Solvability of a Model for the Vibration of a Beam with a Damping Tip Body

We consider a model for the vibration of a beamwith a damping tip body that appeared in a previous article. In this paper we derive a variational form for the motion of the beam and use it to prove that the model problem has a unique solution. The proofs are based on existence results for a general linear vibration model problem, in variational form. Finite element approximation of the solution is discussed briefly.


Introduction
In article [1], the authors model and analyze the damped vibration of a cantilever beam with an attached hollow tip body that contains a granular material.The Euler-Bernoulli theory for a beam with Kelvin-Voigt damping is used.The beam is clamped at one end and the tip body is attached to the other end.The authors state that "the problem contains more complicated boundary conditions" than problems "treated previously" (and provide references).This is due to the fact that the model is more realistic-as we explain below.It is an interesting model for more than one reason.The dynamics of the rigid body is treated in a realistic way: the fact that the center of mass of the rigid body is not at the endpoint of the beam is taken into account.The damping mechanism is also explained unlike other papers where vibration models with boundary damping are considered.There are other articles with realistic models of a beam with damping, for example [2].However, [1] provides the only realistic model for the relevant beam-body configuration.
The existence of a unique solution for the model problem is established in [1].To obtain the result, the problem is written as an abstract differential equation and an abstract existence result from a previous paper [3] is applied.
In this paper, we also prove the existence of a unique solution.Our approach differs from that in [1]: we write the model problem in variational form and use results from [4] where general linear vibration models in variational form are considered.The existence theorems in [4] were applied to the vibration of a complex plate beam system in [5].It should be noted that the same variational form can be used for finite element approximations (see Section 7).Since our approach differs from that in [1], it is natural to consider differences regarding assumptions and results.There are indeed some differences but these are not substantial (see Section 6).
The mathematical model is considered in Section 2. In Section 3, we write the model problem in variational form and present the weak variational form in Section 4. Auxiliary results are proved in Section 5.The existence theorems are stated and proved in Section 6 where different methods are also compared.In Section 7, natural frequencies and modes are discussed as well as finite element approximation.

Model Problem
The Euler-Bernoulli model for the transverse vibration of a beam is derived from the equation of motion for the beam deflection : and the relation In these equations,  denotes the density,  the area of the cross section,  the shear force,  the bending moment, and  a load (beam models are treated in [6, pages 323-324], [7, pages 337-338], and [8, pages 392-395]).
The usual constitutive equation is  =  2  , where  is an elastic constant (Young's modulus) and  is the area moment of inertia.Due to Kelvin-Voigt damping, it changes to where  denotes the damping parameter.The partial differential equation (which we do not use) is The constitutive equation for the moment  and the relation between the moment and shear force  are also used to model the interface conditions.
The left endpoint of the beam is clamped where the boundary conditions are the usual  (0, ) =    (0, ) = 0. ( The interface conditions at the other endpoint are determined by the interaction between the beam and the rigid body.This is explained in [1] in some detail.It is necessary to consider the equations of motion for the rigid body carefully when deriving these conditions.
The position of the center of mass of the tip body relative to the endpoint of the beam is where  is the angle of the neutral plane with the horizontal (see Figure 1).Therefore, the velocity k  and acceleration a  of the center of mass are given by v  =    (ℓ, ) j −  θ sin i +  θ cos j, where ℓ denotes the length of the beam.For the linear approximation, it is assumed that the term θ 2 sin j may be neglected, θ ≈     (ℓ, ), θ ≈  2    (ℓ, ), and cos  ≈ 1.Using these approximations, we have the following expressions for the vertical components of the velocity and acceleration: In [1], the term     (ℓ, ) in the expressions for the vertical component of the velocity is neglected.In our opinion, this should not be done and we motivate our point of view in the next section where we discuss the decay of energy for the system.
In the equations below,  and  * denote damping parameters,  the mass, and  the moment of inertia of the rigid body.Using Newton's second law for the motion of the center of mass, we have where   () is an external force that may act on the rigid body, for example, gravity.Taking moments about the center of mass, we have Following [1], we combine ( 9) and ( 10) and find that It is convenient to rewrite ( 9) and ( 11) as follows: Model Problem.The mathematical model consists of equations of motion ( 1) and ( 2) and constitutive equation (3) for the beam, boundary conditions (5), and interface conditions (12).Initial conditions (⋅, 0) =  0 and   (⋅, 0) =  1 need to be specified.

Variational Form
Multiply (1) by an arbitrary smooth function V and integrate.Using integration by parts and (2) yields V  is integrable, and for each V ∈ [0, ℓ].
We now use the constitutive equation  =  2   +    2    and the interface conditions (12) to derive the variational form of the model problem.
It is convenient to introduce the following bilinear forms: We now have the variational form of the model problem.
Remark 1.Note that the bilinear forms , , and  are symmetric.The additional term   (ℓ)V(ℓ) in the definition of  is necessary for symmetry.Problem  may be used to compute finite element approximations.
Mechanical Energy.The mechanical energy (kinetic energy plus elastic potential energy) of the system is Using the symmetry of  and  and assuming that  is sufficiently smooth, we have for the homogeneous case.It is obvious that (, ) is nonnegative and not difficult to show that (, ) and (, ) are nonnegative: As a result,   () ≤ 0. This result is to be expected from Physics.The fact that  is symmetric and (, ) is nonnegative is due to the additional term.
The following product spaces are necessary for the abstract problem: An element  ∈  is written as  = ⟨ 1 ,  2 ,  3 ⟩.An obvious inner product for  is and we denote the corresponding norm by ‖ ⋅ ‖  .
Definition 2 (bilinear forms).For  and V in , and for  and Remarks.
(2) For  and V in , (, 3) It is essential to use product spaces since the bilinear form  is not defined on L 2 (0, ℓ).
For the weak variational form of the model problem, we need to show that the bilinear forms  and  are inner products for  and , respectively.We use the well-known Poincare type inequalities given in the proposition below.The boundedness of Γ is also required.
By the definition of the moment of inertia of a rigid body,  ≥  2 for some  > 0. Now choose 0 <  < 1 and  < (1 + ) −1 and the desired result follows.
Corollary 5.The bilinear form  is an inner product for the space .
Definition 6 (inertia space).The norm ‖ ⋅ ‖  is defined by ‖‖  = √(, ).We refer to the vector space  equipped with this norm as the inertia space and denote it by .Proposition 7.There exists a constant  such that Proof.We use Proposition 3 and the definition of the bilinear form : Corollary 8.The bilinear form  is an inner product for .
Definition 9 (energy space).The space  equipped with the inner product  is referred to as the energy space.The norm ‖ ⋅ ‖  is defined by ‖‖  = √(, ).
Remark 10.It is natural to think that ũ0,2 = Γ 0 and so forth are the correct initial conditions.This is discussed in Section 6.

Auxiliary Results
We need the results of this section to apply Theorems 15 and 16 in Section 6. Proposition 11.Space  is a dense subset of .
Proposition 13.There exists a constant  such that for each  and V in .
Proof.We use Proposition 3. Consider Now use Proposition 7.
The result above is true for  ≥ 0. If  > 0, the bilinear form  is positive definite on  and this has implications for existence results.

Existence
In this section, we apply the existence results from [4].
For convenience, we formulate the general linear vibration problem and present the relevant existence theorems.Let , , and  be real Hilbert spaces with  ⊂  ⊂ .Spaces , , and  have inner products (⋅, ⋅)  , , and  and norms ‖ ⋅ ‖  , ‖ ⋅ ‖  , and ‖ ⋅ ‖  , respectively.Consider also a bilinear form  defined on .
In Theorems 15 and 16, the following is assumed. Assumptions then problem PG has a unique solution: Theorem 16 (see [4,Theorem 3]).Suppose that assumptions (A1), (A2), (A3), and (A4) are satisfied.If (39) The approach in [4] is relatively new and therefore we discuss briefly how it is related to semigroup theory.Problem G is equivalent to a first order differential equation in the product space  =  × .A linear operator  with domain D() ⊂  ×  is constructed in [4] and problem G is equivalent to an initial value problem of the form The pair ⟨ 0 ,  1 ⟩ ∈ D() if and only if  0 and  1 in  satisfy condition () in Theorem 15.The properties of  are determined by the properties of the three bilinear forms , , and .Under the assumptions in Theorem 15  is the infinitesimal generator of a  0 semigroup of contractions and under the assumptions in Theorem 16  is the infinitesimal generator of an analytic semigroup.

Application
7.1.Natural Frequencies.In the second half of Section 4 in [1], the sequence of natural frequencies of the undamped system is considered.The conjecture on p1041 concerning the eigenvalues for the undamped system is indeed correct.For each  ∈ , the problem (, V) = (, V), for each V ∈ , has a unique solution  ∈ .The mapping , defined by  = , is a symmetric linear operator.Since  is bounded as a mapping from  into  and the embedding of  into  is compact,  is compact.Considering  as a bilinear form in , its eigenvalues are real and if  is an eigenvalue, then  −1 is an eigenvalue of .The corresponding eigenvectors are the same.It follows that the sequence of eigenvalues tends to infinity and the sequence of eigenvectors is complete in  (see, e.g., [12,Theorem 4.A, p.232]).
For the model problem with the damping tip body, the situation is different.To the best of our knowledge, there is no general spectral theory for systems with boundary damping.However, results are known for specific model problems.In [13], it is proved that the sequence of eigenfunctions for an Euler-Bernoulli beam with boundary damping has the Riesz basis property, but there is no attached body.
Galerkin Approximation for Problem .Let  ℎ be a finite dimensional subspace of .
Problem  ℎ is equivalent to a system of ordinary differential equations:   +   +  =  () . (52 The system can be used to approximate the solution of problem PW.How to construct the relevant matrices is explained in [14]. The quadratic eigenvalue problem  2  +   +  = 0 (53) can be used to calculate approximations for the natural frequencies (see [14]).

Figure 1 :
Figure 1: The tip body at the end of the beam.
. (A1)  is dense in  and  is dense in .(A2) There exists a constant   such that ‖V‖  ≤   ‖V‖  for each V ∈ .(A3) There exists a constant   such that ‖V‖  ≤   ‖V‖  for each V ∈ .(A4) The bilinear form  is symmetric, nonnegative, and bounded on ; that is, |(, V)| ≤ ‖‖  ‖V‖  for each  and V in . 0 ∈ ,  1 ∈  and there exists a  ∈  such that