7.3.3.1 The Model

Figure 7.18 shows a schematic drawing of the ACLD treatment of a sandwiched beam divided into N finite elements. It is assumed that the shear strains, in the piezoelectric sensor/actuator layers and in the base beam, are negligible.

It is also assumed that the transverse displacements w of all points on any cross section of the sandwiched beam are considered to be equal. Furthermore, the piezoelectric sensor/actuator layers and the base beam are assumed to be elastic and to dissipate no energy, whereas the core is assumed to be linearly viscoelastic. In addition, the piezoelectric sensor and the base beam are considered to be perfectly bonded together such that they can be reduced to a single equivalent layer. Accordingly, the original four‐layer sandwiched beam reduces to an equivalent three‐layer beam. Hence, the modeling approach presented in Section 4.3 for beams treated with PCLD can be extended to account for the effect of the piezo‐sensing and actuation.

The piezo‐sensor can take the general, uniform, or linear shapes shown in Figures 7.19a–c, respectively.

The complete description of the beam/ACLD assembly model includes:

Potential Energy

This remains as described by Eq. (4.60) as follows:

(7.21)

where u₁, u₃, w, and w_,x denote the axial displacement of the constraining layer, axial displacement of the beam, as well as the transverse and rotational deflections of beam, respectively. Also, γ defines the shear strain of the VEM given by Eq. (4.58). Note that E_iA_i and E_iI_i denote the longitudinal and flexural rigidity of the ith layer. Also, G₂A₂ denotes the shear rigidity of the VEM with G₂ defining its shear modulus.

Kinetic Energy

This is also as given by Eq. (4.62) as follows:

(7.22)

with ρ_i denoting the density of the ith layer.

Piezo‐Control Forces and Moments

1. Piezo‐Actuator. The strain ε_p induced in the piezoelectric actuator is given by (Crawley and de Luis 1987):

(7.23)

where d₃₁ is the piezoelectric strain constant resulting from the application of the voltage V_c across the piezo‐actuator layer. In Eq. (7.23), V_c is assumed constant over the length of the beam element. The voltage V_c is generated from the proper manipulation of the piezo‐sensor voltage V_s.

2. Piezo‐Sensor. The strain ε_s induced in the piezo‐sensor is proportional to the beam curvature (w_,xx) and is given by:

(7.24)

where h is the distance from the beam neutral axis to the sensor surface.

The induced strain ε_s integrated over the entire length of the sensor due to its distributed nature, generates an output voltage V_s given by (Appendix 7.A):

(7.25)

where f_i(x) is a spatial distribution function that defines the shaping of the sensor over the ith element. For uniform sensor f_i(x) = 1 and for a linearly shaped sensor f_i(x) = 1 − x/L as shown in Figure 7.19c. In Eq. (7.25), the sensor is extended between elements i_s and i_f. Also, is the electromechanical coupling factor, g₃₁ is the piezoelectric voltage constant and C is the capacitance of the sensor, which is given by:

(7.26)

where A is the sensor surface area and K_3t is the dimensionless dielectric constant.

3. Control Law. The manipulation of the piezo‐sensor voltage V_s to generate the actuator voltage V_c is governed by the following proportional and derivative control law:

(7.27)

where K_p and K_d are the proportional and derivative control gains, respectively.

4. Control Forces and Moments. The vector {F_c} of the control forces and moments generated by the piezo‐constraining layer on the treated beam element can be expressed in the following matrix form:

(7.28)

where F_pj, F_pk, M_pj, and M_pk denote the control forces and moments generated at nodes j and k, which are given by:

For a uniform sensor

and

(7.29)

For a shaped sensor

and

(7.30)

where , is the distance between a neutral axis of entire sandwiched beam and piezo‐actuator, and p is the d/dt operator. Also, G_c = (K_P + K_d p) denotes the controller gain.

7.3.3.2 Equations of Motion

The stiffness matrix [K_i], the mass matrix [M_i], and the control force vector {F_c} are combined to describe the dynamics of the ACLD‐treated beam element as follows:

(7.31)

The effect of the proportional and derivative control actions on the performance of the assembled closed‐loop system, given by Eq. (7.31), is determined by computing the eigenvalues (i.e., natural frequencies and damping ratios) of the closed‐loop system and comparing these eigenvalues with those of the open‐loop system. Note that {Δ_i} denotes the nodal deflection vector of the ith element that is bounded between nodes j and k. The nodal deflection vector {Δ_i} is given by:

(7.32)

Also, stiffness matrix [K_i] and the mass matrix [M_i] are extracted from the potential energy (PE) and the kinetic energy (KE) such that:

(7.33)

Solution

The equation of motion of the beam/ACLD system, which is divided into N finite elements, is as given in Sections 4.3 and 7.3.3 as follows:

(7.34)

where , {C_e}_1 × 4N = [0   0  .........0  1   0]^T, and {C_C}_1 × 4N = [0  0  ..  …   0   1]^T.

Also, F_e and F_c denote the external and control forces, respectively. The matrices [M_t], [C_t], and [K_t] are as described in Section 4.3.3 of Chapter 4.

The first natural frequencies of the open‐loop beam/ACLD system are listed in Table 7.4 along with comparisons with the predictions obtained by ANSYS. Close agreement is evident between the predictions of the FEM and ANSYS models.

Mode	1	2	3
FEM (Hz)	48.88	248.4	668.78
ANSYS (Hz)	46.35	253.52	677.49

The control force F_c is generated using the following velocity feedback control law:

(7.35)

where K_v is the derivative control gain.

Combining Eqs. (7.34) and (7.35) yields the following state‐space representation:

State Equations:

(7.36)

and

Output:

where , C = [0_1 × 4N 0_{1 × (4N − 1)}1 ], D = 0, and u = F_c.

The root locus of the system can be plotted using the following MATLAB command:

Figure 7.20 displays the root locus of the system. The figure indicates the locations of the first three poles of the system that also coincide with the predictions of the FEM.

For maximum closed‐loop damping, the velocity feedback gain is selected to be 7.4 as displayed in Figure 7.20.

The frequency response of the beam/ACLD system is shown in Figure 7.21a. The corresponding control voltage is also displayed in Figure 7.21b.

The displayed results indicate that the amplitude of vibration at the first mode is attenuated from 3.22E‐5m for the uncontrolled case to 9.93E‐6m when the system is controlled by a 38.15 control volts.

Solution

Figure 7.23a displays the finite element mesh of the beam/ACLD assembly. Also, Figures 7.23b through 7.23d show the modes and mode shapes of the first three natural frequencies of the assembly. Table 7.7 lists a comparison between the natural frequencies of the beam/ACLD assembly as obtained by the finite element model and ANSYS predictions.

Method	Mode 1	Mode 2	Mode 3	Mode 4
ANSYS	22.85 Hz	143.10 Hz	400.53 Hz	784.70 Hz
MATLAB	21.80 Hz	136.50 Hz	383.30 Hz	754.90 Hz

Figure 7.24 displays the frequency response characteristics of the beam/ACLD with a uniform sensor when the proportional (P) and derivative (D) control gains K_p = 700 N m⁻¹ and k_r = 0.15 where k_r = K_d/K_p. The figure displays also the response of the beam/PCLD for comparison purposes.

The displayed response indicates that the use of the ACLD has resulted in reducing the maximum amplitude of vibration at the first mode from 0.0724 to 0.002 42 m and required a control voltage of 78.87 V.

In Figure 7.25, the time response characteristics of the beam/ACLD with uniform sensor are displayed for proportional‐derivative (PD) control gains K_p = 700 N m⁻¹ and k_r = 0.15 where k_r = K_d/K_p. The figure displays also the response of the beam/PCLD for comparison purposes.

The displayed response indicates that the use of the ACLD has also resulted in reducing the maximum amplitude of vibration to 0.0298 μm in about 0.1 s. Such a fast attenuation is achieved with a maximum control voltage of only 0.5 V.

Figures 7.26 and 7.27 display the frequency and time response characteristics of the beam/ACLD with a linearly shaped sensor using the same proportional (P) and derivative (D) control gains as those used with the uniform sensor. The figures also display the response of the beam/PCLD for comparison purposes.

The displayed frequency response emphasizes that the use of the ACLD has attenuated the maximum amplitude of vibration at the first mode to 0.0178 m with a control voltage of 336.3 V. Also, the displayed time response suggests that using the ACLD has reduced the maximum amplitude of vibration to 0.476 μm in about 0.1 s. Such a fast attenuation is achieved with a maximum control voltage of only 1.0 V.

Table 7.8 summarizes comparisons between the performance characteristics of the ACLD with uniform and linearly shaped sensors at the first mode of vibration. It can be seen that the use of ACLD with a shaped sensor has resulted in significant attenuation in the amplitudes of vibration and required lower voltage to achieve such attenuation as compared to the case of uniform shaped sensor.

Sensor type	Sensor shape	Measured parameter	Frequency response		Time response
			Max. amplitude (m)	Max. volt	Max. amplitude (μm)	Max. volt
Uniform		w,x(L)	0.002 42	78.873	0.0585	0.5
Linearly Shaped		w(L)/L	0.001E‐3	4.76	0.0217	0.5

7.3.4.1 Overview

This section focuses on developing a distributed‐parameter model (DPM) using Hamilton's principle to describe the dynamics of beams that are fully treated with ACLD (Baz 1997a,b). The variational formulation, being energy‐based, is much simpler to employ than the classical force equilibrium‐based shear models (Mead and Markus 1969; DiTaranto 1965). Furthermore, the variational formulation provides directly the boundary conditions associated with the ACLD treatment.

For the active control purposes, the variational model provides also direct means for devising a globally stable boundary control strategy that is compatible with the operating nature of the ACLD treatments. In this manner, the instability problems associated with the simple proportional and/or derivative controllers are completely avoided. More importantly, as the control strategy is based on a DPM, the classical spillover problems resulting from using “truncated” finite element models are eliminated. Accordingly, the devised boundary controller enables the control of all the modes of vibration of the ACLD‐treated structures.

7.3.4.2 The Energies and Work Done on the Beam/ACLD Assembly

The potential and kinetic energies as well as the external work done by the piezoelectric actuator and external loads, as described in Section 7.2.3, are recast as follows:

Potential Energy

Equation (7.21) is rewritten to include only the conservative component of the energy associated with the VEM as follows:

(7.37)

where denotes the shear storage modulus of the VEM. Also, K₁ = E₁A₁, K₃ = E₃A₃, and D_t = (E₁I₁ + E₃I₃).

Kinetic Energy

Equation (7.22) is simplified to only include the contribution of the transverse motion to the kinetic energy as the components associated with the longitudinal motions can be assumed negligible. Hence, Eq. (7.22) reduces to:

(7.38)

where .

External Work Done

The different components that account for the work done by the non‐conservative loads include:

1. Work done W₁ by piezoelectric control forces. This is given by

(7.39)

where ε_p denotes the piezo‐strain that is induced in the piezoelectric actuator and given by Eq. (7.23). This strain ε_p is assumed constant over the entire length of the constraining layer in order to maintain and emphasize the simplicity and practicality of the ACLD treatment.

2. Work W₂ dissipated in the viscoelastic core. This is given by

(7.40)

where τ_d is the dissipative shear stress developed by the viscoelastic core. It is given by:

(7.41)

where η, ω, and i denote the loss factor of the viscoelastic core, the frequency and , respectively. In Eq. (7.41), the behavior of the viscoelastic core is modeled using the common complex modulus approach, which is a frequency domain‐based method.

7.3.4.3 The Distributed‐Parameter Model

The equations and boundary conditions governing the operation of the beam/ACLD assembly are obtained by applying Hamilton's principle (Meirovitch 2010):

(7.42)

where δ(.) denotes the first variation in the quantity inside the parentheses.

The resulting equations of the beam/ACLD assembly are:

(7.43)

(7.44)

and

(7.45)

where is the complex modulus of the VEM.

For a cantilevered beam, Hamilton's principle yields the following boundary conditions:

At x = L:

(7.46)

and

At x = L:

(7.47)

Equations (7.43) and (7.44) give:

(7.48)

and

(7.49)

where z = (u₁ − u₃) and g = G₂/h₂[(K₁ + K₃)/(K₁K₃)].

Combining Eqs. (7.45) and (7.49) yields:

(7.50)

or,

(7.51)

where and . Note that g and Y are defined as the shear parameter and geometrical factors, respectively, as introduced by DiTaranto (1973).

Differentiating Eq. (7.51) partially with respect to x, substituting into Eq. (7.22), and rearranging the terms yields a sixth order partial differential equation describing the dynamics of the beam/ACLD assembly as follows:

(7.52)

The associated boundary conditions given by Eqs. (7.46) and (7.47) simplify to:

At x = 0:

(7.53)

and

At x = L:

(7.54)

Further simplification of the boundary conditions can be obtained by using Eq. (7.51) and its spatial derivative with respect to x to yield:

At x = 0:

(7.55)

and

At x = L:

(7.56)

It is important here to note that the sixth order partial differential equation describing the beam/ACLD system, given by Eq. (7.52) is the same as that describing a beam treated with conventional PCLD as obtained by Mead and Markus (1969). However, the boundary condition given by Eq. (7.56) is modified to account for the control action generated by the strain ε_p induced by the active constraining layer at the free end of the beam (i.e., at x = L). Therefore, the particular nature of operation of the beam/ACLD system implies the existence of boundary control action ε_p.

7.3.4.4 Globally Stable Boundary Control Strategy

The globally stable control strategy presented in this section aims at ensuring that the total energy E_n = PE + KE of the beam/ACLD system is maintained as a strictly non‐increasing function of time.

The total energy E_n is given by:

(7.57)

Differentiating the total energy expression given by Eq. (7.57) with respect to the time and using Eqs. (7.52) through (7.54), yielding:

(7.58)

Equation (7.58) indicates that the second term is strictly negative, hence a globally stable boundary controller with a continuously decreasing energy norm (i.e., ) is obtained when the control action ε_p takes the following form:

(7.59)

where K_g is the gain of the boundary controller. This form of a controller makes the rate of change of the total energy of the entire assembly strictly negative.

Equation (7.59) indicates that the control action is a velocity feedback of the longitudinal displacement of the piezoelectric constraining layer. It is important also here to note that when the active control action ε_p ceases or fails to operate for one reason or another (i.e., when ε_p = 0), the beam system remains globally stable as indicated by Eq. (7.58). Such inherent stability is attributed to the second term in the equation that quantifies the contribution of the Passive Constrained Layer Treatment (PCLD).

7.3.4.5 Implementation of the Globally Stable Boundary Control Strategy

The boundary control strategy can be implemented in one of the following two ways:

1. in terms of the longitudinal displacement u₁ of the piezo‐actuator by using Eq. (7.57) as follows:

(7.60)

where s is the Laplace complex number.

2. in terms of the longitudinal displacement u₃ of the base beam by using Eqs. (7.47) and (7.48), in order to relate u₃ to u₁ as follows:

(7.61)

Eliminating u₁ between Eqs. (7.60) and (7.61) gives the control action ε_p in terms of u₃ as follows:

(7.62)

7.3.4.6 Response of the Beam/ACLD Assembly

The response w of the beam/ACLD assembly can be determined by solving Eq. (7.52) using the classical separation of variables method such that:

(7.63)

where W(x) and T(t) denote spatial and temporal functions. Substituting Eq. (7.63) into Eq. (7.52) and setting yields the following characteristic equation:

(7.64)

where λ is a differential operator with respect to x. Let ±δ_i with i = 1,…,3 be the roots of the characteristics of the equation, then the spatial function W(x) can be written as:

(7.65)

where the coefficients C_i's are to be determined from the boundary conditions given by Eqs. (7.55) and (7.56).

Solution

Note that when the beam/ACLD system is excited by a transverse end load F, the boundary condition at x = L, Eq. (7.56), must be modified to:

At x = L:

(7.66)

Accordingly, the transverse deflection W(x) of beam can be determined using Eq. (7.65) subjected to the boundary conditions (7.55) and (7.56) after including the modifications imposed by Eq. (7.66).

Figure 7.28 displays a comparison between the frequency response characteristics of the open‐loop beam/ACLD system (i.e., K_D = 0) as predicted by the FEM and DPM approaches.

It can be seen that there is close agreement between the two approaches.

Table 7.9 lists also a comparison between the first three natural frequencies of the beam/ACLD system as computed using the FEM, ANSYS, and DPM approaches when K_D = 0.

Mode	1	2	3
FEM (Hz)	48.88	248.4	668.78
ANSYS (Hz)	46.35	253.52	677.49
DPM –(Hz)	47.00	256.00	683.00

It is evident that the predictions of the DPM are very close to those of the ANSYS model.

Figure 7.29 displays the compliance and control voltage characteristics of the beam/ACLD system for control gain K_D = 7.4 along with a comparison of the corresponding characteristics of the beam/PCLD system, that is, K_D = 0.

Note that there are discrepancies between the predictions of the DPM and the FEM of the closed‐loop response characteristics of the bean/ACLD system. These discrepancies can be attributed to the fact that in the DPM, the control action is a linear force due to the strain ε_p applied to the constraining layer whereas in the FEM, the control action is a moment due to the piezo‐strain applied to the entire assembly.

The energy dissipated in the beam/ACLD system is given by Eqs. (7.40) and (7.41) as:

(7.67)

where the shear strain γ can be determined from Eq. (7.51) as follows:

(7.68)

Figure 7.30 shows the corresponding energy dissipation of the beam/ACLD system in comparison with the corresponding characteristics of the beam/PCLD system, that is, K_P = 0. It is evident that the energy dissipation with the ACLD becomes higher than that of the PCLD particularly at high frequencies.

Table 7.10 summarizes the performance characteristics of the beam/ACLD as compared with that of the beam/PCLD system.

Mode	Uncontrolled			KD = 7.4
	Frequency (Hz)	Amplitude (μm)	Energy dissipated (μJ)	Frequency (Hz)	Amplitude (μm)	Energy dissipated (μJ)	Volts (V)
1	47	187.4	187.8	42	0.1823	0.124	64.28
2	256	5.366	5.231	262	1.626	3.729	32.98
3	683	1.181	1.121	702	0.903	7.030	47.21

7.4.1.1 The In‐Plane Piezoelectric Forces

These in‐plane forces (F_px, F_py, and F_pxy) at the jth node, are given by:

(7.69)

with K_p and K_d denoting the proportional and derivative control gains. Also, p and V_s denote the operator d/dt and the sensor voltage. The constants d₃₁ and d₃₂ define the piezoelectric strain constants in the x and y directions. Also, [B_1p] and [D_1p] are given by (Section 4.8):

where

(7.70)

where N_u1 and N_v1 define the axial shape functions, respectively. Also, E₁ and ν₁ denoting Young's modulus and Poisson's ratio of the piezoelectric constraining layer, respectively.

7.4.1.2 The Piezoelectric Moments

These piezoelectric moments {M_pi} due to the bending of the piezoelectric constraining layer are given by:

(7.71)

where .

7.4.1.3 Piezoelectric Sensor

The voltage, V_s, developed by the piezo‐sensor is obtained from (Appendix 7.A, and Lee, 1987):

(7.72)

where

(7.73)

Also, h is the distance from the plate neutral plane to the sensor surface and b(x, y) is a distribution shape function of the sensor [b (x,y) = 1 for a uniform sensor]. The sensor is extended between elements i_sx and i_fx in the x direction and i_sy and i_fy in the y direction. Also, is the electromechanical coupling factor, g₃₁ is the piezoelectric voltage constant and C is the capacitance of the sensor that is given by:

(7.74)

where A is the sensor surface area and k_3t is the dielectric constant.

Note also that , , and [N_w] define the in‐plane and transverse shape functions, respectively, as defined in Section 4.8.

7.4.1.4 Control Voltage to Piezoelectric Constraining Layer

The voltage V_A applied to the piezoelectric constraining layer is given by:

(7.75)

Solution

Figure 7.32 displays the modes and mode shapes of the plate/ACLD patches.

The frequency response of the plate tip and the corresponding control voltage are displayed in Figure 7.33 when the ACLD patches are controlled with a velocity feedback that has a gain K = 1 Ns m⁻¹. Also displayed in the figure, for comparison purposes, is the response of the plate when K = 0 that corresponds to the open‐loop condition. In this case, it is evident that the controller effectiveness is only limited to higher order modes.

When the controller gain is increased to K = 1000 Ns m⁻¹, the resulting plate response and the control voltage are shown in Figure 7.34. The effectiveness of the controller, in this case, in attenuating the vibration is clear over a wide frequency band and is extended to the low frequency modes. Note that the maximum control voltage required in this case is only 13.37 V.

Table 7.11 summarizes the performance characteristics of the plate/ACLD patches, at the first mode of vibration, for different control gains.

The time response of the plate tip and the corresponding control voltage are displayed in Figure 7.35 when the ACLD patches are controlled with a velocity feedback that has a gain K = 1 Ns m⁻¹. The displayed characteristics are computed when the plate is subjected, at its mid tip point, to a transient force of 1 N for a duration of 0.01 s. The closed‐loop response is compared also with that of the open‐loop, that is, when K = 0.

The obtained comparison indicates that the damping ratio is increased from 0.00196 for the open loop to 0.00710 for the closed‐loop condition. Note that this 3.5‐times increase in the damping ratio is achieved with only 0.158 V.

When the controller gain is increased to K = 1000 Ns m⁻¹, the resulting plate time response and the control voltage are shown in Figure 7.36. Such an increase in the control gain results in increasing the damping ratio from 0.001 96 for the open loop to 0.051 for the closed‐loop condition. Note that this 25‐times increase of the damping ratio is achieved with only 1.424 V.

Table 7.12 summarizes the closed‐loop damping ratio and control voltage of the plate/ACLD patches, for different control gains in comparison with the corresponding characteristics of the open‐loop system.

Solution

Figure 7.39a displays the frequency response of the shell/ACLD assembly, as predicted by the FEM approach from Sections 4.7 and 7.5, when using a proportional control gain K_P = 1.5E6 Nm m⁻¹. The corresponding control voltage is shown in Figure 7.39b. The effectiveness of the controller is very limited at the first and second modes (60 and 62 Hz). These modes and corresponding mode shapes are displayed in Table 4.12.

The effectiveness of the controller becomes more apparent at the 125 Hz mode and higher.

The associated control voltage reaches 11 V at the first mode (60 Hz) and attains a maximum value of 24 V at the 119 Hz mode.

Increasing the control gain K_P to 3E6 Nm m⁻¹ considerably improves the performance of the controller as can be seen in the frequency response displayed in Figure 7.40a. The effectiveness of the controller becomes evident at the 62 Hz mode as well at the higher order modes.

In this case, the control voltage increases by 30 V at the first mode (60 Hz) and attains a maximum value of 120 V at the 119 Hz mode.

Figure 7.41 displays the ANSYS finite element model of the shell/ACLD assembly. The ANSYS model is used to predict the time response characteristics of the assembly at different control gains.

Figure 7.42a shows the time response of the shell/ACLD assembly when using a proportional control gain K_P = 1.5E6 Nm m⁻¹. The corresponding control voltage is shown in Figure 7.42b. The effectiveness of the controller is limited as it has only attenuated the first and second modes (60 and 62 Hz) but the 119 and 120 Hz modes are still dominating the response.

A peak control voltage of 18 V is observed with this control gain.

Figure 7.43a shows the time response of the shell/ACLD assembly when the proportional control gain K_P is increased to 3E6 Nm m⁻¹. The corresponding control voltage is shown in Figure 7.43b. The effectiveness of the controller has improved considerably as demonstrated by the significant attenuation of the 119 and 120 Hz modes of vibrations. In this case, the peak control voltage required reaches 40 V.

Table 7.14 summarizes the performance of the shell/ACLD assembly for different control gains as compared with that of the shell/PCLD assembly. The performance is quantified by the maximum deflection and control voltage at the different modes of vibration.

Problems

1. 7.1 Consider a mass = 100 kg supported on a VEM that is modeled on the following GHM model with one mini‐oscillator such that its stiffness is given by:
   

   that is, K_o = 100, α₁ = 7, ζ₁ = 1, and ω₁ = 1000 (see Problem 4.1). Determine the state‐space representation of the system. Plot the root locus of the system when the control law relies on:

   1. a position sensor with F_c = − k_gy, where y = [1  0  0  0]{Δ}.
   2. a velocity sensor with F_c = − k_gy, where y = [0  0  1  0]{Δ}.

   with kg = control gain and where x and z are the DOF of the mass and the internal DOF of the VEM. Note that F_c is the control force as shown in Figure P7.1.

   

   Identify the poles and zeros of the structure and of the VEM. Also determine the maximum possible damping ratio of the closed‐loop system when the controller is provided with position or velocity sensor.

2. 7.2 Consider the three‐layer composite rod that is mounted in a cantilevered configuration as shown in Figure P7.2. The rod consists of a base structure treated with a VEM layer that is constrained by an active piezoelectric constraining layer. The assembly experiences longitudinal vibration along the x direction.

   

   Derive expression for a globally stable boundary controller that ensures that the piezoelectric strain ε_p is generated to keep the rate of the total energy of the assembly strictly negative where E_n is given by:

   

   where T and U denote the kinetic and PE of the assembly such that:

   

   and

   

   where h_i, E_i, m_i, and u_i denote the thickness, Young's modulus, mass per unit length, and deflection of the ith layer (i = 1,…,3, such that 1 = piezo‐layer, 2 = VEM layer, and 3 = base structure). Also, b and L denote the width and length of the rod, respectively. The shear storage modulus and shear strain of the VEM are G^′ and γ, respectively, with γ = (u₁ − u₃)/h₂.

   Using Hamilton's principle show first that the equations of motion and the boundary conditions of the assembly are given by:

   

   and

   

   subject to the boundary conditions:

   • at x = 0: u_{1, x} = ε_P and u_{3, x} = 0
   • at x = L: u₁ = 0 and u₃ = 0.

   where G^* = G^′(1 + ηi) with η denoting the loss factor of the VEM.

   Using the equations of motion and the boundary conditions show that:

   

   Hence, for , show that the globally stable boundary control law is given by:

   ε_P = − K_G u_1,t with K_G = control gain > 0

3. 7.3 Derive the finite element model for a single element of the cantilevered three‐layer composite rod of Figure P7.2 such that:
   

   Assuming that the VEM is described by a single GHM mini‐oscillator using the modeling approach given by Eq. (7.5) such that:

   where , , and .

   with [M₀] and are given by:

   

   and

   where nodal deflection vector. The spatial distribution of u₁(x) and u₃(x) are assumed to be given by the following shape functions:

   

   where = wave number of piezo‐layer and

   

   where = wave number of base structure with ω and ρ_i denote the excitation frequency and density of the ith layer.

   Assume that the rod is made of aluminum with width of 0.025 m, thickness of 0.025 m, and length of 1 m. The rod is treated with a constrained damping treatment that has a width of 0.025 m, thickness = 0.025 m, and density of 1100 kg m⁻³. The storage modulus and loss factor of the VEM are predicted by the GHM model with one mini‐oscillator with E₀ = 15.3MPa, α₁ = 39, ζ₁ = 1, ω₁ = 19, 058rad/s. The physical and geometrical properties of the piezoelectric constraining layer are listed in Table 7.15 Physical and geometrical properties of the piezoelectric constraining layer.

   Determine the frequency response of the treated rod when subjected to a longitudinal force F = 10 N at its free end. Compare the results with the response of the rod assembly when its finite element is extracted using classical linear shape functions.

   Length (m)	Thickness (m)	Width (m)	Density (kg m−3)	Young's modulus (GPa)	d31 (m V−1)	k31	g31 (mV N−1)	k3t
   1.0	0.0025	0.025	7600	63	186E‐12	0.34	116E‐2	1950

4. 7.4 Consider a beam mounted in a fixed‐free configuration as shown in Figure P7.3. The beam is made of aluminum with width of 0.025 m, thickness of 0.025 m, and length of 1 m. The beam is treated with an unconstrained damping treatment that has a width of 0.025 m, thickness = 0.025 m, and density of 1100 kg m⁻³. The storage modulus and loss factor of the VEM are predicted by GHM model with one mini‐oscillator with E₀ = 15.3MPa, α₁ = 39, ζ₁ = 1,  ω₁ = 19, 058rad/s. For simplicity purposes, the structural system is modeled by one finite element model.

   

   Plot the pole‐zero map for the treated beam identifying the structural modes and VEM modes. Compare the results with the pole‐zero map of the untreated beam.

   Also determine the frequency response of the treated beam when subjected to a transverse force F = 10 N at its free end for a duration of 0.01 s. Compare the results with the response of the untreated beam.

5. 7.5 For the beam considered in Problem 7.3, the VEM is constrained by an aluminum constraining layer that is 0.025 m wide and 0.0025 m thick as shown in Figure P7.4.

   

   Plot the pole‐zero map for the treated beam identifying the structural modes and VEM modes. Compare the results with the pole‐zero map of the untreated beam.

   Also determine the time response of the treated beam when subjected to a transverse force F = 10 N at its free end for a duration of 0.01 s. Compare the results with the response of the untreated beam.

6. 7.6 Consider the beam system of Example 7.1 that has the physical and geometrical properties listed in Tables 7.2 and 7.3. The beam system is modeled by a single finite element as shown in Figure P7.5. The beam is controlled using a proportional‐derivative (PD) controller such that K_G(1 + 0.01s) where K_G is the control gain and s is the Laplace complex number. The input to the controller is the transverse deflection of the free end of the beam w₁.

   Plot the root locus map for the actively treated beam identifying the structural modes and VEM modes for different control gains K_G. Select the control gain that ensures the maximum closed‐loop damping ratio.

   Also determine the time response of the controlled beam when subjected to a transverse force F = 10 N at its free end for a duration of 0.01 s. Compare the results with the response of the uncontrolled beam. Also plot the associated time history of the control effort.

   

7. 7.7 For the distributed‐parameter modeling of the beam/ACLD assembly described in Section 7.3.4, develop a globally stable boundary control law if the beam is mounted in a simply‐supported configuration at both of its ends.

   Assume the physical and geometrical parameters of the beam/ACLD system with the VEM are the same as those described in Example 7.1.

   Determine the compliance and control voltage characteristics of the beam/ACLD system using a velocity feedback controller with a control gain K_D = 7.4. Compare these characteristics with those of the beam/PCLD system using the DPM approach outlined in Section 7.3.4. Also, compare the predictions of the DPM against the predictions of the finite element model when the beam is excited by a unit sinusoidal load at its free end.

   Also determine the energy dissipation characteristics of the beam/ACLD system as compared with that of the beam/PCLD system.

8. 7.8 Consider the aluminum cantilever plate that is fully treated with an ACLD patch placed as shown in Figure P7.6. The dimensions of the plate and the PCLD patches are shown in the figure with h₁ = h₂ = h₃ = 0.005m.

   The VEM is modeled using a GHM model with three mini‐oscillators such that:

   

   The parameters α_i, ζ_i, and ω_i are listed in Table 4.7. Assume that the constraining layer is made of piezoelectric material (PZT‐4) with d₃₁ = d₃₂ = −123 × 10⁻¹² m V⁻¹ and . Determine the frequency response of the plate, at different control gains, when it is excited by a unit force acting at the middle of the free end. Assume a proportional controller that feeds the transverse deflection of the free end back to control the piezoelectric constraining layer.

   Also determine the corresponding control effort.

   

9. 7.9 Consider the aluminum cantilever plate that is partially treated with an ACLD patch placed as shown in Figure P7.7. The dimensions of the plate and the PCLD patches are shown in the figure with h₁ = h₂ = h₃ = 0.005m. The physical properties of the VEM and the piezoelectric constraining layers are the same as listed in Problem 7.6.

   Determine the frequency response of the plate, at different control gains, when it is excited by a unit force acting at the middle of the free end. Assume a proportional controller that feeds the transverse deflection of the free end back to control the piezoelectric constraining layer.

   Also determine the corresponding control effort. Compare the results with the corresponding results obtained when the plate is fully treated with the ACLD treatment as in Problem 7.8.

   

10. 7.10 Consider the clamped‐free shell/ACLD assembly shown in Figure P7.8a. The main physical and geometrical parameters of the shell and PCLD are listed in Table 7.13. The shell has an internal radius R is 0.1016 m. The ACLD treatment consists of two patches as displayed in Figure P7.8b. The patches are bonded 180° apart on the outer surface of the cylinder with each of which subtending an angle of 90° angle at the center of the shell.

    Determine the frequency response of the system when excited at the free end by a unit force. Also determine the time response of system when a unit pulse force of duration 0.10 ms is applied at the free end. Compare the response when the response is predicted using ANSYS and the finite element approach outlined in Sections 4.7 and 7.5.

    Compare the results with those displayed in Example 7.5 for the partially treated shell/ACLD system.