The modal strain energy (MSE) method has been widely accepted as an effective and practical means for predicting the modal parameters of complex structures treated with viscoelastic damping treatments. The method is based on estimating the modal strain energies of the structure and the viscoelastic material (VEM) by using the undamped (real) mode shapes of the structure/VEM assembly instead of the exact damped (complex) mode shape. Such an approximation makes it easy to integrate the MSE with commercial finite element codes that generally do not use complex eigenvalue problem solvers. The theoretical basis of the original MSE method and several of its modified versions are presented. Application of the method to various types of viscoelastic damping treatments is discussed and compared with the predictions obtained by using the exact complex eigenvalue problem solvers and the Golla–Hughes–McTavish Model (GHM) approach discussed in Chapter 4.
The MSE method was originally introduced by Kerwin and Ungar (1962) and then extended by Johnson and Kienholz (1982) as an approximate means for predicting the modal parameters of complex structures treated with viscoelastic damping treatments. The method is based on describing the viscoelastic material by the “complex modulus” approach and therefore, it is limited to frequency domain analysis. To account for the variation of the VEM properties with frequency, the MSE method becomes iterative in nature. In spite of these two limitations, the MSE method has been widely accepted because of its practicality and ease of integration with finite element models of structures treated with VEM without increasing the size of models as in the case of the GHM method (Golla and Hughes 1985).
The theory behind the MSE method starts by describing the dynamics of the structure/VEM system by the following finite element equation:
where {X} is the nodal deflection vector of the structure, [M] its mass matrix (real), and [K] is its stiffness matrix, which is complex to account for the VEM.
The stiffness matrix can be written as
where [Ke] and [Kv] are the stiffness matrices of the elastic structure and VEM, respectively. Also, and are the stiffness matrices corresponding to the storage and the dissipative components of the VEM stiffness, respectively. In Eq. (5.2), ηv denotes the loss factor of the VEM. Also, [KR] and [KI] denote the total elastic stiffness matrix of the structure/VEM and the dissipative stiffness matrix of the VEM. These matrices are given by
To cast the finite element model of Eq. (5.1) as an eigenvalue problem, the solution {X} can be written as
where and denote the eigenvector (mode shape) and eigenvalue (natural frequency) of the structure/VEM at mode n. These two quantities are complex and can be described as follows
where and are the real and imaginary components of the eigenvector . Also, ωn and ηn are the natural frequency and the loss factor of the nth mode.
Substituting Eq. (5.3) into Eq. (5.1) gives the following complex eigenvalue problem
Solution of this problem requires an iterative approach to account for the variation of both the storage and dissipative components of the stiffness matrix [K] with the frequency.
Substituting Eqs. (5.3) and (5.5) into Eq. (5.6) yields the following
Equating the real and imaginary parts on the two sides of Eq. (5.5), gives
and
Pre‐multiplying Eqs. (5.8) and (5.9) by gives
and
Dividing Eq. (5.11) by Eq. (5.10) and substituting for [KI] using Eq. (5.3) gives the modal loss factor ηn as follows
The MSE method simplifies Eq. (5.12) by replacing the exact damped (complex) mode shape by the undamped (real) mode shapes of the structure/VEM assembly . Such simplification makes it easy to integrate the MSE with commercial finite element codes that generally do not use complex eigenvalue problem solvers.
This yields the following expression
Physically, Eq. (5.13) means
that is, the loss factor of a structure/VEM assembly at the nth mode is equal to the loss factor of the VEM at the same frequency multiplied by the ratio of the MSE of the VEM to that of the structure/VEM assembly.
Implementation of MSE method is carried out according to the iterative scheme shown in Figure 5.1 in order to account for the variation of the VEM properties with frequency, the MSE method becomes in nature.
Table 5.1 Comparison between the predictions of GHM and MSE methods for the rod/unconstrained VEM system.
Method | GHM | MSE | ||
Mode | Freq. (Hz) | Damping ratio | Freq. (Hz) | Damping ratio |
1 | 1103.50 | 0.002 42 | 1102.93 | 0.00 241 |
2 | 3869.42 | 0.001 57 | 3865.52 | 0.001 56 |
Table 5.2 Convergence of the iterative solution of the MSE method for the rod/unconstrained VEM system.
Iteration | Mode 1 (Hz) | Mode 2 (Hz) | ζ1 | ζ2 |
1 | 1101.41 | 3847.69 | 0.000 17 | 0.000 036 |
2 | 1102.93 | 3865.51 | 0.002 40 | 0.001 56 |
3 | 1102.93 | 3865.52 | 0.002 41 | 0.001 56 |
Table 5.1 lists the natural frequencies and the corresponding modal damping ratios (MDRs) for the rod/unconstrained VEM system as obtained by the GHM, and the MSE methods. Table 5.2 lists the results of the iterative solution and its convergence when using the MSE method.
The displayed results suggest that the MSE method accurately predicts the modal parameters of the considered case. Furthermore, the MSE method converges after three iterations to the final modal parameters.
Table 5.3 Comparison between the predictions of the GHM and MSE methods for the rod/constrained VEM system.
Method | GHM | MSE | ||
Mode | Frequency (Hz) | Damping ratio | Frequency (Hz) | Damping ratio |
1 | 1113.06 | 0.0489 | 1103.81 | 0.0179 |
2 | 3757.96 | 0.0123 | 3740.42 | 0.0136 |
3 | 5191.08 | 0.0869 | 5071.87 | 0.0959 |
4 | 6417.20 | 0.0341 | 6395.56 | 0.0353 |
Table 5.3 lists the natural frequencies and the corresponding MDRs for the rod/constrained VEM system as obtained by the GHM and the MSE methods.
The displayed results show that the predictions of the modal parameters by the MSE are inaccurate for the considered case as compared to the predictions of the exact or the GHM methods. Such inaccuracy is attributed to the fact that approximating the exact (complex) eigenvectors by the undamped (real) eigenvectors is far from accurate because the loss factor is high in this case.
For this reason, several modified versions of the original MSE method have been developed to improve the estimates of the eigenvectors.
Four modified versions of the original MSE method will be discussed in this section. These methods aim at developing improved eigenvectors to account for the imaginary component that has been neglected in the original MSE method. These methods range from heuristic methods, as the Weighted Stiffness Matrix (WSM) method (Hu et al. 1995) and the Weighted STorage Modulus method (WSTM) (Xu et al. 2002) to the more rigorous methods as the Improved Reduction System method (O'Callahan 1989; Scarpa et al. 2002) and the low frequency approximation (LFA) method (Scarpa et al. 2002).
This method is based on substituting Eq. (5.5) into Eq. (5.5) to give
Equating the real part on both sides of this equation gives
Assume a vector to be defined such that:
Then, Eq. (5.15) reduces to
or
where [KM] = [KR] + β[KI] , β = a/b, and .
Eq. (5.16) represents a modified eigenvalue problem that has real eigenvalue and eigenvector . Note that [KM] is a modified stiffness matrix that augments the elastic stiffness matrix [KR] with a weighted contribution of the imaginary component of the stiffness [KI]. The weighting parameter β is calculated from the following empirical formula proposed by Hu et al. (1995)
If β = 0, the modified eigenvalue problem reduces to that used in computing the real eigenvectors employed by the MSE method. If β ≠ 0, the modified eigenvalue problem attempts to heuristically account for the imaginary component of the stiffness in order to generate a better estimate of the real eigenvector. This estimate is used to compute the loss factor ηn of the nth mode as follows
In this method, the shear modulus of the VEM as described by
has a storage modulus G′ and loss factor ηv. Hence, to generate the real eigenvectors only G′ is used because it directly affects the real stiffness matrix of the VEM. However, a better estimate of the real eigenvectors can be obtained if the storage modulus is modified to account for the dissipative part. Xu et al. (2002) proposed to modify the storage modulus as follows
In this manner, the modified storage modulus is the magnitude of the shear modulus that augments the storage modulus by the contribution of the loss modulus. Such a heuristic modification was motivated by the fact that increasing the storage modulus increases the natural frequencies and the observations that the natural frequencies increase with increasing the loss factor of the VEM as reported by Xu and Chen (2000) when using the exact complex eigenvalue problem solvers.
The loss factor of the structure/VEM system is then determined from
where is the elastic component of the stiffness matrix of the VEM as modified by the weighted storage modulus, that is, .
Also, is the eigenvector of the following eigenvalue problem:
This method is based on assuming the following modal transformation
where [Φ] is the eigenvector matrix for the undamped part of Eq. (5.1), that is,
such that
Also, in Eq. (5.22), {q} denotes the modal displacement vector, which is given by
But, for the damped system we have
Substituting Eqs. (5.22) and (5.24) into Eq. (5.25) gives
Pre‐multiplying Eq. (5.26) by [Φ]Tand using Eq. (5.23) gives
where .
Equating the real and imaginary parts on both sides of Eq. (5.23), gives the following matrix equation
or
Using static condensation, the second row of Eq. (5.28) gives
and
Hence, the condensed system can be obtained by combining Eqs. (5.28) and (5.30) to yield the following
where
and
Solution of the eigenvalue problem given by Eq. (5.31) yields the eigenvalue ω and the eigenvector {qr}. The full complex eigenvector can be extracted as follows
This eigenvector can be used to compute the modal loss factor of the structure/VEM assembly as follows
This method is based on expanding the second row of Eq. (5.28) to give
or
For low frequencies, the Taylor series expansion of Eq. (5.29) in terms of ω is given by
Note that the first term of Eq. (5.35) is corresponding to the static condensation Eq. (5.29). In this manner, Eq. (5.35) includes the contribution of the inertia terms and accordingly presents a dynamic condensation of {qi) in terms of {qr}.
Expanding the first row of Eq. (5.28) gives:
Combining Eqs. (5.35) and (5.36) gives
Note that β is given by Eq. (5.17) as suggested by Hu et al. (1995).
Equation (5.37) presents a condensation equation with the transformation
Then, solving the original system given by Eq. (5.28) reduces to the following condensed system:
where
and
Solution of the eigenvalue problem given by Eq. (5.39) yields the eigenvalue ω and the eigenvector {qr}. The full complex eigenvector can be reconstructed as follows
This eigenvector can be used to compute the modal loss factor of the structure/VEM assembly as follows
Table 5.4 Comparison between the natural frequency predictions of the different modified MSE methods for the rod/constrained VEM system.
Mode | GHM | MSE | Weighted stiffness | Weighted storage | IRS | LFA |
1 | 1113.06 | 1103.81 | 1105.26 | 1109.67 | 1111.89 | 1112.33 |
2 | 3757.96 | 3740.42 | 3740.81 | 3748.69 | 3746.69 | 3744.90 |
3 | 5191.08 | 5071.87 | 5072.19 | 5118.93 | 5059.11 | 5055.82 |
4 | 6417.20 | 6395.56 | 6395.32 | 6409.19 | 6391.55 | 6396.75 |
Table 5.5 Comparison between the damping ratio predictions of the different modified MSE methods for the rod/constrained VEM system.
Mode | Exact | MSE | Weighted stiffness | Weighted storage | IRS | LFA |
1 | 0.0049 | 0.0179 | 0.0078 | 0.0049 | 0.0062 | 0.0346 |
2 | 0.0123 | 0.0136 | 0.0124 | 0.0124 | 0.0132 | 0.0135 |
3 | 0.0869 | 0.0959 | 0.0943 | 0.0942 | 0.0964 | 0.0952 |
4 | 0.0341 | 0.0353 | 0.0352 | 0.0352 | 0.0354 | 0.0361 |
Tables 5.4 and 5.5 list the natural frequencies and the corresponding MDRs for the rod/constrained VEM system as obtained by the exact eigenvalue problem solver of MATLAB, the MSE method, and four modified MSE methods.
It is clear that the four modified methods have improved the accuracy of the MSE. All the methods provide adequate predictions of the natural frequencies. Also, all the methods have predicted accurately the damping ratios except for the first mode with the exception of the LFA method.
Table 5.6 summarizes the basic equations that are used to compute the modal loss factor using the conventional or the modified MSE methods. The table also lists the different forms of the eigenvectors needed to predict the modal loss factors for the considered MSE methods. Note that the MSE, WSM, and WSTM methods all use real eigenvectors while the IRS and LFA methods use imaginary eigenvectors.
Table 5.6 The basic equations used to determine the modal loss factor using the conventional or the modified MSE methods.
Method | Modal loss factor | Eigenvectors | |
MSE | The real eigenvector { ϕn} is solution of: |
||
MODIFIED MSE | Weighted Stiffness Matrix Method | The real eigenvector is solution of: where β = trace[KI]/trace[KR], |
|
Weighted Storage Modulus Method | The real eigenvector is solution of: where |
||
Improved Reduction System Method (IRS) |
The imaginary eigenvector { ϕ*} is given by: {ϕ*} = [Φ][[I] + i[S]]{qr} where , , with [Λ]and [Φ] are the eigenvalues and vectors of and {qr} = eigenvector of { [Kc] − ω2[Mc] } {qr} = 0 with [Mc] = [I] + [S]T[S], |
||
Low Frequency Approximation Method (LFA) |
The imaginary eigenvector is given by: where , β = trace[KI]/trace[KR], and {qr} = eigenvector of with , |
The MSE, as described in Sections 5.1 through 5.4, serves as an important design metric for selecting the optimal design parameters (Lepoittevin and Kress 2009; Sainsbury and Masti 2007); location (Ro and Baz 2002); and topology of damping treatments (Ling et al. 2010).
Figure 5.2 summarizes the basic concept behind using the MSE as a design metric. For a given base structure (i.e., known [Ke] and [M]), an initial guess of the design parameters and/or topology (i.e., [Kv]), of the VEM is input to the MSE module to determine the modal loss factors ηn for the first N modes. These factors can be maximized for a particular mode or a group of critical modes by adjusting the design parameters and/or topology (i.e., [Kv]), of the VEM, in a rational manner, using available optimization tools such as the MATLAB Optimization Toolbox. This process is repeated until an optimal configuration of the VEM is attained while satisfying a set of design constraints.
In this section, the MSE will be utilized in selecting the optimal thickness of unconstrained damping layers which are used to treat rods undergoing longitudinal vibrations.
In order to illustrate the utility of the MSE as a design metric, consider the following example:
The design problem is formulated mathematically as follows:
In this optimum design problem, the objective function is written as the ratio of the sum of the modal loss factor for the first five modes to the total weight of the treatment. In this manner, maximizing F will simultaneously ensure maximizing the modal loss factor and minimizing the total weight. The constraints imposed on the design problem ensure that all the design variables tvi are positive and each is bounded from below and from above by tvmin and tvmax, respectively.
Note that the lower bound is selected to ensure adequate damping and avoid reaching the unrealistic trivial solution where all the design variables tvi vanish, which makes the mass of the treatment minimum (= zero) and the objective function maximum (= ∞). However, the upper bound is selected to avoid an impractically thick VEM.
Consider the following objective functions:
Two sets of constraints are imposed:
Set 1: tvmin = 0.001 and tvmax = 0.01 m
Let the initial guess of the thickness distribution is [tv1, tv2, …, tv5] = [0.005 0.005 0.005 0.005 0.005]. The MATLAB solution of the optimization problem using the “fmincon” subroutine of the Optimization Toolbox gives optimal thicknesses = [0.01 0.01 0.01 0.01 0.01] as displayed in Figure 5.4. The corresponding value of the objective function F1 = 0.002 043.
Set 2: tvmin = 0.001 and tvmax = 0.025 m
If the initial guess of the thickness distribution is maintained at [tv1, tv2, …, tv5] = [0.005 0.005 0.005 0.005 0.005]. The MATLAB solution of the optimization problem using the “fmincon” subroutine of the Optimization Toolbox gives optimal thicknesses = [0.025 0.025 0.025 0.025 0.025] as displayed in Figure 5.5. The corresponding value of the objective function F1 = 0.005 44.
In both cases, the optimum is attained when the thickness of each element attains the allowable upper bound in order to maximize the sum of the modal loss factor for the first five modes. Accordingly, the optimization algorithm pushes the VEM thickness to its maximum limit without any regard to the weight.
In this case, the objective function F2 puts a penalty on the weight of the VEM.
Two sets of constraints are imposed:
Set 1: tvmin = 0.001 and tvmax = 0.025 m
If the initial guess of the thickness distribution is [tv1, tv2, …, tv5] = [0.005 0.005 0.005 0.005 0.005]. MATLAB Optimization Toolbox gives optimal thicknesses = [0.025 0.001 0.001 0.001 0.001] as displayed in Figure 5.6. The corresponding value of the objective function F2 = 0.002 28.
Set 2: tvmin = 0.01 and tvmax = 0.025 m
If the initial guess of the thickness distribution is [tv1, tv2, …, tv5] = [0.01 0.01 0.01 0.01 0.01]. MATLAB Optimization Toolbox gives optimal thicknesses = [0.025 0.012 0.01 0.01 0.01] as displayed in Figure 5.7. The corresponding value of the objective function F2 = 0.001 73.
Figure 5.8a,b show the time response of the rod when treated with the optimal damping treatments corresponding to constraint sets 1 and 2, which are imposed on objective function F2. The rod in both cases is subjected to a unit impulse at its free end.
It is important to note that although the optimal objective function for the first set of constraints is F2 = 0.002 28 and that for the second set is F2 = 0.001 73, the vibration damping characteristics for the second set is better than the first. This is attributed to the fact that the weight of the VEM for the second set is 2.39 times that of the first set. Hence, the sum of the modal loss factors for the second set is almost twice that of the first set. Accordingly, more damping is achieved with the second set of constraints in spite of the lower value of the objective function.
Engineered Damping Treatments (EDTs) that have high damping characteristics per unit volume are presented in this section. The EDTs under consideration consist of cellular viscoelastic damping matrices with optimally selected cell configuration, size, and distribution. These perforated EDTs are intended to replace conventional viscoelastic damping treatments to improve their damping characteristics and reduce their weight at the same time. Examples of such perforated EDTs are shown in Figure 5.9. The number, shape, and spacing of the perforations are critical to the effective damping characteristics of the treatment and to the minimization of its weight. Figure 5.9a–c shows a conventional damping treatment, treatment with square holes that has a positive Poisson's ratio, and treatment with reentrant hexagon holes that has a negative Poisson's ratio.
The cellular topologies of the EDTs are modeled using the finite element method in an attempt to determine the optimal topologies that maximize the strain energy, maximize the damping characteristics, and minimize the total weight. The damping characteristics of the manufactured EDTs are evaluated and compared with the corresponding characteristics obtained by conventional solid damping treatments in order to emphasize the importance of using optimally configured damping treatment to achieve high damping characteristics.
Consider the configuration shown in Figure 5.10. It consists of a base plate covered from one side with a layer of VEM. The base plate is isotropic and linearly elastic with density, elasticity modulus, and Poisson's ratio of ρp, Ep, vp, respectively. The VEM layer properties are denoted by ρv, Ev, vv where the complex elasticity modulus Ev = E0(1 + jη) is used to describe the viscoelastic properties of the layer. The viscoelastic layer is conventionally solid with constant thickness or it can have a variable thickness to maximize the damping behavior.
Figure 5.10b shows a finite element of this composite. The element is a four‐noded rectangular element with dimensions 2a × 2b while the thicknesses of the base layer and the treatment are hp and hv, respectively. The plate element is aligned to the x–y‐plane. The displacement components of the base plate are u, v, and w in the x, y, and z directions. Also, θx and θy are the rotational components about the x and y directions. The displacement vector u of the base plate is given by: u = {u v w θx θy}T. Accordingly, each node has five degrees of freedom corresponding to the five displacement components. Furthermore, the base plate is assumed to be thin and hence the strain components εxz, εyz, and εzz vanish. According to this assumption, the rotation degrees of freedom can be expressed in terms of the gradients of the lateral deflection such that
The other strain components are
Within the finite element, the displacement vector is approximated as
where q = {p1 p2 p3 p4}T= nodal deflection vector of the four nodes and N represents the appropriate shape function. Using bilinear interpolation functions ϕ1 for the variation of in‐plane displacement components (u and v) and bi‐cubic interpolation functions ϕ2 for the variation of lateral displacement and rotation components (w, θx, or θy) such that:
and
Then, C is a 5 × 20 matrix that includes the basis functions for all the five variables, that is:
Hence, the shape function N is also a 5 × 20 matrix, which can be expressed as:
For the VEM layer, the five degrees of freedom are as follows:
where T is the transformation matrix relating the degrees of freedom of the VEM to those of the base plate.
The total kinetic energy T of the composite plate is the summation of the kinetic energies of the plate and VEM, which are denoted by Tp and Tv respectively. T is given by:
For the ith layer in the eth element:
Similarly, the total potential energy of the composite plate is the summation of the plate and VEM elastic energies.
For the ith layer in the eth element:
In Eq. (5.55), the stress–strain relationship is given by:
The equation of motion for plate/VEM system can then be reduced to:
where KR is the real part of global stiffness matrix and KI is the imaginary part of global stiffness matrix. Therefore, the nth MDR can be written:
where ϕn is the nth eigenvector.
Table 5.7 Comparison between the natural frequencies, modal damping ratios (MDR), modal strain energy (MSE) in VEM and modal strain energy MSE per unit volume for the different viscoelastic treatments.
Mode number | Natural frequencies (Hz) | ||
Conventional | Squares | Reentrant hexagons | |
1 | 24 | 23 | 23 |
2 | 52 | 50 | 51 |
3 | 114 | 113 | 113 |
4 | 145 | 143 | 143 |
Mode number | MDR (%) | ||
Conventional | Squares | Reentrant hexagons | |
1 | 0.006015 | 0.005551 | 0.005734 |
2 | 0.006070 | 0.005350 | 0.005627 |
3 | 0.008950 | 0.007724 | 0.008188 |
4 | 0.009249 | 0.007556 | 0.008196 |
Mode Number | MSE in VEM (mJ cycle−1) | ||
Conventional | Squares | Reentrant hexagons | |
1 | 0.48 | 0.45 | 0.46 |
2 | 1.03 | 0.99 | 1.01 |
3 | 2.26 | 2.22 | 2.24 |
4 | 2.86 | 2.82 | 2.83 |
Mode Number | MSE in VEM/Treatment Volume (GJ m−3) | ||
Conventional | Squares | Reentrant hexagons | |
1 | 4.74 | 5.54 | 5.33 |
2 | 11.02 | 12.19 | 11.71 |
3 | 22.32 | 27.32 | 25.97 |
4 | 28.25 | 34.71 | 32.81 |
Average | 16.58 | 19.94 | 18.96 |
%Gain | 0 | 20.26% | 14.32% |
Figure 5.12 displays the finite element models of the considered three damping treatment configurations.
The results in Table 5.7 indicate that the natural frequencies and MSE in the VEM of the three treatments are nearly the same. However, the MDRs of the conventional treatment are slightly higher than those of treatments with square or reentrant hexagon holes. But, the results reveal that the treatments with square perforations has the highest MSE in VEM (i.e. dissipated energy) per unit volume of VEM followed by the treatments with reentrant hexagon holes and then the conventional treatments came last. The percentage gains in the dissipated energy relative to the conventional treatments are 20.26 and 14.32% for treatments with square holes and treatments with reentrant hexagon holes, respectively.
The energy dissipation of plate/constrained layer damping comes from the shearing deformation in VEM layer, it is reasonable to consider the MDR or modal loss factor of damping structures as the objective function of topology optimization (El‐Sabbagh and Baz 2014). Therefore, the objective function is written:
where f denotes the objective function of the optimization problem in present study. m represents the number of the considered MDR. Define the relative density of each VEM element as the design variable vector:
In addition, the constraint is considered to limit the consumption of VEM, the volume fraction is considered. The optimization problem is formulated as follows:
where N is the number of elements, Φj is the eigenvector, M and Kare the global mass and stiffness matrices, and V/V0 = α = volume fraction of viscoelastic material on the plate.
According to the Solid Isotropic Material with Penalization (SIMP) topology optimization method, the element mass and stiffness matrices can be expressed as the product of variables density and the entity element mass and stiffness matrices. The penalty factors p, q; p, q ≥ 1 are put in to accelerate the convergence of iteration results, that is:
where ρe is variable density of each VEM element; it is a relative quantity and 0 ≤ ρe ≤ 1. Note that if, ρe = 0 then there is no VEM treatment for this element or equivalently, the thickness of the VEM is equal to zero. Similarly, if ρe = 1, then the thickness of VEM in this element is equal to the assigned thickness. Also, , are the mass and stiffness matrices of VEM element. Keep the base plate and constrained layer unchanged, the global mass, and stiffness matrices can be calculated as follows:
where p and q are penalty factors, p = 1, q = 3. The layout of VEM on the plate can be determined by searching the optimal relative density of each VEM element. In order to solve the presented optimum problem, Method of Moving Asymptote (MMA) method is employed.
According to MSE method, approximate expressions of the ith MDR can be obtained. The equation of motion for plate/VEM system is given:
where KR is the real part of global stiffness matrix, KI is the image part of global stiffness matrix. Therefore, the nth MDR can be written:
where ϕr is the nth eigenvector, the derivatives of Eq. (5.66) to design variables are:
The derivatives of the stiffness matrix are obtained by solving the following sensitivity equations:
and
when p = 1, q = 3, the derivative of MDR can be expressed as:
The dynamic constraint function is
The derivative of the dynamic constraint function is calculated as follows:
and
When p = 1, q = 3, the sensitivity of constrained functions become:
Figure 5.13 presents the flow chart of the topology optimization using the MMAs proposed by Svanberg (1987, 2002) to determine the optimal distribution of damping material over structures. It is noted that the computation time is mainly determined by building/solving the finite element model with the complex stiffness matrices and using the MMA approach.
A finite element code is developed to describe the dynamics of a plate treated with VEM as outlined in Section 5.6.2. The finite element model is used to extract the strain energy in order serve as a quantitative measure for computing the MDR. Then, the MMA topology optimization method outlined in Sections 5.6.2.2 and 5.6.2.3 is employed to determine the optimal topology of surface damping treatments.
Figures 5.15–5.17 present the optimal topologies of the VEM for volume ratios of 0.25, 0.5, and 0.75 in order to maximize the MDR of the first mode. The figures display also the effect of optimization iteration number on the MDRs of the first four modes.
The figures indicate that the optimum MDRs for the first mode attain their peak values nearly after 2000 iterations. These optimal damping ratios are 0.0002, 0.00026, and 0.00024 for volume ratios of 0.25, 0.5, and 0.75 respectively. These damping ratios as well as those for higher order modes increase with increasing volume ratios. The figures indicate also that the obtained optimal topologies tend to concentrate the VEM near the fixed end of the plate with additional treatments placed at the mid‐section of the plate near its free end.
Furthermore, it is observed that the MMA algorithm converges faster as the volume ratio of the VEM is increased.
The approach adopted in solving Problem 5.6 is used. However, the simply supported boundary condition is imposed on the finite element model. Solving the topology optimization problem of the VEM, for volume ratios of 0.25, 0.5, and 0.75, yields the optimal topologies shown in Figures 5.18–5.20, respectively. The figures display also the effect of optimization iteration number on the MDRs of the first four modes.
The figures indicate that the optimum MDRs for the first mode attain their peak values nearly after 2000 iterations. These optimal damping ratios are 0.00032, 0.00042, and 0.00044 for volume ratios of 0.25, 0.5, and 0.75 respectively. These damping ratios as well as those for higher order modes increase with increasing volume ratios. The figures indicate also that the obtained optimal topologies tend to concentrate the VEM near the fixed end of the plate with additional treatments placed at the mid‐section of the plate near its free end.
This chapter has presented the MSE method with its original and modified forms. The theories behind the MSE method and its modifications are presented. Numerical examples are given to quantify the accuracy of the different method in predicting the modal parameters of rods treated with VEM.
The MSE has also been shown to serve as an important design metric for selecting the optimal design parameters and/or determining the optimal topologies of damping treatments based on rational design objectives.
Assume the dimensions and the material properties of the two systems are as given in Example 4.3.
Assume also that the two systems are modeled using three finite elements.
Assume the dimensions and the material properties of the two systems are as given in Example 4.4.
Assume also that the two systems are modeled using three finite elements.
where
where , with K21 = k2/k1 and K23 = k2/k3.
That is, MSE over estimates the exact loss factor.
where [M] = mass matrix, [KR] = real stiffness matrix, [KI] = imaginary stiffness matrix, and
Assume that the system is modeled by a two element finite element model and that tv is constrained such that: 0.005 ≤ tv ≤ 0.05 m.
Comment on the optimum results obtained in items a and b.
Determine the sum of the MDRs of the beam/stripped VEM assembly for the first four modes of vibration.
Determine such sum for three arrangements shown in Figure P5.8.
Table P5.1 Physical and geometrical parameters of a plate/PCLD system.
Layer | Length (m) | Width (m) | Thickness (mm) | Density (kg m−3) | Modulus (MPa) |
Base Plate | 0.32 | 0.125 | 0.50 | 2700 | 7100a |
Viscoelastic | stripped | 0.125 | 0.50 | 1140 | 20b |
Constraining Layer | stripped | 0.125 | 0.25 | 2700 | 7100a |
a Young's modulus.
b Shear modulus, η = 0.5.
Using the topology optimization approach, determine the optimal distribution of the VEM over the plate surface in order to maximize the sum of the MDRs of the plate/VEM assembly for the first four modes of vibration.
Determine such optimal topologies for volume fractions of the VEM of 0.2, 0.4, 0.6, and 0.8 (Ling et al. 2010).
Determine such optimal topologies for volume fractions of the VEM of 0.2, 0.4, 0.6, and 0.8.