Dynamic Response of a Beam With Absorber Exposed to a Running Force: Fractional Calculus Approach

2012 ◽  
Author(s):  
Ibrahim Abu-Alshaikh ◽  
Anas N. Al-Rabadi ◽  
Hashem S. Alkhaldi
Keyword(s):  
Fractional Calculus ◽  
Dynamic Response ◽  
Fractional Derivative ◽  
Initial Conditions ◽  
Damping Ratio ◽  
Experimental Models ◽  
Damping Characteristics ◽  
Damping Behavior ◽  
The Laplace Transform ◽  
Vibration Parameters

This paper analyzes the transverse vibration of Bernoulli-Euler homogeneous isotropic simply-supported beam. The beam is assumed to be fractionally-damped and attached to a single-degree-of-freedom (SDOF) absorber with fractionally-damping behavior at the mid-span of the beam. The beam is also exposed to a running force with constant velocity. The fractional calculus is introduced to model the damping characteristics of both the beam and absorber. The Laplace transform accompanied by the used decomposition method is applied to solve the handled problem with homogenous initial conditions. Subsequently, curves are depicted to measure the dynamic response of the utilized beam under different set of vibration parameters and different values of fractional derivative orders for both of the beam and absorber. The results obtained show that the dynamic response decreases as both the damping-ratio of the absorber and beam increase. The results reveal that there are critical values of fractional derivative orders which are different from unity. At these optimal values, the beam behaves with less dynamic response than that obtained for the full-order derivatives model of unity order. Therefore, the fractional derivative approach provides better damping models for fractionally-damped structures and materials which may allow researchers to choose suitable mathematical models that precisely fit the corresponding experimental models for many engineering applications.

scholarly journals Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

Vibration of a Beam-Oscillator System Subjected to a Moving Vehicle: Fractional Derivative Approach

2012 ◽  
Author(s):  
Hashem S. Alkhaldi ◽  
Ibrahim Abu-Alshaikh ◽  
Anas N. Al-Rabadi
Keyword(s):  
Dynamic Response ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Moving Load ◽  
Moving Vehicle ◽  
Beam Dynamic ◽  
Damping Characteristics ◽  
Born Series ◽  
The Laplace Transform ◽  
Derivatives Of

The dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam is investigated. The beam is appended at its mid-span by a single-degree-of-freedom (SDOF) fractionally-damped oscillator. The beam is further subjected to a vehicle modeled as a spring-dashpot system moves with a constant velocity over the beam. Hence, the damping characteristics of the beam and SDOF attached-oscillator are formally described in terms of fractional derivatives of arbitrary orders. In the analysis, the beam, SDOF oscillator, and the vehicle are assumed to be initially at rest. A system of three coupled differential equations is produced. These equations are handled by combining the Laplace transform with the Born series. Thereafter, curves are plotted to show the effect of the moving vehicle and the fractional derivatives behavior on the dynamic response of the beam. The numerical results show that the dynamic response decreases as the damping-ratios of the used absorber and beam increase. However, there are some optimal values of fractional derivative orders which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivative model. A comparison between the moving load and moving vehicle shows a significant reduction in the beam dynamic response in the case when vehicle is compared with the running load.

The Effects of Fiber Volume Fraction on the Vibration Damping Characteristics of Three-Layer-Connected Biaxial Weft Knitted Fabric Reinforced Composite Materials

2012 ◽  
Vol 602-604 ◽  
pp. 49-52
Author(s):  
Jing Xue Liu ◽  
Jia Lu Li
Keyword(s):  
Fiber Volume Fraction ◽  
Volume Fraction ◽  
Damping Ratio ◽  
Vibration Damping ◽  
Knitted Fabric ◽  
Vibration Modes ◽  
Fiber Volume ◽  
Damping Behavior ◽  
Weft Knitted Fabric ◽  
Vibration Parameters

The paper presents an analysis of the vibration damping properties of three-layer-connected biaxial weft knitted fabric (TBWK), which are constituted of carbon fibers as inserted yarns and polyester yarns as knitted yarns impregnated in an epoxy matrix with resin transfer molding (RTM) technique. Damping parameters were investigated using beam test specimens and an impulse technique. Several vibration parameters were varied to characterize the damping behavior in different amplitudes, natural frequencies and vibration modes. The results obtained show that the damping ratio of TBWK composites decreases with the increasing of fiber volume fraction in all the three vibration modes. The vibration test also indicates that the natural frequency of the TBWK composites increases with the increasing of fiber volume fraction (Vf) in all the three modes.

Experimental Investigation of Vibration Damping Behavior of Magneto-Mechanical Coated AISI321 Stainless-Steel

2019 ◽  
Author(s):  
Hafiz Muhammad Ashraf ◽  
Farhan Ali
Keyword(s):  
Stainless Steel ◽  
Dynamic Response ◽  
High Speed ◽  
High Cycle Fatigue ◽  
Damping Ratio ◽  
Base Material ◽  
Rotating Machinery ◽  
Cycle Fatigue ◽  
Damping Characteristics ◽  
Aisi 321

Abstract High speed rotating machineries usually operate under severe conditions and enormous loadings and thus, are susceptible to several problems. One such problem that has caught the attention in recent decades is known as High Cycle Fatigue. More than 60 percent of rotating machinery failures has been attributed to this High cycle Fatigue. Along with High Cycle Fatigue, Vibration, an inherent phenomenon in machineries, also share its part in failure of rotating machineries. Rotating machinery components suffer from high amplitude vibrations when they pass through resonance. Stresses are developed as a result of these vibrations and fatigue in mechanical structures, providing a conducive environment for the development of cracks at Surface. When these surface cracks reach critical size, crack nucleation starts, which ultimately leads to catastrophic failures. So, in order to avoid the disastrous consequences, damping is needed. Damping keeps material’s integrity in case of impact forces, stresses due to thermal shocks in turbo machinery and earth quakes in huge structures. Thin layer of magneto elastic coating can be applied on substrate surface that acts as first line of defense. Large number of coating Processes are available around the globe. The optimized combination of coating material, substrate material and coating technique according to specific application is necessary. These coatings have the capability to combat the phenomenon of oxidation, wear and fatigue acting as a barrier between substrate and hostile environments. Further, they enhance the damping characteristics, and thus allows the high-speed rotating machinery to reach its operational speed without any failure at resonance. In this way, they not only enhance the performance of components in aggressive environments, but also improve the life cycle, saving assets of millions of dollars’ worth. This research is an endeavor to experimentally investigate effect of magneto mechanical coating on damping of AISI 321 Stainless steel. AISI 321 was selected as base material because of its wide applications in engine components of gas turbines, heat exchangers and in different chemical industries. Two types of Air plasma sprayed magneto-mechanical powder (NiAl & CoNiCrAlY) were coated on base material and thickness was maintained up to 250μm in both the cases. Experiments were designed and performed on cantilever beam specimens for dynamic response measurement. Dynamic response of the system was measured to investigate the modal parameters of natural frequencies, damping ratio and time of vibration decay. For damping ratio, vibration analyzer mode was adjusted in time domain and beam was excited by using a hammer. Vibration analyzer showed the vibration decay as a function of time. Logarithmic decrement method was used to calculate the damping ratio in both cases. Dynamic response of all the three cases (NiAl coating, CoNiCrAlY and uncoated AISI321) were compared. Results were very reassuring and showed a significant improvement in damping characteristics.

scholarly journals Vibration Control of Fractionally-Damped Beam Subjected to a Moving Vehicle and Attached to Fractionally-Damped Multiabsorbers

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Hashem S. Alkhaldi ◽  
Ibrahim M. Abu-Alshaikh ◽  
Anas N. Al-Rabadi
Keyword(s):  
Dynamic Response ◽  
Fractional Derivatives ◽  
Single Degree Of Freedom ◽  
Moving Vehicle ◽  
Single Degree ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
Fractional Differential ◽  
Damped Systems ◽  
Special Case

This paper presents the dynamic response of Bernoulli-Euler homogeneous isotropic fractionally-damped simply-supported beam. The beam is attached to multi single-degree-of-freedom (SDOF) fractionally-damped systems, and it is subjected to a vehicle moving with a constant velocity. The damping characteristics of the beam and SDOF systems are described in terms of fractional derivatives. Three coupled second-order fractional differential equations are produced and then they are solved by combining the Laplace transform with the decomposition method. The obtained numerical results show that the dynamic response decreases as (a) the number of absorbers attached to the beam increases and (b) the damping-ratios of used absorbers and beam increase. However, there are some critical values of fractional derivatives which are different from unity at which the beam has less dynamic response than that obtained for the full-order derivatives model. Furthermore, the obtained results show very good agreements with special case studies that were published in the literature.

Experimental Investigation of Vibration Damping Behavior of Magneto-Mechanical Coated AISI321 Stainless-Steel

2020 ◽  
Author(s):  
Hafiz Muhammad Ashraf ◽  
Farhan Ali ◽  
Muhammad Imran Sadiq
Keyword(s):  
Stainless Steel ◽  
Dynamic Response ◽  
High Cycle Fatigue ◽  
Damping Ratio ◽  
Base Material ◽  
Rotating Machinery ◽  
Cycle Fatigue ◽  
Damping Characteristics ◽  
Aisi 321 ◽  
Rotating Machineries

Abstract High speed rotating machineries usually operate under severe conditions and enormous loadings and thus, are susceptible to several problems. One such problem that has caught the attention in recent decades is known as High Cycle Fatigue. More than 60 percent of rotating machinery failures has been attributed to this High cycle Fatigue. Along with High Cycle Fatigue, Vibration, an inherent phenomenon in machineries, also share its part in failure of rotating machineries. Rotating machinery components suffer from high amplitude vibrations when they pass through resonance. Stresses are developed as a result of these vibrations and fatigue in mechanical structures, providing a conducive environment for the development of cracks at Surface. When these surface cracks reach critical size, crack nucleation starts, which ultimately leads to catastrophic failures. So, in order to avoid the disastrous consequences, damping is needed. Damping keeps material’s integrity in case of impact forces, stresses due to thermal shocks in turbo machinery and earth quakes in huge structures. Thin layer of magneto elastic coating can be applied on substrate surface that acts as first line of defense. Large number of coating Processes are available around the globe. The optimized combination of coating material, substrate material and coating technique according to specific application is necessary. These coatings have the capability to combat the phenomenon of oxidation, wear and fatigue acting as a barrier between substrate and hostile environments. Further, they enhance the damping characteristics, and thus allows the highspeed rotating machinery to reach its operational speed without any failure at resonance. In this way, they not only enhance the performance of components in aggressive environments, but also improve the life cycle, saving assets of millions of dollars’ worth. This research is an endeavor to experimentally investigate effect of magneto mechanical coating on damping of AISI 321 Stainless steel. AISI 321 was selected as base material because of its wide applications in engine components of gas turbines, heat exchangers and in different chemical industries. Two types of Air plasma sprayed magneto-mechanical powder (NiAl & CoNiCrAlY) were coated on base material and thickness was maintained up to 250μm in both the cases. Experiments were designed and performed on cantilever beam specimens for dynamic response measurement. Dynamic response of the system was measured to investigate the modal parameters of natural frequencies, damping ratio and time of vibration decay. For damping ratio, vibration analyzer mode was adjusted in time domain and beam was excited by using a hammer. Vibration analyzer showed the vibration decay as a function of time. Logarithmic decrement method was used to calculate the damping ratio in both cases. Dynamic response of all the three cases (NiAl coating, CoNiCrAlY and uncoated AISI321) were compared. Results were very reassuring and showed a significant improvement in damping characteristics.

The Effect of Longitudinal Center of Gravity Position on the Sway Stability of a Small Cargo Trailer

2008 ◽  
Author(s):  
Michael G. Gilbert ◽  
Daniel A. Godrick ◽  
Richard H. Klein
Keyword(s):  
Dynamic Response ◽  
Damping Ratio ◽  
Leaf Spring ◽  
Loading Conditions ◽  
Novice Drivers ◽  
Training Requirements ◽  
Mandatory Training ◽  
Damping Characteristics ◽  
The Stability ◽  
The Impact

Small and mid-sized cargo trailers are often used to transport goods by people with limited experience in loading trailers and driving vehicles with trailers attached. This paper examines the effect of front to rear load position on the stability of a trailer by measuring its dynamic response to a variety of steer inputs at several different highway speeds. Additionally, tests with varying steers and speeds were performed with a simulated suspension malfunction to study the trailer’s dynamic response to this condition. Trailer sway has been a well-documented trailer characteristic for decades. However there are no special driver’s licensing or mandatory training requirements for even large trailers and campers. The trailer chosen for this test was a lightweight double axle cargo trailer commonly rented by people with limited to no towing experience. This consumer is likely to be unfamiliar with the best practices of trailer loading. This consumer is also likely a non-professional driver with little to no towing experience in the event of encountering unexpected trailer sway. Therefore it was the goal of the authors to determine how the stability of this type of trailer varies with different front to rear loading conditions and speeds to see if it is safe to operate on the highways by novice drivers. Trailer sway stability was determined by measuring the trailer sway (articulation) response during repeated, pulse steer tests. The trailer sway damping characteristics were measured, as a “damping ratio”, for six different hitch loads that corresponded to six different longitudinal loading conditions. These conditions, expressed as % load forward of the trailer centerline / % load aft of the trailer centerline were: 65/35, 60/40, 55/45, 50/50, 45/55, and 40/60. These loading conditions were tested per SAE J2664 [1] protocol. The resulting trailer sway characteristics for each loading condition then were compared to published trailer sway stability criteria [2, 3] to determine the suitability of this particular tow vehicle-trailer combination for use by the public in a rental market. The impact of a suspension malfunction on the trailer stability was also studied. This consisted of a detachment of one rear leaf spring hanger.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

scholarly journals The Laplace transform in discrete fractional calculus

2011 ◽  
Vol 62 (3) ◽  
pp. 1591-1601 ◽  
Author(s):  
Michael T. Holm
Keyword(s):  
Fractional Calculus ◽  
Laplace Transform ◽  
Discrete Fractional Calculus ◽  
The Laplace Transform
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