ONE TO ONE RESONANCE PHENOMENON ON A NONLINEAR
QUADRANGLE CANTILEVER BEAM
Myoung-Gu Kim Changdu Cho and Heung-Shik Lee
Inha University Mechanical Engineering
iasmeLab(2N374B) Department of Mechanical Engineering, Inha University, 253, Yonghyun-Dong,
Nam-Ku, Incheon city, South Korea.
ABSTRACT
The response characteristics of one to one resonance on the quadrangle cantilever beam in which basic harmonic excitations
are applied by nonlinear coupled differential-integral equations are studied. These equations have 3-dimensional non-linearity
of nonlinear inertia and nonlinear curvature. Galerkin and multi scale methods are used for theoretical approach to one-to-one
internal resonance. Nonlinear response characteristics of 1st, 2nd modes are measured from the experiment for basic
harmonic excitation. From the experimental result, geometrical terms of nonlinearity display light spring effect and these terms
play an important role in the response characteristics of low frequency modes. Dynamic behaviors in the out of plane are also
studied.
Introduction
The non-linear dynamics has been studied to analyze the dynamic phenomena in flexible structures such as robotics or in
large space structures. Many non-linear dynamic response phenomena, such as modal coupling and saturation phenomena,
may occur in mechanical systems composed, in part, of flexible elements. When external forces with the above properties are
applied to the large space structures with high flexibility, these non-linear phenomena can occur easily and it’s hard to avoid
them systematically when they really occur. Thus, the non-linear phenomena should be predicted theoretically and proved in
the structures that resemble the beams showing elastic motion. When theoretical models can show and analyze all the
dynamic phenomena that can occur in flexible structures consistently, it is possible to design and construct safe and credited
flexible structures.
Haight and King [1] were among the first researchers who attempted experimentally to verify the response characteristics of
the structures using the results of their non-linear elastodynamic analyses. They studied the flexural-flexural single mode nonlinear response of a cantilever beam subjected to base harmonic excitation. They based this study on a set of differential
equations, where the non-linear contributions from both torsion and curvature were neglected a priori and poor correlation was
obtained between theory and experiment. Haddow, Barr and Mook [3] performed an experimental and theoretical investigation
of the primary resonances of a two-degree-of-freedom structure possessing an internal resonance. They obtained qualitative
agreement between their experimental results and the theoretical predictions of a perturbation based on the method of multiple
time scales. Nayfeh [4], meanwhile, has addressed the non-linear response of a cantilever under the action of a parameter
excitation both experimentally and analytically using the method of multiple time scales. Crespo da Silva [7] studied one-to-one
non-linear coupling, the characteristics of the dynamic response of cantilever beams excited by a periodic transverse base
displacement, both experimentally and analytically. He obtained excellent agreement between the experimental results and the
analytical predictions based on a set of differential equations considering the non-linear effects of curvature and inertia. In this
paper, the planar motion and the non-linear motion of a beam that occur in the one-to-one resonance of cantilever beams of
square cross-section excited by a transverse base harmonic oscillation are compared and investigated.
Nonlinear phenomena of a beam in a one to one resonance
One-to-one resonant excitation cause the in-plane and out-of-plane motion in a beam, whose frequency in torsional mode is
higher than in bending mode, excited by a sinusoidal transverse base motion. The length of a beam excited by the in-plane
motion is L, the mass per unit length is m, and the bending stiffness are Dη , Dς . The non-linear equations of motion are given
2
by the following integro-partial differential equations with the external excitation Qv ( s, t ) = eΩ cos(Ωt ) [7], [11].
⎧
v + cv + β y v ′′′′ = ⎨(1 − β y ) ⎡ w′′ ∫ v′′w′′ds − w′′′ ∫ v′′w′ds ⎤ −
s
⎣
⎩
s
1
⎦
0
{
(1 − β y )
βy
{
}
2
(
w′′ ∫
s
0
}
′
s
s
′ 1
2
2
− β y v ′ ( v′v ′′ + w′w′′ )′ − v′∫ ⎡ ∫ (v′ + w′ )ds ⎤ ds
1 ⎣ 0
⎦
2
ii
∫
s
1
′
′⎫
′′
′′
v w dsds ⎬
)⎭
+ eΩ cos(Ωt ),
2
2
′
(1 − β y )
⎧
s
s
s
s
′⎫
⎡
⎤
w + cw + β y w = − ⎨(1 − β y ) v′′ ∫ v′′w′′ds − v′′′∫ v′w′′ds +
v′′ ∫ ∫ v′′w′′dsds ⎬
0
0 1
⎣ 1
⎦
βy
⎩
⎭
(
''''
{
{
}
(1a)
)
}
′
(1b)
s
s
′ 1
2
2
− w′ ( v ′v′′ + w′w′′ )′ −
w′∫ ⎡ ∫ (v′ + w′ )ds ⎤ ds .
1 ⎣ 0
⎦
2
ii
The boundary conditions for equations (1a, b) are
v (0, t ) = w(0, t ) = v′(0 , t ) = w′(0 , t ) = 0 ,
v′′(1 , t ) = w′′(1 , t ) == v′′′(1 , t ) = w′′′(1 , t ) = 0 .
Note that e in equation (1a) is the amplitude of the excitation normalized by the length of the beam, and that t denotes nondimensionalized time. Also s denotes arc-length along the beam, normalized by L,
(
( )
i
= ∂(
) / ∂t
and
( )′ = ∂ ( ) / ∂s .
)
When the above equations are formulated, the inertial ( xˆ , yˆ , zˆ ) and the local ξˆ, ηˆ , ςˆ unit vector triads are used. The base
harmonic motion is directed along the y axis and the quantities v( s, t ) and w( s, t ) denote elastic deflections of the beam’s midline in the ŷ and ẑ directions, respectively, normalized by the length of the beam. For the symmetric beams, equations (1a,
b) were used to analyze non-linear one-to-one resonance between a planar and a non-planar bending mode for the beams for
which Ω ≈ ω and β y ≈ 1 , where β y is the ratio between the in-plane and out-of-plane bending stiffness and ω is the
undamped natural frequency of the w-motion. The in-plane and out-of-plane deflection variables
v ( s, t ) and w( s, t ) are
expanded in power series for α.
α ( s, t0 , t 2 ; ε ) = εα 1 ( s, t0 , t 2 ) = ε α 3 ( s, t0 , t 2 ) +
3
.
α 1 ( s, t0 , t 2 ) = F ( s ) Aα (t 2 ) cos [ω t 0 + Bα (t 2 ) ] .
ti = ε t (i = 0 and 2) .
i
With the above equations in power series involving a perturbation parameter ε, the following conditions for elimination of
secular terms in the perturbation analysis are obtained.
2α 5 Av + ε c2ω Av − α 5ε Av Aw sin ψ − ε f3 sin φ = 0,
2
2
2
2
(2a)
2α 5 Aw + ε c2ω Aw + α 5ε Av Aw sin ψ = 0,
2
2
2
(2b)
α 5 Av [ −2φ + ε Av (1 − α 6 ) + ε Aw (cosψ − α 6 )] + 2ω ε Av (σ 2 − ∆ 2 ) + ε f3 cos φ = 0,
2
2
⎡
2ω
⎣
α5
Aw ⎢ −2φ + ψ +
2
2
2
2
2
2
2
(2c)
⎤
ε σ 2 + ε Aw (1 − α 6 ) + ε Av (cos ψ − α 6 ) ⎥ = 0.
2
2
2
2
2
⎦
(2d)
In the above equations,
f = eΩ
2
∫
1
F ( s )ds ≈ eω
0
2
∫
1
0
ε f3 ,
3
F ( s )ds
(3a)
α 3 = ∫ F ( s ){F ′( s )[ F ′( s ) F ′′( s )]′ }′ds,
1
(3b)
0
α 4 = ∫ F ( s ) ⎡ F ′( s ) ∫
1
⎣
0
ω α4
2
α5 =
s
1
−
2
α3
∫
s
′
2
F ′ ( s )dsds ⎤ ds ,
⎦
0
α6 =
,
4
ψ = 2( Bw − Bv ),
α3
2α 5
(3c)
,
φ =ε σ 2ω t − Bv ,
(3d, e)
2
,
F ( s ) = cosh( rs ) − cos( rs ) − [(cosh r + cos r ) /(sinh r + sin r )][sinh( rs ) − sin( rs )].
(3f, g)
(3h)
The eigenfunction F ( s ) satisfies the differential equation F ′′′′ − r F = 0, with F (0) = F ′(0) = F ′′(1) = F ′′′(1) = 0 , and r is
4
2
3
determined from the characteristic equation 1 + (cosh r ) cos r = 0 . The parameters ε c2 and ε f3 appear in equations (2a-d) as
a result of letting c = ε 2 c2 and f = ε 3 f3 . There are the effects of the small damping and of the resonant excitation when
frequency Ω is near ω = r 2 . The quantities ε 2 ∆ 2 and ε 2σ 2 are detuning parameters defined by equations (4a, b).
β y = 1 + ε 2∆2 +
,
(4a)
Ω =ω (1+ε σ 2 ).
2
(4b)
The numerical values of α 3 through α 6 for the first and second mode of a cantilever beam is given in reference [7]. Two
equilibrium solutions Aα = constant
≈ Aα e are obtained by equations (2a-d).
c2ω Ave − f3 sin φe = 0, ,
Awe = 0,
α 5 Ave (1 − α 6 ) + 2ω Ave (σ 2 − ∆ 2 ) + f3 cos φe = 0,
3
2
c2ω Ave − α 5 Ave Awe sin ψ e − f3 sin φe = 0,
2
Awe [c2ω + α 5 Ave sin ψ e ] = 0,
2
2
α 5 Ave [ Ave (1 − α 6 ) + Awe (cos ψ e − α 6 )] + 2ω Ave (σ 2 − ∆ 2 ) + f3 cos φe = 0,
2
2
2
⎡ 2ω
⎤
2
2
⎢ α σ 2 + Awe (1 − α 6 ) + Ave (cos ψ e − α 6 ) ⎥ = 0. ,
⎣ 5
⎦
(5a, b)
(5c)
(6a)
(6b)
(6c)
2
2
Awe
(6d)
An equilibrium solution to equations (2a-d) coincides with a periodic approximate solution to equations (1a, b). The stability of
motions of a cantilever beam can be determined by perturbing the equilibrium state as x(t ) = xe + xs (t ) , where x = Av , Aw ,ψ
and φ , and using the Routh-Hurwitz stability criterion on the linearized form of equations (2a-d). Since the conditions for
elimination of secular terms for the excitation modes the natural frequencies of which are not near the excitation frequency Ω
are the same as equations (2a-d) with f3 = 0 , these equations indicate that the amplitudes of such modes decay to 0 because
of the damping in the system. For f3 ≠ 0 , the steady state motion corresponding to the equilibrium solution governed by
equations (5a-c) consists of only a planar component: that is, the beam responds in the plane where it is excited. The effects
of the non-linear inertia and curvature are represented by the terms related to the coefficients α 5 and α 6 . Depending on the
mode being excited, the amplitude-frequency response can have either hardening or softening characteristics. When the
excitation frequency is near the frequency of the first mode, the effects of the non-linear inertia and curvature practically
disappear and the response of the beam becomes nearly linear. The steady state motion defined by equations (6a-d) is given
ε Ave and ε Awe by the non-linear phenomenon. The response of the beam is non-planar along the z axis when the beam is
excited along the inertial y axis. Since the in-plane and out of plane natural frequencies of a beam for which β y ≈ 1 are close
to one another, some of the energy supplied to the beam through the planar excitation is transferred to the non-planar motion
due to non-linear phenomenon, which is usually called a one to one internal resonance.
(a) in-plane motion of the first mode
(b) out of plane motion of the first mode
Figure 1. The analytical amplitude frequency response of the first mode for ε f = 0.0466 (e = 0.61mm ) ,
3
*
*
e 2 c2 = 0.0625 ,
ε ∆ 2 = 0.05 . (－ stable motion, … unstable motion)
2
(a) in-plane motion of the second mode
(b) out of plane motion of the second mode
Figure 2. The analytical amplitude frequency response of the second mode for ε f = 0.0472 , (e = 0.13mm ) ,
3
*
*
e c2 = 0.0936 , ε ∆ 2 = 0.05 . (－ stable motion, … unstable motion)
2
2
Analysis of the nonlinear variables for the cantilever beam
To find the correlation between the experimental results and the analytical predictions, the viscous damping coefficient c in
equations (1a, b) and the natural frequency ω for each mode should be determined by the experiments. The damping
coefficient c of a cantilever beam was measured using the logarithmic decrement method. The natural frequency for each
mode of the beam was measured using Pseudo-random signal as an excitation function, and using FRF method after selecting
rectangular weighting function as a time weighting function.
This technique was used to determine the time normalization factor
for each beam and the three natural
4
Dη /( mL )
frequencies for the planar motion of a cantilever beam can be represented as in Table 1, where r1 = 1.875 , r2 = 4.694 . The
measured damping coefficients were affected by many factors. These coefficients showed a little sensitive response to the
location of the accelerometers along the span of the beams. Thus, care was taken to have the accelerometers placed in the
same locations when the damping coefficients were measured. The effects of the base stiffness on ε 3 f3 for the first, second
2
mode of a cantilever beam in case of Ω ≈ ω = r for each mode are given in Table 2. The value of k z for the shaker was
3
*
obtained from the linear range of the test results. For this, k z was varied until the theoretical curves generated for ε f3 ,
3
instead of ε f3 , agreed closely with the “essentially linear” portion of the solid curve. Thus, the test results obtained from the
second mode responses of the beam were used to determine the stiffness of the torsional base spring.
Table 1. Measured eigenfrequencies and calculation of the damping coefficient
Mode
Meas
(Hz)
D
=
η
Theory
(non-dimensional)
4
mL
2πω
c = ε 2 c2
meas
ω
meas
(rad/s)
1
2
8.75
55.25
3.5160
22.0346
15.63
15.75
0.06250
0.09375
Table 2. Comparison of the experimental and analytical values for frequency of the bending mode
Frequency order no
Analytical values(Hz)
Experimental
values(Hz)
Damping coefficient(c)
1
2
9.664
56.8189
8.75
55.25
0.06250
0.09375
Measurement Procedure for the non-linear response
Aluminum(AL 6063), a uniform elastic material, was used for a cantilever beam of square cross-section. The following are the
dimensions of AL 6063; the elastic coefficient E=69Gpa, the stiffness coefficient G=25Gpa, Poisson’s ratio ν = 0.3333 , and
length per unit mass of square cross-section beam m = 70 × 10 −3 kg/m. The square cross-section of the beam was 5 × 5 mm, the
length was 665mm and the accelerometers were fixed at s = 125 mm. Also, since the aluminum alloy is a ductile material, it had
structural damping slightly. The cantilever beam was excited using sine sweep for the base harmonic excitation. The jig that
was used to fix the shaker amateur and the beam was made of AL 2024, which had lightweight and excellent mechanical
properties. The shape of the jig was designed and manufactured to satisfy the boundary conditions of a cantilever beam and to
be suitable for taking the transverse excitation. The natural frequencies of the jig were measured with FRF method and
confirmed not to exist along the x, y, and z-axis within 200Hz. The natural frequency of the cantilever beam was calculated
analytically using its dimensions and the boundary conditions. The beam was excited by the shaker to determine the natural
frequency of the beam experimentally [2], [12].
Figure 3. Instrument for experiential device
Figure 4. The one to one resonance in first mode and second mode of the nonlinear quadrangle cantilever beam.
A flexible square cross-section beam was fixed to satisfy the fix-free boundary conditions. The first, second, and third mode of
a square cross-section beam were all measured. The measurement was completed using the B&K 4374 accelerometers, each
of which was attached along the planar direction (the x axis) and the non-planar direction (the z axis), respectively, to find the
response characteristics of the beam precisely. Each of B&K 4374 accelerometers had lightweight of about 0.65grams and the
measurement range of frequency was 1-25kHz. The accelerometers were attached to the surface of the beam at the height of
125mm from the fixed base using a strong adhesive. The signals from the accelerometers of Charge type were transformed
into voltage signals by the Charge Amp(B&K 2635) and then stored in SONY DAT. When the voltage supplied for the shaker
was kept constant, the speed element in the harmonic oscillation generated by the shaker was constant regardless of the
change of the excitation frequency. The excitation frequency, whose type was sine sweep, was used to increase of reduce the
magnitude of the frequency, and its rate of change was 0.03Hz/s. The measurement range of the frequency was set to be
within 3Hz from each resonant frequency. Thus, the measurement time in both increasing and decreasing the excitation
frequency was 1 minute and 40 seconds. The signals stored in DAT were sent to the level recorder (B&K 2307) and the
voltage level of the signal was recorded on the record paper. The transmission rate of the paper was 1mm/sec and the moving
rate of the pen was 251mm/sec. The values recorded on the paper were transformed into RMS level of vibration. Using the
graph recorded on this paper, the non-linear dynamic response was analyzed.
The graph shown in Figure 5, which was drawn by the level recorder, shows the in-plane and out of plane response: the upper
curve indicates the in-plane motion and the lower curve indicates the out of plane motion. It implies that the vibration energy is
transferred to the out of plane to cause the out of plane motion when nonlinear phenomena occur in the in-plane motion. Thus
it can be seen that the vibration energy in the in-plane motion decreases when the out of plane motion occurs. The graph in (a)
was drawn while the excitation frequency was increasing from 7.25Hz to 10.25Hz, when the increase rate of the excitation
frequency was 0.030Hz/sec. The graph in (b) was drawn while the frequency was decreasing. The nonlinear response of the
beam is seen in those two graphs. Figure 6 shows the nonlinear phenomena in the second mode and tells that the vibration
energy in the in-plane motion is most actively transferred to the out of plane motion and that the in-plane motion response
never decays even when the response amplitude of the out of plane motion increases. This means that the one to one
resonant excitation occur most actively in the second mode. Also, it can be seen that the jump phenomenon of the softening
spring occurred in the in-plane motion and that the jump phenomenon of the hardening spring occurred in the out of plane
motion. Also, it can be seen that those jump phenomena occurred in both increasing and decreasing the excitation frequency.
(a)Response curve for forward direction
(b) Response curve for backward direction
Figure 5. Frequency response curve for in-plane and out of plane response curve on the first mode
(a)Response curve for forward direction
(b) Response curve for backward direction
Figure 6. Frequency response curve for in-plane and out of plane response curve on the second mode
Conclusions
The response of a cantilever beam with non-linear characteristics of one to one resonant excitation was investigated both
experimentally and theoretically and the results from those two cases were compared and analyzed. In analysis of the motion
of a beam, Crespo da Silva’s motion equations of beams and method of analysis was used. In the experimental investigation,
we applied the experimental variables found in the first and second mode of a cantilever beam to the theoretical equations and
performed the theoretical analysis. The characteristics of the first mode of a cantilever beam with one to one resonant
excitation obtained both in experiment and theory was almost same as each another quantitatively. However, there was a
slight difference between the experiment and the theory qualitatively. It seems to be a cause that the experiment showed lower
response characteristics than the theory because of the damping effects in the system. It can be seen that in the first mode the
response of one to one resonant excitation was affected more by the effects of non-linear spring than by the effects of nonlinear inertia.
Acknowledgments
This work was supported by BK21 Project in Korea.
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