Free vibration studies of classical beams/
rods with lumped masses at boundaries
using an approach based on wave vibration
C. Mei
Department of Mechanical Engineering, The University of Michigan – Dearborn, 4901
Evergreen Road, Dearborn, MI 48128, USA
E-mail: [email protected]
Abstract Vibrations in beams/rods with lumped masses at boundaries are normally not covered
in introductory vibration textbooks. An effort is made here to formulate a concise and systematic
approach to the study of the relatively complex vibration problems pertaining to such structures. An
exact analytical solution is obtained for bending, torsional, and longitudinal vibration analysis based
on a wave vibration approach. The reflection matrices corresponding to bending, torsional, and
longitudinal incident vibration waves at the end mass are derived from classical vibration theories;
these are then assembled with propagation matrices and reflection matrices at classical boundaries to
form a systematic approach to the analysis of free vibration. Numerical examples are presented,
comparisons with results available in the literature are made and good agreements are reached.
Keywords
lumped mass; vibration; wave analysis
Introduction
Many engineering structures can be modeled as beams/rods with lumped end masses,
such as mast antenna structures and robot arms. However, vibrations in beams/rods
with lumped masses at boundaries are normally not covered in introductory vibration
textbooks. In this paper, bending, torsional, and longitudinal vibrations are analyzed
from a wave vibration standpoint, in which vibrations are described as waves
propagating along a uniform waveguide and being reflected and/or transmitted at
discontinuities [1–3].
The bending vibrations of such structures have been studied by researchers. In a
simplified model, the end mass has been modeled as a dimensionless point attachment
[4]. In an extended model, the dimension of the end mass has been included [5]. The
axial vibrations of such structures have also been considered [6]. In this paper, both
the dimension and the rotary inertia of the end mass are taken into account in bending,
torsional, and longitudinal vibration analysis. The analysis procedure is concise and
systematic. It is hoped that this paper offers a simple approach to analyzing a relatively complex vibration problem in such distributed engineering structures.
This paper is organized as follows. In the following section, the equations of
motion are presented. Next, the propagation and reflection relations of various wave
components are obtained. The wave approach is then applied for free vibration
analysis of a beam with a lumped end mass, and numerical examples are given, with
comparisons to results available in the literature. Conclusions are drawn in the last
section.
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Free vibration studies of beams/rods
Fig. 1
257
A uniform beam with lumped mass end attachment.
Equations of motion and wave propagation
Consider a uniform beam element lying along the x-axis, as shown in Fig. 1. When
applying classical beam/rod-related theories, the equations of motion for bending,
longitudinal, and torsional vibrations of the uniform beam/rod are [7]:
EI z
∂ 4 y( x, t )
∂ 2 y( x, t )
+ ρA
= q( x , t )
4
∂x
∂t 2
(1)
ρA
∂ 2u( x, t )
∂ 2u( x, t )
−
EA
= p( x, t )
∂t 2
∂x 2
(2)
∂ 2θ ( x, t )
∂ 2θ ( x, t )
− GI x
= τ ( x, t )
2
∂t
∂x 2
(3)
Jx
where x is the position along the beam axis; t the time; A the cross-sectional area;
y(x, t), u(x, t), and θ(x, t) are the transverse, longitudinal, and torsional deflections
of the centerline of the beam/rod, respectively; q(x, t), p(x, t), and τ(x, t) are the
externally applied transverse forces, longitudinal forces, and torques, respectively;
E and G are Young’s modulus and the shear modulus; Iz, Ix, and Jx are the area
moment of inertia of cross-section about the z-axis, the area moment of inertia of
cross-section about the x-axis, and the polar mass moment of inertia about the x-axis
per unit length, respectively. ρ is the volume mass density of the beam/rod.
The shear force, V(x, t), bending moment, M(x, t), and longitudinal force, F(x, t),
and torque, T(x, t), at any section of the beam are related to the transverse deflection,
y(x, t), the bending slope, ψ(x, t), the longitudinal deflection, u(x, t), and the torsional
deflection, θ(x, t), by:
V = − EI z
where ψ =
∂3 y
∂ψ
∂u
∂θ
, M = EI z
, F = EA , T = GI x
3
∂x
∂x
∂x
∂x
(4)
∂y
according to the classical Euler–Bernoulli beam theory.
∂x
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258
C. Mei
Wave components and the propagation of waves
Wave components
Bending waves
First, consider the free bending vibration problem. When no external force is applied
to the beam, the differential equation of motion becomes:
EI z
∂ 4 y( x, t )
∂ 2 y( x, t )
+ ρA
=0
4
∂x
∂t 2
(5)
Assuming time harmonic motion and using separation of variables, the solution
to equation 5 can be written in the form y(x, t) = y0e−ikxeiω t, where ω is the frequency
and k the wavenumber. Substituting this into equation 5 gives a set of wavenumbers
that are functions of the frequency, ω, as well as of the properties of the structure:
k = ± 4 ρ Aω 2 EI z
(6)
The ± sign in equation 6 indicates the direction of propagation of the waves along
the beam.
With the time dependence, eiω t, suppressed, the solutions to equation 5 can be
written as:
y( x) = a1+ e − ik1x + a2+ e − k2 x + a1− eik1x + a2− e k2 x
(7)
where the bending wavenumbers
k1 = k2 = 4 ρ Aω 2 EI z
(8)
Longitudinal waves
Second, consider the free longitudinal vibration problem. When no external force is
applied to the beam, the differential equation for free longitudinal motion is:
ρA
∂ 2u( x, t )
∂ 2u( x, t )
− EA
=0
2
∂t
∂x 2
(9)
Again, assuming time harmonic motion and using separation of variables, the
solution to equation 9 can be written in the form u(x, t) = u0e−ikxeiω t, where ω is the
frequency and k the wavenumber. Substituting this into equation 9 gives the longitudinal wavenumber, which is also a function of the frequency, ω:
k=±
ρ
ω
E
(10)
Again, the ± sign indicates that longitudinal waves in the beam travel in both the
positive and negative directions.
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Free vibration studies of beams/rods
259
With the time dependence, eiω t, suppressed, the solutions to equation 9 can be
written as
u( x) = a3+ e −ik3 x + a3− eik3 x
(11)
where the longitudinal wavenumber
k3 =
ρ
ω
E
(12)
Torsional waves
Last, consider the free torsional vibration problem. When no external torque is
applied to the beam, the differential equation for free torsional motion is:
Jx
∂ 2θ ( x, t )
∂ 2θ ( x, t )
GI
−
=0
x
∂t 2
∂x 2
(13)
Again, assuming time harmonic motion and using separation of variables, the
solution to equation 13 can be written in the form θ(x, t) = θ0e−ikxeiω t, where ω is the
frequency and k the wavenumber. Substituting this into equation 13 gives the torsional wavenumber, which is also a function of the frequency ω:
k=±
Jx
ω
GI x
(14)
Again, the ± sign indicates that longitudinal waves in the beam travel in both the
positive and negative directions. For a uniform beam/rod, Jx = ρIx.
With the time dependence, eiω t, suppressed, the solutions to equation 13 can be
written as:
θ ( x) = a4+ e −ik4 x + a4− eik4 x
(15)
where the torsional wavenumber
k4 =
Jx
ω.
GI x
Propagation relations
Consider two points, A and B, on a uniform beam/rod a distance x apart, as shown
in Fig. 2. Waves propagate from one point to the other, with the propagation being
determined by the appropriate wavenumber. Denoting the positive- and negativegoing bending wave vectors at points A and B as A+b and A−b and B+b and Bb−, respectively, they are related by:
Ab− = fb ( x)Bb− ,
(16a)
Bb+ = fb ( x)Ab+
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260
C. Mei
a+
b+
a–
b–
x
point A
point B
Fig. 2 Wave propagation relation.
where
⎡e −ik1x
fb ( x) = ⎢
⎣ 0
0 ⎤
e ⎥⎦
− k2 x
(16b)
is the bending propagation matrix for a distance x and
⎡ a1+ ⎤
⎡ a1− ⎤
⎡b1+ ⎤
⎡b1− ⎤
Ab+ = ⎢ + ⎥ , Ab− = ⎢ − ⎥ , Bb+ = ⎢ + ⎥ , Bb− = ⎢ − ⎥
⎣ a2 ⎦
⎣ a2 ⎦
⎣b2 ⎦
⎣b2 ⎦
(16c)
Similarly, the positive- and negative-going longitudinal and torsional waves at
points A and B are related as:
a3− = fl ( x)b3− , b3+ = fl ( x)a3+
(17)
and
a4− = ft ( x)b4− , b4+ = ft ( x)a4+
(18)
where fl(x) and ft(x) are the longitudinal and torsional propagation matrices for a
distance x, respectively.
Reflections at boundaries with lumped masses
Waves incident upon a structural boundary are reflected. In this section, reflections
of bending, longitudinal, and torsional incident waves at boundaries with lumped
masses are obtained.
At a boundary, the incident waves, A+i (i = b, l, and t for bending, longitudinal,
and torsional waves, respectively), give rise to reflected waves, A−i, which are related
by:
Ai− = ri Ai+
(19)
The reflection matrix, ri, can be determined by considering the corresponding
equilibrium at the boundary.
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Free vibration studies of beams/rods
261
h
M
Lumped end mass
Beam
V
yM
y
uM
y
yM
u
Fig. 3 Free body diagram at the boundary with lumped mass attachment.
Reflection of bending waves
Fig. 3 shows the bending vibration related free body diagram (FBD) at the boundary
with the lumped mass attachment. From the FBD, the equations of motion at the
boundary are obtained as:
−V = mym , − M + V
h
= J m − xyψm
2
(20)
where m is the mass of the lumped mass, which is modeled as a rigid body. Jm−xy is
the mass moment of inertia of the lumped end mass about an axis normal to x and
h
∂y
y and passing its center of mass, ym = y + ψ m , and ψ m = ψ = .
2
∂x
Substituting equations 4 and 7 into equation 20 gives the relation between the
incident and reflected bending waves, which is written in matrix form as:
h ⎤
−
2 ⎥ ⎡ a1 ⎤
⎥⎢ −⎥
h a
−k22 EI z + k2 J mω 2 − k23 EI z ⎥ ⎣ 2 ⎦
2 ⎥⎦
h
⎡ −ik 3 EI − mω 2 + ik mω 2 h
k23 EI z − mω 2 + k2 mω 2 ⎤⎥ +
1
z
1
⎢
2
2 ⎡ a1 ⎤
=⎢
⎥⎢ +⎥
⎢ −k12 EI z + ik1 J mω 2 + ik13 EI z h k22 EI z + k2 J mω 2 − k23 EI z h ⎥ ⎣ a2 ⎦
⎢⎣
2
2 ⎥⎦
⎡ −ik 3 EI + mω 2 + ik mω 2
1
⎢ 1 z
⎢
⎢ k12 EI z + ik1 J mω 2 + ik13 EI z
⎢⎣
h
2
h
2
k23 EI z + mω 2 + k2 mω 2
(21)
The reflection matrix of incident bending waves is then obtained from equations
19 and 21 as:
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C. Mei
⎡ −ik 3 EI + mω 2 + ik mω 2
1
⎢ 1 z
rb = ⎢
⎢ k12 EI z + ik1J mω 2 + ik13 EI z
⎣
−1
h ⎤
2 ⎥
⎥
h
− k22 EI z + k2 J mω 2 − k23 EI z ⎥
2⎦
h
h
⎡ −ik 3 EI − mω 2 + ik mω 2
k23 EI z − mω 2 + k2 mω 2 ⎤⎥
z
1
1
⎢
2
2
×⎢
⎥
h
h
⎢ − k12 EI z + ik1J mω 2 + ik13 EI z
k22 EI z + k2 J mω 2 − k23 EI z ⎥
⎣
2⎦
2
h
2
h
2
k23 EI z + mω 2 + k2 mω 2
(22)
Reflection of longitudinal and torsional waves
Following a similar procedure, the reflection coefficients of longitudinal and
torsional waves incident upon a boundary with a lumped mass are obtained as
follows:
rl =
ik3 EA + mω 2
ik3 EA − mω 2
(23)
rt =
ik4GI x + J m − yzω 2
ik4GI x − J m − yzω 2
(24)
where Jm−yz is the mass moment of inertia of the lumped end mass about an axis
normal to y and z and passing its center of mass.
The reflection relations at classical boundaries can also be easily obtained, and
they are found to be as follows:
−1 − i ⎤
⎡ −i
rb−c = ⎢
, rl−c = −1, rt −c = −1
i ⎥⎦
⎣ −1 + i
⎡ −i 1 + i ⎤
,
rl −f = 1, rt −f = 1
rb−f = ⎢
i ⎥⎦
⎣1 − i
(25)
⎡ −1 0 ⎤
rb−s = ⎢
⎥
⎣ 0 −1⎦
for clamped, free, and simply supported boundary conditions, respectively.
Wave vibration analysis
With the propagation and reflection relations derived, free vibration analysis of
beams/rods with lumped end masses can be obtained by assembling these relations.
Fig. 4 shows a beam/rod structure with a clamped and a lumped mass attached
boundary at ends A and B, respectively. Now consider the free response of this
clamped beam/rod with end mass. Denoting the incident and reflected waves at
boundaries A and B as a−, a+, b+ and b− respectively, the relationships between the
different waves are given as:
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Free vibration studies of beams/rods
263
L
a+
b+
–
b–
a
Boundary A
Boundary B
Fig. 4
TABLE 1
Free vibration analysis of a clamped beam with end mass.
Correction factors for torsional vibrations of beams of rectangular
cross-sections [1]
l/s
(ωn)rectangular/(ωn)calculated
1
0.92
1.5
0.85
2
0.74
3
0.56
6
0.32
10
0.19
a+ = rA ab- = rB b+
(26)
a - = f ( L ) bb+ = f ( L ) a +
where, rA and rB are the reflection matrices at boundaries A and B respectively, and
f(L) is the propagation matrix between A and B, which are distance L apart.
Solving equation 26 gives:
[rA f (L )rB f (L ) − I ] a+ = 0
(27)
where I denotes the identity matrix. For non-trivial solution, it follows that:
rA f ( L )rB f ( L ) − I = 0
(28)
Equation 28 is the characteristic equation from which the natural frequencies of
the beam can be found. Once the natural frequencies are identified, the modeshapes
can be obtained from equation 27 by simply deleting one row of the coefficient
matrix and solving all the other wave components in terms of any chosen wave
component.
It should be pointed out that although the torsional equation was derived for a
shaft, or a rod, it can be used to approximate the torsional motion of other types of
cross-section quite accurately. Table 1 lists the correction coefficients for obtaining
natural frequencies of a rectangular beam (ωn)rectangular from the calculated natural
frequencies (ωn)calculated, where the lengths of the longer and shorter sides of the
rectangle are denoted l and s respectively.
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C. Mei
Numerical examples
Numerical results corresponding to a beam and a rod, that is, to rectangular and
circular cross-sections, respectively, are presented. The cross-section of the attached
end mass is assumed to be of the same type as that of the beam/rod, though it does
not necessarily have to be so. The end without a mass attachment is assumed to be
clamped, for the convenience of comparison with published results.
The material properties of the example steel beam and rod are as follows: Young’s
modulus E = 210 GN/m2 volume mass density ρ = 7850 kg/m3. The dimensions
of the beam are: height 1.27 × 10−2 m, depth 2.54 × 10−2 m, and length 0.9144 m.
For the rod, the radius of the cross-section is 0.05 m and the length is
6.00 m.
The mass of the lumped end mass is assumed to be equal to the total mass of
the beam/rod. The cross-sectional dimension of the lumped end mass is assumed
to be twice that of the beam/rod. The width, h, of the lumped end mass is
1.27 × 10−2 m and 0.05 m for the rectangular and circular end attachments,
respectively.
The natural frequencies of the example beam and rod are listed in Tables 2 and
3, respectively. The mode-shapes of the first 12 bending modes of the rod are shown
in Fig. 5.
For comparison, modal analysis was also performed using the commercially available finite element analysis (FEA) software package HyperWorks, from Altair.
Eigenvalue solution was carried out using the Lanczos method.
TABLE 2 Natural frequencies (up to 5000 Hz) of the example beam with a lumped end mass
Natural frequencies (Hz) obtained from
wave analysis
Natural frequencies (Hz) obtained
from finite element analysis
Bending
Bending
In plane
Out of plane
Torsion
Axial
In plane
Out of plane
Torsion
Axial
5.6
57.8
180.4
371.1
628.1
948.1
1326.7
1759.2
2244.2
2786.9
3396.8
4081.1
4841.9
11.3
115.6
360.8
742.2
1256.1
1896.1
2653.4
3518.4
4488.3
198.4
1330.0
2612.3
3904.8
774.6
3083.9
5.6
58.0
180.8
371.3
627.0
944.0
1317.1
1741.1
2213.2
2736.5
3317.8
3962.0
4669.6
11.2
115.2
355.4
717.4
1181.8
1725.5
2343.3
3059.4
3891.4
199.6
1339.2
2630.9
3933.7
776.2
3096.2
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Free vibration studies of beams/rods
TABLE 3
265
Natural frequencies (up to 1500 Hz) of the example rod with a lumped end
mass
Natural frequencies (Hz) obtained from wave
analysis
Natural frequencies (Hz) obtained
from finite element analysis
Bending
Torsion
Axial
Bending
Torsion
Axial
0.9
9.3
28.8
59.2
100.1
151.2
211.5
280.2
356.9
442.1
538.0
645.8
765.9
898.3
1042.7
1198.9
1366.7
40.9
273.9
538.0
804.2
1071.0
1337.9
118.1
470.0
883.2
1307.4
0.9
9.1
28.5
58.5
98.9
149.0
208.0
274.0
348.0
429.0
519.0
619.0
729.0
848.0
977.0
1110.0
1260.0
40.4
270.0
530.0
792.0
1050.0
1320.0
118.0
469.0
881.0
1300.0
Apart from the above verifications using the FEA software package, further comparisons are made below between the results obtained using the wave analysis
approach and those obtained using other approaches that are available in the
literature.
The natural frequencies of the torsional and longitudinal vibrations of a shaft with
one end clamped and the other end attached to a lumped mass can be found using
the formulas below [8]:
Torsional :
ωL
JxL
= α tan(α ), where α = n
J m − yz
G ρ
Longitudinal :
ρ AL
ωL
= β tan( β ), where β = n
m
E ρ
(29)
(30)
The torsional and longitudinal natural frequencies of the shaft determined numerically using formulas 29 and 30 are as follows (in Hz):
Torsional : 40.9, 273.9, 538.0, 804.2, 1070.9, 1337.9
Longitudinal : 118.1, 470.0, 883.2, 1307.4
Comparing these results with those in Table 3, it can be seen that the wave
analysis approach offers very accurate results.
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C. Mei
1
1
0
0
-1
-1
-2
2
0 1 2 3 4 5 6
0
-2
2
2
0 1 2 3 4 5 6
0 1 2 3 4 5 6
-2
2
-2
2
0 1 2 3 4 5 6
-2
2
0 1 2 3 4 5 6
-2
2
-2
0 1 2 3 4 5 6
0
0 1 2 3 4 5 6
-2
2
0
0 1 2 3 4 5 6
0 1 2 3 4 5 6
0
0
0
-2
2
0
0
0
-2
-2
2
0 1 2 3 4 5 6
0
0 1 2 3 4 5 6
x
-2
0 1 2 3 4 5 6
Fig. 5 Modeshapes of the first 12 bending modes of the rod.
ωn2 ρ A in Table 2 of reference
L
EI
[5] were obtained using the wave approach and are shown in Table 4. The natural
frequency values in the present study are obtained through a self-written program
in MATLAB environment. Though the accuracy can be up to the precision of the
computer in use, it is kept to be at the second decimal place (in general, higher
accuracy requires longer computing time). Excellent agreement has been observed.
The dimensionless bending natural frequencies
4
Conclusions
An exact wave-based analytical solution has been found for bending, torsional, and
longitudinal vibrations in beams with lumped masses at boundaries. The reflection
matrices corresponding to bending, torsional, and longitudinal incident vibration
waves at the end mass are derived from classical vibration theories. The propagation
and reflection matrices are assembled for free bending, torsional, and longitudinal
vibration analysis. Natural frequencies and modeshapes are obtained. Numerical
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1
2
3
4
5
0.93735
2.24496
5.02740
8.03827
11.12765
Ref. [5]
0.6
0.4
0.94
2.25
5.03
8.04
11.13
Wave
0.89457
2.08261
4.97253
8.00717
11.10656
Ref. [5]
0.8
0.4
0.89
2.08
4.97
8.01
11.11
Wave
0.6
0.85068
1.98013
4.94508
7.99216
11.09652
Ref. [5]
1.0
0.85
1.98
4.95
7.99
11.10
Wave
0.84425
2.15522
5.02002
8.04124
11.13224
Ref. [5]
0.8
(End mass gyration / beam length)
0.4
(0.5 [End mass width]) / (beam length)
0.84
2.16
5.02
8.04
11.13
Wave
0.81048
2.04543
4.97823
8.01500
11.11347
Ref. [5]
1.0
0.6
0.81
2.05
4.98
8.02
11.11
Wave
0.77280
2.10370
5.01576
8.04301
11.13504
Ref. [5]
1.0
0.8
0.77
2.10
5.02
8.04
11.14
Wave
TABLE 4 Dimensionless bending natural frequencies determined using the wave approach and those listed in reference [5] (where mass ratio
m/ρAL = 1)
Free vibration studies of beams/rods
267
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268
C. Mei
examples are given, and good agreements with available published results as well
as FEA results have been reached. It should be pointed out that the results obtained
in this paper are applicable to relatively low frequencies. When the effects of rotary
inertia and shear distortion become important, which normally happens at higher
frequencies, advanced theories (such as Timoshenko theory for bending vibrations)
have to be adopted.
Acknowledgements
The author gratefully acknowledges the support from the CMMI Division of the
NSF through grant #0825761.
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