Command Shaping Control of a Vertically Rotating Flexible Beam

2019 ◽  
Author(s):  
Khaled A. Alhazza
Keyword(s):  
Initial Conditions ◽  
Equation Of Motion ◽  
Wave Form ◽  
Flexible Beam ◽  
Beam Length ◽  
Rotating Beam ◽  
Double Step ◽  
Higher Modes ◽  
Command Shaping ◽  
System Equation

Abstract Vertically rotating flexible beam under the effect of gravity adds some challenges to the existing input- and command shaping techniques. Classical shapers usually have zero initial conditions while vertically rotating beam may have initial or final deflections. These conditions may introduce large residual vibrations at the end of rest-to-rest maneuvers. Furthermore, the system contains some nonlinearities that may reduce the effectiveness of classical input- and command-shapers. In this work, a waveform command shaping profile is used initially and then optimized to reduce rest-to-rest residual vibrations of a vertically rotating flexible beam. The system equation of motion is determined, discretized, and then linearized to find an initial command shaper. The parameters of the proposed command shaper is then optimized to find a better performance. Only the first mode is considered in this work since higher modes usually have negligible amplitudes. To show the importance of the proposed work, comparisons between the uncontrolled shaper, double step shaper, smooth wave form command shaping, and the optimized command shaping are performed. The proposed technique is tested numerically using two cases, with different maximum velocities, flexible beam length, and acceleration times. Results show that the effectiveness of the proposed technique in reducing residual vibrations in rest-to-rest maneuvers.

Adjustable maneuvering time wave-form command shaping control with variable hoisting speeds

2016 ◽  
Vol 23 (7) ◽  
pp. 1095-1105 ◽  
Author(s):  
Khaled A Alhazza
Keyword(s):  
Continuous Wave ◽  
Equation Of Motion ◽  
Approximation Technique ◽  
Wave Form ◽  
Input Shaping ◽  
Overhead Crane ◽  
Algorithm Optimization ◽  
Optimization Scheme ◽  
Command Shaping ◽  
Pendulum Model

Classical input shaping is based on convolving a general input signal with a sequence of timed impulses. These impulses are chosen to match certain modal parameters of the system under control to eliminate residual vibrations in rest-to-rest maneuvers. This type of input shaping is strongly dependent on the system period. In this work, an adjustable maneuvering time wave form command shaper is presented. The equation of motion of a simple pendulum model of a crane is derived and solved in order to eliminate residual vibrations at the end of motion. Several cases are simulated numerically and validated experimentally on an experimental model of an overhead crane. Results show that the proposed command shaper is capable of eliminating residual vibrations effectively with a single continuous wave form command. The work is extended to include the effect of hoisting on the shaper performance. Several functions are used to simulate hoisting. To overcome the added complexity of hoisting on the system, an approximation technique is used to determine initial shaped command parameters, which are later used in a genetic algorithm optimization scheme. Numerical and experimental results prove that the proposed command shaper can effectively eliminate residual vibrations in rest-to-rest maneuvers.

A Continuous Modulated Wave-Form Command Shaping for Damped Overhead Cranes

2011 ◽  
Author(s):  
Khaled A. Alhazza ◽  
Asmahan H. Al-Shehaima ◽  
Ziyad N. Masoud
Keyword(s):  
Control Strategy ◽  
Damping Ratio ◽  
Harmonic Oscillators ◽  
Wave Form ◽  
Double Step ◽  
Command Shaping ◽  
Overhead Cranes ◽  
Input Shaper ◽  
Displacement Profile ◽  
System Properties

A new command-shaping control strategy for oscillation reduction of damped harmonic oscillators is derived and implemented on damped overhead cranes. The effect of damping on the shaper frequency and duration is investigated. The performance of the proposed simper is simulated numerically and compared with the classical double-step input-shaper for different system properties. It was shown that, the proposed wave-form command profiles are capable of eliminating the travel and residual oscillations for systems with different damping ratio. Further, unlike traditional impulse and step command shapers, the proposed command profiles have smoother intermediate acceleration, velocity, and displacement profile.

Particle motion with air resistance

2012 ◽  
Vol 8 (1) ◽  
pp. 1-15
Author(s):  
Gy. Sitkei
Keyword(s):  
Basic Equation ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Terminal Velocity ◽  
Dimensionless Form ◽  
Wood Industry ◽  
Acceptable Error ◽  
General Validity ◽  
Practical Applications ◽  
Air Resistance

Motion of particles with air resistance (e.g. horizontal and inclined throwing) plays an important role in many technological processes in agriculture, wood industry and several other fields. Although, the basic equation of motion of this problem is well known, however, the solutions for practical applications are not sufficient. In this article working diagrams were developed for quick estimation of the throwing distance and the terminal velocity. Approximate solution procedures are presented in closed form with acceptable error. The working diagrams provide with arbitrary initial conditions in dimensionless form of general validity.

Nonlinear Vibrations of a Hinged Beam Including Nonlinear Inertia Effects

1973 ◽  
Vol 40 (1) ◽  
pp. 121-126 ◽  
Author(s):  
S. Atluri
Keyword(s):  
Nonlinear Equation ◽  
Nonlinear Vibrations ◽  
Initial Conditions ◽  
Equation Of Motion ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
Nonlinear Ordinary Differential Equations ◽  
Inertia Effects ◽  
Modal Expansion ◽  
General Response

This investigation treats the large amplitude transverse vibration of a hinged beam with no axial restraints and which has arbitrary initial conditions of motion. Nonlinear elasticity terms arising from moderately large curvatures, and nonlinear inertia terms arising from longitudinal and rotary inertia of the beam are included in the nonlinear equation of motion. Using a Galerkin variational method and a modal expansion, the problem is reduced to a system of coupled nonlinear ordinary differential equations which are solved for arbitrary initial conditions, using the perturbation procedure of multiple-time scales. The general response and frequency-amplitude relations are derived theoretically. Comparison with previously published results is made.

On the Vibration of a Rotating Disk

1972 ◽  
Vol 39 (4) ◽  
pp. 1143-1144 ◽  
Author(s):  
S. Barasch ◽  
Y. Chen
Keyword(s):  
Approximate Calculation ◽  
Initial Conditions ◽  
Rotating Disk ◽  
Equation Of Motion ◽  
Outer Radius ◽  
Order Equation ◽  
System Of Differential Equations ◽  
Fourth Order Equation ◽  
First Order ◽  
Inner Radius

The equation of motion of a rotating disk, clamped at the inner radius and free at the outer radius, is solved by reducing the fourth-order equation of motion to a set of four first-order equations subject to arbitrary initial conditions. A modified Adams’ method is used to numerically integrate the system of differential equations. Results show that Lamb-Southwell’s approximate calculation of the frequency is justified.

scholarly journals ON OSCILLATIONS DESCRIBING A GENERALIZED DIFFERENTIAL RELAY EQUATION

2020 ◽  
pp. 53-60
Author(s):  
Vasiliy Olshansky ◽  
Stanislav Olshansky ◽  
Oleksіі Tokarchuk
Keyword(s):  
Differential Equation ◽  
Initial Conditions ◽  
Oscillatory System ◽  
Equation Of Motion ◽  
Resistance Force ◽  
Oscillatory Process ◽  
Initial Deviation ◽  
Approximate Analytical Solutions ◽  
Positive Exponent ◽  
Generalized Differential

The motion of an oscillatory system with one degree of freedom, described by the generalized Rayleigh differential equation, is considered. The generalization is achieved by replacing the cubic term, which expresses the dissipative strength of the equation of motion, by a power term with an arbitrary positive exponent. To study the oscillatory process involved the method of energy balance. Using it, an approximate differential equation of the envelope of the graph of the oscillatory process is compiled and its analytical solution is constructed from which it follows that quasilinear frictional self-oscillations are possible only when the exponent is greater than unity. The value of the amplitude of the self-oscillations in the steady state also depends on the value of the indicator. A compact formula for calculating this amplitude is derived. In the general case, the calculation involves the use of a gamma function table. In the case when the exponent is three, the amplitude turned out to be the same as in the asymptotic solution of the Rayleigh equation that Stoker constructed. The amplitude is independent of the initial conditions. Self-oscillations are impossible if the exponent is less than or equal to unity, since depending on the initial deviation of the system, oscillations either sway (instability of the movement is manifested) or the range decreases to zero with a limited number of cycles, which is usually observed with free oscillations of the oscillator with dry friction. These properties of the oscillatory system are also confirmed by numerical computer integration of the differential equation of motion for specific initial data. In the Maple environment, the oscillator trajectories are constructed for various values of the nonlinearity index in the expression of the viscous resistance force and a corresponding comparative analysis is carried out, which confirms the adequacy of approximate analytical solutions.

scholarly journals Determine of the wave equation in the task of electrical oscillations

2018 ◽  
Vol 184 ◽  
pp. 01023
Author(s):  
Gordana V. Jelić ◽  
Vladica Stanojević ◽  
Dragana Radosavljević
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Mathematical Physics ◽  
Equation Of Motion ◽  
Independent Variables ◽  
Equations Of Mathematical Physics ◽  
Basic Equations ◽  
Derivatives Of ◽  
Two Independent Variables ◽  
Transverse Oscillations

One of the basic equations of mathematical physics (for instance function of two independent variables) is the differential equation with partial derivatives of the second order (3). This equation is called the wave equation, and is provided when considering the process of transverse oscillations of wire, longitudinal oscillations of rod, electrical oscillations in a conductor, torsional vibration at waves, etc… The paper shows how to form the equation (3) which is the equation of motion of each point of wire with abscissa x in time t during its oscillation. It is also shown how to determine the equation (3) in the task of electrical oscillations in a conductor. Then equation (3) is determined, and this solution satisfies the boundary and initial conditions.

scholarly journals Vibration Based Damage Detection of Rotating Beams

2018 ◽  
Vol 148 ◽  
pp. 14008 ◽  
Author(s):  
Stanislav Stoykov ◽  
Emil Manoach ◽  
Maosen Cao
Keyword(s):  
Damage Detection ◽  
Selection Criteria ◽  
Excitation Frequency ◽  
Real Life ◽  
Equation Of Motion ◽  
Damage Localization ◽  
Rotating Beam ◽  
Free Boundary Conditions ◽  
Wide Range ◽  
Detection And Localization

The early detection and localization of damages is essential for operation, maintenance and cost of the structures. Because the frequency of vibration cannot be controlled in real-life structures, the methods for damage detection should work for wide range of frequencies. In the current work, the equation of motion of rotating beam is derived and presented and the damage is modelled by reduced thickness. Vibration based methods which use Poincaré maps are implemented for damage localization. It is shown that for clamped-free boundary conditions these methods are not always reliable and their success depends on the excitation frequency. The shapes of vibration of damaged and undamaged beams are shown and it is concluded that appropriate selection criteria should be defined for successful detection and localization of damages.

Deformation and Fatigue Analysis of Micro Thermal-Electrostatic Actuator Devices

Volume 3 ◽  
2004 ◽  
Author(s):  
Jeng-Nan Hung ◽  
Meng-Ju Lin ◽  
Chung-Li Hwan
Keyword(s):  
Applied Voltage ◽  
High Cycle Fatigue ◽  
Fatigue Analysis ◽  
Flexible Beam ◽  
Cycle Fatigue ◽  
Fatigue Cycle ◽  
Electrostatic Actuator ◽  
Beam Length ◽  
The Difference ◽  
Cold Arms

Micro thermal-electrostatic actuator devices are widely used in MEMS. However, the effect of structure sizes on deformation and fatigue is seldom discussed. In this work, the effect of structure sizes on deformation and fatigue is investigated. In this device, two beams called hot and cold arms with different width under applied voltage will have different elongation for there different width and the structure will cause the structure laterally bent. Theoretical solutions of deformation and stresses are derived. And numerical methods of finite element are used to analyze for details. The stresses obtained from the finite element are used in fatigue analysis. In the fatigue analysis, high-cycle fatigue model is used as the load in the elastic regime. Considering the accumulation of damage by fatigue being linear, Miner theory is used to estimate the life of the thermal-electrostatic devices under high-cycle fatigue. The result shows with the same length and flexible beam length connecting the hot and cold arms, the large width will cause larger displacement and stresses. However, the difference is not significant. It is also found that as the applied voltage increasing, the displacement and stresses will increase nonlinearly. With the same width and flexible beam length, the larger length will cause larger stresses and small displacement. For fatigue analysis, as the gap increasing and the length and width decreasing, the fatigue cycle increases. It shows when the length and gap are 220 and 5 μm, the fatigue cycle of 50 μm width is more than ten times of 90 μm width. When the width and gap are 50 and 5 μm, the fatigue cycle of 220 μm length is more than ten times of 260 μm length. When the length and width are 220 and 50 fatigue cycles of 50 μm width are more than ten times of 90 μm width, the difference of fatigue cycle between gap 9 and 5 μm is more than 10 times. However, the most significant effect on fatigue is the applied voltage. It shows the fatigue cycle decays very fast as the applied voltage increasing. When the applied voltages are 2 and 8 volts, the fatigue cycles will decrease from 1018 to less than 108. As the applied voltage being 25 volt, the fatigue cycle near zero. Therefore, the limit applied voltage is about 25 volt.

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