- Email: [email protected]
Chaotic Behavior of Hydraulic Engine Mount
Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 96 (2016) 1597 – 1608
20th International Conference on Knowledge Based and Intelligent Information and Engineering Systems, KES2016, 5-7 September 2016, York, United Kingdom
Chaotic Behavior of Hydraulic Engine Mount Hormoz Marzbani, M. Fard, Reza Jazar * School of Engineering, RMIT University, Bundoora East Campus, Melbourne, VIC, 3083, Australia 0F
Abstract
The rubber materials that act as the main spring of a hydraulic mount obey nonlinear constitutive relationships. In addition to the material nonlinearity, further nonlinearities may be introduced by mount geometry, turbulent fluid behavior, boundary conditions, temperature, decoupler action, and hysteretic behavior. While all influence the behavior of the system only certain aspects are realistically considered using the lumped parameter approach employed in this research. The nonlinearities that are readily modeled by the lumped parameter approach constitute the geometry and constitutive relationship induced nonlinearity, including hysteretic behavior, noting that these properties all make an appearance in the load-deflection relationship for the mount and may be readily determined via experiment or finite element analysis. In this paper we will shoe that under certain conditions, the nonlinearities involved in the hydraulic engine mounts can show a chaotic response. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review underresponsibility responsibility KES International. Peer-review under ofof KES International Keywords: Hydraulic Engine Mount, Nonlinear Analysis, Vibration Isolation, Averaging Method
* Corresponding author. Tel.: +61400693496. E-mail address: [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of KES International doi:10.1016/j.procs.2016.08.207
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1. INTRODUCTION
The constitutive relationships of the rubber materials that act as the main spring of the hydraulic mount are nonlinear. In this investigation only certain aspects are realistically considered using the lumped parameter approach employed in this research. Turbulent fluid behavior is neglected as all flow is assumed laminar in nature for this modeling procedure. Boundary conditions are assumed constant such that the mount is perfectly mounted. Lastly, temperature-induced effects are neglected; however, the later assumption is validated noting that temperature effects have been shown to be negligible throughout a large spectrum of values by Kim and Singh [1]. The nonlinearities that are readily modeled by the lumped parameter approach constitute the geometry and constitutive relationship induced nonlinearity, including hysteretic behavior, noting that these properties all make an appearance in the load-deflection relationship for the mount and may be readily determined via simple experiment or finite element analysis. As an example, consider the load-deflection relationship illustrated in Figure 1. This load-deflection relationship is the result of a finite element analysis on the upper structure of the hydraulic mount. 2000 1500
Load [N]
1000 500 0 -500 -1000 -1500 -2000 -8
-6
-4
-2
0
2
4
6
8
10
Deflection [mm] FEM
Poly. (FEM)
Figure 1. Load-deflection relationship.
The load-deflection relationship illustrated in Figure 1 considers positive deflection and load to be tensile type loading and hence negative deflection and load to be compressive. With this scheme, a third order polynomial, determined via a least-squares regression (F=0.448x3-2.193x2+231.349x), fits the entire spectrum of the function with an r-squared value of 0.9997. Notice the behavior of the load-deflection relationship is practically linear throughout the majority of the region illustrated. However, at the extremities of the spectrum investigated, the load-deflection relationship strays from linear behavior, inducing some nonlinearity as noted by the nonzero coefficients of the third order polynomial. Additionally, notice that the polynomial exhibits a nonzero term associated with the squared deflection term indicating nonsymmetrical behavior about the unloaded configuration. Such phenomenon is simply explained by considering the constitutive relationships illustrated in Figure 2. Figure 2 illustrates the difference in behavior of the rubber material under compressive loadings as compared to tensile loadings; therefore, the appearance of a nonzero term associated with the squared term in the load-deflection polynomial is appropriate.
Hormoz Marzbani et al. / Procedia Computer Science 96 (2016) 1597 – 1608
Figure 2. Rubber uniaxial loading constitutive relationship.
With the load-deflection relationship established for the structural component of the hydraulic mount selection of the most appropriate analytical model becomes paramount. To this end Chang describes several nonlinear models utilizing Kelvin and Voigt chains and suggests replacing linear elements for the appropriate nonlinear elements [2]. Muravskii also investigated nonlinear models to describe the loaddeflection relationship for materials exhibiting hysteretic type damping. It was suggested that there exists a damping coefficient that is proportional to the restoring force term such that DCD represents the restoring force and therefore, DgDCD represents the damping term with gD representing the damping coefficient [3]. For the hydraulic mount structural behavior consider a standard Voigt model consisting of a linear spring and damper in parallel that has been used on practically all previous research regarding hydraulic mounts. However, instead of utilizing a linear constitutive relationship, it has already been shown that a nonlinear relationship is necessary, consider a similar approach to that of Chang and simply replace the linear elements with appropriate nonlinear elements. To utilize this approach the linear spring in the traditional Voigt model is replaced with a nonlinear spring exhibiting the load-deflection relationship illustrated in Figure 1. In addition, consider the linear damper, Muravskii has discussed how the hysteretic damping can be approximated by using a damping coefficient that is proportional to the linearized restoring force; therefore, force the damper in the Voigt model to exhibit a damping force proportional to the coefficient of the linear term in the load-deflection polynomial previously discussed.
F
gD k3 x k1 x3 k2 x 2 k3 x
(1) Equation (1) represents the viscoelastic constitutive relationship for the structural component of the hydraulic mount without preload. However, in the real system the hydraulic mount experiences a static load due to the mass of the engine. Such static load induced deformations cause the appearance of a static deflection term in the constitutive relationship in (1). To avoid the complications this causes due to the nonlinearity, the zero point of the coordinate system is shifted along the load-deflection curve and the coefficients of the polynomial describing the load-deflection behavior are recomputed. For a static deflection of 2 mm the coefficients after the coordinate shift are expressed in Table 1.
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Table 1. Coefficients of the load-deflection polynomial Parameter k1 k2 k3
Value 0.439 N/mm3 -0.016 N/mm2 226.315 N/mm
The mathematical model for nonlinearity due to decoupler motion has been introduced in [4-6] and haz been utilized to develop a frequency response. Optimization of the system is discussed in [7-9].
2. Modeling
With the parameters given in Table 1 equation (1) can be implemented directly without further modification into the equations of motion for a three degree of freedom isolator.
Cq Kq f Mq
(2)
where
M
ªM «0 « «¬ 0
0 Md 0
0 º 0 »» C M i »¼
ª Br « «0 « «0 ¬«
0 Bd E
2 d 2
x '
0
0º » 0» » Bi »¼»
ª 2 Ap2 A A AA º d p i p» « k1 z k2 z k3 C1 C1 C1 » « « » A A K « d p Ad2 K Ad Ai K » C1 « » « » Ai Ap « Ad Ai K Ai2 K » «¬ »¼ C1 My½ z½ 1 1 ° ° ° ° Br gD k3 f ® 0 ¾ q ® xd ¾ K C C 1 2 ° ° °x ° ¯ 0 ¿ ¯ i¿ Noting the highly nonlinear nature of both the damping and stiffness matrices of equation (2) analysis of the system is most readily performed on a nondimensional equivalent to (2).
:t , : 2
W xd
'ud , xi
Ad2 K , z Yu , Md 'ui , w
Z :
Using the parameters of (3) the equations of motion in (2) are transformed.
(3)
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u cc ] u c n N1u 3 N 2u 2 N 3u b 2 fu bnf Uu d abnf Uu i
(4)
w 2 sin w W
bf
udcc ] d eud2 udc ud aui uicc ] i uic
U
a a2 abf ud ui u m m mU
0
u
(5)
0
(6)
where
k2Y k3 Bi Bd k1Y 2 ]d , N2 , N3 , ]i , 2 2 2 M i: Md: Ad K Ad K Ad K Ap E Ai ' Mi 1 e , b , a , U , m , f Y Md C1 K Ad Md: Ad In order to describe the nonlinearity effectively a small parameter H is introduced along with the following
]
Br , n M:
Md , N1 M
simplifying nondimensional terms.
H
a, Hd
]
,
H s1
nN1 , H s2
nN 2 , H s3
nN 3 , J
a , F m
Q
bf
U
,
H dd
] d , H di
] i , HD
nbf
e, (7)
Using the parameters in (7), equations (4) through (6) are redefined and illustrated in the following equations.
u cc H du c H s1u 3 s 2u 2 s 3u Fbu FUu d HFUu i
(8)
w 2 sin w W
udcc H d d D ud2 udc ud H ui J u uicc H di uic Q ud HQ ui QJ u
0
(9)
0
(10)
To determine the equilibrium conditions for the hydraulic mount system, allow the time derivatives in (8) through (10) to equal zero.
H s1u 3 s2u 2 s3u F bu FU ud HFU ui ud H ui J u 0 Q ud HQ ui QJ u 0
0
(11) (12) (13)
Notice that equations (12) and (13) are merely scalar multiples of one another. In addition, notice that said equations also consider the behavior of the fluid inside the hydraulic mount, and for the mount to function properly, the pressure differential between the two chambers at an equilibrium condition must be zero; therefore, the only nontrivial and acceptable equilibrium solution for the decoupler and inertia track degrees of freedom is ud = ui = 0. With this realization, determination of the equilibrium states for the system rests on determining a solution to (11) with ud and ui set to equal zero.
u
0,
H s2 r H 4 s1 H s3 F b H s22 2H s1
(14)
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With the equilibrium states of the system determined by (14), the stability of said equilibrium conditions must be investigated. For this, consider expressing equation (8) as a system of first order differential equations as
f1 f2
u c ua
(15)
H dua H s1u s2u s3u F bu 3
uac
2
(16)
To determine stability it is necessary to obtain the determinant of the Jacobian matrix of the system; therefore, the Jacobian and its corresponding determinant are defined.
J
ª wf1 « wu « « wf 2 « wu ¬
wf1 º wua » » wf 2 » wua »¼
J
0 1 º ª « » 2 «¬ H 3s1u 2 s2u s3 F b H d »¼
H 3s1u 2 2s2u s3 F b
(17)
(18)
3. Analysis
In order to utilize the mathematical foundation set forth thus far, a specific system must be defined. Additionally, due to the nonlinearity of the system, accurate analysis is made possible only through the use of numerical analysis; therefore, consider the parameters in Table 2 [9]. These parameters are obtained from a real hydraulic engine mount, with a floating-decoupler design. Such design is thoroughly discussed in [9]. Table 2. Hydraulic mount parameters Value Unit Property Ad 8.107E-05 m2 Ai 6.668E-05 m2 Ap 3.663E-03 m2 Bd 1.031E-04 Ns/m Bi 2.852E-02 Ns/m Br 0.50E+03 Ns/m C1 5.70E-11 m5/N C2 2.923E-09 m5/N k1 0.439 N/mm3 k2 -0.016 N/mm2 k3 226.315 N/mm Md 5.220E-04 kg Mi 1.773E-03 kg M 50 kg E 0.50 mm 0.5 ' Y 1.0 mm
Figure 3 illustrates the equilibrium diagrams resulting from equation (14) for varying values of s2 considering both positive and negative values of s1 (hardening or softening springs). In Figure 3.a. the
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value of s1 is positive and the equilibrium appears asymptotic about the u-axis. However, investigating the conditions for a saddle-node bifurcation (equations (19) through (22)) indicate that such a bifurcation exists at the location u = +/- 32.096, s2 = 3.045E-03 [10].
a.
b.
Figure 3. Equilibrium plot for +/- s1 (a. +s1 and b. –s1).
f2
H dua H s1u 3 s2u 2 s3u F bu wf 2 wu
0
(19)
H 3s1u 2 2s2u s3 F b 0
(20)
wf 2 ws2 u
H u 2 z 0
(21)
r32.096
w2 f2 H 6s1u 2s2 z 0 wu 2 u r32.096
(22)
Figure 3.b. illustrates the equilibrium diagram for negative values of s2. Here no bifurcation in the equilibrium diagram can be found with the outer two branches of the diagram providing a bounds on the possible amplitude of motion, u. Consider the phase portraits of Figure 4 to illustrate the stability of the equilibrium branches illustrated in Figure 3. Figure 3a illustrates the free response of the existing system in an “as-is” condition, and Figure 3b illustrates the free response of the system for s2 = 0 (i.e. a symmetric system). Notice both systems die to zero as the solution progresses indicating the zero branch of the equilibrium diagram is stable. Similarly it can be deduced for the determinant of the Jacobian matrix that the middle equilibrium branch illustrated in Figure 3 is unstable and the upper equilibrium branch is stable.
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a.
b.
Figure 4. Phase portraits (a. s2 =-3.045E-03 and b. s2 =0).
Figure 5 illustrates the equilibrium diagram resulting from holding s1 constant and altering s3 for positive and negative values of s2 (Figure 5a and 5b, respectively). Both figures exhibit a classical pitch-fork type bifurcation for small negative values of s3. A negative value of s3 indicates a negative slope of the loaddeflection relationship, while physically meaningless, it is interesting from a mathematical perspective to investigate the bifurcation; therefore, equations (23) through (28) describe the necessary six conditions to define the pitch-fork type bifurcation [10].
f2 wf 2 wu
H dua H s1u 3 s2u 2 s3u F bu
H 3s1u 2 2s2u s3 F b 0 wf 2 ws3
H u
0
w2 f2 H 6s1u 2s2 z 0 wu 2 w2 f2 H z 0 wuws3 w3 f2 wu 3
6H s1 z 0
0
(23) (24) (25) (26) (27) (28)
Figure 6 illustrates contains multiple phase portraits for different values of s3 exploring the stability of the equilibrium branches illustrated in Figure 5. According to Figure 6a, the solution is stable for positive values of s3 where the zero solution branch is the only available solution. After s3 becomes negative, Figures 6c and 6d illustrate that the outer solution branches are stable and the inner zero branch is unstable. Figure 7a illustrates the phase portrait for the forced system at its initial configuration with the first 80% of the time-domain solution eliminated to kill transient effects. Figure 7b is the corresponding spectral density of the time domain solution.
Hormoz Marzbani et al. / Procedia Computer Science 96 (2016) 1597 – 1608
u
u
s3
s3
a.
b.
Figure 5. Equilibrium plot for +/- s2 (a. +s2 and b. –s2).
a.
b.
c.
d.
Figure 6. Phase portraits (a. s3 =2.444E-02, b. s3 =0, c. s3=-5.0E-03, and d. s3=-2.444E-02).
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a. b. Figure 7. Forced response of the system (a. phase portrait and b. spectral density).
The solution illustrated in Figure 7 is periodic oscillating with its corresponding forcing frequency of w = 1. To investigate if this system is capable of chaotic motion, the bifurcation diagram is generated by altering s3 between 0 and 0.2 by sectioning the phase portrait at the horizontal axis and noting the intersection of the resulting plane with the solution trajectory. Figure 8 illustrates the bifurcation diagram for the system by varying the s3 parameter. Clearly, the system exhibits only periodic responses for this combination of values, a fact that is reinforced by Figure 7.
Figure 8. Bifurcation diagram for s3 = 0…0.2
As illustrated in Figure 8, the system remains periodic for all values of s3 investigated (the initial configuration has s3 = 2.444E-02). Noting the similarity of this system to a Duffing oscillator, which does exhibit chaotic behavior [64-68], there should be some values of parameters that induce chaotic motion. Therefore, consider the system with the value of s1 set to positive one (currently s1 = 4.744E-05) and alter s2 instead of s3. In the above analysis, s3 was altered noting it is the parameter that is most controllable in regards to the structural behavior of the engine mount. Figure 9a illustrates the bifurcation diagram arrived at by the phase portrait method previously discussed
Hormoz Marzbani et al. / Procedia Computer Science 96 (2016) 1597 – 1608
showing that chaotic motion is possible for a range of values for s2. Figure 9b is a magnification of the region of s2 between 1 and 2.25 to better illustrate the transitions between periodic and chaotic motion.
a.
b. b. Figure 9. Bifurcation diagrams for s2 (a. s2 = 0..3 and b. s2 = 1..2.25).
REMARKS Noting the similarity of the model to a Duffing system with the addition of an asymmetric term associated with the load-deflection relationship, and of course the decoupler and inertia track behaviors, chaotic motion is of concern [64-68]. However, for the real system considered no chaotic motion was detectable for the region of parameters investigated. While advantageous in terms of a real situation, little mathematical information is obtained on the effects of asymmetric load-deflection terms in the governing equations; therefore, after adjusting the parameters chaotic motion was detectable for a large range of asymmetric terms indicating the addition of nonsymmetrical behavior to a vibrating system can indeed cause random and chaotic responses. References [1] Kim G., Singh R., “Nonlinear Analysis of Automotive Hydraulic Engine Mount,” ASME Journal of Dynamic Systems, Measurement and Control (1993) 115, 482-487. [2] Chang, S.I., “Bifurcation Analysis of a Non-Linear Hysteretic Oscillator Under Harmonic Excitation,” Journal of Sound and Vibration (2004) 276, 215-225. [3] Muravskii, G.B., “On Frequency Independent Damping,” Journal of Sound and Vibration (2004) 274, 653-668. [4] Golnaraghi M.F., Jazar G.N., “Development and Analysis of a Simplified Nonlinear Model of a Hydraulic Engine Mount,” Journal of Vibration and Control (2001) 7, 495-526. [5] Jazar R., Golnaraghi M.F., “Nonlinear Modeling, Experimental Verification, and Theoretical Analysis of a Hydraulic Engine Mount,” Journal of Vibration and Control (2002) 8, 87-116. [6] Jazar R., Golnaraghi M.F., “Engine Mounts for Automotive Application. A Survey,” The Shock and Vibration Digest (2002) 34, 363-379. [7] Jazar R., Narimani A., Golnaraghi M. F., and Swanson D. A., "Practical Frequency and Time Optimal Design of Passive Linear Vibration Isolation Mounts,” Journal of Vehicle System Dynamics (2003) 39, 437-466. [8] Alkhatib R., Jazar R., Golnaraghi M.F., “Optimal Design of Passive Linear Suspension Using Genetic Algorithm,” Journal of Sound and Vibration (2004) 275, 665-691.
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[9] Christopherson J., Jazar R., "Optimization of Classical Hydraulic Engine Mounts Based on RMS Methods,” To be published in The Shock and Vibration Digest (2005). [10] Nayfeh A. H., Balachandran B., “Applied Nonlinear Dynamics,” John Wiley (1995), New York.
ScienceDirect Procedia Computer Science 96 (2016) 1597 – 1608
20th International Conference on Knowledge Based and Intelligent Information and Engineering Systems, KES2016, 5-7 September 2016, York, United Kingdom
Chaotic Behavior of Hydraulic Engine Mount Hormoz Marzbani, M. Fard, Reza Jazar * School of Engineering, RMIT University, Bundoora East Campus, Melbourne, VIC, 3083, Australia 0F
Abstract
The rubber materials that act as the main spring of a hydraulic mount obey nonlinear constitutive relationships. In addition to the material nonlinearity, further nonlinearities may be introduced by mount geometry, turbulent fluid behavior, boundary conditions, temperature, decoupler action, and hysteretic behavior. While all influence the behavior of the system only certain aspects are realistically considered using the lumped parameter approach employed in this research. The nonlinearities that are readily modeled by the lumped parameter approach constitute the geometry and constitutive relationship induced nonlinearity, including hysteretic behavior, noting that these properties all make an appearance in the load-deflection relationship for the mount and may be readily determined via experiment or finite element analysis. In this paper we will shoe that under certain conditions, the nonlinearities involved in the hydraulic engine mounts can show a chaotic response. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review underresponsibility responsibility KES International. Peer-review under ofof KES International Keywords: Hydraulic Engine Mount, Nonlinear Analysis, Vibration Isolation, Averaging Method
* Corresponding author. Tel.: +61400693496. E-mail address: [email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of KES International doi:10.1016/j.procs.2016.08.207
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1. INTRODUCTION
The constitutive relationships of the rubber materials that act as the main spring of the hydraulic mount are nonlinear. In this investigation only certain aspects are realistically considered using the lumped parameter approach employed in this research. Turbulent fluid behavior is neglected as all flow is assumed laminar in nature for this modeling procedure. Boundary conditions are assumed constant such that the mount is perfectly mounted. Lastly, temperature-induced effects are neglected; however, the later assumption is validated noting that temperature effects have been shown to be negligible throughout a large spectrum of values by Kim and Singh [1]. The nonlinearities that are readily modeled by the lumped parameter approach constitute the geometry and constitutive relationship induced nonlinearity, including hysteretic behavior, noting that these properties all make an appearance in the load-deflection relationship for the mount and may be readily determined via simple experiment or finite element analysis. As an example, consider the load-deflection relationship illustrated in Figure 1. This load-deflection relationship is the result of a finite element analysis on the upper structure of the hydraulic mount. 2000 1500
Load [N]
1000 500 0 -500 -1000 -1500 -2000 -8
-6
-4
-2
0
2
4
6
8
10
Deflection [mm] FEM
Poly. (FEM)
Figure 1. Load-deflection relationship.
The load-deflection relationship illustrated in Figure 1 considers positive deflection and load to be tensile type loading and hence negative deflection and load to be compressive. With this scheme, a third order polynomial, determined via a least-squares regression (F=0.448x3-2.193x2+231.349x), fits the entire spectrum of the function with an r-squared value of 0.9997. Notice the behavior of the load-deflection relationship is practically linear throughout the majority of the region illustrated. However, at the extremities of the spectrum investigated, the load-deflection relationship strays from linear behavior, inducing some nonlinearity as noted by the nonzero coefficients of the third order polynomial. Additionally, notice that the polynomial exhibits a nonzero term associated with the squared deflection term indicating nonsymmetrical behavior about the unloaded configuration. Such phenomenon is simply explained by considering the constitutive relationships illustrated in Figure 2. Figure 2 illustrates the difference in behavior of the rubber material under compressive loadings as compared to tensile loadings; therefore, the appearance of a nonzero term associated with the squared term in the load-deflection polynomial is appropriate.
Hormoz Marzbani et al. / Procedia Computer Science 96 (2016) 1597 – 1608
Figure 2. Rubber uniaxial loading constitutive relationship.
With the load-deflection relationship established for the structural component of the hydraulic mount selection of the most appropriate analytical model becomes paramount. To this end Chang describes several nonlinear models utilizing Kelvin and Voigt chains and suggests replacing linear elements for the appropriate nonlinear elements [2]. Muravskii also investigated nonlinear models to describe the loaddeflection relationship for materials exhibiting hysteretic type damping. It was suggested that there exists a damping coefficient that is proportional to the restoring force term such that DCD represents the restoring force and therefore, DgDCD represents the damping term with gD representing the damping coefficient [3]. For the hydraulic mount structural behavior consider a standard Voigt model consisting of a linear spring and damper in parallel that has been used on practically all previous research regarding hydraulic mounts. However, instead of utilizing a linear constitutive relationship, it has already been shown that a nonlinear relationship is necessary, consider a similar approach to that of Chang and simply replace the linear elements with appropriate nonlinear elements. To utilize this approach the linear spring in the traditional Voigt model is replaced with a nonlinear spring exhibiting the load-deflection relationship illustrated in Figure 1. In addition, consider the linear damper, Muravskii has discussed how the hysteretic damping can be approximated by using a damping coefficient that is proportional to the linearized restoring force; therefore, force the damper in the Voigt model to exhibit a damping force proportional to the coefficient of the linear term in the load-deflection polynomial previously discussed.
F
gD k3 x k1 x3 k2 x 2 k3 x
(1) Equation (1) represents the viscoelastic constitutive relationship for the structural component of the hydraulic mount without preload. However, in the real system the hydraulic mount experiences a static load due to the mass of the engine. Such static load induced deformations cause the appearance of a static deflection term in the constitutive relationship in (1). To avoid the complications this causes due to the nonlinearity, the zero point of the coordinate system is shifted along the load-deflection curve and the coefficients of the polynomial describing the load-deflection behavior are recomputed. For a static deflection of 2 mm the coefficients after the coordinate shift are expressed in Table 1.
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Table 1. Coefficients of the load-deflection polynomial Parameter k1 k2 k3
Value 0.439 N/mm3 -0.016 N/mm2 226.315 N/mm
The mathematical model for nonlinearity due to decoupler motion has been introduced in [4-6] and haz been utilized to develop a frequency response. Optimization of the system is discussed in [7-9].
2. Modeling
With the parameters given in Table 1 equation (1) can be implemented directly without further modification into the equations of motion for a three degree of freedom isolator.
Cq Kq f Mq
(2)
where
M
ªM «0 « «¬ 0
0 Md 0
0 º 0 »» C M i »¼
ª Br « «0 « «0 ¬«
0 Bd E
2 d 2
x '
0
0º » 0» » Bi »¼»
ª 2 Ap2 A A AA º d p i p» « k1 z k2 z k3 C1 C1 C1 » « « » A A K « d p Ad2 K Ad Ai K » C1 « » « » Ai Ap « Ad Ai K Ai2 K » «¬ »¼ C1 My½ z½ 1 1 ° ° ° ° Br gD k3 f ® 0 ¾ q ® xd ¾ K C C 1 2 ° ° °x ° ¯ 0 ¿ ¯ i¿ Noting the highly nonlinear nature of both the damping and stiffness matrices of equation (2) analysis of the system is most readily performed on a nondimensional equivalent to (2).
:t , : 2
W xd
'ud , xi
Ad2 K , z Yu , Md 'ui , w
Z :
Using the parameters of (3) the equations of motion in (2) are transformed.
(3)
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u cc ] u c n N1u 3 N 2u 2 N 3u b 2 fu bnf Uu d abnf Uu i
(4)
w 2 sin w W
bf
udcc ] d eud2 udc ud aui uicc ] i uic
U
a a2 abf ud ui u m m mU
0
u
(5)
0
(6)
where
k2Y k3 Bi Bd k1Y 2 ]d , N2 , N3 , ]i , 2 2 2 M i: Md: Ad K Ad K Ad K Ap E Ai ' Mi 1 e , b , a , U , m , f Y Md C1 K Ad Md: Ad In order to describe the nonlinearity effectively a small parameter H is introduced along with the following
]
Br , n M:
Md , N1 M
simplifying nondimensional terms.
H
a, Hd
]
,
H s1
nN1 , H s2
nN 2 , H s3
nN 3 , J
a , F m
Q
bf
U
,
H dd
] d , H di
] i , HD
nbf
e, (7)
Using the parameters in (7), equations (4) through (6) are redefined and illustrated in the following equations.
u cc H du c H s1u 3 s 2u 2 s 3u Fbu FUu d HFUu i
(8)
w 2 sin w W
udcc H d d D ud2 udc ud H ui J u uicc H di uic Q ud HQ ui QJ u
0
(9)
0
(10)
To determine the equilibrium conditions for the hydraulic mount system, allow the time derivatives in (8) through (10) to equal zero.
H s1u 3 s2u 2 s3u F bu FU ud HFU ui ud H ui J u 0 Q ud HQ ui QJ u 0
0
(11) (12) (13)
Notice that equations (12) and (13) are merely scalar multiples of one another. In addition, notice that said equations also consider the behavior of the fluid inside the hydraulic mount, and for the mount to function properly, the pressure differential between the two chambers at an equilibrium condition must be zero; therefore, the only nontrivial and acceptable equilibrium solution for the decoupler and inertia track degrees of freedom is ud = ui = 0. With this realization, determination of the equilibrium states for the system rests on determining a solution to (11) with ud and ui set to equal zero.
u
0,
H s2 r H 4 s1 H s3 F b H s22 2H s1
(14)
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With the equilibrium states of the system determined by (14), the stability of said equilibrium conditions must be investigated. For this, consider expressing equation (8) as a system of first order differential equations as
f1 f2
u c ua
(15)
H dua H s1u s2u s3u F bu 3
uac
2
(16)
To determine stability it is necessary to obtain the determinant of the Jacobian matrix of the system; therefore, the Jacobian and its corresponding determinant are defined.
J
ª wf1 « wu « « wf 2 « wu ¬
wf1 º wua » » wf 2 » wua »¼
J
0 1 º ª « » 2 «¬ H 3s1u 2 s2u s3 F b H d »¼
H 3s1u 2 2s2u s3 F b
(17)
(18)
3. Analysis
In order to utilize the mathematical foundation set forth thus far, a specific system must be defined. Additionally, due to the nonlinearity of the system, accurate analysis is made possible only through the use of numerical analysis; therefore, consider the parameters in Table 2 [9]. These parameters are obtained from a real hydraulic engine mount, with a floating-decoupler design. Such design is thoroughly discussed in [9]. Table 2. Hydraulic mount parameters Value Unit Property Ad 8.107E-05 m2 Ai 6.668E-05 m2 Ap 3.663E-03 m2 Bd 1.031E-04 Ns/m Bi 2.852E-02 Ns/m Br 0.50E+03 Ns/m C1 5.70E-11 m5/N C2 2.923E-09 m5/N k1 0.439 N/mm3 k2 -0.016 N/mm2 k3 226.315 N/mm Md 5.220E-04 kg Mi 1.773E-03 kg M 50 kg E 0.50 mm 0.5 ' Y 1.0 mm
Figure 3 illustrates the equilibrium diagrams resulting from equation (14) for varying values of s2 considering both positive and negative values of s1 (hardening or softening springs). In Figure 3.a. the
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value of s1 is positive and the equilibrium appears asymptotic about the u-axis. However, investigating the conditions for a saddle-node bifurcation (equations (19) through (22)) indicate that such a bifurcation exists at the location u = +/- 32.096, s2 = 3.045E-03 [10].
a.
b.
Figure 3. Equilibrium plot for +/- s1 (a. +s1 and b. –s1).
f2
H dua H s1u 3 s2u 2 s3u F bu wf 2 wu
0
(19)
H 3s1u 2 2s2u s3 F b 0
(20)
wf 2 ws2 u
H u 2 z 0
(21)
r32.096
w2 f2 H 6s1u 2s2 z 0 wu 2 u r32.096
(22)
Figure 3.b. illustrates the equilibrium diagram for negative values of s2. Here no bifurcation in the equilibrium diagram can be found with the outer two branches of the diagram providing a bounds on the possible amplitude of motion, u. Consider the phase portraits of Figure 4 to illustrate the stability of the equilibrium branches illustrated in Figure 3. Figure 3a illustrates the free response of the existing system in an “as-is” condition, and Figure 3b illustrates the free response of the system for s2 = 0 (i.e. a symmetric system). Notice both systems die to zero as the solution progresses indicating the zero branch of the equilibrium diagram is stable. Similarly it can be deduced for the determinant of the Jacobian matrix that the middle equilibrium branch illustrated in Figure 3 is unstable and the upper equilibrium branch is stable.
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a.
b.
Figure 4. Phase portraits (a. s2 =-3.045E-03 and b. s2 =0).
Figure 5 illustrates the equilibrium diagram resulting from holding s1 constant and altering s3 for positive and negative values of s2 (Figure 5a and 5b, respectively). Both figures exhibit a classical pitch-fork type bifurcation for small negative values of s3. A negative value of s3 indicates a negative slope of the loaddeflection relationship, while physically meaningless, it is interesting from a mathematical perspective to investigate the bifurcation; therefore, equations (23) through (28) describe the necessary six conditions to define the pitch-fork type bifurcation [10].
f2 wf 2 wu
H dua H s1u 3 s2u 2 s3u F bu
H 3s1u 2 2s2u s3 F b 0 wf 2 ws3
H u
0
w2 f2 H 6s1u 2s2 z 0 wu 2 w2 f2 H z 0 wuws3 w3 f2 wu 3
6H s1 z 0
0
(23) (24) (25) (26) (27) (28)
Figure 6 illustrates contains multiple phase portraits for different values of s3 exploring the stability of the equilibrium branches illustrated in Figure 5. According to Figure 6a, the solution is stable for positive values of s3 where the zero solution branch is the only available solution. After s3 becomes negative, Figures 6c and 6d illustrate that the outer solution branches are stable and the inner zero branch is unstable. Figure 7a illustrates the phase portrait for the forced system at its initial configuration with the first 80% of the time-domain solution eliminated to kill transient effects. Figure 7b is the corresponding spectral density of the time domain solution.
Hormoz Marzbani et al. / Procedia Computer Science 96 (2016) 1597 – 1608
u
u
s3
s3
a.
b.
Figure 5. Equilibrium plot for +/- s2 (a. +s2 and b. –s2).
a.
b.
c.
d.
Figure 6. Phase portraits (a. s3 =2.444E-02, b. s3 =0, c. s3=-5.0E-03, and d. s3=-2.444E-02).
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a. b. Figure 7. Forced response of the system (a. phase portrait and b. spectral density).
The solution illustrated in Figure 7 is periodic oscillating with its corresponding forcing frequency of w = 1. To investigate if this system is capable of chaotic motion, the bifurcation diagram is generated by altering s3 between 0 and 0.2 by sectioning the phase portrait at the horizontal axis and noting the intersection of the resulting plane with the solution trajectory. Figure 8 illustrates the bifurcation diagram for the system by varying the s3 parameter. Clearly, the system exhibits only periodic responses for this combination of values, a fact that is reinforced by Figure 7.
Figure 8. Bifurcation diagram for s3 = 0…0.2
As illustrated in Figure 8, the system remains periodic for all values of s3 investigated (the initial configuration has s3 = 2.444E-02). Noting the similarity of this system to a Duffing oscillator, which does exhibit chaotic behavior [64-68], there should be some values of parameters that induce chaotic motion. Therefore, consider the system with the value of s1 set to positive one (currently s1 = 4.744E-05) and alter s2 instead of s3. In the above analysis, s3 was altered noting it is the parameter that is most controllable in regards to the structural behavior of the engine mount. Figure 9a illustrates the bifurcation diagram arrived at by the phase portrait method previously discussed
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showing that chaotic motion is possible for a range of values for s2. Figure 9b is a magnification of the region of s2 between 1 and 2.25 to better illustrate the transitions between periodic and chaotic motion.
a.
b. b. Figure 9. Bifurcation diagrams for s2 (a. s2 = 0..3 and b. s2 = 1..2.25).
REMARKS Noting the similarity of the model to a Duffing system with the addition of an asymmetric term associated with the load-deflection relationship, and of course the decoupler and inertia track behaviors, chaotic motion is of concern [64-68]. However, for the real system considered no chaotic motion was detectable for the region of parameters investigated. While advantageous in terms of a real situation, little mathematical information is obtained on the effects of asymmetric load-deflection terms in the governing equations; therefore, after adjusting the parameters chaotic motion was detectable for a large range of asymmetric terms indicating the addition of nonsymmetrical behavior to a vibrating system can indeed cause random and chaotic responses. References [1] Kim G., Singh R., “Nonlinear Analysis of Automotive Hydraulic Engine Mount,” ASME Journal of Dynamic Systems, Measurement and Control (1993) 115, 482-487. [2] Chang, S.I., “Bifurcation Analysis of a Non-Linear Hysteretic Oscillator Under Harmonic Excitation,” Journal of Sound and Vibration (2004) 276, 215-225. [3] Muravskii, G.B., “On Frequency Independent Damping,” Journal of Sound and Vibration (2004) 274, 653-668. [4] Golnaraghi M.F., Jazar G.N., “Development and Analysis of a Simplified Nonlinear Model of a Hydraulic Engine Mount,” Journal of Vibration and Control (2001) 7, 495-526. [5] Jazar R., Golnaraghi M.F., “Nonlinear Modeling, Experimental Verification, and Theoretical Analysis of a Hydraulic Engine Mount,” Journal of Vibration and Control (2002) 8, 87-116. [6] Jazar R., Golnaraghi M.F., “Engine Mounts for Automotive Application. A Survey,” The Shock and Vibration Digest (2002) 34, 363-379. [7] Jazar R., Narimani A., Golnaraghi M. F., and Swanson D. A., "Practical Frequency and Time Optimal Design of Passive Linear Vibration Isolation Mounts,” Journal of Vehicle System Dynamics (2003) 39, 437-466. [8] Alkhatib R., Jazar R., Golnaraghi M.F., “Optimal Design of Passive Linear Suspension Using Genetic Algorithm,” Journal of Sound and Vibration (2004) 275, 665-691.
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[9] Christopherson J., Jazar R., "Optimization of Classical Hydraulic Engine Mounts Based on RMS Methods,” To be published in The Shock and Vibration Digest (2005). [10] Nayfeh A. H., Balachandran B., “Applied Nonlinear Dynamics,” John Wiley (1995), New York.