Axisymmetric free vibration of thick annular plates

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Application of partition method to vibration problems of plates

The “partition method” or “sub-domain method” consists of expressing the solution of a governing differential equation, partial or ordinary, in terms of functions which satisfy the boundary conditions and setting to zero the error in the differential equation integrated over each of the sub-domains into which the given domain is partitioned. In this paper, the use of this method in eigenvalue problems with particular reference to vibration of plates is investigated. The deflection of the plate is expressed in terms of polynomials satisfying the boundary conditions completely. Setting the integrated error in each of the subdomains to zero results in a set of simultaneous, linear, homogeneous, algebraic equations in the undetermined coefficients of the deflection series. The algebraic eigenvalue problem is then solved for eigenvalues and eigenvectors. Convergence is examined in a few typical cases and is found to be satisfactory. The results obtained are compared with existing results based on other methods and are found to be in very good agreement.

On quasi-degeneracies in plate vibration problems

In plotting the variation of frequencies with geometric parameters such as side ratio, skew angle, thickness taper, etc. in detailed studies of the vibration characteristics of plates, situations are encountered such as crossing of the frequency curves or the tendency of these curves to come close together and veer away from each other. These have been generally referred to as “frequency crossings” and “transitions” respectively. The latter may preferably be referred to as “quasi-degeneracies”. In the literature there appears to be some ambiguity in the analysis and interpretation of these features. In this paper, a clarification of some of these questions as regards rectangular and skew plates is presented by making use of concepts from physics dealing with molecular vibrations.

Vibration spectra of ferroelectric alkali trihydrogen selenites and the nature of the hydrogen bond potential in these crystals

Raman spectra of the ferroelectric LiH3 (SeO3)2 and NaH3(SeO3)2 and the anti-ferroelectric KH3 (SeO3)2 have been recorded at room temperature using a He-Ne and also an Ar-ion laser source. The infrared absorption spectra of these crystals and their deuterated analogues have been recorded in the region 400–4000 cm−1 both below and above the Curie temperature. From an analysis of the spectrum in the region 400–900 cm−1 it is concluded that (i) in LiH3 (SeO3)2 the protons are ordered in an asymmetric double minimum potential with a low barrier and the spectrum can be interpreted in terms of HSeO3− and H2SeO3 vibrations, (ii) in NaH3 (SeO3)2 all three protons occupy a single minimum potential at room temperature and below the transition temperature the groups HSeO3− and H2SeO3 are present, (iii) the proton at the inversion centre in KH3(SeO3)2 is in a broad troughed potential well and the low temperature spectrum is more likely to be due to H3SeO3+ and SeO32− species. This deviation of the spectrum from that of the previous two crystals is attributed to the difference in H-bond scheme and hence the absence of any cooperative motion of protons in this crystal.

Active vibration suppression of non-linear beams using optimal dynamic inversion

Euler–Bernoulli beams are distributed parameter systems that are governed by a non-linear partial differential equation (PDE) of motion. This paper presents a vibration control approach for such beams that directly utilizes the non-linear PDE of motion, and hence, it is free from approximation errors (such as model reduction, linearization etc.). Two state feedback controllers are presented based on a newly developed optimal dynamic inversion technique which leads to closed-form solutions for the control variable. In one formulation a continuous controller structure is assumed in the spatial domain, whereas in the other approach it is assumed that the control force is applied through a finite number of discrete actuators located at predefined discrete locations in the spatial domain. An implicit finite difference technique with unconditional stability has been used to solve the PDE with control actions. Numerical simulation studies show that the beam vibration can effectively be decreased using either of the two formulations.

Application of partition method to vibration problems of plates

The “partition method” or “sub-domain method” consists of expressing the solution of a governing differential equation, partial or ordinary, in terms of functions which satisfy the boundary conditions and setting to zero the error in the differential equation integrated over each of the sub-domains into which the given domain is partitioned. In this paper, the use of this method in eigenvalue problems with particular reference to vibration of plates is investigated. The deflection of the plate is expressed in terms of polynomials satisfying the boundary conditions completely. Setting the integrated error in each of the subdomains to zero results in a set of simultaneous, linear, homogeneous, algebraic equations in the undetermined coefficients of the deflection series. The algebraic eigenvalue problem is then solved for eigenvalues and eigenvectors. Convergence is examined in a few typical cases and is found to be satisfactory. The results obtained are compared with existing results based on other methods and are found to be in very good agreement.

Analysis and prevention of extreme vibration in high speed craft - A research framework

Extreme vibration has been reported for small, high speed craft in the maritime sector, with performance and health threatening effects on boat operators and crew. Musculoskeletal injuries are an enduring problem for high speed craft passengers. Spinal or joint injuries and neurological disorders may occur from repetitive pounding over rough water, continued vibration and single impact events. The risk from whole body vibration (WBV) induced through the small vessels mainly depends on time spent on the craft, which can’t be changed in a military scenario; as well as the number of shocks and jolts, and their magnitude and frequency. In the European Union for example, physical agents directives require all employers to control exposure to a number of physical agents including noise and vibration. The EC Vibration Directive 2002/44/EC then sets out regulations for the control of health and safety risks from the exposure of workers to hand arm vibration (HAV) and WBV in the workplace. Australia has exposure standards relating to WBV, AS 2670.1-2001 – Evaluation of human exposure to whole body vibration. This standard is identical to the ISO 2631-1:1997, Mechanical vibration and shock – Evaluation of human exposure to whole-body vibration. Currently, none of the jurisdictions in Australia have specific regulations for vibration exposures in workplaces. However vibration is mentioned to varying degrees in their general regulations, codes of practice and guidance material. WBV on high speed craft is normally caused by “continuous 'hammering' from short steep seas or wind against tide conditions. Shock on High Speed Craft is usually caused by random impacts. Military organisations need the knowledge to make informed decisions regarding their marine operations, compliance with legislation and potentially harmful health effects, and develop and implement appropriate counter-measures. Marine case studies in the UK such as published MAIB (Marine Accident Investigation Branch) reports show injuries that have occurred in operation, and subsequent MCA (Maritime Coastguard Agency) guidance is provided (MGN 436 (M+F), WHOLE-BODY VIBRATION: Guidance on Mitigating Against the Effects of Shocks and Impacts on Small Vessels. MCA, 2011). This paper proposes a research framework to study the origin, impact and pathways for prevention of WBV in small, high speed craft in a maritime environment.

Solution of a class of vibration problems by flowgraph methods

A 4-degree-of-freedom single-input system and a 3-degree-of-freedom multi-input system are solved by the Coates', modified Coates' and Chan-Mai flowgraph methods. It is concluded that the Chan-Mai flowgraph method is superior to other flowgraph methods in such cases.

Footing Response to Horizontal Vibration

For the prediction of response of footings subjected to horizontal vibration, different types of contact shear distributions and displacement conditions are to be considered. Solutions using elastic half-space theory are not available for all the cases of shear distribution and displacement conditions. In this paper, solutions are obtained for the cases in which solutions are not available and the relevant coefficients are presented in tables which could be used in the appropriate equations for the prediction of dynamic response. Spring constants are evaluated and tabulated for different displacement and shear distribution conditions.

Vibration of generally orthotropic skew plates

Vibration problem of generally orthotropic plates with particular attention to plates of skew geometry is studied. The formulation is based on orthotropic plate theory with arbitrary orientation of the principal axes of orthotropy. The boundary conditions considered are combinations of simply supported, clamped, and free-edge conditions. Approximate solution for frequencies and modes is obtained by the Ritz method using products of appropriate beam characteristic functions as admissible functions. The variation of frequencies and modes with orientation of the axes of orthotropy is examined for different skew angles and boundary conditions. Features such as "crossings" and "quasi-degeneracies" of the frequency curves are found to occur with variation of the orientation of the axes of orthotropy for a given geometry of the skew plate. It is also found that for each combination of skew angle and side ratio, a particular orientation of the axes gives the highest value for the fundamental frequency of the plate.

Vibration of skew plates

The vibration problems of skew plates with different edge conditions involving simple support and clamping have been considered by using the variational method of Ritz, a double series of beam characteristic functions being employed appropriate to the combination of the edge conditions. Natural frequencies and modes of vibration have been obtained for different combinations of side ratio and skew angle. These detailed studies reveal several interesting features concerning the frequency curves and nodal patterns. The results presented should, in addition, be of considerable value and practical significance in design applications.

Waves in branched hydraulic pipes

The paper deals with the classical problem of axi-symmetric transmission of low amplitude waves through a circular pipe containing a viscous liquid. Exact governing equations are identified and solved, the radial as well as the axial component of the velocity being considered. Attention is drawn to certain fallacies underlying the conventional approach. The parameters required in the formulation of the transfer matrix for a pipe have been evaluated. In order to evaluate the response at the terminal point of a branched system for a sinusoidal input at one of the ends, a general algorithm has been developed.

Random vibration of a second order non-linear elastic system

A method is presented for obtaining, approximately, the response covariance and probability distribution of a non-linear oscillator under a Gaussian excitation. The method has similarities with the hierarchy closure and the equivalent linearization approaches, but is different. A Gaussianization technique is used to arrive at the output autocorrelation and the input-output cross-correlation. This along with an energy equivalence criterion is used to estimate the response distribution function. The method is applicable in both the transient and steady state response analysis under either stationary or non-stationary excitations. Good comparison has been observed between the predicted and the exact steady state probability distribution of a Duffing oscillator under a white noise input.

Vibration of Non-Uniform Thin-Walled Beams of Arbitrary Shape

A generalised theory for the natural vibration of non-uniform thin-walled beams of arbitrary cross-sectional geometry is proposed. The governing equations are obtained as four partial, linear integro-differential equations. The corresponding boundary conditions are also obtained in an integro-differential form. The formulation takes into account the effect of longitudinal inertia and shear flexibility. A method of solution is presented. Some numerical illustrations and an exact solution are included.