Exact solutions of in-plane natural vibration of a circular plate with outer edge restrained elastically

We solve exactly the equations of in-plane natural vibration for a circular plate whose outer edge is restrained elastically. The mode shapes are represented by trigonometric functions with a number of nodal diameters in the circumferential direction and mode functions in the radial direction. We present the exact frequency equations and mode functions and tabulate the frequency parameters satisfying the frequency equations. The corresponding mode functions and two-dimensional mode shapes are illustrated when both radial and tangential stiffness are zero (free edge), infinity (clamped edge), or medium. Comparisons with previous reported results confirm the accuracy of the present work.     

方板阶梯辐射体辐射声场的计算及测试

研究了一种同相弯曲振动方板阶梯辐射体。以边长50 mm的方形平板为例,材料为钢,厚度为6mm。其第四阶本征振型的频率为19332 Hz,用同频率的纵振换能器激励其中心,会出现节线与边长成45°夹角的响应(本征振型中无)。为了获得辐射体辐射声场的高指向性,在四个板角与四条节线围成的等腰直角三角形区域加阶梯,可将平板辐射体改进成阶梯辐射体。计算得到阶梯辐射体的指向性比平板辐射体的指向性尖锐,与方板活塞辐射体的指向性相似。实验测试辐射体指向性与计算结果基本一致。进一步计算发现,方板阶梯辐射体与线性尺寸相同的矩形、圆盘阶梯辐射体比较,其频率低,指向性同样尖锐,可望应用于超声测距、料位测试中。      

On operating deflection shapes of the violin body including in-plane motions

Earlier investigations have assumed only "out-of-plane" vibrations of the plates of the violin. The violin body can, however, be described as a thin-walled, double-arched shell structure and as such it may very well elongate in one direction as it contracts in another. Therefore, at least two orthogonal vibration components have to be included to describe the vibrations. The operating deflection shapes (ODSs) of a good, professionally made and carefully selected violin were therefore measured in several directions by TV holography to determine both "in-plane" and out-of-plane vibration components of the ODSs. The observations were limited to the frequency range 400-600 Hz, as this interval includes two most-prominent resonance peaks of bridge mobility and sound radiation as well as a third poorly radiating resonance. These three peaks clearly showed orthogonal vibration components in the ODSs. The vibration behavior of the violin body, sectioned in the bridge plane, was interpreted as the vibrations of an "elliptical tube" with nodal diameters. The number of nodal diameters increases from two to three in the selected frequency range. The TV holography measurements were supported by electrodynamical point measurements of bridge mobility, of air volume resonances, and by reciprocity, of radiation properties. Furthermore, a fourth mode, the air mode, A1, is involved indirectly in the sound radiation via influence on the body vibrations.     

Radiation efficiency of a guitar top plate linked with edge or corner modes and intercell cancellation

This paper was based on a theoretical framework to determine strong and weak radiation by a guitar top plate, vibrating through deflections hard to analyze: multipolar mode shapes. The air-structure interaction was examined in terms of edge modes or corner modes, and considering even or odd number modes. A numerical model was implemented and experimentally calibrated, exhibiting several advantages exploring the coupling between vibratory and acoustic waves in a top plate. Two analyses were applied detecting high or low radiation efficiency for the structure. First, the addition of volume velocity for odd numbers of poles and cancellation for even numbers were examined. In fact, both behaviors can happen at the same time, as it was shown for a corner radiator case used as an example. Second, the ratio between bending and acoustic wavenumbers was explored. To illustrate the importance of this ratio, some theoretical features of a more efficient radiator than the corner mode were exposed in an edge mode example. Labeling multipolar mode shapes as efficient or inefficient radiators showed to be a useful alternative analyzing the top plate behavior. It can be applied knowing the nodal lines of the vibration pattern and estimating the bending and acoustic wavelengths.     

Natural frequencies and vibration modes of laminated composite plates reinforced with arbitrary curvilinear fiber shape paths

The development of tow-placement technology has made it possible to control fiber tows individually and place fibers in curvilinear distinct paths in each layer of a laminated plate. This paper presents an analytical method for determining natural frequencies and vibration modes of laminated plates having such curvilinear reinforcing fibers. Spline functions are employed to represent arbitrarily shaped fibers, and Ritz solutions are used to derive frequency equations using series type shape functions. The strain energy is evaluated by numerical integration involving the fiber orientation angle, and is calculated using the derivative of the spline function in minute intervals. The results show that the natural frequencies obtained by the present method agree well with results from finite element analyses. The vibration mode shape contour plots of the plates are seen to reflect clear influences of the fiber shapes.     

Vibration analysis of annular-like plates

The existence of eccentricity of the central hole for an annular plate results in a significant change in the natural frequencies and mode shapes of the structure. In this paper, the vibration analysis of annular-like plates is presented based on numerical and experimental approaches. Using the finite element analysis code Nastran, the effects of the eccentricity, hole size and boundary condition on vibration modes are investigated systematically through both global and local analyses. The results show that analyses for perfect symmetric conditions can still roughly predict the mode shapes of “recessive” modes of the plate with a slightly eccentric hole. They will, however, lead to erroneous results for “dominant” modes. In addition, the residual displacement mode shape is verified as an effective parameter for identifying damage occurring in plate-like structures. Experimental modal analysis on a clamped-free annular-like plate is performed, and the results obtained reveal good agreement with those obtained by numerical analysis. This study provides guidance on modal analysis, vibration measurement and damage detection of plate-like structures.     

Free vibrations of a vessel consisting of circular plates and a shell of revolution having varying meridional curvature

An analytical solution procedure is presented for the free vibration of vessels consisting of a shell of revolution having varying meridional curvature and circular plate lids. The Lagrangian of vibration of the combined system is obtained in quadratic forms of boundary values. The frequency equation and the relations among the boundary values are obtained from minimizing conditions of the Lagrangian with respect to the unknown boundary values. The natural frequencies and the mode shapes of vessels having elliptical and hyperbolical meridians have been obtained by carrying out numerical calculations. Effects of various parameters upon natural frequencies and mode shapes are illustrated in discussions of numerical results.     

Sensitivity analysis and optimization of nodal point placement for vibration reduction

A method is developed for sensitivity analysis and optimization of nodal point locations in connection with vibration reduction. A straightforward derivation of the expression for the derivative of nodal locations is given, and the role of the derivative in assessing design trends is demonstrated. An optimization process is developed in which added lumped masses on the structure are used as design variables to move the node to a preselected location—for example to where a low response amplitude is required or to a point which makes the mode shape nearly orthogonal to the force distribution, thereby minimizing the generalized force. The optimization formulation leads to values for added masses that adjust a nodal location while minimizing the total amount of added mass required to do so. As an example, the node of the second mode of a cantilever box beam model of a rotor blade is relocated to coincide with the centroid of a prescribed force distribution, thereby reducing the generalized force substantially without adding excessive mass. A comparison with an optimization formulation that directly minimizes the generalized force indicates that nodal placement gives essentially a minimum generalized force when the node is appropriately placed.     

Vibration and buckling of rectangular plates under in-plane hydrostatic loading

Numerical solutions are presented for the fundamental natural frequency and mode shape of a rectangular plate loaded by in-plane hydrostatic forces for a wide variety of aspect ratios, boundary conditions, and load magnitudes. All six possible combinations of simply supported and clamped edges are considered. The limiting conditions of unloaded vibration and buckling are discussed in detail, with emphasis on the preferred mode shape. Design curves and approximate formulae are presented which provide a simple means of determining the fundamental frequency parameter.     

EXPERIMENTAL AND THEORETICAL INVESTIGATION OF THE LINEAR AND NON-LINEAR DYNAMIC BEHAVIOUR OF A GLARE 3 HYBRID COMPOSITE PANEL

A theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of hybrid composite plates made of Glare 3, a new aircraft structural material. It consists of alternating layers of metal- and fibre-reinforced composites. In previous work, the theoretical model has been used to calculate the first non-linear mode of fully clamped rectangular composite fibre-reinforced plastic (CFRP) laminated plates. This study concerns determination of the linear dynamic properties of the Glare 3 hybrid composite rectangular panel (G3HCRP) such as natural frequencies and mode shapes. The theoretical model is used to calculate the fundamental non-linear mode shape and associated flexural behaviour of the fully clamped G3HCRP. A series of experimental investigations have been conducted using a G3HCRP in order to determine linear dynamic properties. The response due to random excitation was investigated and the experimental measurements are analyzed and discussed. Comparisons are made with finite element predictions and response estimates given by the ESDU method, the latter being a “design guide” approach used by industry. Concerning the non-linear analysis, the results are given for various plate aspect ratios and vibration amplitudes, showing a higher increase of the induced bending stress near the clamps at large deflections. Comparisons between the dynamic behaviour of an isotropic plate and G3HCRP at large vibration amplitudes are presented and good results are obtained.     

挖掘机驾驶室低频结构噪声分析与优化*

针对某小型液压挖掘机驾驶室低频噪声过大问题,对驾驶室结构振动特性进行分析。基于有限元模型计算各工况下驾驶室噪声传递函数,运用统计学方法确定主要噪声峰值频率及相应工况;通过模态声学贡献度计算,确定危险工况下噪声贡献量较大的模态阶数,参照模态振型确定驾驶室振动变形最大的车身板件;并对该板件进行形貌优化处理,提高其一阶固有频率,进而降低驾驶室内噪声。优化结果表明,驾驶室噪声传递函数在危险频率下的峰值下降了2~4 dB。     

Dynamic stiffness method for inplane free vibration of rotating beams including Coriolis effects

The paper addresses the in-plane free vibration analysis of rotating beams using an exact dynamic stiffness method. The analysis includes the Coriolis effects in the free vibratory motion as well as the effects of an arbitrary hub radius and an outboard force. The investigation focuses on the formulation of the frequency dependent dynamic stiffness matrix to perform exact modal analysis of rotating beams or beam assemblies. The governing differential equations of motion, derived from Hamilton's principle, are solved using the Frobenius method. Natural boundary conditions resulting from the Hamiltonian formulation enable expressions for nodal forces to be obtained in terms of arbitrary constants. The dynamic stiffness matrix is developed by relating the amplitudes of the nodal forces to those of the corresponding responses, thereby eliminating the arbitrary constants. Then the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm. Numerical results for an individual rotating beam for cantilever boundary condition are given and some results are validated. The influences of Coriolis effects, rotational speed and hub radius on the natural frequencies and mode shapes are illustrated.     

Influence of initial geometrical imperfections on vibrations of axially compressed stiffened cylindrical shells

The vibrations of stiffened cylindrical shells having axisymmetric or asymmetric initial geometrical imperfections and axial preload are analyzed. The analysis is based on a solution of the von Kárman-Donnell non-linear shell equations, an “exact” solution of the compatibility equation, and a first order approximation by the Galerkin method of the equilibrium equation. The stiffeners are closely spaced and “smeared” stiffener theory is employed. The results of an extensive parametric study carried out on shells similar to those used in vibration and buckling tests at the Technion show that stiffening of the shell will lower the imperfection-sensitivity of its free vibrations, but the decrease depends on the type of stiffening (stringers or rings), the mode shapes of the vibration and the imperfection, the stiffener strength and eccentricity. The imperfection-sensitivity decrease, caused by the stiffeners, is greater for vibration mode shapes with high imperfection-sensitivity than for other vibration mode shapes. The sensitivity differences between stringer and ring-stiffened shells depend especially on the vibration and the imperfection mode shapes, and on their coupling. Small imperfections change the natural frequencies of stiffened shells in the same directions as for isotropic shells, but to a smaller extent. The frequency dependence on the external load is also strongly affected by the imperfection mode shape. The results correlate well with earlier ones for isotropic shells.     

An automatic mode pairing strategy using an enhanced modal assurance criterion based on modal strain energies

In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches.In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.     

FREE VIBRATION ANALYSIS OF CANTILEVER PLATES WITH STEP DISCONTINUITIES IN PROPERTIES BY THE METHOD OF SUPERPOSITION

Utilizing the superposition method, an analytical type solution is obtained for the free vibration eigenvalues and mode shapes of a cantilever plate with step discontinuities in plate properties. Property discontinuity lines run parallel to the clamped edge of the plate. Verification tests are performed for limiting cases by comparing computed eigenvalues with known eigenvalues for plates with uniform properties. Very good agreement is also obtained when computed results are compared with those obtained experimentally utilizing a test plate with discontinuities in thickness. Computed eigenvalues and mode shapes are presented for the benefit of other researchers. Besides the general interest, the problem has an application in the modelling of certain multi-story buildings during seismic studies.     

Effects of driving frequency of longitudinal transducer on the vibration characteristics of a stepped plate

The flexural vibration characteristics of a stepped plate, driven at its center by different frequency of longitudinal transducer with a certain area are investigated. The variation in the nodal circle, fundamental frequency and displacement distribution of the stepped plate are calculated by using finite element method (FEM) under different driving frequencies. The results show that the fundamental frequency and nodal circle of the flexural-vibration stepped plate (FVSP) increase with an increase in the driving frequency of the longitudinal vibration ultrasonic transducer (LVUT), before the second-order flexural vibration occurs. When the driving frequency is f = 28 kHz, the displacement amplitude of the stepped plate can achieve the maximum, and the nodal circle radius of the stepped plate is 2.61 cm which fits evenly the edge of stepped profile. Meanwhile, the directivity and radiation efficiency of the FVSP would be greatly improved in a special driving frequency. The conclusions agree with the experimental ones and are significant for both design and applications of the stepped plate.     

Vibration tailoring of inhomogeneous rod that possesses a trigonometric fundamental mode shape

In this study, a special class of closed-form solutions for inhomogeneous rod is investigated. Namely, the following problem is considered: determine the distribution axial rigidity when the material density is given of an inhomogeneous rod so that the postulated fundamental trigonometric mode shape serves as an exact vibration mode. In this study, the associated semi-inverse problem is solved that results in the distributions of axial rigidity that together with a specified law of material density satisfy the governing eigenvalue problem. For comparison, the obtained closed-form solutions are contrasted with approximate solutions based on an appropriate polynomial shapes, serving as trial functions in an energy method. The obtained results are utilized for vibration tailoring, i.e. construction of the rod with a given natural frequency.     

Wall compliance and violin cavity modes

Violin corpus wall compliance, which has a substantial effect on cavity mode frequencies, was added to Shaw's two-degree-of-freedom (2DOF) network model for A0 ("main air") and A1 (lowest length mode included in "main wood") cavity modes. The 2DOF model predicts a V(-0.25) volume dependence for A0 for rigid violin-shaped cavities, to which a semiempirical compliance correction term, V(-x(c)) (optimization parameter x(c)) consistent with cavity acoustical compliance and violin-based scaling was added. Optimizing x(c) over A0 and A1 frequencies measured for a Hutchins-Schelleng violin octet yielded x(c) approximately 0.08. This markedly improved A0 and A1 frequency predictions to within approximately +/- 10% of experiment over a range of about 4.5:1 in length, 10:1 in f-hole area, 3:1 in top plate thickness, and 128:1 in volume. Compliance is a plausible explanation for A1 falling close to the "main wood" resonance, not increasingly higher for the larger instruments, which were scaled successively shorter compared to the violin for ergonomic and practical reasons. Similarly incorporating compliance for A2 and A4 (lowest lower-/upper-bout modes, respectively) improves frequency predictions within +/-20% over the octet.     

A note on the damping of large amplitude beam vibrations

This paper presents a new series-type method for solving the eigenvalue problems of irregularly shaped plates clamped at all edges. An irregularly shaped plate is formed on a simply supported rectangular plate by rigidly fixing several segments. With the reaction forces and moments acting on all edges of an actual plate of irregular shape regarded as unknown harmonic loads, the stationary response of the plate to these loads is expressed by the use of the Green function. The force and moment distributions along the edges are expanded into Fourier series with unknown coefficients, and the homogeneous equations for the coefficients are derived by restraint conditions on the edges. The natural frequencies and the mode shapes of the actual plate are determined by calculating the eigenvalues and eigenvectors of the equations. The method is applied to a cross-shaped, an I-shaped and an L-shaped plate clamped at all edges, the natural frequencies and the mode shapes of the plates are calculated numerically and the effect of the shape is discussed.     

Free vibration analysis of stiffened plates using finite difference method

Free vibration characteristics of rectangular stiffened plates having a single stiffener have been examined by using the finite difference method. A variational technique has been used to minimize the total energy of the stiffened plate and the derivatives appearing in the energy functional are replaced by finite difference equations. The energy functional is minimized with respect to discretized displacement components and natural frequencies and mode shapes of the stiffened plate have been determined as the solutions of a linear algebraic eigenvalue problem. The analysis takes into consideration inplane deformation of the plate and the stiffener and the effect of inplane inertia on the natural frequencies and mode shapes. The effect of the ratio of stiffener depth to plate thickness on the natural frequencies of the stiffened plate has also been examined.