a
study of active suspensions by authors Türkay and
Akҫay. They used their quarter-car model derived
equation with a “ct” value of zero and got the sprung
mass acceleration response over a wide range of
road vibration frequency input. Then they took a “ct”
value of 10% of “cs” as a realistic estimate, and
again used their equation. Comparison of the two
sets of sprung mass acceleration results indicated a
general reduction in acceleration for the “ct = 0.1 cs”
over the “ct = 0” model of about 3% for a traditional
passive suspension system.
(Türkay, Semiha; and Hüseyin Akҫay, “Influence of Tire Damping on the Ride Potential of
Quarter-Car Active Suspensions”, Eskişehir, Turkey; Anadolu University, 2009.)

Having considered the basic aspects of the sprung
mass response to road input, we now move on to a
consideration of the sprung mass response to “body”
input. The sprung mass response to body input is quite
different from its response to road input, not just
because the gain function is different, but because the
input is fundamentally different. Input at the body
level results from such things as power plant
torque/inertial fluctuations, gear mesh imperfections,
driveline imbalance and/or misalignment, and exhaust
system pulsation. Such inputs are much more regular
(periodic) in nature, with their frequency and power
varying in direct relation to power plant rpm. The only
possible exception to this characterization of body
inputs is the input due to aerodynamic buffeting.

How the mass ratio affects the sprung mass
response gain to body input may be seen from the
following plot:

Now consider the effect of the spring rate ratio on
the sprung mass response gain to body input:

Also, there is the effect of the damping ratio
variation on the sprung mass response gain to body
input:

Lastly, the sprung mass response to
unsprung mass (a.k.a. “axle” in the
literature) input is yet another story. Input
at the “axle” level is the result of some tire
/wheel/hub/brake imbalance, misalignment,
out-of-round conditions, and/or radial
stiffness variations. Such inputs are like
body inputs in that they are more regular in
nature (not random) than road input, but
the inputs vary in frequency and power in
direct relation to vehicle speed, not engine
speed.

As an illustrative example, suppose that a vehicle
has tires with a rolling radius of 14.09 inches
(0.3579 m), and there is a simple imbalance of a
rotating assembly (wheel, tire, brake disk, etc.) of
1 lb (0.00259 lb-sec2/in, or 0.4536 kg) at 12 inches
(0.3048 m) radius from the hub. As the vehicle
accelerates the imbalance causes, at a
correspondingly increasing frequency, an axle
level input to the sprung mass of ever increasing
magnitude:

This force translates into an acceleration (a = F/mus) for a
corresponding oscillating acceleration input into the sprung
mass. The sprung mass response is the input acceleration times
the appropriate gain value over the range of frequencies
considered. When plotted, the results look like this:

THE THREE BASIC TRANSMISSIBILITY
EQUATIONS AS PRESENTED WERE OBTAINED
FROM:
Gillespie, Thomas D.; Fundamentals of
Vehicle Dynamics, Warrendale, PA; SAE R-
114, 1992.
ALTERNATIVE FORMULATION FOR TWO OF
THOSE TRANSMISSIBILITY EQUATIONS MAY BE
OBTAINED FROM:
Dukkipati, Rao V.; and Jian Pang, Mohamad
S. Qatu, Gang Sheng, Zuo Shuguang; Road
Vehicle Dynamics, Warrendale, PA; SAE R-
366, 2008.

THE INSTRUCTOR OF THIS DYNAMICS
COURSE ATTEMPTED TO UTILIZE PROF.
GILLESPIE’S FORMULATIONS IN THE
COURSE OF A REWORK OF AN EXISTING
SUSPENSION DESIGN. THIS DID NOT
YIELD ANY REASONABLE RESULTS;
TYPOGRAPHICAL ERRORS IN THE
GILLESPIE EQUATIONS ARE SUSPECTED.
CONSEQUENTLY, A FORCED RELIANCE
ON PROF. DUKKIPATI’S EQUATIONS
PRODUCED FOLLOWING EXCELLENT
RESULTS...

The Road and the Body (“Internal”) Transmissibility
of Vibration to the Sprung Mass at the Front
Suspension, Base Vehicle vs. “Advanced”:

of Vibration to the Sprung Mass at the Front with
ARB, Base Vehicle vs. “Advanced”:

of Vibration to the Sprung Mass at the Rear
Suspension, Base Vehicle vs. “Advanced”:

PROF. DUKKIPATI’S TRANSMISSIBILITY EQUATIONS:
ROAD TO SPRUNG MASS TRANSMISSION:
BODY (INTERNAL) TO SPRUNG MASS TRANSMISSION:
Where “P” and “Q” are…

PROF. DUKKIPATI’S TRANSMISSIBILITY EQUATIONS,
“P” AND “Q”:
ω = angular frequency, radians per sec (ω = 2π Hz).
ms = sprung mass (lb-sec2/in).
mus = the unsprung mass (lb-sec2/in).
Cs = suspension damper damping coeff (lb-sec/in).
Ks = spring rate for the suspension spring (lb/in).
Kt = spring rate for the tire (lb/in).
WHERE:

There is a sprung mass rotational (pitch) oscillation,
in addition to the previously considered translational
(bounce) oscillations. Study of this new motion
necessitates a change from the previous quarter-car
models to a 2-DOF half-car model:

The simple uncoupled equation for the
pitch motion is analogous to the simple
sprung mass bounce equation:
The quantity “K” is the pitch radius of
gyration of the sprung mass. The
quantity “Ms K2” is equivalent to the
sprung mass pitch moment of inertia “J”
(also can be symbolized as “Iy”).

Heterodyning is when two different frequencies
interact to produce a “beat” frequency which can
be depicted:
For two frequencies of equal amplitude “A” the
beat “motion” can be written as:

When the pitch frequency is equal to the bounce
frequency the following equivalency of the above
simple sprung pitch equation to the simple
sprung bounce equation can be made:
This may be further reduced:
The quantity “K2/(lf x lr)”, known as the Dynamic
Index in Pitch (DIP), is significant as an indicator
of when the pitch frequency will be the same as
the bounce frequency, which eliminates the
possibility of “heterodyning”.

Note that resulting amplitude is now
an oscillating amplitude, and that the
resulting beat frequency is equal to the
difference in the interacting frequencies
because the sine function will reach an
absolute maximum twice every cycle,
and “2 x [(𝝎1 – 𝝎2)/2]” is equal “𝝎1 – 𝝎
2”. So a frequency of 14 cps and a
frequency of 13.5 cps could interact to
form a rather languid beat frequency of
0.5 cps.

The possible heterodyning of pitch and
bounce frequencies can produce a motion
much like being at sea in a small boat, so
it should not be surprising that, in vehicles
with poorly designed suspensions,
passengers may be able to enjoy all the
sensations of “mal de mare” while still on
dry land. Note that the long slow cycle of
the “beat” is the problem, and that
ordinary long slow vibrations under 1 cps
that have nothing to do with heterodyning
can have the same physiological effect.

The reality is such that the Dynamic Index in
Pitch is seldom exactly equal to “1”; the
automotive designer is left with getting as close
to “1” as possible. For vintage sports cars of
conventional design (four wheels, front engine,
etc.) where the engine was situated well aft of a
front beam axle, and there was no significant
body overhang of the wheelbase (i.e., “K” tended
to be relatively small), the value of “K2/(lf x lr)”
tended to be about 0.6 (circa 1954) with
consequent significant beat motions. Many later
sports cars tended to be closer to 0.8 (circa
1969). Even more modern designs may be around
0.9 (circa 1980).

The more general 2-DOF model, as opposed to
the previous special 2-DOF model (where the c.g.
was centrally located and the suspension spring
rates equal front to rear), may be depicted as:

This general model is readily amenable to simple,
classical, non-computerized analysis. Taking the
equilibrium attitude of the sprung mass and its
c.g. position as the reference, the equations of the
two small (thus eliminating non-linear effects)
amplitude free vibratory motions, one linear and
one rotary, are:

These equations of motion may be rearranged to
express the two accelerations:
“J” is the pitch mass moment of inertia at the c.g.,
which relates to our old friend “K”, the pitch radius
of gyration: “J = Ms K2”; this may be substituted
into the second equation:

These equations, and the motions they represent,
are linked by one term they have in common:
“(krlr-kflf)/Ms”, which is the “Coupling Coefficient”
(CC). When this term is zero, i.e. “krlr = kflf”, the
motions of the sprung mass become uncoupled,
and the sprung mass can rotate about the c.g.
without the c.g. bouncing, and the c.g. can bounce
without the mass rotating:

The way this is explained is that the pitch node
is at the c.g. while the “bounce” is really just
pitch about some node at an infinite distance
away from the c.g. In reality there will always
be some coupling of bounce and rotation of the
sprung mass because the uncoupled situation
requires an exact point in an infinite spectrum
of possibilities. If for no other reason, this will
be because the location of the sprung mass
c.g. is always varying as the vehicle is
operated (fuel is consumed, loads are varied).
Vehicle behavior always has to be examined
throughout the full range of its possible
loading.

It is important to be able to find the natural
frequencies and normal modes of vibration
for both uncoupled and coupled cases. For
the special uncoupled (krlr = kflf) condition
the equations are:
THE PRINCIPAL MODES OF VIBRATION

For the usual coupled (“krlr ≠ kflf”) condition the
general equations of the two angular frequencies
(each a combined bounce and pitch) for the two
principle modes of vibration are:
To simplify calculations, note that certain
parametric expressions were condensed into
various equation coefficients…

…those equation coefficients are:

The node points for the uncoupled frequencies
(“fz” & “fθ”) are, as noted, at infinity and at the c.g.
respectively, but what about the node points for
the more general coupled case? Something called
the “amplitude ratio equation” is used for finding
the node point locations for the coupled
frequencies (“f1” & “f2”) of the Principal Modes of
vibration:

Along with the c.g. and the nodes (for the Principal
Modes) there are a number of other special points
located along the 2-DOF bounce/pitch model
longitudinal axis. E.g., if the two principle modes
are in a coupled state (krlr ≠ kflf), there will always
be some point “SC” (the “Spring Center”) which
will seem to be a node of uncoupled motion: a
force gradually applied at that point will produce
only vertical motion, no rotation. Yet when said
force is suddenly removed the resulting vibratory
motion will be both vertical and rotational, due to
the reaction moment about “SC” caused by the
inertia at the c.g. times the arm from the c.g. to the
“SC”.

The “SC” point is where “krβ = kfα”:
THE SPRING CENTER

Since “α + β = l” (“l” is the wheelbase) it’s easy to
solve for the SC locating dimensions “α” and “β”
(two unknowns, two equations); the results are:
There are just two more special points other than
the SC, known as the “Conjugate Centers of
Percussion”, but also known as “Double
Conjugate Points”. There are always a set of two
such points, and the special thing about them is
that a force applied at one will produce only a
rotation at the other.

The procedure for locating these points, whether
they are at the wheel centers or not (but if they’re
not close to the wheel centers they are of little
interest), and determining the associated
frequencies is…
THE CONJUGATE CENTERS OF PERCUSSION

1 - Find the “Spring Center” locating dimensions “α” and “β”.
2 - Find the quantity “c2”:
3 - Find the quantity “e”:
4 - Solve for “r” (locates point “H” forward of the c.g.):
5 - Solve for “s” (locates point “J” aftward of the c.g.):
6 - Solve for the frequencies about points “H” and “J” :
…as follows:

EXAMPLE RIDE MOTION ANALYSIS
1958 Jaguar XK150S sprung weight is about 3118
lb (1414.3 kg, sprung mass is 3118/32.174 = 96.91
lb-sec2/ft) with an “lf” of 4.32 ft (1.32 m), an “lr” of
4.18 ft (1.27 m), and a wheelbase “l” of 8.5 ft (2.59
m). The pitch mass moment of inertia “J” is 1130.5
lb-ft-sec2 (1532.75 kg-m2), so the pitch radius of
gyration squared is 11.6654 ft2 (“K” = 3.42 ft, or
1.0424 m). The spring constants at the axles front
and rear are 2364.0 lb/ft (3518.0 kg/m) and 2841.6
lb/ft (4228.8 kg/m), respectively (spring constants
at an axle are double the spring constants at a
wheel, and in bounce there are no ARB
complications).

The equations for coupled motion are employed as
they will degenerate into the simpler uncoupled
equations if the Coupling Coefficient (“CC”) is
zero:
a = (kr+ kf)/Ms = (2841.6+2364.0)/96.91
= 53.72
b = CC = (krlr - kflf)/Ms = (11877.9-10212.5)/96.91
= 17.185
c = (krlr
2+kflf
2)/J = (49649.6+44117.9)/1130.5
= 82.94
(c – a) = 29.23
DI = Dynamic Index = K2/(lf lr) = 3.422/(4.32 x 4.18)
= 0.65

The principal modes of vibration frequencies are:
f1 = √c+(b2/(K2(c-a))) = √82.94 + (17.1852/(11.6654(29.23))
= 9.15 rad/sec, or 1.46 cps
f2 = √a-(b2/(K2(c-a))) = √53.72-(17.1852/(11.6654(29.23))
= 7.27 rad/sec, or 1.16 cps
To find the node points these principal mode
frequencies are input (as “radians/sec”) into:
X = b/(𝜔2-a)
X1 = 17.185/((9.15)2-53.72) = 0.57 ft (0.1737 m)
X2 = 17.185/((7.27)2-53.72) = -19.84 ft (-6.0472 m)

The Spring Center (“SC”) location is determined:
α = krl / (kf + kr) = 2841.6 (8.5) / (2364.0+2841.6)
= 4.64 ft (1.4143 m)
β = kfl / (kf + kr) = 2364.0 (8.5) / (2364.0+2841.6)
= 3.86 ft (1.1765 m)
The Conjugate Point “H” and “J” locations from
the c.g. are “r” and “s” respectively:
c2 = α x β = 4.64 x 3.86 = 17.91 ft2 (1.6639 m2)
e = α-lf = 4.64-4.32 = 0.32 ft (0.0975 m)
r = (K2-e2-c2)/2e + √ ((K2-e2-c2)2+4K2e2) / 2e
= (11.6654-0.322-17.91)/0.64+√((11.6654-.322-17.91)2+
4(11.6654)0.322)/2(0.32) = 0.57 ft (0.1737 m)
s = K2/r = 11.6654/0.57 = 20.41 ft (6.2210 m)

The frequencies about points “H” and “J” are:
fh = 1/2π √(2kf(α-e-r)2 + 2kr(β+e+r)2)/(Ms(K2+r2))
= ((2(2364.0)(4.64-0.32-0.57)2+
2(2841.6)(3.86
+0.32+0.57)2)/(96.91(11.6654+0.572)))0.5/2π
= 2.06 cps
fj = 1/2π √(2kf(α-e+s)2 + 2kr(s-β-e)2)/(Ms(K2+s2))
= ((2(2364.0)(4.64-
0.32+20.41)2+2(2841.6)(20.41
-3.86-0.32)2)/(96.91(11.6654+20.412)))0.5/2π
= 1.64 cps

When plotted against a side elevation of the
Jaguar, all these points would look as
follows:

What we have learned from all this is that the
Jaguar’s principal vibrations are a bit stiff, as we
would expect for a sports car of this era, at 1.46
cps pitch and 1.16 cps bounce. The Coupling
Coefficient was not zero, so the two motions are
linked to a certain extent (the motions are only
slightly coupled, and some authorities claim that a
little coupling is desirable in special cases such as
when the DI is equal to 1), which is generally not
desirable. Also, the Dynamic Index is 0.65, which is
appropriate for 1958, but means that some
heterodyning can occur, and that the Conjugate
Points of Percussion will not play any role in this
design.

PITCH NODE PLACEMENT SIGNIFICANCE
1931 Cadillac V12 limos would undergo motion like shown, and
note that it is the patrician passenger, seated way back over the
rear axle (!), who is getting the worst of the ride. The chauffeur,
who is seated near the oscillation node point, is experiencing
relatively moderate amplitude in his bouncing up and down. His
Lordship, however, is further away from the node and is
experiencing a motion three to four times that which his
employee is enduring.

Maurice Olley (1889-1972) was one of the greatest
automotive dynamicists of all time, specializing
in suspensions and steering. Olley was a
proponent of what he called the “flat ride”, which
has been alluded to elsewhere in this course and
would constitute Olley’s ultimate “rule”. What he
meant by “flat ride” is indicated by his famous
quote (verbatim):
“I think the solution is that for a flat ride at,
say, 40 mph, you use a front end much softer
than the rear. For a flat ride at 100 mph you
use almost equal front and rear deflections”
16 July 1959

“Flat ride” means minimizing pitch motion
initiated by the front wheels encountering a bump
in the road. As soon as contact is made the front
begins to rise in accord with its spring-mass
system natural frequency causing a growing pitch
angle. Depending on vehicle speed “V” and
wheelbase “l” the rear wheels will encounter the
same bump at some time lag “tθ” after the front
wheel contact. The time lag determination is:

Attainment of a “flat ride” is dependent upon the
velocity which means that a target velocity must
chosen in accord with the character of the vehicle
considered, a relatively lower target velocity for an
economy car and a relatively higher target velocity
for a sports car. For a conventional passive
suspension the optimum flat ride condition will
only be attained at the target velocity, and will get
further away from that optimum as velocity varies
with respect to that design speed. So, it would
seem that the best target choice would be near
midpoint of the subject vehicle speed range.

To illustrate the matter, let’s consider the familiar
case of the 1958 Jaguar XK150S. The Jaguar had
a top speed of about 133 mph, so it might seem
reasonable to set the flat ride target speed at 66.5
mph (107.0 kph). The equation for damped simple
harmonic motion used to give the mass position
“z” as a function of time “t” is:
The front to rear spring rate relationship was
varied while plotting the consequent body height
differential “zf – zr” and pitch angle “θ” changes:

The results clearly show the benefit of Maurice
Olley’s “ ” (“ ”) when plotted using the
characteristic parameters of a 1958 Jaguar XK150
, with a of (107.0 kph),
encountering a of (5.1 cm) in
height. Increasing the wheelbase “ ” would
increase the lag time “ ”, but would also
increase the pitch inertia “ ” and decrease the
pitch angle “ ”. Overall, increasing the
wheelbase, like increasing the rolling radius,
makes for a smoother riding vehicle. This is a
tendency that does much to explain the
enormous size of many early luxury type vehicles
(Bugatti Royal, etc.).

This course section has attempted to demonstrate
that, although the vehicle suspension is a
complicated dynamic system with many degrees of
freedom, there is much that can be learned from
simplified models. The key to using simplified
models successfully is to know the limitations of
each model and which is the most appropriate for
the intended purpose. Despite the current trend to
use ever more expensive and complicated
modeling, such as a FEM (Finite Element Model), to
determine vehicle dynamic behavior, there still is
some opportunity for the small company or
individual, short on bucks but long on brains, to
operate successfully in this field.

•NO COMPUTER PROGRAM FOR AUTOMOTIVE
SUSPENSION ANALYSIS HAS BEEN WRITTEN BY
THE INSTRUCTOR (TBD). HOWEVER, A COPY
OF “MASS PROPERTIES AND AUTOMOTIVE
VERTICAL ACCELERATION” WILL BE PROVIDED
EVERY SURVIVING STUDENT.
•MANY COMMERCIAL SUSPENSION ANALYSIS
PROGRAMS ARE AVAILABLE:
SuspensionSim, Mechanical Simulation
Corporation; Ann Arbor, MI, 2015

4- AUTOMOTIVE VERTICAL DYNAMICS (damping, shock, vibration, pitch & bounce, flat ride)

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