Existence, Nonexistence, and Stability of Solutions for a Delayed Plate Equation with the Logarithmic Source

In this work, we study a plate equation with time delay in the velocity, frictional damping, and logarithmic source term. Firstly, we obtain the local and global existence of solutions by the logarithmic Sobolev inequality and the Faedo-Galerkin method. Moreover, we prove the stability and nonexistence results by the perturbed energy and potential well methods.

A Novel Test Rig for the Basic Nonlinear Characterization of Bolted Joints

The paper aims at performing a comprehensive experimental study on the peculiar properties of a bolted joint, and investigates the damping induced at different interfaces (between flanges, bolt head/nut and flange, threads) during vibrations. A novel, simplified, single-bolt system joining a two-beam structure is designed and tested. Experimental results under different boundary conditions are presented, and the influence of the harmonic excitation force, as well as the bolt tension, is investigated. The test results show how the contact interface between the clamped flanges plays an important role in terms of frictional damping provided to the system during vibration, while the contact interfaces between the head/nut and flange, and secondarily between the threads, affect the system response at a less, but not negligible, extent. The test setup and test procedure can provide a database to validate single bolt contact models to be included in a more complex structure.

Exponential decay for the semilinear wave equation with localized frictional and Kelvin–Voigt dissipating mechanisms

In the present paper, we are concerned with the semilinear viscoelastic wave equation in an inhomogeneous medium Ω subject to two localized dampings. The first one is of the type viscoelastic and is distributed around a neighborhood ω of the boundary according to the Geometric Control Condition. The second one is a frictional damping and we consider it hurting the geometric condition of control. We show that the energy of the wave equation goes uniformly and exponentially to zero for all initial data of finite energy taken in bounded sets of finite energy phase-space.

Frictional damping in hollow beam structures joined by bolts, rivets, and adhesive

The objective of this work was to investigate how different joining techniques affect the level of damping in structures. Beams were constructed from four different joining techniques, bolting, riveting, adhesive bonding, and brazing by joining two lengths of steel each with a ‘U’-shaped cross-section. They were joined such that the edges of the ‘U’ overlapped to form a tube. The damping of each beam was determined by flexural vibration. The bolted beam had a series of bolts along its length. The effect of removing bolts was investigated. It was found that removing bolts increased damping. When bolts were removed successively from holes at the end of the beam, the damping increased more than when bolts were removed from holes in the middle of the beam. A further objective of this project was to investigate the effect of introducing penetrant between two surfaces. WD-40 was introduced between the contacting surfaces for the beams joined by mechanical fastening. The penetrant had the effect of increasing damping. This may be because the penetrant has the effect of increasing the relative displacement between the two beams, leading to greater energy dissipation. Introducing penetrant also changed the order of which beam had the greatest damping, with the bolted beam now having greater damping than the riveted beam. The effect of increasing bolt tension on the bolted beam was also investigated. When the beams were dry, increasing bolt tension reduced the damping, but when penetrant was introduced increasing the bolt tension increased the damping. A comparison between the damping properties from different joining techniques was made. The conclusions could be applied in industry by engineers constructing beams of a similar fashion.

General decay results for a viscoelastic wave equation with a variable exponent nonlinearity

In this paper we consider a viscoelastic wave equation with a very general relaxation function and nonlinear frictional damping of variable-exponent type. We give explicit and general decay results for the energy of the system depending on the decay rate of the relaxation function and the nature of the variable-exponent nonlinearity. Our results extend the existing results in the literature to the case of nonlinear frictional damping of variable-exponent type.

Existence and general decay of Balakrishnan-Taylor viscoelastic equation with nonlinear frictional damping and logarithmic source term

<p style='text-indent:20px;'>In this paper, we consider a Balakrishnan-Taylor viscoelastic wave equation with nonlinear frictional damping and logarithmic source term. By assuming a more general type of relaxation functions, we establish explicit and general decay rate results, using the multiplier method and some properties of the convex functions. This result is new and generalizes earlier results in the literature.</p>

Decay result in a problem of a nonlinearly damped wave equation with variable exponent

<abstract><p>In this work we study a wave equation with a nonlinear time dependent frictional damping of variable exponent type. The existence and uniqueness results are established using Fadeo-Galerkin approximation method. We also exploit the Komornik lemma to prove the uniform stability result for the energy associated to the solution of the problem under consideration.</p></abstract>

Global existence, energy decay and blow-up of solutions for wave equations with time delay and logarithmic source

In this paper, we study the wave equation with frictional damping, time delay in the velocity, and logarithmic source of the form $$ u_{tt}(x,t) - \Delta u (x,t) + \alpha u_{t} (x,t) + \beta u_{t} (x, t- \tau ) = u(x,t) \ln \bigl\vert u(x,t) \bigr\vert ^{\gamma } . $$ u t t ( x , t ) − Δ u ( x , t ) + α u t ( x , t ) + β u t ( x , t − τ ) = u ( x , t ) ln | u ( x , t ) | γ . There is much literature on wave equations with a polynomial nonlinear source, but not much on the equations with logarithmic source. We show the local and global existence of solutions using Faedo–Galerkin’s method and the logarithmic Sobolev inequality. And then we investigate the decay rates and infinite time blow-up for the solutions through the potential well and perturbed energy methods.