Vibration Analysis of Beams Using Alternative Admissible Functions With Penalties

Assumed mode methods are often used in vibrations analysis, where the choice of assumed mode affects the stability and useability of the method. System eigenfunctions are often used for these expansions, however a change in the boundary conditions usually results in a change in eigenfunction. This paper investigates the use of Alternative Admissible Functions (AAF) with penalties for the vibration analysis of an Euler-Bernoulli beam for different boundary conditions. A key advantage of the proposed approach is that the choice of AAF does not depend on the boundary conditions since the boundary conditions are modelled via penalty functions. The mathematical formulation of the system matrices, and the effect of beam geometry changes on the computed natural frequencies and modeshapes are presented. The computed natural frequencies and mode shapes show an excellent agreement when compared with closed-form Euler-Bernoulli beam values. The study reveals that with an increase in the stiffness of the beam, the values of the penalties need to be increased. The results of this study suggest that boundary conditions, as well as beam geometrical parameters should be considered when selecting appropriate values of the penalties.

Parametric Stability Analysis of a Spring Attached, Pre-Twisted, Rotating Sandwich Beam with Tip Mass and Viscoelastic Support

This paper inspects the influence of a spring attachment provided on the top elastic layer on the stability of a pre-twisted, rotating sandwich beam having viscoelastic supports at the root under the impact of a periodically varying axial load. The spring is deployed on the beam to achieve more strength to weight ratio without compromising the stability. The beam is exponentially tapered, and a tip mass is at the free end to represent the rotating members in various types of machinery as closely as possible. The ruling equations and inter-related boundary conditions are attained by applying Hamilton’s principle. To obtain the solution, a matrix equation was developed through the assumed-mode variational method. The resulting matrix equation was converted to a coupled Hill’s equation of parametric vibration through the modal matrix corresponding to the free vibration problem. Finally, static and dynamic stability graphs were obtained for several system parameters such as position and length of the attached spring on the top elastic layer, the mass of the spring attachment, stiffness of the spring attachment, angle of pre-twist, tip mass, taper parameter, temperature gradient parameter, setting angle, viscoelastic spring stiffness, etc. to analyze their impact on the system’s stability. Saito and Otomi conditions were used to obtain dynamic stability plots. Greater stability is achieved due to the spring attachment on the top of the top elastic layer.

Development of Polynomial Mode Shape Functions for Continuous Shafts with Different End Conditions

Common methods used to determine the solutions for vibration response of continuous systems are assumed mode method, Rayleigh-Ritz method, Galerkin Method, finite element method, etc. Each of these methods requires the shape functions which satisfy the boundary conditions. Shape functions derived in most of the classical textbooks are simple trigonometric functions for some end conditions but are very complex transcendental functions for many end conditions. It is very difficult to determine the vibration response of a continuous system analytically by using such transcendental shape functions. Hence this paper presents a method to develop polynomial shape functions required to solve the vibration of continuous shafts with different end conditions. The natural frequencies obtained from the developed polynomial shape functions are compared to those obtained from the classical transcendental shape functions and found very close for the first three modes.

A composite controller for manipulator with flexible joint and link under uncertainties and disturbances

In this article, a composite controller is proposed for the manipulator with the flexible joint and link under uncertainties and time-varying disturbances. The dynamic of the system is developed by the Euler–Lagrange and assumed mode method, which is a nonlinear, strong coupling, and underacted system. Therefore, based on the singular perturbation theory, the dynamic is decomposed into a slow and fast subsystem. For the slow dynamic, a novel adaptive-gain super-twisting sliding mode controller is designed to guarantee joint tracking under the uncertainties and disturbances. For the fast dynamics, adaptive dynamic programming is used to deal with the uncertainty. The simulation result shows that the proposed composite controller can effectively track the trajectory and suppress the vibration simultaneously.

Nonlinear Controller Design for a Flexible Spacecraft

This paper combines the nonlinear Udwadia-Kalaba control approach with the Assumed Mode Method to model flexible structures and derives an attitude controller for a spacecraft. The study case of this paper is a satellite with four flexible cantilever beams attached to a rigid central hub. Two main topics are covered in this paper. The first one is the formulation of the equation of motion and the second one is the nonlinear controller design. The combination of these two techniques is able to provide a controller that damps the vibration of a flexible structure while achieving the desired rigid-motion state.

Vibration analysis and pull-in instability behavior in a multiwalled piezoelectric nanosensor with fluid flow conveyance

In this work, surface/interface effects for pull-in voltage and viscous fluid velocity effects on the dimensionless natural frequency of fluid-conveying multiwalled piezoelectric nanosensors (FC-MWPENSs) based on cylindrical nanoshells is investigated using the Gurtin–Murdoch surface/interface theory. The nanosensor is embedded in a viscoelastic foundation and subjected to nonlinear van der Waals and electrostatic forces. Hamilton’s principle is used to derive the governing and boundary conditions and is also the assumed mode method used for changing the partial differential equations into ordinary differential equations. The influences of the surface/interface effect, such as Lame’s constants, residual stress, piezoelectric constants and mass density, are considered for analysis of the dimensionless natural frequency with respect to the viscous fluid velocity and pull-in voltage of the FC-MWPENSs.

Static Nodes of an Axially Moving String With Time-Varying Supports

By investigating the transverse vibrations of an axially moving string with time-varying supports, the existence and the pattern of static nodes are studied based on the assumed mode method and the linear superposition method. The explicit expressions for the response of the system with five different boundary conditions are illustrated. Traditional excited static strings show nodes when resonance occurs. However, it is found in this study that the static nodes of axially moving strings appear under arbitrary frequency even far away from resonance, if the excitation frequency is higher than the fundamental frequency. The varying nodes and frequencies under different time-varying supports are revealed.

Vibration analysis and pull-in instability behavior in multi walled piezoelectric nano-sensor with fluid flow conveyance: influences of surface/interface energy

In this work, surface/interface effects for pull-in voltage and viscous fluid velocity effects on dimensionless natural frequency (DNF) of fluid-conveying multi walled piezoelectric nanoresonator (FC-MWPENS) based on cylindrical nanoshell is investigated using the Gurtin–Murdoch surface/interface theory. The nano-sensor is embedded in viscoelastic foundation, nonlinear van der Waals and electrostatic forces. Hamilton’s principle is used for deriving of the governing equations and boundary conditions and also the assumed mode method is used for changing the partial differential equations into ordinary differential equation. The influences of the surface/interface effect such as Lame’s constants, residual stress, piezoelectric constants and mass density are considered for analysis of dimensionless natural frequency respect to viscous fluid velocity and pull-in voltage of FC-MWPENS.

Effects of surface energy on vibration characteristics of double-walled piezo-viscoelastic cylindrical nanoshell

In this paper, vibration analysis of double-walled piezo-viscoelastic cylindrical nanoshell integrated with piezoelectric layers is investigated using Gurtin–Murdoch surface/interface theory and Donnell's theory. Three parameters namely, shear modulus, damp coefficient, and Winkler modulus are used for simulation of visco-Pasternak model. Hamilton's principle is used for deriving the governing equations and boundary conditions and also the assumed mode method is used for changing the partial differential equations into ordinary differential equation. The effects of the surface energy, length and thickness of nanoshell and piezoelectric layer, boundary condition, van der Waals force, and visco-Pasternak effects on the undamped and damped natural frequencies of piezo-viscoelastic cylindrical nanoshell is studied. Also, the results show that on considering surface effects in the nanoscale system without considering the surface density, the maximum frequency will be obtained and this case will be considered as the critical state of the system. As a result, controlling the frequency of the system in this case is essential and it is quite clear that considering the effects of the surface energy will have a remarkable effect on the natural frequency of the piezo-viscoelastic nanoshell.