模态应变能法计算方法

Revised Modal Strain Energy Method for Finite Element Analysis of Viscoelastic Damping Treated Structures
Yanchu Xu, Yanning Liu, and Bill Wang
Maxtor Corporation, 500 McCarthy Blvd, Milpitas, CA 95035
ABSTRACT
In traditional modal strain energy method, the real eigen-vector of each mode obtained from finite element analysis of the corresponding undamped structure is used to calculate modal strain energy in each material layer, and an iterative approach is used in dealing with the frequency dependency of viscoelastic materials. In this paper, a revised modal strain energy method is presented to significantly improve analysis accuracy of the structural natural frequencies and modal loss factors when the material loss factor is high, and a simplified approach is recommended to replace the iterative analysis to avoid tremendous amount of computational effort.
Keywords: modal strain energy method, finite element analysis, damping, modal analysis
1. INTRODUCTION
In structural noise and vibration control, damping treatments have found more and more applications. Among a variety of damping mechanisms, such as from fluid viscosity, from friction in fibrous materials, etc., viscoelastic materials remain to be the favorite choices for the most effective damping treatments. Numerous successful cases applying viscoelastic damping to control structural noise and vibration can be found in applications from aerospace structures to hard disk drives. These applications normally involve bonding a relatively stiff thin member, also called constraining layer, to sheet metal structure with a soft viscoelastic material such that strain is induced in the adhesive during vibration. To analyze the dynamic performance of the damping treatments, considerable effort has been devoted to the studies of dynamic characteristics of viscoelastically damped structures [1]. There are two major approaches in the analysis of damping effect: analytical and numerical.
The analytical approach is usually applicable to relatively simple structures, such as sandwich beams and plates, etc. The earliest analytical work on damping analysis can be found mostly related to the viscoelastic material property characterization. To develop an understanding of the parameters in the constrained layer damper, Ross, Kerwin and Unger [2] outlined the dominant design parameters for the case where all layers vibrate with the same sinusoidal spatial dependence. The outer layers are assumed to deform as Eular-Bernoulli beams and the adhesive is assumed to deform only in shear, which leads to a single fourth order beam equation where the equivalent complex bending stiffness depends on the properties of the three layers. To extend Ross, Ungar and Kerwin’s analysis to beams with general boundary conditions in which sinusoidal spatial dependence cannot be assumed, Mead etc.[3] obtained a sixth order equation of motion. It is assumed that the beam’s deflection is small and uniform across a section, the axial displacements are continuous, the base and constraining layers bend according to the Eular hypothesis, the damping layer deforms only in shear, and the longitudinal and rotary inertia effects are insignificant. The validity of the analysis is therefore limited to some upper range of core stiffness. Miles etc.[4] obtained a sixth order equation of motion by using Hamilton’s principal. The assumptions were equivalent to those of Mead except that relative transverse deflection is permitted between the outer layers and longitudinal inertia is included.
Though analytical methods are useful for predicting damping characteristics of some simple structures, a numerical approach, mainly finite element method, remains to be the method of choice when complex physical systems are analyzed. In the finite element analysis of structures with visco-elastic damping material treatment, there are two issues making the analysis a tough task. One is that the modulus of a viscoelastic material is normally complex, however, most commercial finite element packages are not designed to deal with complex modulus efficiently and accurately. The other Smart Structures and Materials 2002: Damping and Isolation, Gregory S. Agnes,
one is that the material properties of viscoelastic material are frequency dependent that creates a non-linear eigenproblem for the dynamic analysis. To deal with the complex modulus of the viscoelastic material, several different techniques have been developed, of which modal strain energy method has become a commonly used approach. In the modal strain energy method, the structure is first assumed to be undamped and modeled using the real part of complex modulus as the modulus of the damping layer. The real eigen-vectors of each mode are obtained from finite element analysis and strain energies in all layers of the structure are calculated. The dissipative energy of the structure is calculated proportional to the strain energy in the damping layer and the material loss factor, and the modal loss factor is obtained by calculating the ratio of the dissipative energy to the total structural energy. However, modal strain energy method becomes quite inaccurate when the damping of the structure becomes high. To consider the frequency and temperature dependence of elastic modulus of viscoelastic material, an iterative method is normally combined with commercial finite element software, which requires tremendous amount of computational effort since for each mode, eigen-solutions need to be repeated until converged results are obtained.
In this paper, a revised modal strain energy method is presented. An equivalent modulus, the magnitude of the complex modulus, is used for the finite element modal analysis to obtain real eigen-vector. The strain energy and dissipative strain energy are calculated proportional to that of equivalent modulus, and the modal loss factor is calculated accordingly. The results are compared to direct complex eigen-solution and the accuracy of the modal strain energy method is found improved significantly. In replacing the iterative analysis, a simplified approach is proposed to avoid tremendous amount of computational effort, which has the most significant advantage in viscoelastic material selection. .
2. MODAL STRAIN ENERGY METHOD
When a structure with viscoelastic damping treatment is to be analyzed, finite element modeling procedure can be used to establish its mass matrix [M], and stiffness matrix [K]. The structural eigen-value problem can be written as,
[]{}[]
{}0=+x K x M (1)
where []M is a real matrix, and [][][]
i r K i K K += a complex matrix due to the complex modulus of the viscoelastic damping material used in the structure.
However, there are two issues associated with the eigen-problem of (1). One is that most commercial finite element software does not have the corresponding solver for the complex eigen-solution for a damped structure. Another one is that the modulus and loss factor of the viscoelastic material are frequency/temperature dependent, which results in the eigen-problem of (1) being a non-linear one.
Modal strain energy method is one of the economical approaches in dealing with the complex modulus of the damping material. It assumes that the damped structure has the same natural frequencies and modal shapes as the undamped structure, thus the eigen-problem of the undamped structure is written as,
[]{}[]{}0=+x K x M r
(2)
By solving (2), eigen-values and eigen-vectors, {},...3,2,1,,=r f r r φ can be obtained. For the rth mode, the dissipated and strain energies are defined as,
{
}[]{}{}[]{}
r
r
T
r
S
r
r i T
r D r K E
K E φφφφ== (3)
The damping loss factor for the rth mode, r ζ, therefore becomes,
{}[]
{}{}[]
{}r
r T r r i T
r S r D r r K K E E φφφφζ==
(4)
Since damping modulus can be expressed as G i )1(η+, where G is the shear storage modulus of the viscoelastic material, and η is the material loss factor, and in finite element analysis, the strain energy in the viscoelastic material
layer, V
r
E , can be also calculated, thus the damping loss factor of the rth mode can be estimated [5] as,
S
r V
r r r E E ηζ=
(5)
where r ηis the material loss factor at the natural frequency of the rth mode. Xu [6] etc. compared the result for a cantilever sandwich beam using above mentioned modal strain energy method with that from direct complex eigen-solution using compound beam element, and found that results from both methods are very close when the material loss factor is low, however, significantly different when the material loss factor becomes high.
However, due to viscoelastic materials ’ frequency dependent feature of the storage shear modulus G and loss factor η, as shown in Figure 1, the structural stiffness matrix in (1) is not only a complex one, but also in theory a function of frequency. Therefore, the dynamic characterization of a damped structure has not completed yet by the simple application of the modal strain energy method as outlined above. The storage shear modulus G and loss factor η of viscoelastic material are also temperature dependent, however, it is not going to be considered here since in most dynamic analysis, constant temperature could be assumed.
Then the []
r K in (2) is varies with the frequency of the interested mode. The modal analysis of the non-linear eigen-problem (2) can be normally simplified to an iterative process. For the modal parameters ,,r r f ηand {}r φof the rth
mode, the method can be summarized as,
Figure 1. Material Moduls and Loss Factor vs. Frequency at Different Temperature 1010010001000010100100010000Frequency (Hz)S h e a r S t o r a g e M o d u l u s - G (p s i )
0.4
0.711.3
L o s s F a c t o r - η
Shear Storage Modulus (70F)Shear Storage Modulus (100F)
Loss Factor (70 F)Loss Factor (100 F)
Initialize: 0f f =, find the corresponding ()0f G G = , and calculate []()[]
0f K K r r = For ,...3,2,1=k
Solve: []{}[]
{}0=+x K x M r Î ,,)()(k r k r f η and {})
(k r φ If ε≤−)()(/k r k r f f f Î Stop
Update: )(k r f f =, find the corresponding ())(k r f G G =, and calculate []()[]
)(k r r r f K K =
As the iteration continues, the estimated ,,)()(k r k r f η and {})
(k r φwill converge to their exact solution ,,r r f ηand {}r φ. Similarly, modal parameters of other modes can be determined. This iterative process requires tremendous amount of computational effort. Especially in the process of viscoelastic material selection, this process needs to be repeated for each material trial.
3. REVISED MODAL STRAIN ENERGY METHOD AND SIMPLIFIED PROCESS
As mentioned above, the traditional modal strain energy method uses the real eigen-vector of each mode obtained from finite element analysis of the corresponding undamped structure to calculate strain energy in each material layer. The dissipative energy is calculated proportional to the strain energy in the viscoelastic damping material layer and the material loss factor. The modal loss factor is then obtained by calculating the ratio of the dissipative energy to the total structural strain energy. The problem associated with this approach is that the errors in natural frequency and modal loss factor estimation increase dramatically when the material loss factor increases. The reason is that traditional modal strain energy method uses real part of the material modulus in the finite element analysis such that the natural frequencies don ’t change with material loss factor. A revised modal strain energy method will be discussed here first.
In order to consider the effect of material loss factor on the structural natural frequencies, it is suggested to use an equivalent modulus, the magnitude of the viscoelastic material modulus, i.e. 21'η+=G G instead of G , in the undamped structural modal analysis, and use the resulting natural frequencies as the ones of the damped structure.
When the equivalent modulus as shown in Figure 2 is used, the natural frequencies of the structure will increase with the loss factor even when the storage modulus of the viscoelastic material keeps the same, which agrees with what was
Figure 2. Material Modulus, Loss Factor and Equivalent Modulus 1010010001000010100100010000Frequency (Hz)S h e a r S t o r a g e M o d u l u s - G (p s i )E q u i v a l e n t M o d u l u s (p s i )
0.40.711.3L o s s F a c t o r - η
Shear Storage Modulus (70F)Equivalent Modulus (70 F)
Loss Factor (70 F)
verified by Xu [6] etc. using direct complex eigen-solution. After the modal analysis of the undamped system finished, the strain energies in different materials can be calculated accordingly. To estimate the modal loss factor, the strain energy and dissipative energy in the viscoelastic material need to be obtained differently as follows,
V
r VD r V
r VS r E E E E 2
2
111ηηη+=
+=
(6)
where, V
r
E is the total strain energy of the rth mode in the viscoelastic material by assuming its modulus to be 'G . If the strain energy in all other material is O r E , then the modal loss factor of the rth mode can be estimated by,
VS
r
O r VD
r r E E E +=ζ (7)
Also, due to the frequency dependency, the dynamic characterization of the damped structure normally needs an iteration process as stated in Section 2. To avoid the tremendous computational effort in solving the non-linear eigen-problem, a simplified process is proposed as follows,
1. For viscoelastic materials to be evaluated, estimate the maximum and minimum modulus, /
max G and /min G
2. Starting from /
min
G , perform FEM analysis to obtain all natural frequencies and strain energies in all layers for different equivalent modulus in incremental 'G ∆ until /
max G .
3. Plot structural dynamic characteristic curves: natural frequency curves of all interested modes against the equivalent modulus as shown in Figure 3a); strain energy curves in different materials of all modes verse frequency as shown in Figure 3b).
4. For different materials or material at different temperatures, repeat the following: a. Plot curves of material equivalent modulus and loss factor against frequency onto Figure 3a), as shown in Figure 3c). b. Find intersection of the material modulus curve with the natural frequency curve of each mode, ,...3,2,1,,'=r G f r r , to determine the natural frequency and the material loss factor from the corresponding frequency, ,...3,2,1,,=r f r r η, as shown in Figure 3c)
c. Determine the strain energies in different materials for each natural frequency, ,...3,2,1,,=r E E V
r O r as shown in Figure 3d).
d. Calculate modal loss factor by,
,...3,2,1,12=++=
+=r E E E E E E V
r r O r V r r VS r O r VD
r r ηηζ (8)
It is obvious that the simplified process requires only limited number of structural FEM analysis, so it can avoid the tremendous amount of computational effort due to the iterative process. Furthermore, after the limited FEM analysis, it doesn ’t require any more FEM analysis when other materials or the same material at different temperatures need to be evaluated as in material selection, which will result in significant computational cost saving.
a)
b)
c)
d)
Figure 3. Simplified Process for Modal Analysis of Viscoelastically Damped Structures
a) Structural dynamic characteristic curves: f r , r=1,2,3,… b) Strain energy curves: E r V , E r O , r=1,2,3,…
c) Finding natural frequency, material modulus and loss factor for specific
viscoelastic material
d) Determining strain energies in viscoelastic material and other materials
and calculating modal loss factor
f 1
f 2
f 3
f 4
f 5
f 6
f 7
10
100
1000
10000
10
100
1000
10000
Frequency (Hz)
V i s c o e l a s t i c S h e a r M o d u l u s (p s i )
E 1O
E 2O
E 3O
E 4O E 5O
E 6O
E 7O
E 1V
E 2V E 3V
E 4V
E 5V
E 6V
E 7V
00.2
0.4
0.6
0.8
1
10
100
1000
10000
Frequency (Hz)
M o d a l S t r a i n E n e r g y
G'
G
(f , G ')
(f ,G ')
η
f 1=22.480
1
=1.18
f 2=190.04
2
=1.10
10
100
10001000010100
1000
10000
Frequency (Hz)
M o d u l u s (p s i )
00.5
1
1.5
M a t e r i a l L o s s F a c t o r
f 1=22.4801=0.149
E 1V =0.182
E 1O =0.818
f 2=190.042=0.167
E 2V =0.211
E 2O =0.789
0.2
0.4
0.6
0.8
1
10100100010000
Frequency (Hz)
M o d a l S t r a i n E n e r g y
4. CASE STUDY
To demonstrate how much the revised modal strain method improves the accuracy of natural frequency and modal loss factor estimation of a damped structure, two sandwich beam samples of 10” long with different boundary conditions are given in Figure 4. Layers 1 and 3 are identical metal beams of 0.06” thick with Young ’s modulus of 30x106 psi, layer 2 is a soft core of 0.002” thick with material property as shown in Figure 1. Only one metal layer of beam (a) is clamped at one end while both metal layers of beam (b) are clamped at one end. The two samples are analyzed using direct complex eigen-analysis based on the compound beam element [6], the conventional modal strain energy method, and the revised modal strain energy method proposed in this paper.
Both beams are evaluated for a viscoelastic material at temperatures of 70F and 100F. Figure 3 indicates the process for beam (a) at 70F using revised modal strain energy method, where the viscoelastic material equivalent modulus is from Figure 2. Table 1 shows the comparison of three methods for beam (a) and Table 2 the comparison for beam (b). In the tables, the solutions from direct complex eigen-solution are used as reference, to which the results from conventional and revised modal strain energy methods are compared. It is shown that the revised method significantly improves the accuracy for modal loss factor estimations.
Note: Values after / in both Table 1 and 2 indicate percentage differences of both modal strain energy methods relative to direct complex eigen-solution
10"10"(a)
(b)
1
23Figure 4. Sandwich beams with different boundary conditions Table 1. Natural frequency and modal loss factor comparison for sandwich beam (a) Table 2. Natural frequency and modal loss factor comparison for sandwich beam (b) f r ζr f r ζr f r ζr f r ζr f r ζr f r ζr
1
22.1730.163120.7850.113121.603/-2.57%0.1888/15.74%20.351/-2.09%0.1220/7.87%22.480/1.38%0.1487/-8.85%20.871/0.41%0.1033/-8.66%2187.330.1710161.310.2450180.83/-3.47%0.2654/55.18%153.63/-4.76%0.3079/25.65%190.04/1.45%0.1674/-2.12%164.06/1.71%0.2277/-7.08%3
529.160.2040445.800.2574510.94/-3.44%0.2811/37.83%425.86/-4.47%0.2941/14.28%538.12/1.69%0.1987/-2.57%454.20/1.88%0.2391/-7.10%41,025.40.2168866.790.2500989.68/-3.48%0.2745/26.63%829.04/-4.36%0.2719/8.77%1,040.0/1.42%0.2096/-3.31%879.41/1.46%0.2322/-7.11%5
1,647.40.22011,400.20.23851,605.3/-2.55%0.2611/18.61%1,362.8/-2.67%0.2523/5.80%1,679.8/1.97%0.2111/-4.11%1,438.0/2.70%0.2219/-6.95%6
2,397.30.21792,116.10.22602,345.0/-2.18%0.2456/12.69%2,023.6/-4.37%0.2348/3.90%2,443.1/1.91%0.2074/-4.84%2,121.4/0.25%0.2095/-7.29%7
3,284.80.21162,892.80.21303,199.0/-2.61%0.2291/8.25%2,805.4/-3.02%0.2170/1.87%3,318.6/1.03%0.1999/-5.54%2,926.4/1.16%0.1969/-7.57%Mode
Direct Complex Eigen-solution Conventional Modal Strain Energy Method Revised Modal Strain Energy Method 70F 100F 70F 100F 70F 100F
f r ζr f r ζr f r ζr f r ζr f r ζr f r ζr
134.0810.204629.1280.269132.576/-4.41%0.3347/63.61%27.923/-4.14%0.3480/29.32%34.757/1.98%0.2011/-1.70%29.974/2.90%0.2590/-3.76%2199.270.2425166.170.2486
189.69/-4.81%0.3253/34.17%159.12/-4.24%0.2892/16.31%202.66/1.70%0.2312/-4.64%169.68/2.11%0.2340/-5.89%3552.850.2338459.200.2590
530.23/-4.09%0.3041/30.06%439.19/-4.36%0.2904/12.12%561.92/1.64%0.2262/-3.26%468.37/2.00%0.2430/-6.18%4
1,053.00.2342883.050.2511
1,015.6/-3.55%0.2879/22.94%847.84/-3.99%0.2689/7.08%1,070.7/1.68%0.2262/-3.41%899.22/1.83%0.2342/-6.73%51,690.20.23041,439.00.2378
1,638.3/-3.07%0.2683/16.43%1,389.4/-3.45%0.2488/4.64%1,717.2/1.60%0.2212/-4.01%1,464.5/1.77%0.2215/-6.84%6
2,445.00.22422,145.50.22492,384.6/-2.47%0.2496/11.32%2,057.6/-4.10%0.2318/3.09%2,486.5/1.70%0.2136/-4.73%2,155.9/0.49%0.2089/-7.09%73,297.50.21562,918.90.2116
3,245.5/-1.58%0.2314/7.33%2,847.4/-2.45%0.2142/1.24%3,368.4/2.15%0.2039/-5.43%2,968.7/1.71%0.1960/-7.36%100F 70F 100F Direct Complex Eigen-solution Mode 70F 100F 70F Conventional Modal Strain Energy Method Revised Modal Strain Energy Method
5. CONCLUSION
Viscoelastic damping treatment has found more and more applications as an effective means of passive noise and vibration control. However, dynamic characterization of the damped structures has been a difficult task, especially in the finite element analysis, due to both complex modulus and frequency dependency of viscoelastic material property. In dealing with the complex modulus, conventional modal strain energy method uses the real eigen-vector of the undamped structure to calculate modal strain energy, and then the modal loss factor is calculated accordingly, this approach can’t give an accurate estimation when the material loss factor is high. To consider the frequency dependency, an iterative process is normally required when commercial finite element analysis software is used, which takes tremendous amount of computational effort.
The revised modal strain energy method presented in this paper significantly improves analysis accuracy of the structural natural frequencies and modal loss factors when the material loss factor is high. And the simplified approach for dealing with the frequency dependency can be utilized to avoid tremendous amount of computational effort, which is especially powerful in viscoelastic material selection.
REFERENCES
1. A. D. Nashif, D. I. G. Jones, J. P. Henderson, Vibration Damping, John Wiley & Sons, New York, 1985
2. D. Ross, E. E. Ungar and Jr. E. M. Kerwin, “Damping of Flexural Vibrations by Means of Viscoelastic Laminates”,
Structural Damping, ASME, New York, 1959
3. D. J. Mead, S. Markus, “The Forced Vibration of a Three-Layer, Damped Sandwich Beam with Arbitrary Boundary
Conditions,” 1969, Journal of Sound and Vibration, Vol. 10(2), pp. 163-175
4. R. N. Miles, and P. G. Reinhall, “An analytical model for the vibration of laminated beams including the effects of
both shear and thickness deformation in the adhesive layer”, Journal of Vibration, Acoustics, Stress, and Reliability in Design, Vol. 108, pp. 56-64, 1986
5. ANSYS Theory Manual for Revision 5.5.
6. Y. Xu, D. Chen, “Finite Element Modeling for the Flexural Vibration of Damped Sandwich Beams Considering
Complex Modulus of the Adhesive Layer,” Proceedings of SPIE, vol. 3989, Damping and Isolation, pp. 121-129, Newport Beach, California, 2000。