An Application of Floquet Theory to Prediction of Mechanical Instability

Hammond, C. E.

doi:10.4050/jahs.19.14

Cited by 55 publications

(39 citation statements)

References 4 publications

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“…Clearly, the effects of fluid loading (including forward flight) are complicated, but suffice it to say here that there are practical situations in which there is elastic coupling between flapping and lagging motion [49,50]. Given the periodic nature of rotating systems certain special mathematical techniques can be used including Floquet theory (see Chapter 14) [52]. This is an important design consideration for helicopters, and stability boundaries have been developed to take into account the various parameters of the problem [51].…”

Section: Rotating Beamsmentioning

confidence: 99%

Continuous Struts

Virgin¹

2007

Vibration of Axially-Loaded Structures

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In this chapter, we consider the dynamics of a thin elastic strut including axial effects, arising primarily from one of two situations:• the axial load is applied externally (including postbuckling), or • deformation is sufficient to cause coupling between axial and bending behavior (the membrane effect).In the first section, attention is focused on a traditional approach to setting up the equations of motion (by means of D'Alembert's principle) for the simple case based on engineering beam theory (Euler-Bernoulli) with the addition of axial loads. The resulting partial differential equation of motion is then separated into temporal and spatial ordinary differential equations and the response analyzed for various magnitudes of the axial load [1,2]. Then an energy approach is used together with Rayleigh's method [3]. In this case, additional terms are retained in the potential energy to allow postbuckled effects to be analyzed [4] and the effect of initial geometric imperfections are included. An alternative approach is developed based on a simple application of Hamilton's principle, and in this instance stretching effects are also included and a solution developed by use of Galerkin's method. This approach will be similar to that used in the previous chapter on strings but now bending strain energy enters into the analysis (as well as compressive axial loading). In the final part of the chapter we consider the dynamics of struts that are loaded by gravity through self-weight. The next chapter will then continue the study of axially loaded members but with the scope opened to include a wider class of problem. Basic FormulationIn this section, we develop the governing equation of motion for a thin, elastic, prismatic beam subject to a constant axial force. In Fig. 7.1(a) a schematic of the beam is shown. It has mass per unit length m, constant flexural rigidity EI, and is subject to an axial load P. The length is L, the coordinate along the beam is x, and the lateral (transverse) deflection is w(x, t). In part (b) is shown an element of the beam between locations x and x + x, which is subject to the D'Alembert forces R(x, t) = m∂ 2 w/∂t 2 . (7.1) 111Cambridge Books Online

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Section: Rotating Beamsmentioning

confidence: 99%

Continuous Struts

Virgin¹

2007

Vibration of Axially-Loaded Structures

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“…viewed as a natural extension of the stability of equilibria introduced in Chapter 3), we again make use of linearization before proceeding to Floquet theory [4][5][6][7][8][9][10][11].…”

Section: The Variational Equationmentioning

confidence: 99%

Harmonic Loading: Parametric Excitation

Virgin¹

2007

Vibration of Axially-Loaded Structures

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“…In order to clearly identify the effects of the damper, a simplified mathematical model of the helicopter is considered for the numerical studies in this paper (Hammond, 1974). The rigid fuselage undergoes x-and y-hub motions, and the rigid blades undergo lag motions.…”

Section: Ground Resonancementioning

confidence: 99%

Performance Comparison between Piezoelectric and Elastomeric Lag Dampers on Ground Resonance Stability of Helicopter

Kim

Yun

2001

Journal of Intelligent Material Systems and Structures

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In this study, a piezoelectric lag damper using damping characteristics in piezoelectric ceramics is proposed to alleviate ground resonance instability. The piezoelectric passive damper consists of piezoelectric elements and electrical circuits such as resistors and inductors in a series. By bonding the piezoelectric ceramics to the rotor flexures, piezoelectric strain energy is converted to electrical energy and dissipated through an electrical network. To verify the effect of the piezoelectric damper on rotating blades, experiments for the aeroelastic stability of the rotor blades were carried out. The experiment results for rotor blades with the piezoelectric damper show that the piezoelectric damper has a strong stabilizing effect on the inherently weakly lag-mode damping of the blades. The performance of piezoelectric and elastomeric lag dampers on ground resonance instability are compared and modal frequency and modal damping with respect to rotating speed are shown.

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An Application of Floquet Theory to Prediction of Mechanical Instability

Abstract: The problem of helicopter mechanical instability is considered for the case where one blade damper is inoperative.

Cited by 55 publications

References 4 publications

Continuous Struts

Continuous Struts

Harmonic Loading: Parametric Excitation

Performance Comparison between Piezoelectric and Elastomeric Lag Dampers on Ground Resonance Stability of Helicopter

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