A.A. Al-Qaisia

Comparison of analytical techniques for nonlinear vibrations of a parametrically excited cantilever

Abstract This paper is concerned with second-order approximations to the steady-state principal parametric resonance response of a vertically mounted flexible cantilever beam subjected to a vertical harmonic base motion. The unimodal form of the nonlinear equation describing the in-plane large amplitude parametric response of the beam, derived in Krishnamurthy (Ph.D. Thesis, Department of Mechanical Engineering, Washington State University, 1986) based on the previous analysis in Crespo da Silva and Glynn (Journal of Structural Mechanics 1978; 6:437–48), is analysed using the harmonic balance (HB) and the perturbation method of multiple time scales (MMS). Single term HB, two terms HB, and second-order MMS with reconstitution version I (Nayfeh and Sanchez, Journal of Sound and Vibration 1989; 24:483–97) and version II (Rahman and Burton, Journal of Sound and Vibration 1989; 133:369–79) approximations to the steady-sate frequency–amplitude curves of the principal parametric resonance for each of the first four natural modes of the cantilever beam are compared with each other and with those obtained by numerically integrating the unimodal equation of motion. The time transformation T= Ω t is used in obtaining these approximations; also detuning is used in obtaining the square of the forcing MMS approximations. The obtained results show that, for the problem under consideration, the MMS version II is, in comparison with MMS version I, simpler to apply and leads to qualitatively more accurate second-order results. These results, however, show that the MMS version II tends to produce appreciable over corrections to the first-order results and may breakdown at relatively low response amplitudes, whereas the two terms HB solutions tend to improve the first-order results and lead to fairly accurate results even for relatively large response amplitudes.

Identification and cascade control of servo-pneumatic system using Particle Swarm Optimization

Abstract This paper presents a cascade control methodology for pneumatic systems using Particle Swarm Optimization (PSO). First, experimental data is collected and used to identify the servo-pneumatic system where an Auto-Regressive Moving-Average (ARMA) model is formulated using PSO algorithm. Then, cascaded Proportional–Integral–Derivative (PID) controller with PSO tuning is proposed and implemented on real system using Hardware-In-the-Loop (HIL). The identified model is validated experimentally and the performance of the cascaded-PID controller is tested under various conditions of speed variation. Experimental results show that cascaded-PID with PSO tuning performs better than single-PID, especially in disturbance rejection (a practical challenge in industrial pneumatic systems). Results also show that cascaded-PID with PSO-tuning performs better than cascaded-PID with self-tuning in the transient and steady-state responses.

Robust Estimation-Based Control of Chaotic Behavior in an Oscillator with Inertial and Elastic Symmetric Nonlinearities

In this paper, we study the dynamics of a forced nonlinear oscillator with inertial and elastic symmetric nonlinearities using modern nonlinear, bifurcation and chaos theories. The results for the response have shown that, for a certain combination of physical parameters, this oscillator exhibits a chaotic behavior or a transition to chaos through a sequence of period doubling bifurcations. The main objective of this paper is to control the chaotic behavior for this type of oscillator. A nonlinear estimation-based controller is proposed and the transient performance is investigated. The design of the parameter update mechanism is analyzed while discussing ways to extend its performance to further account for other types of uncertainties. We examine robustness problems as well as ways to tune the controller parameters. Simulation results are presented for the uncontrolled and controlled cases, verifying the effectiveness and the capability of the proposed controller. Finally, a discussion and conclusions are given with possible future extensions.

Effect of Fluid Mass on Non-Linear Natural Frequencies of a Rotating Beam

The non-linear natural frequencies of the first three modes of a beam clamped to a rigid rotating hub and carrying a distributed fluid along its span are investigated. The mathematical model is derived using the Lagrangian method and the continuous system is discretized using the assumed mode method. The resulted unimodal nonlinear equation of motion was solved using two methods; harmonic balance (HB) and time transformation (TT), to obtain approximate analytical expressions for the nonlinear natural frequencies. Results have shown that the two terms harmonic balance method (2THB) is more accurate than the time TT method. Results for the effect and type of distribution, i.e. uniform or linearly distributed, on the variation of the nonlinear natural frequency with the rotational speed of the system and how they affect the stability are studied and presented in non-dimensional form.Copyright

Primary Resonance Response of a Beam with a Differential Edge Settlement Attached to an Elastic Foundation

In this work, we study, through one-mode simplification, the influence of frequency curve veering on the primary resonance response of an elastic Euler-Bernoulli attached to a linear Winkler elastic foundation. The beam is a hinged-hinged with one torsional spring at one end, has an initial 1/4 sine deflection shape due to a constant differential edge settlement and is subjected to a uniformly distributed vertical load which is harmonically varying with a time part and a large mean part. A combined numerical-analytical procedure which accounts for the nonlinear interdependence between the vertical deflection and induced axial force due to mid-plane stretching was used to determine the beam static deflection. The assumed single mode approach is used to obtain the reduced nonlinear temporal equation of motion about the static equilibrium deflection which contains quadratic and cubic nonlinear terms. The results of numerical simulation indicate that the coefficients of the quadratic and cubic nonlinear terms in the temporal problem can, depending on the selected range of system parameters, vary widely and take positive and negative values, and thus change the number and stability of equilibrium positions as well as the system behavior which can be a hardening or softening type. The harmonic balance and, for comparison purposes, the method of multiple-scales are used to obtain approximate analytical solution for the primary resonance response and its stability. The obtained primary frequency response results are presented over a selected range of system parameters near and away from veering points which show a significant change in response behavior. The obtained approximate-analytical results were compared with those obtained numerically.

On the Steady State Response of a Cantilever Beam Partially Immersed in a Fluid and Carrying an Intermediate Mass

This paper presents a study on the nonlinear steady state response of a slender beam partially immersed in a fluid and carrying an intermediate mass. The model is developed based on the large deformation theory with the constraint of inextensible beam, which is valid for most engineering structures. The Lagrangian dynamics in conjunction with the assumed mode method is utilized in deriving the non-linear unimodal temporal equation of motion. The distributed and concentrated sinusoidal loads are accounted for in a consistent manner using the assumed mode method. The non-linear equation of motion is, analytically, solved using the single term harmonic balance (SHB) and the two terms harmonic balance (2HB) methods. The stability of the system, under various loading conditions, is investigated. The results are presented, discussed and some conclusions on the partially immersed beam nonlinear dynamics are extracted.

Crack Localization in Stepped Beams

A method for crack localization, based on so-called Local Modal Crack Sensitivity (LMCS), has recently been proposed by one of the Authors. As a major feature, the method only involves measuring eigenfrequencies. The method has already been used for beams of uniform cross section and homogeneous boundary conditions. In this paper, the method is extended to stepped beams, with both homogeneous and non-homogeneous boundary conditions. Experimental results confirm its usefulness for crack localization. Satisfactory localization of a second crack was also possible, so confirming the effectiveness of the method.

Dynamic Model of a Rotating Flexible Arm-Flexible Root Mechanism Driven by a Shaft Flexible in Torsion

This paper presents a dynamic model of a rotating flexible beam carrying a payload at its tip. The model accounts for the driving shaft and the arm root flexibilities. The finite element method and the Lagrangian dynamics are used in deriving the equations of motion with the small deformation theory assumptions and the Euler-Bernoulli beam theory. The obtained model is a nonlinear-coupled system of differential equations. The model is simulated for different combinations of shaft and root flexibilities and arm properties. The simulation results showed that the root flexibility is an important factor that should be considered in association with the arm and shaft flexibilities, as its dynamics influence the motor motion. Moreover, the effect of system non-linearity on the dynamic behavior is investigated by simulating the equivalent linearized system and it was found to be an important factor that should be considered, particularly when designing a control strategy for practical implementation.

Crack localization in non‐rotating shafts coupled to elastic foundation using sensitivity analysis techniques

The problem of damage and crack detection in structural components has acquired an important role in recent years. Since the presence of cracks in a structure may alter its vibrational characteristics, the estimation of such variations can be used to detect cracks and damage, and to monitor the integrity of structures. The use of fast, easy and inexpensive non‐destructive testing is thus a major task. In this paper, sensitivity analysis by measurement of the reduction of eigenfrequencies was utilized to localize a crack in a non‐rotating shaft coupled to an elastic foundation. The shaft was modeled by the finite element method and coupled to an experimentally identified foundation model. The detection of a crack with different depths and orientations was verified experimentally and a good agreement between actual and detected crack positions was achieved. Finally easiness, effectiveness, applicability of the method and its extensions are also shown.

Non-Linear Free Vibrations of a Rotating Beam Carrying a Tip Mass With Rotary Inertia

The non-linear, for each of the first three modes, of planar, large amplitude flexural free vibrations of a beam clamped with an angle to a rigid rotating hub and carrying a tip mass with rotary inertia are investigated. The shear deformation and rotary inertia effects are assumed to be negligible, but account is taken of axial inertia, non-linear curvature and the inextensibility condition. The Lagrangian dynamics in conjunction with the assumed mode method, assuming constant hub rotation speed, is utilized in deriving the non-linear unimodal temporal problem. The time transformation method is employed to obtain an approximate solution to the frequency-amplitude relation of the beam-mass free vibration, since the order of the nonlinear terms is not small which includes static and inertial geometric stiffening as well as inertial softening terms. Results in non-dimensional form are presented graphically, for the effect beam root-attachment angle, hub radius and the attached inertia element ratio on the variation of the natural frequency with vibration amplitude.Copyright