Background and Objective In designing a high-performance mechanism for proper dynamic characteristics, its natural frequencies should be a key design factor because they influence its operating speed and the amplitudes and settling times of its vibrations. Natural frequencies of a planar mechanism system is influenced by the inertia and flexibiUty of the links and joints, link offsets, and components comprising the drive and load shafts. The geometry of a kinematically planar mechanism is essentially three dimension (3D) due to the inevitable presence of link offsets dictated by the task and geometric constraints. Link offsets increase the flexibility of the mechanism system, lowers its natural frequency and may cause out-of-plane vibrations to dominate. Furthermore, the inherent compliance and inertia of the drive and load train components such as the driving unit, couphngs, transmission mechanisms, and connecting shafts may influence linkage response by lowering its natural frequencies. Therefore, a reliable model of a planar mechanism should include the influence of all its components rather than its links only

A Smaili

I Bagci

CiteSeerX

A. Smaili 
Mechanical Engineering Department 
Tennessee Tectinolcgical University 
Cookeville, Tennessee, 38505 
ASmaili@tntecti,edu 
I. Bagci 
Fleetguard 
1200 Fleetguard Read 
Cookeville, TN 38506 
ibagcl@fleetguard.com 
Experimental and Finite Element 
Modal Analyses of Planar 
Mechanisms With Three-
Dimensional Geometries 
This article presents the results of numerical and experimental modal analyses of a 
planar four-bar (4R) mechanism having three dimensional geometry due to the offsets 
between its links. An experimental mechanism is built and modern modal testing 
techniques are used. A five-node isoparametric finite element is developed and used 
to model the mass and flexibility of the links and joints, link offsets, and drive and 
load shafts. The results obtained from the finite element solution matched closely the 
experimental results, a testimony to the accuracy of the proposed element. The results 
indicated that link offsets have considerable influence on the dynamic characteristics 
of a mechanism system and should be considered in dynamic modeling of planar 
mechanisms. 
Background and Objective 
In designing a high-performance mechanism for proper dy-
namic characteristics, its natural frequencies should be a key 
design factor because they influence its operating speed and the 
amplitudes and settling times of its vibrations. Natural frequen-
cies of a planar mechanism system is influenced by the inertia 
and flexibiUty of the links and joints, link offsets, and compo-
nents comprising the drive and load shafts. 
The geometry of a kinematically planar mechanism is essen-
tially three dimension (3D) due to the inevitable presence of 
link offsets dictated by the task and geometric constraints. Link 
offsets increase the flexibility of the mechanism system, lowers 
its natural frequency and may cause out-of-plane vibrations to 
dominate. Furthermore, the inherent compliance and inertia of 
the drive and load train components such as the driving unit, 
couphngs, transmission mechanisms, and connecting shafts may 
influence linkage response by lowering its natural frequencies. 
Therefore, a reliable model of a planar mechanism should in-
clude the influence of all its components rather than its links 
only. 
Natural frequencies of a mechanism system can be estimated 
numerically by finite element techniques or experimentally by 
modal testing. Finite element studies available in the literature 
on the dynamics of planar mechanisms using finite element 
techniques dealt primarily with the effect of mass and flexibiUty 
of the links and joints of planar mechanisms and spatial mecha-
nisms. A comprehensive list of references on the subject can 
be found in (Smaili and Bagci, 1996). Only a few experiments 
on mechanism dynamics have been reported in the literature, 
mostly on mechanisms with 2D geometries (Turcic et al , 1984; 
Masurekar and Gupta, 1987; Najarajan and Turcic, 1992; Liou 
and Erdman, 1989; Liou and Peng, 1993; Thompson and Sung, 
1984). Stamps and Bagci (1985) conducted finite element and 
experimental investigations using strain gauges to estimate the 
fundamental natural frequency of a planar mechanism with 3D 
geometry. 
The objective of the current study is to perform modal analy-
sis of a planar 4R mechanism system with 3D geometry by 
finite element and experimental modal testing techniques. An 
experimental 4R mechanism is built for this purpose. A five-
node space-frame finite element is developed to model the links, 
joints, link offsets and drive and load shafts. Modern modal 
testing technology is utilized to perform the experiment. The 
results from both studies are compared to validate the accuracy 
of the finite element results and affirm the importance of model-
ing the mechanism as a system. 
The 5-Node Isoparametric Element 
The study conducted by Smaili et al. (1995) on the effective-
ness of various Timoshenko elements in predicting modal re-
sponse of structures and mechanisms proved that the five-node 
element is simple, accurate, and robust when compared with 
other available elements. Therefore, it will be used in the current 
investigation. Figure 1 shows the 5-node isoparametric space 
frame element and its nodal distribution in terms of spatial and 
natural coordinates, x and 77. The local frame 2^ ( X ^ ^ A ) origi-
nates at the initial node (node 1). The Xe axis represents the 
centroidal axis of a member in its undeformed position. A point 
P along the element has 6 displacements, 3 translations—u, v, 
and w along i\\s x,y, and z axes, and 3 rotations—6^, dy, and 
6^ about iha x,y, and z axes. 
Due to space limitations, details on how the element matrices 
are formulated will be omitted here. Interested readers may refer 
to the reference (Smaili and Bagci, 1996) for details. The final 
form of the mass and stiffness matrices are only given here. 
Using Gauss Quadrature, the mass matrix [m«] and the stiffness 
matrix [m^] of a spatial five node-element e take the following 
forms: 
[mA = p\J\ i Rj[H{nj)Y{Q{r]jmH{^i)^ ( D 
and 
{kA = Ul S RAB{r)j)V[P{'n,mB{r^i)] (2) 
Contributed by the Mechanistns Committee for publication in tlie JOURNAL OF 
MECHANICAL DESIGN . Manuscript received May 1996. Revised May 1998. Associ-
ate Teclinical Editor: K. Kazerounian. 
where n represents the number of Gauss sampling points and 
Rj is the yth weighting factor. | 7 | is the determinant of the 
Jacobian of the element. Matrix [//(r/^)] contains the shape 
functions that relate the displacements at a point on an element 
to the element nodal displacements. Matrix [Q{rij)] is a diago-
nal matrix that contains A, A, A, 7,̂ , lyy, and f^, along its 
Journal of Mechanical Design Copyright © 1998 by ASME SEPTEMBER 1998, Vol. 120 / 401 
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•u„ ©.„ 
""1/ "^2/ ""s/ ""4/ ""s/ 
/ ^/ / 
^4 
/ 
25 
'1 = -/ I 0 
• 7 
^ 3 = " 
OC —zt — * Q 
Fig. 1 Nodal distribution and DOF of the five-node Isoparametric finite 
element 
Table 1 Minimum natural frequencies (Hz) and the corresponding cranio 
position 
oflTset length 
(mm) 
planar 
5.08 
31.75 
63.5 
101.6 
mode 1 
(02*) 
69.64 
(350) 
11.00 
(330) 
10.55 
(330) 
10.08 
(330) 
9.55 
(340) 
Mode 2 
(Q^") 
230.21 
(320) 
56.97 
(320) 
52.63 
(320) 
48.38 
(330) 
43.46 
(330) 
Modes 
(82') 
_ 
98.85 
(320) 
87.18 
(320) 
76.40 
(320) 
76.40 
(320) 
diagonal; matrix [B(??j)] relates the strains at a point on the 
element to the element nodal displacements; and matrix [P(?7;)] 
is a diagonal matrix that contains EA, Elyy, EI^, GAK^y, GAK^^, 
and EI:„ along its diagonal. A, / , and K are the area, area moment 
of inertia, and Timoshenko coefficient of the cross section. E, 
G, and p are the Young's modulus, shear modulus, and density 
of the material. 
Application Example 
A planar 4R mechanism with 3D geometry is built to experi-
mentally determine its natural frequencies by modal testing and 
compare them with those obtained by the 5-node element. The 
experimental 4R mechanism was the subject of the study by 
Stamps and Bagci (1985). It is being used here as the current 
case study for the following reasons: (1) The natural frequen-
cies found by Stamps and Bagci (1985) were the result of 
an experiment that employed strain-gauges. However, using 
modern modal testing hardware and software in the current 
investigation would certainly lead to a more accurate results 
and, in turn, help realize a better finite element model for the 
mechanism system. (2) The finite element and experimental 
results obtained by Stamps and Bagci (1985) did not match. In 
an effort to minimize the difference between the two solutions, 
the authors used two different schemes to model the joints of 
the 4R mechanism, revolute and spherical. The average values 
of the natural frequencies obtained from the two models were 
then compared with the experimental values. This approach, 
however intuitive may be, does not reflect the true physical 
nature of link connections. Consequently, the conclusions made 
by Stamps and Bagci (1985) deserves another look. (3) The 
finite element model used by Stamps and Bagci (1985) is based 
on Euler-Bemoulli thin beam theory and shear deformations and 
rotary inertia of the links were not accounted for. Furthermore, 
lumped mass system instead of consistent mass system was 
used. 
Experimental 4R Mechanism System. Figure 2 shows a 
schematic of the experimental 4R mechanism. The dimensions 
(mm) of this mechanism are IJ = 88.9, JK = 50.8, KA = 63.5, 
AB = 193.675, BC = 101.6 or 5.08, CD = 495.53, DE = 101.6 
or 5.08, EF = 419.1, GF = 101.6, and F / / = 25.4. Link offsets 
BC and DE vary between 5.08 and 101.6 mm. Each link is 
comprised of three segments. The cross section of the main link 
segments is 31.75 mm X 7.9375 mm and the cross section of 
the shaft-support-blocks is 31.75 mm X 25.4 mm. The links 
are made of aluminum having £ = 71 X 10' N/mm^, G = 26.2 
X 10' N/mm^ and p = 2.93 X 10"" kg/mm'. The offsets are 
made of steel having £ = 207 X 10' N/mm", G = 79.3 X 10' 
N/mm^ and p = 8.425 X 10"^ kg/mm'. 
A 3D and a 2D finite element models of the mechanism 
system were developed. Table 1 gives the minimum values of 
the frequencies for the first three modes of the 2D and 3D 
models with several offset dimensions and the crank positions 
at which they occur. The second frequency of the 3D model 
encompasses a frequency range which contains the first in-plane 
frequency. This is similar to the findings reported by Stamps 
and Bagci (1985). These results underscore the importance of 
including the effect of link offsets when modeling a planar 
mechanism. 
Experimental Modal Analysis. Modal testing of the 4R 
mechanism system was performed using impulse excita-
tion hammer (PCB208A03), a pair of accelerometers 
analytical vs. expenmental: mode 1 
analytical 0.2" 
- 0.0 0.0 ô o..?.̂ P.*riT]f̂ ?f̂ ,*tal ,0.2' 
. analytical 4.0" 
+ + + + +;+ experimental 4.0" 
Fig. 2 Schematic of the experimental planar 4R mechanism with 3D 
geometry 
Fig. 3 
100 150 200 250 
angular position (deg) 
Fundamental natural frequency of the experimental 4R mecha-
nism with linl( offsets: Experimental vs. finite element results 
402 / Vol. 120, SEPTEMBER 1998 Transactions of the ASME 
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100 
I" 70 
50,^ 
analytical vs. experimental; mode 2 
/ i o \ 
/ *  \ 
_ Z i ' " \ ' ^ / • x 
- ,/-+ t"'-
/ 
/ 
i* 
! -
i 
i 
1' 
••••1' ' ' 
i 
+ 
1 ' 1 
0 0 o O'O o axperiinental 0.2"-
+ + + + + + experimental 4.0 '̂ 
o o 0 " i 
1 1 
150 200 
angular position (deg) 
250 350 
Fig. 4 Second natural frequency of the experimental 4R mechanism 
with link offset: Experimental vs. finite element results 
Table 2 Comparison of experimental and finite element natural frequen-
cies (Hz) for 5.08 mm link offsets 
crank position mode 5-node element 
180° 
250° 
experimental 
13.12 
99.84 
125.84 
17.18 
69.07 
116.31 
13.58 
37.33 
108.11 
15.01 
96.10 
124,72 
18.22 
64.70 
113.69 
14.87 
32.51 
117.85 
(PCB353B67), Star Modal software, and a Data 6000 spectrum 
analyzer. The accelerometers were mounted at the midpoint of 
the follower link such that one accelerometer measured in-plane 
response and the other captured the response normal to the 
motion plane. The experimental mechanism was mounted on a 
12.7 mm aluminum plate which in turn was fastened to a rigid 
heavy steel frame to eliminate the effects of foundation flexibil-
ity on the response of the mechanism. The mechanism was 
excited by the impact hammer at fifteen different points to 
collect enough data for accurate characterization in the Star 
Modal software. The frequency response functions (FRF) were 
determined by the Data 6000 for 10 deg increments of CCW 
crank rotation for the 5.08 mm offset and for 30 deg increments 
of CCW rotation for the 101.6 mm offset. The data 6000 was 
set for 2048 points and a sampling rate of 500 //s for a frequency 
resolution of 1.0 Hz. 
The first three natural frequencies obtained from the experi-
ment on the 5.08 and 101.6 mm offsets are depicted in Figs. 
3-5 , along with the frequencies obtained from the 5-node finite 
element model. Table 2 compares the finite element and experi-
mental results for the 5.08 mm offset at 0 deg, 180 deg, and 
250 deg crank positions. Figures 3-5 and Table 2 indicate 
analytical vs. experimental: mode 3 
, _,_ I I analytical 0.2" 
j / \ o o O D O p experimental 0.2" 
]/ + ' \ analytical 4.0" 
+',+ + + + + experimental 4.0" 
0 50 100 150 200 250 300 350 
angular position (deg) 
Fig. 5 Third natural frequency of the experimental 4R mechanism with 
link offsets: Experimental vs. finite element results 
that the experimental and finite element results are in close 
agreement, a testimony to the accuracy of the proposed finite 
element model. 
Conclusions 
Numerical and experimental modal analyses of a planar 4R 
mechanism with 3D geometry were presented. An experimental 
mechanism was built and modem modal testing techniques were 
employed. A five-node isoparametric finite element was devel-
oped and used to model the mass and flexibility of the links and 
joints, link offsets and drive and load shafts of the experimental 
mechanism. The finite element and experimental results were 
in close agreement, a testimony to the accuracy of the proposed 
finite element. The results indicated that link offsets have con-
siderable influence on the dynamic characteristics of a mecha-
nism system and that a 2D model of a planar mechanism may 
not reveal its true response. 
References 
Liou, F. W., and Erdman, A. G., 1989, ' 'Analysis of High-Speed Flexible Four-
Bar Linkage: Part II-Analytical and Experimental Results on the Apollo," ASME 
JOURNAL OF VIBRATION, ACOUSTICS, STRESS, AND RELIABILITY IN DESIGN, Vol. 
I l l , pp. 42-47. 
Liou, F. W., and Peng, K. C , 1993, "Experimental Frequency Response Analy-
sis of Flexible Mechanisms," Mechanism and Machine Theory, Vol. 28, No. 1, 
pp. 73-81. 
Masurekar, V., and Gupta, D. A., 1987, "Theoretical and Experimental Kinet-
oelastodynamics Analysis of High-Speed Linkage," Mechanism and Machine 
Theory. Vol. 24, No. 1, pp. 325-334. 
Nagarajan, S., and Turcic, D. A., 1992, "Experimental Verification of Critical 
Speed Ranges for Elastic Closed Loop Linkage Systems," ASME JOURNAL OF 
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Proceedings of the 24"' Biennial Mechanisms Conference, Paper No. DETC/ 
MECH-1180. 
Stamps, F, R„ and Bagci, C , 1985, "Dynamics of Planar, Elastic, High-Speed 
Mechanisms Considering Three-Dimensional Offset Geometry: Analytical and 
Experimental Investigations," ASME JOURNAL OF MECHANICAL DESIGN, Vol. 
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Thompson, B. S., and Sung, C. K., 1984, "A Variational Formulation for the 
Nonlinear Finite Element Analysis of Flexible Linkages: Theory, Implementation, 
and Experimental Results," ASME JOURNAL OF MECHANISMS, TRANSMISSIONS, 
AND AUTOMATION IN DESIGN, Vol. 106, pp. 482-488. 
Turcic, D . A., Midha, A., and Bosnick, J. R., 1984, "Dynamic Analysis of 
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Journal of Mechanical Design SEPTEMBER 1998, Vol. 120 / 403 
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Experimental and Finite Element Modal Analyses of Planar Mechanisms With Three- Dimensional Geometries

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Experimental and Finite Element Modal Analyses of Planar Mechanisms With Three- Dimensional Geometries

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