2014-01-2315

Published 09/30/2014
Copyright © 2014 SAE International
doi:10.4271/2014-01-2315
saecomveh.saejournals.org

Development of a Dynamic Vibration Absorber to

Reduce Frame Beaming
John Anderson
Kenworth Truck Co.

ABSTRACT
This paper describes the development and testing of a Dynamic Vibration Absorber to reduce frame beaming vibration in a
highway tractor.

Frame beaming occurs when the first vertical bending mode of the frame is excited by road or wheel-end inputs. It is
primarily a problem for driver comfort. Up until now, few options were available to resolve this problem.

The paper will review the phenomenon, design factors affecting a vehicle's sensitivity to frame beaming, and the principles
of Dynamic Vibration Absorbers (AKA Tuned Mass Dampers). Finally, the paper will describe simulation and testing that
led to the development of an effective vibration absorber as a field fix.

CITATION: Anderson, J., "Development of a Dynamic Vibration Absorber to Reduce Frame Beaming," SAE Int. J. Commer.
Veh. 7(2):2014, doi:10.4271/2014-01-2315.

INTRODUCTION Despite the best efforts of Engineering Design and Field

Service, some tractors will still exhibit unacceptable vibration in
“Frame beaming” occurs when the first vertical bending mode
the cab. Kenworth, with support from the PACCAR Technical
of the frame is excited by road or wheel-end inputs. It is
Center, undertook a study to better understand this problem
primarily a driver comfort problem. Frame Beaming is
and to look for ways to reduce its severity and occurrence. This
frequently a smooth-road problem that occurs at highway
ultimately led to the development of the Dynamic Vibration
speeds (80 to 100 km/h) when the tire rotational frequency
Absorber described in this paper.
coincides with the frame beaming frequency. Frame beaming
problems occur most often without a trailer, although they
Since the tractor under study (and most encountered in the
sometimes occur with one.
field) had the most pronounced beaming response without a
trailer, the plots in this paper apply to that case. Simulation and
This is not a new problem. William LeFevre describes it in his
testing was also performed with a laden semi-trailer, confirming
“Truck Ride Guide” from 1967 [1] and Thomas Gillespie also
that the vibration in that case was also reduced.
mentions it in his SAE paper from 1985 [2]. Frame beaming
problems are minimized in design by providing isolation of the
cab from the frame, and by limiting the excitation at its source, THE FRAME BEAMING PHENOMENON
wheel end irregularities, particularly radial run-out. Cab/sleeper
The flexible ladder frame that is common to most highway
suspensions were developed primarily to isolate the driver from
tractors will have many modes of vibration that will be excited
the beaming mode of the frame. Wheel-end excitation can be
by inputs from the road, wheel-ends, drivetrain, etc. What is
reduced by limiting radial run-out of wheels and tires, and by
more, the modes of the frame are typically lightly damped, and
match-mounting the rear drive tires. That is, by pairing dual
additional damping is not easy to add. The “frame beaming”
tires of similar radial run-out and clocking them so that the high
mode is the first vertical bending mode of the frame. It will
spots are 180 degrees apart.
typically occur between 7 and 9 Hz in conventional U.S. market
trucks. While this mode can be excited by road roughness,
adversely affecting ride comfort, complaints most frequently

381
382 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014)

arise on smooth pavement when the natural frequency of this cab/sleeper, engine/transmission, cooling module and axles
mode coincides with wheel-end rotation frequency. Figure 1 were included as rigid bodies with the appropriate mass
shows an ODS (Operating Deflection Shape) of the frame properties. Tires are a simple linear model used for ride
beaming of a real bobtail tractor (without semi-trailer) at its comfort evaluation. More information about Kenworth's
problem speed. approach to full vehicle vibration simulation can be found in
reference [3].

Figure 1. Operating Deflection Shape (ODS) of the tractor beaming

mode. Un-deformed shape is shown in black. A line connects the rear
drive axle ends for visualization. The approximate locations of the front
cab supports and sleeper suspension are also indicated.

Several things become clear from looking at the ODS: Figure 2. Finite element model of the test tractor. The hood and trailer
are not shown for clarity.
1. There are two nodal points for the mode (where there is
Since the model was constructed almost entirely of previously
no translational motion)
modeled components, only vehicle level validation was
2. Front cab mounts would ideally located near the forward considered necessary. Data were collected on the road
nodal point to minimize input to the cab. simulator for comparison to the vehicle model and to
3. The rear of the sleeper is far from the rear nodal point. A demonstrate that the frame beaming response was reasonably
suspension will be required for isolation. well reproduced. Model validation consisted of comparing test
4. Rear drive axle is far from a nodal point therefore it and simulation ODS and transmissibility functions from
will be very effective at exciting this mode. Wheel-end individual axle inputs to response points on the chassis. An
irregularities here must be minimized. example transmissibility function between rear drive axle
5. Drive axles move with the tail of frame, so suspension excitation and vertical response at B-pillar is shown in figure 3.
damping has little effect on this mode. Note that “background” 2 Hz sinusoidal symmetric excitation
was present at all other axle ends to ensure that the
Since the frame beaming mode is symmetric (left and right suspensions do not stick. These cause mismatch in
frame rails deform in phase), the tire irregularities that excite it transmissibility at low frequency between the test and
must also be symmetric. Frame beaming can come and go as simulation results.
the vehicle travels through curves. This is because the phase
angle changes between inside and outside wheels. Run to run The location of measurement points and some additional
results can vary. For this reason, much of the experimental work example measurement comparisons are given in the
described in this paper was performed on a road simulator, to appendix. Details of the validation work are outside of the
allow consistent phasing between left and right wheels. scope of this paper.

With a validated vehicle model in hand, it was then possible to

SIMULATION AND TESTING APPROACH investigate the sensitivity of the vehicle to a number of design
To gain a thorough understanding of the frame beaming choices and parameters. The most influential of these were
problem, a combined simulation and test approach was taken. chosen for physical testing. Testing was performed on the
A finite element analysis (FEA) model of a Kenworth T660 test Road Simulator and track at the PACCAR Technical Center,
tractor was created (figure 2). Most flexible components were and on local roads in the area.
included as Abaqus substructures (a.k.a. super-elements). The
 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014) 383

Front Cab Mount Location

As was seen in the operating deflection shape, a nodal point
occurs near the front of the cab. Ideally, one would locate the
front mount here to avoid vertical input, with the mount also
close to the center of rotation to avoid fore/aft input. These are
indeed the recommendations one finds in the literature [1] [2].
In practice, this is not always convenient to do. So how
important is this? To answer this question, simulation work was
performed with the mounts in two locations: One near the
nodal point and one set back approximately 450 mm. The
results are shown in figure 4. It is clear that front mount
location is an important choice.

Figure 3. Transmissibility between rear drive axle symmetric excitation

(at tire contact points) and B-pillar vertical response without semi-
trailer. The peak at about 8 Hz corresponds to the frame beaming
mode.

EFFECTS OF DESIGN CHOICES ON

VIBRATION IN CAB
A number of possible design choices and parameter values
were investigate by simulation. The only one found to have
a significant effect was front cab mount location. Several
Figure 4. Effects of front mount fore/aft location on transmissibility from
other parameters that have been “tweaked” in the past were
rear drive axle symmetric excitation to b-pillar vertical response,
found to have little or no effect in this investigation. They are
without trailer (simulation)
also discussed here in hopes that no more time will be
wasted on them. Testing on the road simulator showed less effect, but the effect
was still clearly visible (figure 5). It is hypothesized that the
discrepancy has to do with the flexibility of the structure used to
cantilever the mounts off the front of the cab for the physical test.
384 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014)

was not achieved with the sample mounts available. To look for
the “sweet spot,” simulation was run with the front engine mount
stiffness and damping varied over a wide range of values. Figure
8 below shows the result: The engine apparently does not work
as a vibration absorber. Rather than have the “sweet spot” that
was sought, there appears to be a “sour” one at about 1200 N/
mm and 0.5 Ns/mm. It is hypothesized that this is so because
the mass of the engine is much greater than that of the frame:
the frame becomes the absorber and has the higher amplitude.
As neither the test nor simulation yielded a useful solution, this
was not investigated further.

Figure 6. Effects of front engine mount tuning on B-pillar acceleration

with drive axle symmetric run-out of 1 mm, without trailer (simulation).
Figure 5. Effects of front mount fore/aft location on transmissibility from Blue = unity (base tuning), Red = ∼1.7X acceleration.
rear drive axle symmetric excitation to B-pillar vertical response,
without trailer (road simulator test) Rear Axle Suspension Damping
Another proposal to reduce frame beaming was to increase
Front Cab Mount Tuning damping in the rear suspension. Unfortunately, simulation
Field engineers sometimes attempt to reduce frame beaming predicts that this will have little effect. This is not surprising
vibration by installing lower durometer (less stiff) front cab when one looks at the ODS (Figure 1): There is little relative
mounts. Simulation showed that softening the mounts will do motion between frame and the rear axles, so the dampers will
little, or possibly make the problem a bit worse. not be working to remove energy from the mode. Some simple
experiments with dampers on the physical vehicle confirmed
this. Damping effects were also evaluated experimentally by
Sleeper Suspension Tuning removing the rear axle dampers. This did provide a small
Simulation showed that tuning of the sleeper suspension could reduction in frame beaming response but would not be
not significantly reduce frame beaming vibration in the cab. In practical or safe for a road going vehicle.
fact, it was seen that changing damping from the base tuning
could make it slightly worse. Less suspension damping
Frame Inserts
reduces the damping in the mode while high damping transmits
more force from the frame into the sleeper. Changes in Another common field fix is to use frame inserts to stiffen the
stiffness within a reasonable range for air springs had almost chassis. This means doubling the mid-frame with C-channels
no effect. nestled the side members. The main effect of this is simply to
move the frame beaming frequency up a few Hz (or km/hr). In
some cases, this is an acceptable fix if the problem occurred at
Front Engine Mount Tuning the upper range of operating speed. Often however it merely
It was suggested that the engine might be used as a vibration shifts the complaint to a different speed while adding a few
absorber. In fact, PACCAR holds a patent on this concept [4]. hundred kg to the vehicle.
Experimentally, this was not successful. It was thought at the
time that perhaps the right combination of stiffness and damping
 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014) 385

THE VIBRATION ABSORBER CONCEPT theoretically only limited by system damping. This is analogous
to our ladder frame's lightly damped first bending mode
The parameters studied within the current design do not offer a
(“beaming”) being excited by wheel-end input.
lot of improvement in frame beaming vibration. The
fundamental problem is still there: A lightly damped vibration
mode whose natural frequency can coincide with wheel-end
excitation in the normal operating range. There is no practical
way to add damping to the system as it stands nor to move the
natural frequency out of the operating range. This leads quite
naturally to the idea of changing the system dynamics with the
addition of a vibration absorber.

Theoretical Background
The concept of a vibration absorber is quite simple, and is
well described in reference [5]. A practical approach for
designing a vibration absorber is described in reference [6].
Here, only a quick overview is given as it pertains to the case
of a highway tractor.

The vibration absorber concept is best understood with simple Figure 8. Transmissibility vs. Frequency Ratio for Several Damping
lumped parameter models. Imagine a lightly damped, 1 degree Ratios
of freedom (DOF) system as shown if figure 7.
Suppose that one then adds a second, smaller mass/spring/
damper system to the original system as shown in figure 9.
Damping of the original system is assumed low and therefore
neglected. This is, after all, why we are adding a vibration
absorber in the first place. Damping of the beaming mode of a
typical tractor frame is low as evidenced by the sharp
resonance peak typically seen at the frame beaming frequency.

Figure 7. Single DOF Mass/Spring/Damper System

The equation for the transmissibility from base motion to mass

motion of this simple system is:

(1)

Where
Figure 9. System with Added Vibration Absorber (neglecting damping
is the frequency ratio (excitation/resonance) in the base system)

With the added vibration absorber, the transmissibility function

is the damping ratio (fraction of critical damping)
from base motion to the primary mass becomes more
complicated:
is the undamped natural frequency

By inspection of equation (1), it can be seen that when the

excitation frequency matches the natural frequency (θ = 1) that
the equation will blow up as the damping ratio ζ approaches
zero. The transmissibility of such a system is shown in figure 8
for several damping ratios. The amplitude of the mass is
(2)
386 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014)

Where In the plot above, a damping ratio ζ2 of 0.2 gives a good result.
The trade-off is that cancellation is not as good at the original
is the resonant frequency ratio resonance frequency. So tuning of damping is a matter of
compromise. When properly tuned, one will replace the original
large resonance peak with two small, highly-damped ones.
is the mass ratio
Of practical concern is the sensitivity of this solution to tuning.
is the damping ratio for absorber alone Figure 11 shows the effects of the resonance frequency ratio
variation. As the system is detuned, the effectiveness of the
Recommended values for the mass and resonance frequency vibration absorber is much less. Notice that as the frequency
ratios are 0.2 and 1/1.2, respectively [5]. We will use these ratio is reduced, the higher frequency peak in the
values for the theoretical model. transmissibility function becomes more prominent. As it is
increased, the lower frequency peak becomes more prominent.
Without damping in the system, equation (2) simplifies to: This insight proved helpful when fine tuning the physical
vibration absorber on the real tractor on the road simulator.

(3)

The denominator of equation (2) is a quadratic equation in θ2,

so instead of one resonance frequency, there will now be two.
This makes intuitive sense, since we now have a two DOF
system. Without damping, these new modes are just two new
potential problem frequencies. In the lower frequency mode,
the two masses move in phase. In the higher frequency mode,
they move 180 degrees out of phase. There is relative motion
between the two masses for both modes, so damping can be
introduced between them, reducing the two peaks to an
acceptable level.

Looking at the transmissibility, the expected two peaks are Figure 11. Transmissibility vs. frequency ratio with Added Vibration
Absorber for several resonance frequency ratios. ζ = 0.2, μ = 0.2
clearly seen when damping is low (figure 10). Between these
two peaks, very low transmissibility is possible with low
Figure 12 shows the effects of mass ratio on transmissibility. It
damping. This would be fine if the tractor only operated at this
appears that 0.20 is indeed the best ratio for this simple system.
frequency. Since it must operate across a range of frequencies
With a smaller mass ratio, the higher frequency peak becomes
(wheel rotation speeds), some damping is necessary.
more pronounced. With a larger mass ratio, the lower peak
becomes slightly higher but the upper peak is significantly
reduced. This comes at the penalty of increased weigh however.

Figure 10. Transmissibility vs. frequency ratio with Added Vibration

Absorber for several damping ratios. μ = 0.2, ψ = 1/1.2 Figure 12. Transmissibility vs. frequency ratio with Added Vibration
Absorber for several mass ratios. ζ = 0.25, ψ = 1/1.2
 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014) 387

Application to the Tractor The resulting B-pillar vertical acceleration with optimal tuning is
For a vibration absorber to be effective, it must be attached shown below (figure 14). Simulation predicts about a 75%
close to an anti-node for the mode shape (point of maximum reduction in peak RMS vibration with the vibration absorber
deflection). Put another way, if it were attached at a nodal point optimally tuned. From this, it was concluded that a physical test
for the mode shape, it will not affect the mode at all as the was warranted.
attachment point is not moving in that mode. Looking back to
the operating deflection shape of figure 1, there are three
candidate locations:

1. At the front of the frame

2. Midway in the frame, under the cab/sleeper
3. At the tail of the frame.

For simplicity of installation and tuning, the end of frame

location was chosen. It also has the advantage of being the
antinode with the largest displacement. Thus, a vibration
absorber here could be quite effective.

An absorber mass of 1/20th of the effective mass at the

attachment point was chosen as recommended in reference
[6]. The effective mass at the tail for the frame was found in the Figure 14. Comfort-weighted RMS Acceleration at B-pillar with rear
simulation model by adding a 200 kg lumped mass to the drive axle run-out of 1 mm, worst-case phase left to right, speeds to
attachment location and then looking at the shift in natural 110 km/h, without semi-trailer (simulation)
frequency. The effective mass was found to be 1270 kg, so the
mass for experimental absorber was chosen to be 1/20th of
that, or 64 kg.
PROOF OF CONCEPT
The next step was to build a Proof of Concept for physical
Using this mass as a starting point, the stiffness and damping of testing. Since it is well understood that our simulation models
the vibration absorber were first estimated by hand for f2 = are not perfectly accurate, a tunable design was wanted,
1/1.2 f1 and ζ2 = 0.2. These parameters were then optimized wherein the stiffness and damping could be independently
using the FE model with simulated rear drive axle tire run-out. varied to find the optimal result. As rear drive axle symmetric
The methods used are described in more detail in reference [3]. excitation was known to be the most effective at exciting frame
beaming, this was chosen for input to the tractor on the road
The contour plot of comfort-weighted acceleration at the simulator. Figure 15 shows the proof-of-concept design. Mass
B-pillar vs. absorber stiffness and damping is shown in figure on a swing-arm is supported by two opposing air-springs. The
13. For this vehicle, there is an obvious “sweet spot” for tuning. mass rests on the upper air-spring, whose height is maintained
by an ordinary height control valve. Pressure in the lower
air-spring can be varied with a remote valve (actually an
air-seat height control valve) to tune the spring rate. An
adjustable damper was included that could also be varied with
a valve.

Of concern of course was the possibility that the absorber

would merely shift the problem to another frequency or
frequencies. Therefore, rather than just excite the tractor at the
original problem frequency, broadband random excitation was
used. Specifically, the symmetric content of a moderately rough
asphalt road to 20 Hz was used. With the system properly
tuned, transmissibility results similar to those shown in figure
10 above are expected: Two well-damped peaks in place of the
original large one. This was in fact the case (figure 16).

Figure 13. Contour plot of RMS acceleration at the driver's side B-pillar
for vibration absorber stiffness and damping with rear drive tire run-out
for speeds 80 to 100 km/h with worst-case tire phasing, without
semi-trailer
388 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014)

SUMMARY/CONCLUSIONS
This paper reviewed parameters of the current design thought
to affect frame beaming vibration in the cab, and the
development of a dynamic vibration absorber to alleviate that
vibration.

Investigation of parameters for the current design shows that

front cab mount placement is critical to good isolation of the
cab from the frame beaming mode. Ideally, one would locate
the front mount close to the forward nodal point of the mode to
avoid vertical input, with the mount also close to the center of
rotation to avoid fore/aft input. Other factors investigated had
little influence on frame beaming vibration in the cab.

Simulation and testing clearly demonstrate that a properly

tuned vibration absorber can be very effective in reducing
Figure 15. Proof-of-Concept design for a tunable vibration absorber.
frame beaming vibration in conventional highway tractors.
Transmissibility from rear drive tire run-out to the B-pillar was
reduced by approximately 60% as measured on the road
simulator, with a similar reduction in RMS acceleration in
smooth-road testing. A mass ratio of 0.20 was found to be
ideal. The effectiveness of the vibration absorber is strongly
affected by tuning, so care must be taken to ensure that the
absorber is accurately tuned to the particular vehicle. The
concept is under review for patent. The next step for Kenworth
is to develop a practical installation and tuning procedure for
problem tractors in the field.

It is hoped that this paper also demonstrates the power of a

combined simulation and testing approach to solve a stubborn
Figure 16. Transmissibility from rear drive tire symmetric excitation to
vibration problem that is inherent to traditional flexible-frame
B-pillar vertical response. tractors.

The real test of course is to put it on the road. Figure 17 below

shows the combined comfort-weighted RMS acceleration vs.
REFERENCES
speed on smooth pavement from 45 to 65 MPH with and
1. LeFevre, W., “Truck Ride Guide,” Rockwell-Standard Corporation,
without the vibration absorber tuned. Results with the detuned 1967
absorber are approximately the same as with no absorber at 2. Gillespie, T., “Heavy Truck Ride,” SAE Technical Paper 850001,
all. It is clear that the frame beaming vibration problem has 1985, doi:10.4271/850001 Also available in “Truck Systems
Design Handbook (Progress in Technology),” Society of
effectively been eliminated by the tuned vibration absorber. Automotive Engineers, Inc., Warrendale, PA, ISBN 1-56091-285-
5, 1992.
3. Anderson, J., “Frequency Domain Vibration Analysis of
Commercial Vehicles Based on a Modular Finite Element Library,”
SAE Technical Paper 2003-01-3415, 2003, doi:10.4271/2003-01-
3415.
4. Stephens, D.L, “Frame Beaming Reduction Assembly,” Dec 30,
1997, US Patent 5701969 A
5. Hunt, J.B., “Dynamic Vibration Absorbers,” First Edition,
Mechanical Engineering Publications, 1979, ISBN-10:0852984170
6. Aubert, A. and Howle, A., “Design Issues in the Use of Elastomers
in Automotive Tuned Mass Dampers,” SAE Technical Paper 2007-
01-2198, 2007, doi:10.4271/2007-01-2198.

Figure 17. Combined comfort-weighted RMS acceleration on smooth

pavement with vibration absorber constrained (blue) and active (red).
 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014) 389

CONTACT INFORMATION
The author may be contacted at:
Johnbrian.anderson@yahoo.com
(Home)

ACKNOWLEDGMENTS
I would like to thank John Olsen, Donald Smith, and the staff of
the PACCAR Technical Center for their support in this
investigation. I would also like to thank Kenworth R&D for
excellent design support, fabrication and installation of the
proof of concept; and Simon de Cock, René Liebregts, and my
colleagues at DAF Trucks in the Netherlands for their input and
advice during this investigation.
390 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014)

APPENDIX

Measurement points are numbered below. Triangles indicate assumed rigid bodies (cab/sleeper, engine, cooling module, tanks, etc.).
These were used for comparing transmissibility functions and operating deflection shapes on the road simulator.
 Anderson / SAE Int. J. Commer. Veh. / Volume 7, Issue 2 (October 2014) 391

Below is a comparison of transmissibility functions between rear drive axle symmetric excitation (at tire contact points) and responses
on vehicle in vertical (3) direction for the tractor without trailer. Blue is from test, red is from simulation. Note that “background” 2 Hz
sinusoidal symmetric excitation was present at all axle ends to ensure that the suspensions do not stick. These cause the excessive
peak in the test data. These plots are given as an indication of model quality. Responses to symmetric and antimetric excitation at all
axles with and without loaded semi-trailer were examined but are not given in this paper.

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