Solution

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.

Solution

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.

5.3.1 Weighted Stiffness Matrix Method (WSM)

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

(5.15)

Assume a vector to be defined such that:

Then, Eq. (5.15) reduces to

or

(5.16)

where [K_M] = [K_R] + β[K_I] , β = a/b, and .

Eq. (5.16) represents a modified eigenvalue problem that has real eigenvalue and eigenvector . Note that [K_M] is a modified stiffness matrix that augments the elastic stiffness matrix [K_R] with a weighted contribution of the imaginary component of the stiffness [K_I]. The weighting parameter β is calculated from the following empirical formula proposed by Hu et al. (1995)

(5.17)

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 η_n of the nth mode as follows

(5.18)

5.3.2 Weighted Storage Modulus Method (WSTM)

In this method, the shear modulus of the VEM as described by

has a storage modulus G′ and loss factor η_v. 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

(5.19)

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

(5.20)

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:

(5.21)

5.3.3 Improved Reduction System Method (IRS)

This method is based on assuming the following modal transformation

(5.22)

where [Φ] is the eigenvector matrix for the undamped part of Eq. (5.1), that is,

such that

(5.23)

Also, in Eq. (5.22), {q} denotes the modal displacement vector, which is given by

(5.24)

But, for the damped system we have

(5.25)

Substituting Eqs. (5.22) and (5.24) into Eq. (5.25) gives

(5.26)

Pre‐multiplying Eq. (5.26) by [Φ]^Tand using Eq. (5.23) gives

(5.27)

where .

Equating the real and imaginary parts on both sides of Eq. (5.23), gives the following matrix equation

or

(5.28)

Using static condensation, the second row of Eq. (5.28) gives

(5.29)

and

(5.30)

Hence, the condensed system can be obtained by combining Eqs. (5.28) and (5.30) to yield the following

(5.31)

where

and

Solution of the eigenvalue problem given by Eq. (5.31) yields the eigenvalue ω and the eigenvector {q_r}. The full complex eigenvector can be extracted as follows

(5.32)

This eigenvector can be used to compute the modal loss factor of the structure/VEM assembly as follows

(5.33)

5.3.4 Low Frequency Approximation Method (LFA)

This method is based on expanding the second row of Eq. (5.28) to give

or

(5.34)

For low frequencies, the Taylor series expansion of Eq. (5.29) in terms of ω is given by

(5.35)

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 {q_i) in terms of {q_r}.

Expanding the first row of Eq. (5.28) gives:

(5.36)

Combining Eqs. (5.35) and (5.36) gives

(5.37)

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

(5.38)

Then, solving the original system given by Eq. (5.28) reduces to the following condensed system:

(5.39)

where

and

Solution of the eigenvalue problem given by Eq. (5.39) yields the eigenvalue ω and the eigenvector {q_r}. The full complex eigenvector can be reconstructed as follows

(5.40)

This eigenvector can be used to compute the modal loss factor of the structure/VEM assembly as follows

(5.41)

Solution

The design problem is formulated mathematically as follows:

(5.42)

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 t_vi are positive and each is bounded from below and from above by t_vmin and t_vmax, respectively.

Note that the lower bound is selected to ensure adequate damping and avoid reaching the unrealistic trivial solution where all the design variables t_vi 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:

1. F₁ = sum of the modal loss factor for the first five modes

   Two sets of constraints are imposed:

Set 1: t_vmin = 0.001 and t_vmax = 0.01 m

Let the initial guess of the thickness distribution is [t_v1, t_v2, …, t_v5] = [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: t_vmin = 0.001 and t_vmax = 0.025 m

If the initial guess of the thickness distribution is maintained at [t_v1, t_v2, …, t_v5] = [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.

2. F₂ = sum of the modal loss factors/total 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: t_vmin = 0.001 and t_vmax = 0.025 m

If the initial guess of the thickness distribution is [t_v1, t_v2, …, t_v5] = [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: t_vmin = 0.01 and t_vmax = 0.025 m

If the initial guess of the thickness distribution is [t_v1, t_v2, …, t_v5] = [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.

5.6.1 Overview

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.

5.6.2 Finite Element Modeling

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 ρ_p, E_p, v_p, respectively. The VEM layer properties are denoted by ρ_v, E_v, v_v where the complex elasticity modulus E_v = 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 h_p and h_v, 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, θ_x and θ_y 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  θ_x  θ_y}^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 ε_xz, ε_yz, and ε_zz vanish. According to this assumption, the rotation degrees of freedom can be expressed in terms of the gradients of the lateral deflection such that

(5.43)

The other strain components are

(5.44)

(5.45)

(5.46)

Within the finite element, the displacement vector is approximated as

(5.47)

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, θ_x, or θ_y) such that:

and

(5.48)

Then, C is a 5 × 20 matrix that includes the basis functions for all the five variables, that is:

(5.49)

Hence, the shape function N is also a 5 × 20 matrix, which can be expressed as:

(5.50)

For the VEM layer, the five degrees of freedom are as follows:

(5.51)

where T is the transformation matrix relating the degrees of freedom of the VEM to those of the base plate.

5.6.2.1 Element Energies

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 T_p and T_v respectively. T is given by:

(5.52)

For the ith layer in the eth element:

(5.53)

Similarly, the total potential energy of the composite plate is the summation of the plate and VEM elastic energies.

(5.54)

For the ith layer in the eth element:

(5.55)

In Eq. (5.55), the stress–strain relationship is given by:

(5.56)

The equation of motion for plate/VEM system can then be reduced to:

(5.57)

where K_R is the real part of global stiffness matrix and K_I is the imaginary part of global stiffness matrix. Therefore, the nth MDR can be written:

(5.58)

where ϕ_n is the nth eigenvector.

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%

Solution

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.

5.6.2.2 Topology Optimization of Unconstrained Layer Damping

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:

(5.59)

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:

(5.60)

In addition, the constraint is considered to limit the consumption of VEM, the volume fraction is considered. The optimization problem is formulated as follows:

(5.61)

where N is the number of elements, Φ_j 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:

(5.62)

where ρ_e is variable density of each VEM element; it is a relative quantity and 0 ≤ ρ_e ≤ 1. Note that if, ρ_e = 0 then there is no VEM treatment for this element or equivalently, the thickness of the VEM is equal to zero. Similarly, if ρ_e = 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:

(5.63)

(5.64)

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.

5.6.2.3 Sensitivity Analysis

According to MSE method, approximate expressions of the ith MDR can be obtained. The equation of motion for plate/VEM system is given:

(5.65)

where K_R is the real part of global stiffness matrix, K_I is the image part of global stiffness matrix. Therefore, the nth MDR can be written:

(5.66)

where ϕ_r is the nth eigenvector, the derivatives of Eq. (5.66) to design variables are:

(5.67)

The derivatives of the stiffness matrix are obtained by solving the following sensitivity equations:

and

(5.68)

when p = 1, q = 3, the derivative of MDR can be expressed as:

(5.69)

The dynamic constraint function is

(5.70)

The derivative of the dynamic constraint function is calculated as follows:

(5.71)

and

(5.72)

When p = 1, q = 3, the sensitivity of constrained functions become:

(5.73)

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.

Solution

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.

Solution

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.

1. 5.1 Consider the fixed‐free rods shown in Figure P5.1. Determine the natural frequencies and the damping ratios of the two rod/VEM systems using the MSE method and its four modified versions discussed in Section 5.3.

   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.

   

2. 5.2 Consider the fixed‐free rods shown in Figure P5.2. Determine the natural frequencies and the damping ratios of the two rod/VEM systems using the MSE method and its four modified versions discussed in Section 5.3.

   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.

   

3. 5.3 Consider the dynamical system shown in Figure P5.3a. The mass m is supported on two springs. One of these springs is an elastic spring with real stiffness k₁ while the third spring is viscoelastic with complex stiffness where η is the loss factor of the VEM. Show that:
   1. the equivalent system shown in Figure P5.3b has:
   

   where

   
   2. the loss factor η_MSE of this system, as calculated by using the MSE concept, is given by:
   

   

4. 5.4 Consider the dynamical system shown in Figure P5.4a. The mass m is supported on three springs. Two are elastic springs with real stiffness k₁ and k₃, respectively, while the third spring is viscoelastic with complex stiffness where η is the loss factor of the VEM. Show:
   1. that this system is equivalent to the system shown in Figure P5.4b such that:

   where , with K₂₁ = k₂/k₁ and K₂₃ = k₂/k₃.

   2. that the loss factor η_MSE of this system as calculated by using the MSE concept is given by:
   
   3. that the error between the exact loss factor η_s and that η_MSE estimated by MSE is:
   

   That is, MSE over estimates the exact loss factor.

   

5. 5.5 Consider the dynamical system shown in Figure P5.5. The vibration of the primary system (m₁–k₁) is controlled by the secondary system (i.e., a dynamic damper system). Assume that m₁ = m₂ = 1kg, k₁ = 1 N m⁻¹, and , then:
   1. Derive the equations of motion of the system and cast in the following form:
   

   where [M] = mass matrix, [K_R] = real stiffness matrix, [K_I] = imaginary stiffness matrix, and

   2. Determine the MDRs of the two modes of vibration of the system using the Original Modal Strain Method (MSE).

   

6. 5.6 Consider the dynamical system shown in Figure P5.6. The mass m is supported on four springs. Three of these springs is an elastic spring with real stiffness k₁ while the fourth spring is viscoelastic with complex stiffness where η is the loss factor of the VEM. Determine the loss factor of the system using:
   1. the equivalent system shown in Figure P5.6 such that K^* = K_R(1 + η_si)
   2. the MSE concept such that the loss factor = η_MSE
   3. Compare the results obtained in a and b.

   

7. 5.7 Consider the fixed‐free rod/VEM system shown in Figure P5.7. The rod is made of aluminum with width of 0.025 m, thickness of 0.025 m, and length of 1 m. The VEM has a width of 0.025 m, thickness = t_v m, and density 1100 kg m⁻³. The VEM is constrained by an aluminum constraining layer that is 0.025 m wide and 0.0025 m thick. Using the GHM modeling approach of the VEM with one mini‐oscillator (E₀ = 15.3MPa, α₁ = 39, ζ₁ = 1,  ω₁ = 19, 058rad/s), determine:
   1. the optimum thickness t_v of the VEM that maximizes the sum of the MDRs for the two modes of vibration the end/VEM (use the MSE as a design criterion).
   2. the optimum thickness t_v of the VEM that maximizes the sum of the MDRs for the two modes of vibration the end/VEM per unit weight of the VEM treatment (use the MSE as a design criterion).

   Assume that the system is modeled by a two element finite element model and that t_v is constrained such that: 0.005 ≤ t_v ≤ 0.05  m.

   Comment on the optimum results obtained in items a and b.

   

8. 5.8 Consider the cantilever beam/passive constrained layer damping (PCLD) system shown as Figure P5.8. The main physical and geometrical parameters of the base beam, the viscoelastic layer and constrained layer are listed in Table P5.1. The beam is made of aluminum and is mounted in a cantilever configuration. The beam is treated with segmented arrangements of viscoelastic material that are constrained by an aluminum layer as shown in Figure P5.8 (Curà et al. 2011).

   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.

   

   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.

9. 5.9 Consider the cantilever plate/PCLD system shown as Figure P5.9. The main physical and geometrical parameters of the base plate, the viscoelastic layer and constrained layer are listed in Table P5.2. The plate is made of aluminum and is mounted in a cantilever configuration. The plate is treated with viscoelastic material, which is constrained by a continuous Polyvinylidene Fluoride (PVDF) piezoelectric layer as shown in Figure P5.9.

   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).

   

   Layer	Length (m)	Width (m)	Thickness (mm)	Density (kg m−3)	Modulus (MPa)
   Base Plate	0.25	0.125	0.5	2700	7,100a
   Viscoelastic	0.25	0.125	0.5	1140	20b
   PVDF	0.25	0.125	0.028	1800	2,250a

   ^a Young's modulus.

   ^b Shear modulus, η = 0.5.

10. 5.10 Using the topology optimization approach, determine the optimal distribution of the VEM over the plate surface of Problem 5.9 when the plate is anchored in a simply‐supported configuration as shown in Figure P5.10. The topology optimization aims at maximizing 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.