模态应变能法计算方法
header for SPIE use
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 ? ,,)()(k r k r f η and {})
(k r φ If ε≤?)()(/k r k r f f f ? 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 1, G 1')
(f 1,G 2')
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
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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
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应力-应变曲线
应力-应变曲线(stress-strain curves) 根据圆柱试件静力拉伸试验所得拉伸图(图a)，对曲线上各对应点用试件原始尺寸除拉伸力与绝对伸长所得出的应力与延伸率的关系曲线(图6)。应力一应变曲线是金属塑性加工工作中最重要的参考资料之一。 应力及应变值按下式计算：
式中σ i 表示拉伸图上任意点的应力值，δ i 为i点的延伸率，P i 及Δl i 为该 点的拉力与绝对伸长值，F 0及l 为试件的断面积和计算长度。 试件受拉伸时，先产生弹性变形，这时应力应变成比例，当出现二者不能保 持线性关系的点时，表示材料已屈服而将发生塑性变形，这时的应力定义为屈服应力或流变应力，用σ s 表示，其求法见屈服点。 拉伸时当试件计算长度上的均匀变形阶段结束而产生细颈时，变形将集中在 细颈部分。出现细颈前材料所能承受的应力名为强度极限或抗拉强度，用σ b 表示 σ b =P max /F 式中P max 为拉伸图上所记录的最大载荷值。 试件出现细颈后很快即断裂，断裂应力σ f σ f =P f /T f 式中P f 是断裂时的拉力，F f 是断口面积。 试件拉断时的延伸率δ f (％)或断面收缩率ψ(％)是表示材料可承受最大塑性变形能力的指标： 矾一牮×100(4)￡fPf=盐≯×100(5)』’0式中厶和Ff是将断开的试件对合后测定的试件长度和断口处的面积。 抗拉强度靠及延伸率d或断面收缩率妒是材料性能的两个基本指标，在工程上有着广泛的应用。屈服应力民(或乱：)是金属塑性加工时变形体开始产生塑性变形所必需的最小应力，它是计算变形力的一个重要参数。 应力-应变曲线表征材料受外力作用时的行为。材料受力后即发生弹性变形，这时应力应变呈简单的线性关系，继续增加作用力至一定大小后材料将出现塑性变形，以后变形与应力的关系复杂，当塑性变形至一定程度以后，试件破断则变
应力应变计算方法
钢筋砼梁应力应变计算方法的探讨 摘要：对于钢筋砼梁应力应变的计算，分别用桥梁规范中弹性体假定的应力计算方法和以砼处于弹塑性阶段的应力计算方法进行分析，通过算例比较两者计算结果的差异，提出一些个人的见解。 关健词：桥梁工程；钢筋砼梁；应力应变值；计算方法；基本假定；弹性；弹塑性 0 前言 钢筋砼梁属于受弯构件。按《公路钢筋砼及预应力砼桥涵设计规范》(以下简称《桥规》)要求，对于钢筋砼受弯构件的设计，首先按承载能力极限状态对梁进行强度计算，从而确定构件的设计尺寸、材料、配筋量及钢筋布置，以保证截面承载能力要大于荷载效应；另外，尚需按正常使用极限状态对构件进行应力、变形、裂缝计算，验算其是否满足正常使用时的一些限值的规定。为检验钢筋砼梁的施工是否满足设计要求，均应对形成该梁的材料（钢筋及砼）进行强度检验，但由于砼的养护环境、工作条件及钢筋的加工、布置等方面，均存在试样与实际构件之间的差异，因而不能完全地说明该构件的工作性能。有时，按需要可对梁进行直接加载试验以量测荷载效应值，通过实测值与理论计算值的比较，以检验其工作性能是否能满足设计和规范的要求。通常情况下，我们不能直接测定梁体的应力值，只能通过实测梁体的应变值，进而求算其应力值。但钢筋砼结构属于非匀质材料，不能直接运用材料力学计算公式进行其应力及应变的计算，因此，本文按弹性阶段应力计算和弹塑性阶段应力计算2种方法进行分析比较。 1 按弹性阶段计算应力的方法 钢筋砼梁在使用阶段的工作状态可认为与施工阶段的工作状态相同，都处于带裂缝工作阶段，因此可按施工阶段的应力计算方法进行计算。 1.1 基本假定 《桥规》规定：钢筋砼受弯构件的施工阶段应力计算，可按弹性阶段进行，并作以下3项假定。 1.1.1 平截面假定 认为梁的正截面在梁受力并发生弯曲变形后，仍保持为平面，平行于梁中性轴的各纵向纤维的应变与其到中性轴的距离成正比，同时由于钢筋与砼之间的粘结力，钢筋与其同一水平线的砼应变相等。其表达式为： εh/x=εh′/(h0-x) εg=εh′ 式中：εh′-为与钢筋同一水平处砼受拉平均应变； εh-为砼受压平均应变； εg-为钢筋平均拉应变； x-为受压区高度； h0-为截面有效高度。 1.1.2 弹性体假定 假定受压区砼的法向应力图形为三角形。钢筋砼受变构件处在带裂缝工作阶段，砼受压区的应力分布图形是曲线形，但曲线并不丰满，与直线相差不大，可以近似地看作呈直线分布，即受压区砼的应力与应变成正比。 σh=εhEh 式中：σh-为砼应力； εh-为砼受压平均应变； E h-为砼弹性模量。 1.1.3 受拉区砼完全不能承受拉应力 在裂缝截面处，受拉区砼已大部分退出工作，但在靠近中和轴附近，仍有一部分砼承担着拉应力。由于其拉应力较小，内力偶臂也不大，因此，不考虑受拉区砼参加工作，拉应力全部由钢筋承担。 σg=εgEg 式中：σg-为钢筋应力； εg-为受拉区钢筋平均应变； E g-为钢筋弹性模量。 1.2采用换算截面计算应力 根据同一水平处钢筋应变与砼的应变相等，将钢筋应力换算为砼应力，则钢筋应力为砼应力的n g 倍（n g=E g/E h）。由上述假定得到的计算图式与材料力学中匀质梁计算图非常接近，主要区别是钢筋砼梁的受拉区不参予工作。因此，将钢筋假想为受拉的砼，形成一种拉压性能相同的假想材料组成的匀质截面，即为换算截面，再按材料力学公式进行应力计算。 1.2.1受压区边缘砼应力
数值分析插值算法源程序
#include
printf("f(%f)=%f\n",x,px); } void Hermite () //Hermite插值 { int i,k,n=2; int flag1=0; printf("Hermite插值多项式H5(x)="); for(i=0;i<=n;i++) { int flag=0; flag1++; if(flag1==1) { printf("y%d[1-2(x-x%d)*(",i,i); } else { printf("+y%d[1-2(x-x%d)*(",i,i); } for(k=0;k<=n;k++) { if(k!=i) { flag++; if(flag==1) { printf("(1/x%d-x%d)",i,k); } else { printf("+(1/x%d-x%d)",i,k);
模态参数识别方法的比较研究
模态参数识别方法的比较研究 发表时间：2017-09-07T14:07:39.937Z 来源：《防护工程》2017年第9期作者：安鹏强[导读] 本文将频域法、时域法和整体识别法识别模态参数的应用范围、存在的优缺点进行对比、分析和说明。 航天长征化学工程股份有限公司兰州分公司甘肃兰州 730050 摘要：本文将频域法、时域法和整体识别法识别模态参数的应用范围、存在的优缺点进行对比、分析和说明，对模态参数识别的研究方向具有指导意义。 关键词：模态参数识别；频域法；时域法；整体识别法 引言 多自由度线性振动系统的微分方程可以表达为[1]： [M]{x ?(t)}+[C]{x ?(t)}+[K]{x(t)}={f(t)} 通过将试验采集的系统输入与输出信号用于参数识别的方法中，进而对系统的模态质量、模态阻尼、模态刚度、模态固有频率及模态振型进行识别，这一过程称为结构的模态参数识别。本文将对模态参数识别的频域法、时域法及整体识别法三者的应用范围、存在的优缺点进行对比、分析和说明。 1频域法 模态参数识别的频域法是结合傅里叶变换理论[1]形成的，这种方法是从实测数据的频响函数曲线上对测试结构的模态参数进行估计。图解法[1]是最早的频域模态参数识别方法，随之，又陆续发展了导纳圆拟合法[2]、最小二乘迭代法[2]、有理式多项式法[2]等多种频域模态参数识别方法。 频域法的优点是直观、简便，噪声影响小，模态定阶问题易于解决。频域法识别模态参数的思路是首先借助实测频响函数曲线对模态参数进行粗略的估计，进而将初步观测的模态估计值作为一些频域识别法的最初输入值，通过反复的迭代获取最终的模态参数。频域识别方法对于实测频响函数的分布容易控制，其输人数据是主观人为的。频域中参数识别方法识别结果的精准度，取决于测试试验中获得的频响函数质量的好坏。判断实测频响函数的质量，就要看其曲线的光滑[2]和曲线的饱满程度[2]，曲线越光滑越饱满的实测频响函数，用其进行参数识别时，识别精度越高。 2时域法 模态参数识别的时域法的研究与应用比频域法晚，时域法可以克服频域法的一些缺陷。时域模态参数识别的技术优点在于无需获得激励力即可进行参数的识别[3-7]。对于一些大型的工程结构如大坝、桥梁等，获取激励荷载不太容易，但容易测得他们在风、地脉动等环境激励下的响应数据，把这些响应数据用于时域中一些参数识别的方法上，即可对测试结构的模态参数进行识别。 时域法的优点不仅在于其无需激励设备、减少测试费用而且可以避免由信号截断而造成对识别精度的影响，并且可实现对大型工程结构的在线参数识别，真实地反映结构的动力特性。但是由于响应信号中含有大量的噪声，这会使得所识别的模态中含有虚假模态。目前，对于如何剔除噪声模态、优化识别过程中的一些参数问题、以及怎样更稳定、可靠地进行模态定阶等成为时域法研究中的重要课题。目前常用的判定模态真假的方法是稳定图方法[8]，该方法的基本思想在于不同阶次的系统模型会对虚假模态的影响比较大，在稳定图中出现次数最多的模态可认为是系统的真实模态。 3整体识别法 结构模态参数识别的单输入单输出类型是针对单个响应点的数据进行相应的计算，从而得到该测点对应的模态频率、阻尼比和振型系数等动力参数，但是对于有多个测点的试验，若要用单输入单输出类型的识别方法对多自由度结构进行参数识别，则需要对各个测点单独计算来识别各个测点对应的模态参数，通过对各个测点分别计算处理，得到每一个测点数据所识别的模态参数，然后求取所有测点响应识别的算术平均值来作为整体结构最终的识别结果。理论上讲，用每个测点数据识别的结果应该是一样的，但实际测试实验中，因测试实验中测点布置位置的不同、测试中其他因素及识别方法上的不完善会使得各个测点的识别结果不同、识别精度不同及错误的识别结果等现象。因此，对于多测点的测试试验，用单输入单输出类型的识别方法进行参数识别不仅会因多次重复导致计算工作量复杂累赘而且识别结果的正确性及精度无法保证。 整体识别的方法避免了单输入单输出类型的一些不足之处。该方法通过将结构上的所有测点的实测数据同时进行识别计算，所识别得到的结果作为结构整体的模态参数，每阶模态的固有频率和阻尼比是唯一的，减小了随机误差，提高了识别进度，并且使得计算工作量大大减少。 4三种识别方法的比较分析 （1）频域内的模态参数识别方法方便、快捷，但在实际运用中人为的主观选择性对识别结果的影响较大； （2）基于环境激励的时域模态参数的识别方法具有测试试验的花费较少、测试相对安全，并且识别精度较高。因此，基于环境激励的时域模态参数的识别方法已成为科研工作者研究的热点问题。 （3）对于多测点的测试试验，用频域和时域的单输入单输出类型识别模态参数不仅会因多次重复导致计算工作量复杂累赘而且识别结果的正确性及精度无法保证。整体识别法将所有测点的数据同时进行处理计算，得到结构的整体识别结果。整体识别方法通过对所有测点数据同时进行识别计算，减小了随机误差，提高了识别进度，使得计算工作量大大减少。 （4）对比时域和频域识别方法对虚假模态的剔除，可以看出，频域中的剔除虚假模态主要依据模态频率在频幅曲线图上会出现峰值的原理，利用该峰值处的幅值角是否为0°或180°来剔除虚假模态；相对频域剔除虚假模态的方法来说，时域中的剔除虚假模态的方法有定量的精度判别指标。总体看来，时域识别方法无法判别是否已将系统的所有模态进行识别且对于阻尼比的确定还有待研究。参考文献 [1] 曹树谦，张德文，萧龙翔. 振动结构模态分析－理论、实验与应用[M]. 天津大学出版社，2001． [2] 王济，胡晓. Matlab在振动信号处理中的应用[M]. 水利水电出版社，2006.
有关模态的知识
什么是模态分析？ 你能为我解释模态分析吗？好，需要花费一点时间，但是这是任何人都能明白的事情…… 你不是第一个要求我用通俗易懂的语言解释模态分析的人，这样一来，任何人都能明白模态分析到底是怎样一个过程。简单地说，模态分析是根据用结构的固有特征，包括频率、阻尼和模态振型，这些动力学属性去描述结构的过程。那只是一句总结性的语言，现在让我来解释模态分析到底是怎样的一个过程。不涉及太多的技术方面的知识，我经常用一块平板的振动模式来简单地解释模态分析。这个解释过程对于那些振动和模态分析的新手们通常是有用的。 考虑自由支撑的平板，在平板的一角施加一个常力，由静力学可知，一个静态力会引起平板的某种静态变形。但是在这儿我要施加的是一个以正弦方式变化，且频率固定的振荡常力。改变此力的振动频率，但是力的峰值保持不变，仅仅是改变力的振动频率。同时在平板另一个角点安装一个加速度传感器，测量由此激励力引起的平板响应。 现在如果我们测量平板的响应，会注意到平板的响应幅值随着激励力的振动频率的变化而变化。随着时间的推进，响应幅值在不同的频率处有增也有减。这似乎很怪异，因为我们对此系统仅施加了一个常力，而响应幅值的变化却依赖于激励力的振动频率。具体体现在，当我们施加的激励力的振动频率越来越接近系统的固有频率（或者共振频率）时，响应幅值会越来越大，在激励力的振动频率等于系统的共振频率时达到最大值。想想看，真令人大为惊奇，因为施加的外力峰值始终相同，而仅仅是改变其振动频率。
时域数据提供了非常有用的信息，但是如果用快速傅立叶变换（FFT）将时域数据转换到频域，可以计算出所谓的频响函数（FRF）。这个函数有一些非常有趣的信息值得关注：注意到频响函数的峰值出现在系统的共振频率处，注意到频响函数的这些峰出现在观测到的时域响应信号的幅值达到最大时刻的频率处。 如果我们将频响函数叠加在时域波形之上，会发现时域波形幅值达到最大值时的激励力振动频率等于频响函数峰值处的频率。因此可以看出，既可以使用时域信号确定系统的固有频率，也可以使用频响函数确定这些固有频率。显然，频响函数更易于估计系统的固有频率。 许多人惊奇结构怎么会有这些固有特征，而更让人惊奇的是在不同的固有频率处，结构呈现的变形模式也不同，且这些变形模式依赖于激励力的频率。 现在让我们了解结构在每一个固有频率处的变形模式。在平板上均匀分布45个加速度计，用于测量平板在不同激励频率下的响应幅值。如果激励力在结构的每一个固有频率处驻留，会发现结构本身存在特定的变形模式。这个特征表明激励频率与系统的某一阶固有频率相等
模态应变能法计算方法
header for SPIE use 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,
各种模态分析方法总结与比较
各种模态分析方法总结与比较 一、模态分析 模态分析是计算或试验分析固有频率、阻尼比和模态振型这些模态参数的过程。 模态分析的理论经典定义：将线性定常系统振动微分方程组中的物理坐标变换为模态坐标，使方程组解耦，成为一组以模态坐标及模态参数描述的独立方程，以便求出系统的模态参数。坐标变换的变换矩阵为模态矩阵，其每列为模态振型。 模态分析是研究结构动力特性一种近代方法，是系统辨别方法在工程振动领域中的应用。模态是机械结构的固有振动特性，每一个模态具有特定的固有频率、阻尼比和模态振型。这些模态参数可以由计算或试验分析取得，这样一个计算或试验分析过程称为模态分析。这个分析过程如果是由有限元计算的方法取得的，则称为计算模记分析；如果通过试验将采集的系统输入与输出信号经过参数识别获得模态参数，称为试验模态分析。通常，模态分析都是指试验模态分析。振动模态是弹性结构的固有的、整体的特性。如果通过模态分析方法搞清楚了结构物在某一易受影响的频率围各阶主要模态的特性，就可能预言结构在此频段在外部或部各种振源作用下实际振动响应。因此，模态分析是结构动态设计及设备的故障诊断的重要方法。 模态分析最终目标是在识别出系统的模态参数，为结构系统的振动特性分析、振动故障诊断和预报以及结构动力特性的优化设计提供依据。 二、各模态分析方法的总结
（一）单自由度法 一般来说，一个系统的动态响应是它的若干阶模态振型的叠加。但是如果假定在给定的频带只有一个模态是重要的，那么该模态的参数可以单独确定。以这个假定为根据的模态参数识别方法叫做单自由度(SDOF)法n1。在给定的频带围，结构的动态特性的时域表达表示近似为： ()[]}{}{T R R t r Q e t h r ψψλ= 2-1 而频域表示则近似为： ()[]}}{ {()[]2ωλωψψωLR UR j Q j h r t r r r -+-= 2-2 单自由度系统是一种很快速的方法，几乎不需要什么计算时间和计算机存。 这种单自由度的假定只有当系统的各阶模态能够很好解耦时才是正确的。然而实际情况通常并不是这样的，所以就需要用包含若干模态的模型对测得的数据进行近似，同时识别这些参数的模态，就是所谓的多自由度(MDOF)法。 单自由度算法运算速度很快，几乎不需要什么计算和计算机存，因此在当前小型二通道或四通道傅立叶分析仪中，都把这种方法做成置选项。然而随着计算机的发展，存不断扩大，计算速度越来越快，在大多数实际应用中，单自由度方法已经让位给更加复杂的多自由度方法。 1、峰值检测 峰值检测是一种单自由度方法，它是频域中的模态模型为根据对系统极点进行局部估计(固有频率和阻尼)。峰值检测方法基于这样的事实：在固有频率附近，频响函数通过自己的极值，此时其实部为零(同相部分最
复合材料螺旋桨应变模态计算与试验
DOI ：１０．７４９５／j ．issn ．１００９‐３４８６．２０１５．０６．００６ 复合材料螺旋桨应变模态计算与试验 收稿日期：２０１５‐０１‐０３；修回日期：２０１５‐０６‐１２。基金项目：国家自然科学基金资助项目（５１１７９１９８）。 作者简介：黄 政（１９８９－），男，博士生，主要研究方向为复合材料螺旋桨性能。通信作者：熊 鹰（１９５８－），男，教授，博士生导师，y ing ＿xiong ２８＠１２６． com 。黄 政１，熊 鹰１，柳 凯２，杨 光１ （１．海军工程大学舰船工程系，武汉４３００３３；２．９１６９７部队，山东青岛２６６４０５） 摘 要：采用经过验证的有限元模态算法计算了金属桨和碳纤维桨模型的固有频率和位移、应变模态振型，在桨叶表面贴附电阻应变片并进行了应变模态测量试验。结果表明：铜桨和碳纤维桨的前三阶固有频率与计算值相差分别在３％和１２％以内；碳纤维桨各阶固有频率均比铜桨要小，应变模态振型相似，结构阻尼前者是后者的４倍左右。 关键词：复合材料螺旋桨；频响函数；应变模态；固有频率 中图分类号：U ６６４．３３ 文献标志码：A 文章编号：１００９－３４８６（２０１５）０６－００２７－０４ Calculation and test of composite propeller ′s strain modal HU ANG Zheng １ ，XIONG Ying １ ，LIU Kai ２ ，YANG Guang １ （１．Dept ．of Naval Architecture Engineering ，Naval Univ ．of Engineering ，Wuhan ４３００３３，China ； ２．Unit No ．９１６９７，Qingdao ２６６４０５，China ） Abstract ：T he natural frequency and displacement strain modal ′s vibration mode are calculated by means of validated finite modal algorithm ．T wo methods are used to test strain modal of propeller model blade to w hich electric resistance strain gauge sticks ．T he result show s that the deviations of first three natural frequency of copper and carbon fibre propeller tested and calculated are within ３％and １０％．T he carbon fibre propeller in similar strain modal has smaller natural frequency and four times greater structure damping than copper propeller ． Key words ：composite propeller ；frequency response function ；strain modal ；natural frequency 复合材料螺旋桨以其高弹质轻、低磁减振、可设计性强、抗冲击和抗腐蚀、阻尼减振降噪等多项优势逐渐被广泛应用。由于螺旋桨模型直径较小，但常规的加速度传感器的大质量会对小模型产生影响，而电阻应变片材薄质轻，粘贴面积小，引起的结构附连刚度也小，且能精确定位桨叶测量位置，更适合轻薄和小型结构的模态测量分析。通常，测量结构固有频率采用目前已经很成熟的位移模态测量方法，而应变模态和位移模态是同一能量平衡状态的两种表现形式，应变模态则是与位移模态相对应的应变分布状态。正如用位移 模态分析方法可建立结构振动位移响应预测模 型，应变模态分析方法也可建立振动应变响应预测模型，从而得到载荷‐应变频响函数，并求取与位移模态相对应的应变模态及有关模态参数。Young ［１］ 通过试验数据 ［２］ 验证了弯扭悬臂板 的模态和固有频率算法可行性，并用于４３８X 系列桨的模态研究，得出桨叶各阶固有频率随侧斜角的增加而下降的结论。熊家敏等 ［３］ 应用FEM ／ BEM 方法对浸入静水域中螺旋桨桨叶固有动力特性进行了理论和试验研究，计算值与试验值基 本吻合。洪毅等［４］ 对五叶复合材料螺旋桨进行了 第２７卷 第６期 ２０１５年１２月 海军工程大学学报 JOURNAL OF NAVAL UNIVERSITY OF ENGINEERING Vol ．２７ No ．６ Dec ．２０１５
钢筋混凝土梁的应力应变计算
钢筋砼梁应力应变计算方法的探讨 余海森 （江西省交通科研院南昌 330038） 摘要：对于钢筋砼梁应力应变的计算，分别用桥梁规范中弹性体假定的应力计算方法和以砼处于弹塑性阶段的应力计算方法进行分析，通过算例比较两者计算结果的差异，提出一些个人的见解。关健词：桥梁工程；钢筋砼梁；应力应变值；计算方法；基本假定；弹性；弹塑性 0 前言 钢筋砼梁属于受弯构件。按《公路钢筋砼及预应力砼桥涵设计规范》(以下简称《桥规》)要求，对于钢筋砼受弯构件的设计，首先按承载能力极限状态对梁进行强度计算，从而确定构件的设计尺寸、材料、配筋量及钢筋布置，以保证截面承载能力要大于荷载效应；另外，尚需按正常使用极限状态对构件进行应力、变形、裂缝计算，验算其是否满足正常使用时的一些限值的规定。为检验钢筋砼梁的施工是否满足设计要求，均应对形成该梁的材料（钢筋及砼）进行强度检验，但由于砼的养护环境、工作条件及钢筋的加工、布置等方面，均存在试样与实际构件之间的差异，因而不能完全地说明该构件的工作性能。有时，按需要可对梁进行直接加载试验以量测荷载效应值，通过实测值与理论计算值的比较，以检验其工作性能是否能满足设计和规范的要求。通常情况下，我们不能直接测定梁体的应力值，只能通过实测梁体的应变值，进而求算其应力值。但钢筋砼结构属于非匀质材料，不能直接运用材料力学计算公式进行其应力及应变的计算，因此，本文按弹性阶段应力计算和弹塑性阶段应力计算2种方法进行分析比较。 1 按弹性阶段计算应力的方法 钢筋砼梁在使用阶段的工作状态可认为与施工阶段的工作状态相同，都处于带裂缝工作阶段，因此可按施工阶段的应力计算方法进行计算。 1.1 基本假定 《桥规》规定：钢筋砼受弯构件的施工阶段应力计算，可按弹性阶段进行，并作以下3项假定。 1.1.1 平截面假定 认为梁的正截面在梁受力并发生弯曲变形后，仍保持为平面，平行于梁中性轴的各纵向纤维的应变与其到中性轴的距离成正比，同时由于钢筋与砼之间的粘结力，钢筋与其同一水平线的砼应变相等。其表达式为： εh/x=εh′/(h0-x) εg=εh′ 式中：εh′-为与钢筋同一水平处砼受拉平均应变； εh-为砼受压平均应变； εg-为钢筋平均拉应变； x-为受压区高度； h0-为截面有效高度。 1.1.2 弹性体假定 假定受压区砼的法向应力图形为三角形。钢筋砼受变构件处在带裂缝工作阶段，砼受压区的应力分布图形是曲线形，但曲线并不丰满，与直线相差不大，可以近似地看作呈直线分布，即受压区砼的 应力与应变成正比。 σh=εhEh 式中：σh-为砼应力； εh-为砼受压平均应变； E h-为砼弹性模量。 1.1.3 受拉区砼完全不能承受拉应力 在裂缝截面处，受拉区砼已大部分退出工作，但在靠近中和轴附近，仍有一部分砼承担着拉应力。由于其拉应力较小，内力偶臂也不大，因此，不考 虑受拉区砼参加工作，拉应力全部由钢筋承担。 σg=εgEg 式中：σg-为钢筋应力； εg-为受拉区钢筋平均应变； E g-为钢筋弹性模量。 1.2采用换算截面计算应力 根据同一水平处钢筋应变与砼的应变相等，将钢筋应力换算为砼应力，则钢筋应力为砼应力的n g 倍（n g=E g/E h）。由上述假定得到的计算图式与材料力学中匀质梁计算图非常接近，主要区别是钢筋砼梁的受拉区不参予工作。因此，将钢筋假想为受拉的砼，形成一种拉压性能相同的假想材料组成的匀质截面，即为换算截面，再按材料力学公式进行应
数值分析常用的插值方法
数值分析报告 班级： 专业： 流水号： 学号： 姓名：
常用的插值方法 序言 在离散数据的基础上补插连续函数，使得这条连续曲线通过全部给定的离散数据点。插值是离散函数逼近的重要方法，利用它可通过函数在有限个点处的取值状况，估算出函数在其他点处的近似值。 早在6世纪，中国的刘焯已将等距二次插值用于天文计算。17世纪之后，牛顿、拉格朗日分别讨论了等距和非等距的一般插值公式。在近代，插值法仍然是数据处理和编制函数表的常用工具，又是数值积分、数值微分、非线性方程求根和微分方程数值解法的重要基础，许多求解计算公式都是以插值为基础导出的。 插值问题的提法是：假定区间[a，b〕上的实值函数f（x）在该区间上n＋1个互不相同点x0，x1……x n处的值是f（x0），……f（x n），要求估算f（x）在[a，b〕中某点的值。其做法是：在事先选定的一个由简单函数构成的有n＋1个参数C0， C1，……C n的函数类Φ(C0，C1，……C n)中求出满足条件P（x i）＝f（x i）（i＝0，1，……n）的函数P(x)，并以P(x)作为f(x)的估值。此处f（x）称为被插值函数，x0，x1，……xn 称为插值结(节)点，Φ(C0，C1，……C n)称为插值函数类，上面等式称为插值条件，Φ（C0，……C n）中满足上式的函数称为插值函数，R（x）＝f（x）－P（x）称为插值余项。
求解这类问题，它有很多种插值法，其中以拉格朗日(Lagrange)插值和牛顿(Newton)插值为代表的多项式插值最有特点，常用的插值还有Hermit 插值，分段插值和样条插值。 一．拉格朗日插值 1.问题提出： 已知函数()y f x =在n+1个点01,,,n x x x L 上的函数值01,,,n y y y L ，求任意一点 x '的函数值()f x '。 说明：函数()y f x =可能是未知的；也可能是已知的，但它比较复杂，很难计算其函数值()f x '。 2.解决方法： 构造一个n 次代数多项式函数()n P x 来替代未知（或复杂）函数()y f x =，则 用()n P x '作为函数值()f x '的近似值。 设()2012n n n P x a a x a x a x =++++L ，构造()n P x 即是确定n+1个多项式的系数 012,,,,n a a a a L 。 3.构造()n P x 的依据： 当多项式函数()n P x 也同时过已知的n+1个点时，我们可以认为多项式函数 ()n P x 逼近于原来的函数()f x 。根据这个条件，可以写出非齐次线性方程组： 20102000 20112111 2012n n n n n n n n n n a a x a x a x y a a x a x a x y a a x a x a x y ?++++=?++++=?? ? ?++++=?L L L L L 其系数矩阵的行列式D 为范德萌行列式： ()20 0021110 2111n n i j n i j n n n n x x x x x x D x x x x x ≥>≥= = -∏L L M M M M L
模态分析与参数识别
模态分析方法在发动机曲轴上的应用研究 xx （xx大学 xxxxxxxx学院 , 山西太原 030051） 摘要：综述模态分析在研究结构动力特性中的应用，介绍模态分析的两大方法：数值模态分析与试验模态分析。并着重介绍目前的研究热点一一工作模态分析。通过发动机曲轴的模态分析这一具体的实例，综述了运行模态分析国内外研究现状，指出了其关键技术、存在问题以及研究发展方向。 关键词：模态分析数值模态试验模态工作模态 Abstract ：Sums up methods of model analysis applied on the research of configuration dynamic;al characteristio. It introduces two methods of model analysis: numerical value model analysis and experimentation model analysis. Then it stresses the hotspot-working model analysis.Some key techniques, unsolved problems and research directions of OMA were also discussed. Key words:Model analysis Numerical value model analysis Experimentation model analysis Working model analysis 1、引言 1.1模态分析的基本概念 物体按照某一阶固有频率振动时，物体上各个点偏离平衡位置的位移是满足一定的比例关系的，可以用一个向量表示，这个就称之为模态。模态这个概念一般是在振动领域所用，你可以初步的理解为振动状态，我们都知道每个物体都具有自己的固有频率，在外力的激励作用下，物体会表现出不同的振动特性。 一阶模态是外力的激励频率与物体固有频率相等的时候出现的，此时物体的振动形态叫做一阶振型或主振型；二阶模态是外力的激励频率是物体固有频率的两倍时候出现，此时的振动外形叫做二阶振型，以依次类推。