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Modal Analysis of A Truck Cab Using the Least
Squares Complex Exponent Test Method
Fan Pingqing Wang Yansong* Zhao Linlin
Shanghai University of Engineering Science, Shanghai
Abstract: This article introduces the basic theory of modal analysis. The modal testing system is
established for a developing truck body. Using a pseudo-random excitation signal along the x, y, z
directions, the modal test is carried out to obtain the dynamic performance by multi-point
excitations. The transfer function set is obtained by averaging all transfer functions, and the set is
processed by order selection to get 28 order modes. Fitting modal parameters and normalizing
modal mass can gain natural frequencies and vibration modes of cab. Then, the modal analysis of
truck cab is calculated based on the finite element method, and the results are compared with the
test results. The improved measures are put forward to enhance the local stiffness, avoid modal
coupling and reduce vibration noise. So this article supplies a reference for the dynamic test for
the large body.
Keywords: Modal test; Dynamic characteristics; Truck body; Transfer function
1 Introduction
With the rapid growth of buildings, long-range transportation and logistics, the heavy lorry
has a huge development in the automotive industry[1] because of the advantage of dead-weight. In
the running process, extrinsic and intrinsic motivations, induced by the uneven surface of road, the
change of speed and direction, the vibration coming from wheel, engine and transmission system,
the impact of gear, and so on, easily cause a strong vibration of cab. The vibration brings many
disadvantages, such as making the driver and passengers feel uncomfortable, bringing noise and
fatigue failure of components, damaging the protective layer and sealing capacity, and weakening
the corrosion resistance of the cab[2]. Therefore, the modal analysis of cab body-in-white is
important. Especially the low order elastic modal reflects the stiffness performance of the whole
vehicle body and is the key parameter to control the vibration of vehicle[3-4].
There are also many studies on the technique of modal parameter identification.
Experimental modal analysis (EMA) or modal testing is an important engineering technique to
determine the structural modal parameters. And EMA can be combined with the computer-aided
engineering (CAE) that provides theoretical simulation to verify the structural theoretical model,
and so forth the virtual testing can be implemented to assist product design and development[5-7].
In the late 1960s, Cole[8]proposed a random decrement technique(RDT) based on the single-point
excitation and single-order modal testing, and applied it to the modal parameter identification of
space shuttle. The RD method is a single channel signal, which cannot be directly used for modal
parameter identification[9]. Using the response signal in time domain, Ibrahim[10]put this
technology into the field of multi-channel signals. With the development of the modal analysis
technology, ITD identification was put forward. This method can identify the modal parameters
only using response without inputs, therefore, it can be only applied to Gaussian distribution
whose mean value of response signal is zero. And it needs many tests, so it is easy to be interfered
by noise and has high signal-to-noise ratio[11]. Box and Jenkins[12]proposed a time-series analysis
method used in modal analysis, and this approach didn’t produce energy leak, had high resolution,
and can be used for real-time online modal analysis, but it can only identify local modal, and
cannot describe the overall modes under white noise excitation[13-14].
Brown[15]put forward a least squares complex exponent method to identify the modal
parameters of multiple responses, which is a single reference point complex exponential method.
Yu Guofei[16]adopted the single-input multi-output analysis method to obtain the modal parameters
of white body, compared with the finite element simulation results, in order to optimize the design
of the body. Because of uneven distribution of energy in the system, the method is applied to the
local mode identification. Van der Auweraer H and Guillaume P[17-18] developed a least squares
complex frequency domain method (LSCF) to identify modal parameters in frequency domain.
This method is more widely, but there are many defects, such as power consumption, truncation
error, frequency aliasing, off-line analysis, etc. In time domain, the modal parameter identification
can be carried out only using response signal, therefore, it can make the real-time online modal
analysis for cars and other structures to reflect the actual dynamic performance.
Based on single-reference point complex exponential method, Leuridan and Vold[19-20]
presented a poly-reference points complex exponential method(PRCE), which extracted
the natural frequencies according to incentive and response signals. Compared with the SRCE, it
expanded the amount of data information, and the energy distribution was more even in the system.
The real and complex modes can be identified, and the extracted modal parameters are more
complete. In addition, this method has the strong ability to recognize the intensive and multiple
root mode, and the identification accuracy is greatly improved. Ma Liming et al.[21]used PRCE to
get modal parameter identification of a car BIW, and the test results were agree with the
simulations.
Due to a large number of truck under long-distance freight work, the drivers have to drive in
a long time with heavy load, so the high request is put forward for the heavy truck comfort[22-23].
The multi-point excitations are widely adopted in order to get more accurate modal parameters in
modal test for vehicle, but the most modal tests are based on the assumption of stationary white
noise excitation. In this paper, considering the pseudo random signal as the incentive in order to
simulate the actual environment accurately, the dynamics parameters of a heavy truck are solved
using the least squares method as parameter identification and adopting multi-point pulsing and
multiple points receiving approach. Then, using the finite element method to solve kinetic
parameters under modal coordinates[24-25], this paper describes the contrast analysis between the
simulation results and the test to check the validity of the experiment. So this article supplies a
reference for the dynamic test for the large body.
2 Theory of modal analysis
The modal test theory is to test the system by the dynamic measuring technology[26-27]. By
processing input and output signals, the testing transfer function is identified to get the inherent
characteristics using modal parameters. The motion equation can be expressed as:
[ M ]{
x (t )}  [C ]{ x (t )}  [ K ]{ x (t )}  { f (t )}
(1)
where [M] is mass matrix; [C] is damping matrix; [K]is stiffness matrix; x is displacement; f
is exciting force; t is time.
Equation(2)was derived from equation(1)based on Laplace transform(s is a variable),
considering the initial displacement and velocity as zero.
( s 2 [ M ]  s[C ]  [ K ])X ( s )  F ( s )
Or
(2)
[ Z ( s)]X ( s)  F ( s)
[ Z ( s)]  X ( s) F ( s)  ( s 2 [ M ]  s[C ]  [ K ])
1
(3)
where [ Z ( s )] is generalized impedance of system.
The modal analysis by finite element method is based on the “Reticulation” material
characteristics to get the mass matrix [M]and stiffness matrix[K]. For the linear or weak damping
system, their damping matrix [C] is considered as proportional damping: C  c K  c M . Then,
based on matrix diagonalization and normal modes, equation(4) can be deduced using equation(2).
k
m
X ( s )  F ( s )( s 2 M  K ) 1
Transfer function matrix [ H ( s )] is defined as:
{X ( s )}  [ H ( s )]{F ( s )}
(4)
(5)
So the equation(6) is expressed as:
[ H ( s )]  [ Z ( s )]1  ( s 2 [ M ]  s[C ]  [ K ]) 1
(6)
Equation(6) is converted to:
[ H ( s )]  [ Z ( s )]1 
adj ([ Z ( s )])
[ Z ( s )]
(7)
Experimental modal analysis is the inverse process of equation(4). To the known input and
output signals, the mass matrix, damping matrix and stiffness matrix can be solved using
formula(6)[25][28].
3 Modal test of truck cab
3.1 Theory of modal test
Using least squares complex index method(LSCE)[17][29], the modal parameters are identified
based on equation(3). In modal test, LSCE is a time domain poly-reference method, which can
identify the complete modes by residue. The formula(3) can be expanded in terms of residue at
pole using inverse Laplace transformation:
N
h(t )   ( Ar e Pr ,t  Ar*e pr t )
*
(8)
r 1
After sampling, Z r  e
Pr t
， hn  h( nt ) , according to the relationship of the modal
vectors V, modal participation factor matrix L, and the residue, the formula(8) is:
N
hnk   (Vkr Z rn Lr  Vkr* Z rn L*r )
*
r 1
where n is the number of sample points; k is degree of freedom system; N is the residue
needing to fit(modal number). The regression equation is defined as:
(9)
Lr ( Z rn I  Z rp 1 A1      Ap )  0
(10)
Based on regression equation(10), the formula(9) can be deduced to:
hn I  hn A1      hn  p Ap  0
(11)
All sample points can constitute Hankel matrix[30], which can be written in form of row
vector:
 h(1)( p 1)

  hp1

 h(1)(0)1
1





 

 

h
  A1    h


h
( Nt  p )(0)1  
( Nt )1

 ( Nt 1)( p 1) 1


A
 
 2 



   

 h(1)( p 1) N0Ni  h(1)(0) N0Ni
  A    hpN0 Ni 

  p   


 



 h( N 1)( p 1)  h( N  p )( p 1) 
 h( Nt ) N N 

N 0 Ni
t
N0 N1 
0 i 

 t
(12)
where the first subscript of h is sample point, from 1 to N t ; the second subscript of h is the
modal order, from p  1 to 0; the third is degree of freedom system, from 1 to N 0 N i ; N 0 is
the output degree of freedom, and N i is the input degree of freedom.
The overdetermined equation(12) is solved by using the least squares method to obtain all the
residue. Then, the system poles can be gotten based on the above residues and equation(10) .
Modal shape, mass, stiffness, damping ratio, and so on are identified according to the relationship
among the modal parameters.
3.2 Establishment of the test system
3.2.1 Composition of the test system
Experimental systems consists of three parts: excitation system, response pick-up system and
modal process analysis systems[31-33]. (1) Excitation system: QDAC signal generation module, the
power amplifier and exciter of LMS SCADAS Ⅲ SC316W; (2) Response pick-up system: ICP
acceleration sensor, ICP force sensor, signal amplification of LMS SCADAS Ⅲ SC316W and
smart data acquisition system; (3) Modal process analysis system: LMS Test.Lab[34]. The test
system is described in Fig.1.
3.2.2 Experiment scheme
In the modal test, the truck cab was in a free state and suspended with rubber rope. The truck
body was hung on the front and rear ends of the left and right longerons using a rubber rope, and it
was in four-point suspension status. The cab body was kept balance as the front hitch point was
located in the installation position of bumper, and the rear hitch point was situated in the holes of
cab stringer. The lower end of the rubber rope was connected to the cab by a hook, and the upper
end was connected to honing car through a pulley[35-36]. The suspension is shown in Fig.2.
The appropriate excitation signal can provide enough energy and excite all the modes in test,
so pseudo-random excitation signal was chosen under the frequency range of 0~200Hz.
Multi-point excitations and multi-point responses (x, y, z three-way responses) can prevent the
loss of important information and avoid the incomplete mode.
Fig.1 The test system
Fig.2 The hitch positions of truck cab in modal test
In experiment, the excitation points were measured by triaxial sensor, and the pick-up points
were tested in seven batches. Each batch included 30 moving sensors to test along x, y, z three
directions. The method of collecting signal increased the experiment workload but improved the
test accuracy. The front end of longeron was defined as X exciting direction, which was shown in
Fig.3. The front end of right threshold beam was considered as Y exciting direction, shown in
Fig.4. The rear end of longeron was Z exciting direction, shown in Fig.5.
Fig.3 The exciting point
in X-direction
Fig.4 The exciting point
in Y-direction
Fig.5 The exciting point
in Z-direction
213 collection points were identified as shown in Fig.6, and their distribution was as follows:
40 points were located in firewall, 64 points lied in side plates, 43 points lied in floorboard, 29
points in roof panel, and 37 points in rear quarter panel.
Fig.6 The layout of test points
3.2.3 Collection of experiment data
During collection, the transfer and coherence functions of test data were simultaneously
processed. The signal was valid when coherent coefficient was above 0.8[37-38]. Collecting signals
while processing them can improve the precision and avoid repetitive trials. Coherence in x, y,
z-directions were shown in Fig.7 to Fig.9 respectively. The coherent coefficient of few frequencies
were below 0.8, and the rest was close to 1. Because of the good correlation between measuring
points and excitation points, the signal-to-noise ratio was high and excitation was effective.
The response acceleration was tested and analyzed in x, y, z three directions. Averaging all
transfer functions to obtain the transfer function set, and the set was processed by order selection
to get 28 order modes, shown in Fig.10.
Fig.7 Coherence in x-direction
Fig.8 Coherence in y-direction
Fig.9 Coherence in z-direction
The response acceleration was tested and analyzed in x, y, z three directions. Averaging all
transfer functions to obtain the transfer function set, and the set was processed by order selection
to get 28 order modes, shown in Fig.10.
Fig.10 The transfer function set
4 Data processing
4.1 Response frequency and vibration modes
Based on the experimental modal analysis theory, using poly-reference points complex
exponential method, the process of modal parameter identification is shown in Fig.11. During
modal trials, error was inevitable. Exciting force should inspire each mode, but too high exciting
force would cause local nonlinear. Due to the large volume of truck cab, the high frequencies
above 100Hz can be detected rarely. The average of all the transfer functions was computed to get
the transfer function set. And the set was processed by the determination of the order to identify
the number of modes. The result was shown in the Figure 10 with adapting the curve of figure in
order to show the peak clearly. Finally 28 steps mode were collected. Using the method of
complex modal single degree to fit the natural modal, and adopting mass normalization
method[39](modal mass is defined as 1) to obtain the vibration mode, all results were described in
Tab.1 and Fig.12.
Fig.11 Identification flow chart for modal parameter
Tab.1 The first eight natural frequencies
Order
Frequency
(Hz)
Modal
Modal stiffness
mass（kg）
(kg/s2)
Modal damping
(kg/s)
Damping
ratio (%)
1
16.499
1
10752.2
6.57552
3.17067
2
17.483
1
12078.5
3.51214
1.59785
3
25.030
1
24742.3
2.18263
0.693793
4
28.076
1
31109.1
2.61957
0.742602
5
29.106
1
33442
1.75744
0.480513
6
30.773
1
37398.2
4.42446
1.14395
7
32.931
1
42815.8
3.45286
0.83435
8
35.680
1
50259.8
2.61931
0.58418
(a) The first torsional mode along X-axis
F=16.499Hz
(c) Local mode in rear side plate
F=25.030Hz
(b) Swing and torsion mode along X-axis
F=17.483Hz
(d) Local mode in rear roof panel
F=28.076Hz
(e) Torsional mode along Y-axis, and local
modes in rear roof panel and the left side plate
F=29.106Hz
(g) The first-order bending vibration around Z-axis
F=30.773Hz
(f) Torsional mode along the Y-axis,
and local mode in roof panel
F=30.773Hz
(h) The mode of rear end of floorboard
F=35.680Hz
Fig.12 The first eight mode shapes
In the modal test, there were many local modes described in Tab.2.
Tab.2 Local modes of cab
Frequency
On the top of
side plate
In the right
In the
rear of
middle of
floorboard floorboard
In the rear
In the left of
In front of
of roof
rear side
roof skylight
skylight
plate
F1(Hz)
25.0340
35.6798
47.3361
28.0706
77.9323
46.7037
F2(Hz)
29.1046
39.2777
――
29.1046
――
47.3361
F3(Hz)
46.7037
59.4272
――
30.7763
――
――
F4(Hz)
50.7271.
63.2192
－－
32.9312
――
――
F5(Hz)
－－
67.0009
－－
40.9036
――
――
F6(Hz)
－－
－－
――
47.3361
――
――
F7(Hz)
－－
－－
54.4701
――
――
－－
4.2 Establishment of the finite element model
Building the finite element model of the truck cab to do modal simulation, the simulating
results are compared with the test. In the process, we should pay attention to the following
points[40-41]:
(1) To save time and cost of analysis, the model should be simplified with deleting the fillets
less than 10mm in diameter, small abdicating steps and transition fillets.
(2) The truck body is made of sheet metal by stamping and spot welding, and its thickness is
relatively thin, so shell element in size of 20-30mm is chosen to discrete the cab. In meshing
process, we can control the element quality by aspect ratio, warpage, skew and jacobian. The finite
model consists of quadrilateral elements in 90%.
(3) The truck cab is assembled together by welding, bolts, riveting and bonding, using the
node coupling method to simulate welding spot, beam elements to simulate the bolts, rigid
elements to simulate welding seam and the spring elements to simulate the bonding
(4) The cab material is low carbon steel, the elasticity modulus is 2.1×105MPa, the ratio of
poisson is 0.3 and density is 7.8×103Kg/m3, adopting isotropic material in simulation.
The finite element model of truck cab shown in Fig.13 includes 211 026 elements and
223 851 nodes. There is 203 212 shell elements, which contain 192 345 quadrilateral elements and
10 867 triangular elements. There are 2152 connecting elements, among them 1389 are rigid
elements, 413 are beam elements and 123 are spring elements.
Fig.13 Finite element model of truck cab
4.3 Comparison analysis between simulation and test results
4.3.1 Results contrast
The simulation and test results were analyzed, as shown in Tab.3.
Tab.3 Comparison between simulation and test modal results
Modes description
The order
Simulation
results (Hz)
Test results
(Hz)
Relative
error (%)
1
16.135
16.499
2.21
The first torsional mode along X-axis
2
17.324
17.483
0.91
Swing and torsion mode along X-axis
3
24.778
25.030
1.01
Local mode in rear side plate
4
26.897
28.076
4.12
Local mode in rear roof panel
5
28.010
29.106
3.76
6
29.512
30.773
4.10
7
33.821
32.931
2.70
8
35.114
35.680
1.59
FEM
Test
Torsional mode along Y-axis, local
modes in rear roof panel and the left
side plate
Torsional mode along the Y-axis, local
mode in roof panel
The first-order bending vibration around
Z-axis
The mode of rear end of floorboard
From Tab.3, the relative error between simulation and experiment results is less than5%, so
the simulation modes are consistent with the test measured. Therefore, the results of simulation
and test are reliable.
4.3.2 Improvements of cab
The low-order modes less than 100Hz of the truck cab are gained in test and simulation.
Combined with the actual condition, some suggestions for improvement are proposed as follows.
Fig.14 shows the local mode in left center of rear side along x-direction. We suggest that the
1 position pointed by arrow in rear part should be stamped into the 2 position pointed by arrow, or
glued, to prevent vibration and noise.
From Fig.15, the vertical beams in both side of roof also exist the local mode, so a bond
between the vertical beams and the roof sides using the foam .
Owing to substandard quality of welding or lack of local stiffness, the right rear part of the
floorboard (the 2 location pointed by arrow in Fig.16) also exists local modal. The position 1 in
left front floor has a elliptical hole compared with the right symmetrical place, and it will result in
insufficient stiffness and local modal, so the floorboard around hole should be stamped into flute
type.
According to Fig.17, the local mode is intensive in rear roof and roof beam, and its frequency
is relatively low, so it is easy to generate modal coupling with passenger compartment and cause
the resonate noise. The roof pointed by arrow should be stamped into wavy stripes along X-axis or
glue; In addition, the local mode is relatively obvious in the rear of floor, and it may be related to
the large opening at the bottom, so it should pay more attention to improvements.
Fig.14 The rear wall of cab
Fig.16 Floorboard of cab
Fig.15 Side panels of cab
Fig.17 Roof of cab
5 Conclusions
The dynamic characteristics of a truck cab are identified using experiments and simulation
methods in this paper. Modal test scheme is constructed and the finite element model of the cab is
established. Below 100Hz, 28 steps mode are extracted and the test and FEM simulation results
are consistent according to comparison. The following observation can be made from the study:
1. The natural frequencies and modal shapes obtained from experiment and simulation are
almost close to each other, so the modal test plan and simulation model are reliable, and the paper
provides the important design basis to improve truck cab.
2. The first frequency of truck body should not be equal or close to the external excitation
frequencies. The external excitations mainly come from road, engine, transmission and wheel etc,
and among these incentives, the effect of engine is greatest. Based on the formula of excitation
frequency[42], the engine idling rotational frequency of truck is about 12.5 Hz. To avoid resonance,
the first modal frequency should be different from the excitation frequency at least 2 Hz, so it
should be larger than 14.5 Hz. Shown in Tab.1, the first-order frequency is 16.5 Hz, so the truck
won’t resonate and the truck body structure is reasonable.
3. The first torsional mode and the first bending mode have the greatest impact on the truck
body, therefore, in order to avoid coupling, the difference between them must be more than 3 Hz.
From Tab.1, the first torsional vibration frequency is 16.5 Hz, and the first bending frequency is
32.9 Hz, without coupling.
4. In the process of test and simulation, there are many local modes, and they are most likely
caused by unreasonable local structure, low-quality welding or insufficient local stiffness, such as
in left center of rear side along x-direction, in the vertical beams, in both side of roof, on the the
right rear part of the floorboard etc. Those local modes tend to generate modal coupling with
passenger compartment and cause the resonate noise, so those positions should be paid more
attention in the future improvements.
Acknowledge
The authors owe a special thank you to Song Shaofang for her suggestions and insights on
test. This work was supported by Natural Science Foundation, under the Grants 51175320.
Corresponding author
*Wang Yansong, Automotive Engineering College, Shanghai University of Engineering
Science, NO 333, Longteng Road, Songjiang District, Shanghai, China.
E-mail: [email protected]
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