Zhuangyi Liu - Academia.edu (original) (raw)

Papers by Zhuangyi Liu

Research paper thumbnail of On coupled transversal and axial motions of two beams with a joint

Journal of Mathematical Analysis and Applications, 2008

In this paper we develop and analyze a mathematical model for combined axial and transverse motio... more In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.

Research paper thumbnail of Stability of the wave equations on a tree with local Kelvin–Voigt damping

Semigroup Forum

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt... more In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalizes the cases of single elastic string with local Kelvin-Voigt damping in [21, 24, 5].

Research paper thumbnail of Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping

The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or vi... more The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or viscoelastic damping and various boundary conditions is proved. This improves the corresponding results by Lagnese by showing that the semigroup is still exponentially stable even without feedback control on the boundary. The proof is essentially based on PDE techniques and the method is remarkable in the sense that it also throws light on applications to other higher-dimensional problems. with a, P, rj, a > 0,7 > 0 being constants, and the prime being time derivative. Various boundary conditions could be imposed on 9 depending on what is assumed about the

Research paper thumbnail of On the time decay of solutions for non-simple elasticity with voids

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2015

In this work we consider the non-simple theory of elastic material with voids and we investigate ... more In this work we consider the non-simple theory of elastic material with voids and we investigate how the coupling of these two aspects of the material affects the behavior of the solutions. We analyze only two kind of different behavior, slow or exponential decay. We introduce four different dissipation mechanisms in the system and we study, in each case, the effect of this mechanism in the behavior of the solutions.

Research paper thumbnail of Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

Journal of Differential Equations, 2015

In this paper, we provide a complete regularity analysis for the following abstract system of cou... more In this paper, we provide a complete regularity analysis for the following abstract system of coupled hyperbolic and parabolic equations      u tt = −Au + γA α w, w t = −γA α u t − kA β w, u(0) = u 0 , u t (0) = v 0 , w(0) = w 0 , where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α, β) ∈ [0, 1] × [0, 1]. We are able to decompose the unit square of the parameter (α, β) into three parts where the semigroup associated with the system is analytic, of specific order Gevrey classes, and non-smoothing, respectively. Moreover, we will show that the orders of Gevrey class are sharp, under proper conditions.

Research paper thumbnail of Analyticity of semigroup associated with a laminated composite beam

We consider a system of coupled partial differential equations that describe the vibrations of la... more We consider a system of coupled partial differential equations that describe the vibrations of laminated beam in which the layers are bonded together by a medium that dissipates energy at a rate proportional to the shear. We show that for the simplest model, in which only transverse inertial energy is accounted for, the associated semigroup is analytic.

Research paper thumbnail of Well-posedness and exponential stability of a thermoelastic Joint–Leg–Beam system with Robin boundary conditions

Mathematical and Computer Modelling, 2009

An important class of proposed large space structures features a triangular truss backbone. In th... more An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint-beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.

Research paper thumbnail of Well-posedness, stability and numerical results for the thermoelastic behavior of a coupled joint-beam PDE-ODE system modeling the transverse motions of the antennas of a space structure

PAMM, 2007

A Mathematical model for both axial and transverse motions of two beams with cylindrical cross-se... more A Mathematical model for both axial and transverse motions of two beams with cylindrical cross-sections coupled through a joint is presented and analyzed. The motivation for this problem comes from the need to accurately model damping and joint dynamics for the next generation of inflatable/rigidizable space structures. Thermo-elastic damping is included in the two beams and the motions are coupled through a joint which includes an internal moment. Thermal response in each beam is modeled by two temperature fields. The first field describes the circumferentially averaged temperature along the beam, and is linked to the axial deformation of the beam. The second describes the circumferential variation and is coupled to transverse bending. The resulting equations of motion consist of four, second-order in time, partial differential equations, four, first-order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. The system is written as an abstract differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections and temperature fields, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove that the system is well-posed, and that with positive damping parameters the resulting semigroup is exponentially stable. Steady states are characterized and several numerical approximation results are presented.

Research paper thumbnail of Surface heating and three-dimensional motion of a thermoelastic beam

Mathematical and Computer Modelling, 2010

ABSTRACT

Research paper thumbnail of Exponential decay of energy of vibrating strings with local viscoelasticity

ABSTRACT In our previous paper (11), we studied the energy decay rate for an one-dimensional line... more ABSTRACT In our previous paper (11), we studied the energy decay rate for an one-dimensional linear wave equation with the Kelvin-Voigt damping presented on a subinterval which models an elastic string with one segment made of viscoelastic material and the other of elastic material. It was proved that the energy of that system does not decay exponentially when each segment is homogeneous, i.e., coe-cient functions are piecewise constant and have discontinuity at the interface. This is a puzzling result since it is not seen for the local viscous damping, where the well-known "geometric optics" condition applies. In order to get a better understanding of the causes for this phenomenon, we study two related problems in this paper. We flrst reconsider the above system with smooth coe-cient functions. Then we replace the Kelvin-Voigt model by the Boltzmann model and allow discontinuity of material properties at the interfaces. Exponential energy decay is proved for both cases. These new results suggest that discontinuity of material properties at the interface and the \type" of the damping can afiect the qualitative behavior of the energy decay.

Research paper thumbnail of Spectrum and Stability for Elastic Systems with Global or Local Kelvin--Voigt Damping

SIAM Journal on Applied Mathematics, 1998

... and strain tensors, mn denotes density, the volume distribution of external forces F is zero,... more ... and strain tensors, mn denotes density, the volume distribution of external forces F is zero, and the area distribution of external force F, is ... Finally, in section 5, we show that the semigroup associated with a one-dimensional wave equation is not exponentially stable, even with KV ...

Research paper thumbnail of Energy decay rate of the thermoelastic Bresse system

Zeitschrift für angewandte Mathematik und Physik, 2009

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes... more In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions. . 74K10.

Research paper thumbnail of Decay Rates for a Beam with Pointwise Force and Moment Feedback

Mathematics of Control, Signals, and Systems (MCSS), 2002

Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with point... more Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0, π) with certain boundary conditions. The energy decay rate in both cases is ...

Research paper thumbnail of Polynomial stability of a joint-leg-beam system with local damping

Mathematical and Computer Modelling, 2007

Recent advances in the design and construction of large inflatable/rigidizable space structures a... more Recent advances in the design and construction of large inflatable/rigidizable space structures and potential new applications of such structures have produced a demand for better analysis and computational tools to deal with the new class of structures. Understanding stability and damping properties of truss systems composed of these materials is central to the successful operation of future systems. In this paper, we consider a mathematical model for an assembly of two elastic beams connected to a joint through legs. The dynamic joint model is composed of two rigid-bodies (the joint-legs) with an internal moment. In an ideal design all struts and joints will have identical material and geometric properties. In this case we previously established exponential stability of the beam-joint system. However, in order to apply theoretical stability estimates to realistic systems one must deal with the case where the individual truss components 1 are not identical and still be able to analyze damping. We consider a problem of this type where one beam is assumed to have a small Kelvin-Voigt damping parameter and the second beam has no damping. In this case, we prove that the component system is only polynomially damped even if additional rotational damping is assumed in the joint.

Research paper thumbnail of On coupled transversal and axial motions of two beams with a joint

Journal of Mathematical Analysis and Applications, 2008

In this paper we develop and analyze a mathematical model for combined axial and transverse motio... more In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.

Research paper thumbnail of A Note on the Exponential Decay of Energy of a Euler–Bernoulli Beam with Local Viscoelasticity

Journal of Elasticity, 2000

In this paper, we consider an Euler–Bernoulli beam equation with one segment of the beam made of ... more In this paper, we consider an Euler–Bernoulli beam equation with one segment of the beam made of viscoelastic material of Boltzmann type and the other segment made of elastic material. Strong stability and exponential stability of the associated semigroup are obtained under certain smoothness conditions imposed on the coefficient functions of the equation.

Research paper thumbnail of Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip

Journal of Computational and Applied Mathematics, 2000

We consider a nonhomogeneous Euler-Bernoulli beam with rotatory inertia at the tip. Uniform bound... more We consider a nonhomogeneous Euler-Bernoulli beam with rotatory inertia at the tip. Uniform boundary stabilization of this system is proved via the frequency-domain multiplier method. We also show that this system is associated with a C 0 group and has a complete set of generalized eigenfunctions.

Research paper thumbnail of Results on Transversal and Axial Motions of a System of Two Beams Coupled to a Joint Through Two Legs

We report some results of an ongoing research related to a model consisting of an assembly of two... more We report some results of an ongoing research related to a model consisting of an assembly of two beams, coupled to a simple joint through two legs. The motivation for this problem comes from the need to simulate the dynamic behavior of the next generation of inflatable/rigidizable space structures and to accurately account for damping and joint effects. We assume Kelvin-Voigt damping in the two beams and coupling through a joint which includes an internal moment. The resulting equations of motions consist of four, second order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. An explicit form for the energy of the system is found and its dissipativeness is proved. The system can be written as a second order ODE in an appropriate Hilbert space, in which well posedness, exponential stability as well as other regularity properties of the solutions can be obtained. A standard finiteelement approximation leads to a second-order differential-algebraic system including joint-forces and an algebraic constraint. A projection approach is used to eliminate joint-forces and enforce compatibility. In a second approach, the compatibility constraint is enforced in the construction of the finite element basis. Several numerical results are shown.

Research paper thumbnail of On coupled transversal and axial motions of two beams with a joint

Journal of Mathematical Analysis and Applications, 2008

In this paper we develop and analyze a mathematical model for combined axial and transverse motio... more In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.

Research paper thumbnail of Stability of the wave equations on a tree with local Kelvin–Voigt damping

Semigroup Forum

In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt... more In this paper we study the stability problem of a tree of elastic strings with local Kelvin-Voigt damping on some of the edges. Under the compatibility condition of displacement and strain and continuity condition of damping coefficients at the vertices of the tree, exponential/polynomial stability are proved. Our results generalizes the cases of single elastic string with local Kelvin-Voigt damping in [21, 24, 5].

Research paper thumbnail of Exponential stability of the Kirchhoff plate with thermal or viscoelastic damping

The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or vi... more The exponential stability of the semigroup associated with the Kirchhoff plate with thermal or viscoelastic damping and various boundary conditions is proved. This improves the corresponding results by Lagnese by showing that the semigroup is still exponentially stable even without feedback control on the boundary. The proof is essentially based on PDE techniques and the method is remarkable in the sense that it also throws light on applications to other higher-dimensional problems. with a, P, rj, a > 0,7 > 0 being constants, and the prime being time derivative. Various boundary conditions could be imposed on 9 depending on what is assumed about the

Research paper thumbnail of On the time decay of solutions for non-simple elasticity with voids

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 2015

In this work we consider the non-simple theory of elastic material with voids and we investigate ... more In this work we consider the non-simple theory of elastic material with voids and we investigate how the coupling of these two aspects of the material affects the behavior of the solutions. We analyze only two kind of different behavior, slow or exponential decay. We introduce four different dissipation mechanisms in the system and we study, in each case, the effect of this mechanism in the behavior of the solutions.

Research paper thumbnail of Regularity analysis for an abstract system of coupled hyperbolic and parabolic equations

Journal of Differential Equations, 2015

In this paper, we provide a complete regularity analysis for the following abstract system of cou... more In this paper, we provide a complete regularity analysis for the following abstract system of coupled hyperbolic and parabolic equations      u tt = −Au + γA α w, w t = −γA α u t − kA β w, u(0) = u 0 , u t (0) = v 0 , w(0) = w 0 , where A is a self-adjoint, positive definite operator on a complex Hilbert space H, and (α, β) ∈ [0, 1] × [0, 1]. We are able to decompose the unit square of the parameter (α, β) into three parts where the semigroup associated with the system is analytic, of specific order Gevrey classes, and non-smoothing, respectively. Moreover, we will show that the orders of Gevrey class are sharp, under proper conditions.

Research paper thumbnail of Analyticity of semigroup associated with a laminated composite beam

We consider a system of coupled partial differential equations that describe the vibrations of la... more We consider a system of coupled partial differential equations that describe the vibrations of laminated beam in which the layers are bonded together by a medium that dissipates energy at a rate proportional to the shear. We show that for the simplest model, in which only transverse inertial energy is accounted for, the associated semigroup is analytic.

Research paper thumbnail of Well-posedness and exponential stability of a thermoelastic Joint–Leg–Beam system with Robin boundary conditions

Mathematical and Computer Modelling, 2009

An important class of proposed large space structures features a triangular truss backbone. In th... more An important class of proposed large space structures features a triangular truss backbone. In this paper we study thermomechanical behavior of a truss component; namely, a triangular frame consisting of two thin-walled circular beams connected through a joint. Transverse and axial mechanical motions of the beams are coupled though a mechanical joint. The nature of the external solar load suggests a decomposition of the temperature fields in the beams leading to two heat equations for each beam. One of these fields models the circumferential average temperature and is coupled to axial motions of the beam, while the second field accounts for a temperature gradient across the beam and is coupled to beam bending. The resulting system of partial and ordinary differential equations formally describes the coupled thermomechanical behavior of the joint-beam system. The main work is in developing an appropriate state-space form and then using semigroup theory to establish well-posedness and exponential stability.

Research paper thumbnail of Well-posedness, stability and numerical results for the thermoelastic behavior of a coupled joint-beam PDE-ODE system modeling the transverse motions of the antennas of a space structure

PAMM, 2007

A Mathematical model for both axial and transverse motions of two beams with cylindrical cross-se... more A Mathematical model for both axial and transverse motions of two beams with cylindrical cross-sections coupled through a joint is presented and analyzed. The motivation for this problem comes from the need to accurately model damping and joint dynamics for the next generation of inflatable/rigidizable space structures. Thermo-elastic damping is included in the two beams and the motions are coupled through a joint which includes an internal moment. Thermal response in each beam is modeled by two temperature fields. The first field describes the circumferentially averaged temperature along the beam, and is linked to the axial deformation of the beam. The second describes the circumferential variation and is coupled to transverse bending. The resulting equations of motion consist of four, second-order in time, partial differential equations, four, first-order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. The system is written as an abstract differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections and temperature fields, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove that the system is well-posed, and that with positive damping parameters the resulting semigroup is exponentially stable. Steady states are characterized and several numerical approximation results are presented.

Research paper thumbnail of Surface heating and three-dimensional motion of a thermoelastic beam

Mathematical and Computer Modelling, 2010

ABSTRACT

Research paper thumbnail of Exponential decay of energy of vibrating strings with local viscoelasticity

ABSTRACT In our previous paper (11), we studied the energy decay rate for an one-dimensional line... more ABSTRACT In our previous paper (11), we studied the energy decay rate for an one-dimensional linear wave equation with the Kelvin-Voigt damping presented on a subinterval which models an elastic string with one segment made of viscoelastic material and the other of elastic material. It was proved that the energy of that system does not decay exponentially when each segment is homogeneous, i.e., coe-cient functions are piecewise constant and have discontinuity at the interface. This is a puzzling result since it is not seen for the local viscous damping, where the well-known "geometric optics" condition applies. In order to get a better understanding of the causes for this phenomenon, we study two related problems in this paper. We flrst reconsider the above system with smooth coe-cient functions. Then we replace the Kelvin-Voigt model by the Boltzmann model and allow discontinuity of material properties at the interfaces. Exponential energy decay is proved for both cases. These new results suggest that discontinuity of material properties at the interface and the \type" of the damping can afiect the qualitative behavior of the energy decay.

Research paper thumbnail of Spectrum and Stability for Elastic Systems with Global or Local Kelvin--Voigt Damping

SIAM Journal on Applied Mathematics, 1998

... and strain tensors, mn denotes density, the volume distribution of external forces F is zero,... more ... and strain tensors, mn denotes density, the volume distribution of external forces F is zero, and the area distribution of external force F, is ... Finally, in section 5, we show that the semigroup associated with a one-dimensional wave equation is not exponentially stable, even with KV ...

Research paper thumbnail of Energy decay rate of the thermoelastic Bresse system

Zeitschrift für angewandte Mathematik und Physik, 2009

In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes... more In this paper, we study the energy decay rate for the thermoelastic Bresse system which describes the motion of a linear planar, shearable thermoelastic beam. If the longitudinal motion and heat transfer are neglected, this model reduces to the well-known thermoelastic Timoshenko beam equations. The system consists of three wave equations and two heat equations coupled in certain pattern. The two wave equations about the longitudinal displacement and shear angle displacement are effectively damped by the dissipation from the two heat equations. Actually, the corresponding energy decays exponentially like the classical one-dimensional thermoelastic system. However, the third wave equation about the vertical displacement is only weakly damped. Thus the decay rate of the energy of the overall system is still unknown. We will show that the exponentially decay rate is preserved when the wave speed of the vertical displacement coincides with the wave speed of longitudinal displacement or of the shear angle displacement. Otherwise, only a polynomial type decay rate can be obtained. These results are proved by verifying the frequency domain conditions. . 74K10.

Research paper thumbnail of Decay Rates for a Beam with Pointwise Force and Moment Feedback

Mathematics of Control, Signals, and Systems (MCSS), 2002

Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with point... more Abstract. We consider the Rayleigh beam equation and the Euler–Bernoulli beam equation with pointwise feedback shear force and bending moment at the position ξ in a bounded domain (0, π) with certain boundary conditions. The energy decay rate in both cases is ...

Research paper thumbnail of Polynomial stability of a joint-leg-beam system with local damping

Mathematical and Computer Modelling, 2007

Recent advances in the design and construction of large inflatable/rigidizable space structures a... more Recent advances in the design and construction of large inflatable/rigidizable space structures and potential new applications of such structures have produced a demand for better analysis and computational tools to deal with the new class of structures. Understanding stability and damping properties of truss systems composed of these materials is central to the successful operation of future systems. In this paper, we consider a mathematical model for an assembly of two elastic beams connected to a joint through legs. The dynamic joint model is composed of two rigid-bodies (the joint-legs) with an internal moment. In an ideal design all struts and joints will have identical material and geometric properties. In this case we previously established exponential stability of the beam-joint system. However, in order to apply theoretical stability estimates to realistic systems one must deal with the case where the individual truss components 1 are not identical and still be able to analyze damping. We consider a problem of this type where one beam is assumed to have a small Kelvin-Voigt damping parameter and the second beam has no damping. In this case, we prove that the component system is only polynomially damped even if additional rotational damping is assumed in the joint.

Research paper thumbnail of On coupled transversal and axial motions of two beams with a joint

Journal of Mathematical Analysis and Applications, 2008

In this paper we develop and analyze a mathematical model for combined axial and transverse motio... more In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.

Research paper thumbnail of A Note on the Exponential Decay of Energy of a Euler–Bernoulli Beam with Local Viscoelasticity

Journal of Elasticity, 2000

In this paper, we consider an Euler–Bernoulli beam equation with one segment of the beam made of ... more In this paper, we consider an Euler–Bernoulli beam equation with one segment of the beam made of viscoelastic material of Boltzmann type and the other segment made of elastic material. Strong stability and exponential stability of the associated semigroup are obtained under certain smoothness conditions imposed on the coefficient functions of the equation.

Research paper thumbnail of Boundary stabilization of a nonhomogeneous beam with rotatory inertia at the tip

Journal of Computational and Applied Mathematics, 2000

We consider a nonhomogeneous Euler-Bernoulli beam with rotatory inertia at the tip. Uniform bound... more We consider a nonhomogeneous Euler-Bernoulli beam with rotatory inertia at the tip. Uniform boundary stabilization of this system is proved via the frequency-domain multiplier method. We also show that this system is associated with a C 0 group and has a complete set of generalized eigenfunctions.

Research paper thumbnail of Results on Transversal and Axial Motions of a System of Two Beams Coupled to a Joint Through Two Legs

We report some results of an ongoing research related to a model consisting of an assembly of two... more We report some results of an ongoing research related to a model consisting of an assembly of two beams, coupled to a simple joint through two legs. The motivation for this problem comes from the need to simulate the dynamic behavior of the next generation of inflatable/rigidizable space structures and to accurately account for damping and joint effects. We assume Kelvin-Voigt damping in the two beams and coupling through a joint which includes an internal moment. The resulting equations of motions consist of four, second order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. An explicit form for the energy of the system is found and its dissipativeness is proved. The system can be written as a second order ODE in an appropriate Hilbert space, in which well posedness, exponential stability as well as other regularity properties of the solutions can be obtained. A standard finiteelement approximation leads to a second-order differential-algebraic system including joint-forces and an algebraic constraint. A projection approach is used to eliminate joint-forces and enforce compatibility. In a second approach, the compatibility constraint is enforced in the construction of the finite element basis. Several numerical results are shown.