Abstract: The objective of this study is to investigate the forced vibration analysis of a planar curved beam lying on elastic foundation by using the mixed finite element method. The finite element formulation is based on the Timoshenko beam theory. In order to solve the problems in frequency domain, the element matrices of two nodded curvilinear elements are transformed into Laplace space. The results are transformed back to the time domain by the well-known numerical Modified Durbin’s transformation algorithm. First, the presented finite element formulation is verified through the forced vibration analysis of a planar curved Timoshenko beam resting on Winkler foundation and the finite element results are compared with the results available in the literature. Then, the forced vibration analysis of a planar curved beam resting on Winkler-Pasternak foundation is conducted.
Abstract: The static and dynamic analyses of hyperboloidal helix having the closed and the open square box sections are investigated via the mixed finite element formulation based on Timoshenko beam theory. Frenet triad is considered as local coordinate systems for helix geometry. Helix domain is discretized with a two-noded curved element and linear shape functions are used. Each node of the curved element has 12 degrees of freedom, namely, three translations, three rotations, two shear forces, one axial force, two bending moments and one torque. Finite element matrices are derived by using exact nodal values of curvatures and arc length and it is interpolated linearly throughout the element axial length. The torsional moments of inertia for close and open square box sections are obtained by finite element solution of St. Venant torsion formulation. With the proposed method, the torsional rigidity of simply and multiply connected cross-sections can be also calculated in same manner. The influence of the close and the open square box cross-sections on the static and dynamic analyses of hyperboloidal helix is investigated. The benchmark problems are represented for the literature.
Abstract: In this study, the free vibration analysis of conical helicoidal rods with two different elliptically oriented cross sections is investigated and the results are compared by the circular cross-section keeping the net area for all cases equal to each other. Problems are solved by using the mixed finite element formulation. Element matrices based on Timoshenko beam theory are employed. The finite element matrices are derived by directly inserting the analytical expressions (arc length, curvature, and torsion) defining helix geometry into the formulation. Helicoidal rod domain is discretized by a two-noded curvilinear element. Each node of the element has 12 DOFs, namely, three translations, three rotations, two shear forces, one axial force, two bending moments and one torque. A parametric study is performed to investigate the influence of elliptical cross sectional geometry and its orientation over the natural frequencies of the conical type helicoidal rod.
Abstract: In this study, out-of-plane free vibrations of a circular
rods is investigated theoretically. The governing equations for
naturally twisted and curved spatial rods are obtained using
Timoshenko beam theory and rewritten for circular rods. Effects of
the axial and shear deformations are considered in the formulations.
Ordinary differential equations in scalar form are solved analytically
by using transfer matrix method. The circular rods of the mass matrix
are obtained by using straight rod of consistent mass matrix. Free
vibrations frequencies obtained by solving eigenvalue problem. A
computer program coded in MATHEMATICA language is prepared.
Circular beams are analyzed through various examples for free
vibrations analysis. Results are compared with ANSYS results based
on finite element method and available in the literature.
Abstract: This paper studies free vibration of functionally
graded beams Subjected to Axial Load that is simply supported at
both ends lies on a continuous elastic foundation. The displacement
field of beam is assumed based on Engesser-Timoshenko beam
theory. The Young's modulus of beam is assumed to be graded
continuously across the beam thickness. Applying the Hamilton's
principle, the governing equation is established. Resulting equation is
solved using the Euler's Equation. The effects of the constituent
volume fractions and foundation coefficient on the vibration
frequency are presented. To investigate the accuracy of the present
analysis, a compression study is carried out with a known data.
Abstract: This paper deals with the design of a periodic output
feedback controller for a flexible beam structure modeled with
Timoshenko beam theory, Finite Element Method, State space
methods and embedded piezoelectrics concept. The first 3 modes are
considered in modeling the beam. The main objective of this work is
to control the vibrations of the beam when subjected to an external
force. Shear piezoelectric sensors and actuators are embedded into
the top and bottom layers of a flexible aluminum beam structure, thus
making it intelligent and self-adaptive. The composite beam is
divided into 5 finite elements and the control actuator is placed at
finite element position 1, whereas the sensor is varied from position 2
to 5, i.e., from the nearby fixed end to the free end. 4 state space
SISO models are thus developed. Periodic Output Feedback (POF)
Controllers are designed for the 4 SISO models of the same plant to
control the flexural vibrations. The effect of placing the sensor at
different locations on the beam is observed and the performance of
the controller is evaluated for vibration control. Conclusions are
finally drawn.
Abstract: Active vibration control is an important problem in
structures. The objective of active vibration control is to reduce the vibrations of a system by automatic modification of the system-s
structural response. In this paper, the modeling and design of a fast
output sampling feedback controller for a smart flexible beam system embedded with shear sensors and actuators for SISO system using
Timoshenko beam theory is proposed. FEM theory, Timoshenko beam theory and the state space techniques are used to model the
aluminum cantilever beam. For the SISO case, the beam is divided into 5 finite elements and the control actuator is placed at finite
element position 1, whereas the sensor is varied from position 2 to 5, i.e., from the nearby fixed end to the free end. Controllers are
designed using FOS method and the performance of the designed FOS controller is evaluated for vibration control for 4 SISO models
of the same plant. The effect of placing the sensor at different locations on the beam is observed and the performance of the
controller is evaluated for vibration control. Some of the limitations of the Euler-Bernoulli theory such as the neglection of shear and
axial displacement are being considered here, thus giving rise to an accurate beam model. Embedded shear sensors and actuators have
been considered in this paper instead of the surface mounted sensors
and actuators for vibration suppression because of lot of advantages. In controlling the vibration modes, the first three dominant modes of
vibration of the system are considered.
Abstract: This paper studies mechanical buckling of
functionally graded beams subjected to axial compressive load that is
simply supported at both ends lies on a continuous elastic foundation.
The displacement field of beam is assumed based on Engesser-Timoshenko beam theory. Applying the Hamilton's principle, the
equilibrium equation is established. The influences of dimensionless geometrical parameter, functionally graded index and foundation
coefficient on the critical buckling load of beam are presented. To investigate the accuracy of the present analysis, a compression study
is carried out with a known data.
Abstract: This paper features the mathematical modeling of a single input single output based Timoshenko smart beam. Further, this mathematical model is used to design a multirate output feedback based discrete sliding mode controller using Bartoszewicz law to suppress the flexural vibrations. The first 2 dominant vibratory modes is retained. Here, an application of the discrete sliding mode control in smart systems is presented. The algorithm uses a fast output sampling based sliding mode control strategy that would avoid the use of switching in the control input and hence avoids chattering. This method does not need the measurement of the system states for feedback as it makes use of only the output samples for designing the controller. Thus, this methodology is more practical and easy to implement.
Abstract: This paper presents the elastic buckling of
homogeneous beams with a pair of piezoelectric layers surface
bonded on both sides of the beams. The displacement field of beam is
assumed based on the Engesser-Timoshenko beam theory.
Applying the Hamilton's principle, the equilibrium equation is
established. The influences of applied voltage, dimensionless
geometrical parameter and piezoelectric thickness on the critical
buckling load of beam are presented. To investigate the accuracy of
the present analysis, a compression study is carried out with a known
data.
Abstract: Active Vibration Control (AVC) is an important
problem in structures. One of the ways to tackle this problem is to
make the structure smart, adaptive and self-controlling. The objective
of active vibration control is to reduce the vibration of a system by
automatic modification of the system-s structural response. This
paper features the modeling and design of a Periodic Output
Feedback (POF) control technique for the active vibration control of
a flexible Timoshenko cantilever beam for a multivariable case with
2 inputs and 2 outputs by retaining the first 2 dominant vibratory
modes using the smart structure concept. The entire structure is
modeled in state space form using the concept of piezoelectric
theory, Timoshenko beam theory, Finite Element Method (FEM) and
the state space techniques. Simulations are performed in MATLAB.
The effect of placing the sensor / actuator at 2 finite element
locations along the length of the beam is observed. The open loop
responses, closed loop responses and the tip displacements with and
without the controller are obtained and the performance of the smart
system is evaluated for active vibration control.
Abstract: This paper deals with a numerical analysis of the
transient response of composite beams with strain rate dependent
mechanical properties by use of a finite difference method. The
equations of motion based on Timoshenko beam theory are derived.
The geometric nonlinearity effects are taken into account with von
Kármán large deflection theory. The finite difference method in
conjunction with Newmark average acceleration method is applied to
solve the differential equations. A modified progressive damage
model which accounts for strain rate effects is developed based on
the material property degradation rules and modified Hashin-type
failure criteria and added to the finite difference model. The
components of the model are implemented into a computer code in
Mathematica 6. Glass/epoxy laminated composite beams with
constant and strain rate dependent mechanical properties under
dynamic load are analyzed. Effects of strain rate on dynamic
response of the beam for various stacking sequences, load and
boundary conditions are investigated.