Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting the peak stresses an extening the fatigue life of structural elements an machine parts subjecte to near-resonant cyclic forces or to suenly applie forces. It is a esirable property if noise reuction is of importance. On the other han, amping is unesirable if internal heating is to be avoie. It also can be a source of ynamic instability of rotating shafts an of error in sensitive instruments. Resonant vibrations of large amplitue are encountere in a variety of moern evices, frequently causing rough an noisy operation an, in extreme cases, leaing to seriously high repeate stresses. Various types of amping may be employe to minimize these resonant vibration amplitues. Design sensitivity analysis usually refers to the stuy of the effect of parameter changes on the result of an optimization proceure or an eigenvalue eigenvector computation. In particular, if a esign change causes a system parameter to change, the eigen solution can be compute without having to recalculate the entire eigenvalue / eigenvector set. This is also referre to as a reanalysis proceure an sometimes falls uner the heaing of structural moification. This section evelops the equations for iscussing the sensitivity of natural frequencies an moe shapes for conservative systems. The motivation for stuying such methos comes from examining the large-orer ynamical systems often use in current vibration technology. Making changes in large systems is part of the esign process. However, large amounts of computer time are require to fin the solution of the reesigne system. It makes sense, then, to evelop efficient methos to upate existing solutions when small esign changes are mae in orer to avoi a complete reanalysis. In aition, this approach can provie insight into the esign process.
Two general types of units are use to specify the amping properties of structural materials: (1) the energy issipate per cycle in a structural element or test specimen an (2) the ratio of this energy to a reference strain energy or elastic energy. Absolute amping energy units are: D T = total amping energy issipate by entire specimen or structural element per cycle of vibration, N.m/cycle D Avg = average amping energy, etermine by iviing total amping energy D T by volume V 0 of specimen or structural element which is issipating energy, N.m/m 3 /cycle D = specific amping energy, work issipate per unit volume an per cycle at a point in the specimen, N.m/m 3 /cycle Of these absolute amping energy units, the total energy D T usually is of greatest interest to the engineer. The average amping energy D Avg epens upon the shape of the specimen or structural element an upon the nature of the stress istribution in it, even though the specimens are mae of the same material an have been subjecte to the same stress istribution at the same temperature an frequency. Thus, quote values of the average amping energy in the technical literature shoul be viewe with some reserve. The specific amping energy D is the most funamental of the three absolute units of amping since it epens only on the material in question an not on the shape, stress istribution, or volume of the vibrating element. However, most of the methos iscusse previously for measuring amping properties yiel ata on total amping energy D T rather than on specific amping energy D. Therefore, the evelopment of the relationships between these quantities assumes importance. If the specific amping energy is integrate throughout the stresse volume, = (6.11) This is a triple integral; V = x y z an D is regare as a function of the space coorinates x, y, z. If there is only one nonzero stress component, the specific amping energy D may be consiere a function of the stress level σ. Then
= (6.12) In this integration, V is the volume of material whose stress level is less than σ. The integration is a single integral, an σ is the peak stress. The integrans may be put in imensionless form by introucing D, the specific amping energy associate with the peak stress level reache anywhere in the specimen uring the vibration (i.e., the value of D corresponing to σ = σ ). Then = (6.13) = (6.14) The average amping energy is = = (6.15) The relationship between the amping energies D T, D Avg, an D epens upon the imensionless amping energy integral α.the integran of α may be separate into two parts: (1) a amping function D/D which is a property of the material an (2) a volume-stress function / which epens on the shape of the part an the stress istribution. Several approaches are available for performing a sensitivity analysis. The one presente here is base on parameterizing the eigenvalue problem. Consier a conservative n-egree-of-freeom system efine by M(α)q (t) + K(α)q(t) = 0 (6.16) where the epenence of the coefficient matrices on the esign parameter α is inicate. The parameter α is consiere to represent a change in the matrix M an/or the matrix K. The relate eigenvalue problem is
M (α)k(α)u (α) = λ (α)u (α) (6.17) Here, the eigenvalue λ (α) an the eigenvector u (α) will also epen on the parameter α. The mathematical epenence is iscusse in etail by Whitesell (1980). It is assume that the epenence is such that M, K, λ (α) an u (α) are all twice ifferentiable with respect to the parameter α. Proceeing, if u (α) is normalize with respect to the mass matrix, ifferentiation of Equation with respect to the parameter α yiels (λ ) = u (K) λ (M) u (6.18) α α α Here, the epenence of α has been suppresse for notational convenience. The secon erivative of λ can also be calculate as α (λ ) = 2u + u α (K) λ (M) u α α (K) α (λ ) α (M) λ α (M) u (6.19) The notation u enotes the erivative of the eigenvector with respect to u. The expression for the secon erivative of λ requires the existence an computation of the erivative of the corresponing eigenvector. For the special case where M is a constant, an with some manipulation (Whitesell, 1980), the eigenvector erivative can be calculate from the relate problem for the eigenvector v i from the formula (V ) = C (i, α)v (6.20) α where the vectors V are relate to u by the mass transformation V = M 1 /2u The coefficients C (i, α) in this expansion are given by
O. C (i, α) =. u u λ λ α (6.21) Where the matrix A is the symmetric matrix M 1/2 KM 1/2 epening on α. Above equations yiel the sensitivity of the eigenvalues an eigenvectors of a conservative system to changes in the stiffness matrix. More general an computationally efficient methos for computing these sensitivities are available in the literature. Ahikari an Friswell (2001) give formulae for ampe systems an reference to aitional methos. Another metho for esign of amping is taken as a laminate metal material, which offers an effective metho to increase the inherent level of amping in sheet-metal components. To assist the prouct esigner consiering the use of laminate metal material in place of traitional sheet metal, various practical moeling techniques are available that can be use both as a amping preiction an esign optimization tool. Optimization of the laminate construction, as with all constraine layer type treatments, is a function of other parameters in aition to the actual properties of the viscoelastic material. This complexity offers more esign flexibility as the thickness an type of the amping core as well as the constraining layers can be altere to optimize effectiveness of the laminate metal prouct. Two specific approaches are available to help assist in the selection an esign of viscoelastic-base amping treatments. Simplifie RKU approach: One approach is to simplify a real worl component own to an equivalent 3- layer beam or plate system. This was first suggeste by Ross, Kerwin, Ungar, an the RKU metho uses a fourth orer ifferential equation for a uniform beam with the sanwich construction of the 3-layer laminate system represente as an equivalent complex stiffness. The equation for the flexural rigiity, EI, of this system has been reporte in many technical references, an is therefore not uplicate here. The most common assumption mae when using this metho is that the moe shapes of the theoretical structure are sinusoial in nature, therefore implying a simply-supporte bounary
conition. When using this approach with other bounary conitions, which may be necessary in working with actual structures, approximations must be mae in the results epening on the moe shape in question. The RKU metho is better suite as a amping inicator as oppose to a precise amping preictor when applie to complex, real worl structures. The goal is to use this simplifie metho to evelop esign trens that will lea to the selection of a amping material, constraining layers, an thickness which yiel optimize amping performance. Moal Strain Energy: Another preiction metho known as the Moal Strain Energy (MSE) approach utilizes a finite element analysis (FEA) representation of a structure as the basis for of moeling the amping effect. This metho has been shown to be an accurate preictor of amping levels in structures comprising layers of elastic an viscoelastic elements. The MSE principle states that the ratio of composite system loss factor to the viscoelastic material loss factor for a given moe of vibration can be estimate from the ratio of elastic strain energy in the viscoelastic elements to the total strain energy in the moel for a given moe. This is shown mathematically in the following equation: Typically, the MSE approach is use in conjunction with an unampe, normal moes analysis to compute the strain energy ratio. The strain energies are etermine from the relative moe shapes. It is assume that the viscoelastic properties are linear in terms of the ynamic strain rate. = (6.22) Where, m s = System amping for nth moe of vibration VEM = material amping for appropriate frequency an temperature m VEM = elastic strain energy store in viscoelastic core m total = total strain energy for n th moe shape Example 2 Consier the system iscusse previously in example 1. Here, take M =I, an K becomes
The eigenvalues of the matrix are λ, K = 3 1 1 3 = K = 2, 4 an the normalize eigenvectors are u1= v1=(1/ 2)[1 1]T an u2=v2=(1/ 2)[ 1 1]T. It is esire to compute the sensitivity of the natural frequencies an moe shapes of this system as a result of a parameter change in the stiffness of the spring attache to groun. To this en, suppose the new esign results in a new stiffness matrix of Then an K(a) = 3 + α 1 1 3 α (M) = 0 0 0 0 α (K) = 1 0 0 0 Following Equations (6.25) an (6.27), the erivatives of the eigen values an eigenvectors become λ α = 0.5, λ α = 0.5, u α = 1 4 2 1 1, u α = 1 4 2 1 1 These quantities are an inication of the sensitivity of the eigen solution to changes in the matrix K. To see this, substitute the preceing expressions into the expansions for λ (α) an u (α) are, λ (α) = 2 + 0.5α λ (α) = 4 + 0.5α u (α) = 0.707 1 1 + 0.177α 1 1 u (α) = 0.707 1 1 0.177α 1 1
This last set of expressions allows the eigenvalues an eigenvectors to be evaluate for any given parameter change α without having to resolve the eigenvalue problem. These formulae constitute an approximate reanalysis of the system. It is interesting to note this sensitivity in terms of a percentage. Define the percentage change in λ λ (α) λ λ (2 + 0.5α) 2 100% = 100% = (25%)α 2 If the change in the system is small, say α = 01, then the eigenvalue λ changes by only 2.5%, an the eigenvalue λ changes by 1.25%. On the other han, the change in the elements of the eigenvector u is 2.5%. Hence, in this case the eigenvector is more sensitive to parameter changes than the eigenvalue is. By computing higher-orer erivatives of λ an u, more terms of the expansion can be use, an greater accuracy in preicting the eigensolution of the new system results. By using the appropriate matrix computations, the subsequent evaluations of the eigenvalues an eigenvectors as the esign is moifie can be carrie out with substantially less computational effort (reportely of the orer of n2 multiplications). The sort of calculation provie by eigenvalue an eigenvector erivatives can provie an inication of how changes to an initial esign will affect the response of the system. In the example, the shift in value of the first spring is translate into a percentage change in the eigenvalues an hence in the natural frequencies. If the esign of the system is concerne with avoiing resonance, then knowing how the frequencies shift with stiffness is critical.