Resonance Characteristics of Laminated Twisted Composite Cantilever Panel

Transcription

1 Resonance Characteristics of Laminated Twisted Composite Cantilever Panel Kalbatharla Anith Pranesh 1, Syed Viqar Malik 2 1 M.Tech (Structural Engineering), 2 Asst. Prof, 1,2 Dept. of Civil Engineering, Prasad Engineering College Abstract The twisted cantilever panels have basic applications in wide concordance turbine sharp edges, compressor edges, and fan bleeding edges and particularly in gas turbines. Helper parts subjected to in-plane incidental forces may provoke parametric resonation, in view of particular mixes of the associated in-plane weight parameters and the standard repeat of transverse vibrations. The instability may happen underneath the essential load of the structure under compressive weights over wide extents of excitation frequencies. Composite materials are continuously used as weight bearing assistant parts in aeronautics and sea structures, autos, weight vessels, turbine forefronts and various other building applications because of their high specific quality, specific strength and tailor ability. In this way, the parametric resonation characteristics of secured composite turned cantilever loads up are of amazing specific centrality for cognizance the dynamic behavior of structures under in-plane infrequent weights. This is a player in static and component soundness behavior of overlaid composite pre twisted cantilever panels is focused on in the present examination. The study revealed that, in view of static piece of weight, the unsteadiness zones tend to move to lower frequencies. The onset of instability happens earlier with augmentation of purpose of bit of board with more broad frailty locale. Not in any way like bowed plates, there is basic deviation of the precariousness behavior of twisted round and empty sheets from that of untwisted barrel formed sheets. Tantamount behavior is also looked for the assortment of precariousness range of bended round and hyperbolic paraboloidal sheets. The excitation repeat lessens from square to rectangular sheets with addition of perspective extent. The handle presentation basically impacts the onset of instability and the width of the zones of wobbliness. Thusly the precariousness behavior of wound cantilever sheets is affected by the geometry, material, handle lay-up and its presentation. This can be used to the upside of fitting in the midst of framework of composite reshaped structures. I. INTRODUCTION The twisted cantilever panels have critical applications in wide harmony turbine cutting edges, compressor sharp edges, fan edges, airplane or marine propellers, helicopter edges, and especially in gas turbines. This scope of viable applications requests a legitimate comprehension of their vibration, static and element solidness attributes. The harm brought about by edges falling flat because of vibratory exhaustion can be calamitous at the very least and at any rate result in extra motor advancement costs because of upgrade and repair. Because of its hugeness, an expansive number of references manage the free vibration of twisted plates. 1.1: Importance of structural stability The blades are frequently subjected to hub occasional strengths because of hub segments of streamlined or hydrodynamic powers following up on the blades. Basic components subjected to inplane intermittent strengths may prompt parametric reverberation, because of specific mixes of the estimations of load parameters. The precariousness may happen beneath the critical load of the structure under compressive loads over wide scopes of excitation frequencies. Composite materials are by and large progressively utilized as a part of turbo-apparatus blades due to their particular quality, firmness and these can be custom-made through the variety of fiber introduction All Rights Reserved 223

2 stacking succession to get a proficient outline. In this manner the parametric reverberation attributes of covered composite twisted cantilever panels are of extraordinary significance for comprehension the frameworks under occasional loads. The refinement amongst great and terrible vibration administrations of a structure, subjected to in-plane occasional loading can be recognized through an examination of element shakiness area (DIR) spectra. The computation of these spectra is frequently given as far as characteristic frequencies and the static clasping loads. In this way, the estimation of these parameters with high accuracy is an essential piece of element security investigation of twisted plates. 1.2: Vibration and buckling of twisted panels With the consistently expanding utilization of turbomachinery at higher execution levels, particularly in air ship, the investigation of vibration issues emerging in twisted blades has turned out to be progressively vital. Free vibration frequencies and mode shapes are vital for the examination of resounding reaction and shudder. Because of its noteworthiness in auxiliary mechanics, numerous specialists have taken a shot at the vibration qualities of turbomachinery blades. 1.3: Dynamic stability of twisted panels Turbine blades are subjected to hub intermittent strengths because of hub parts of aerodynamic or hydrodynamic powers following up on the blades. The expanded use of composite materials in dainty walled basic segments of air ships, submarines, autos and other superior application regions have required a solid need to comprehend their dynamic attributes under various loading conditions. Composite materials are in effect progressively utilized as a part of numerous applications on account of their particular quality and firmness and these can be custom fitted through the variety of fiber introduction and stacking succession to acquire a proficient outline. Auxiliary components subjected to in-plane intermittent strengths may prompt parametric reverberation, because of specific mixes of the estimations of load parameters and irritating recurrence. The above marvel is called dynamic instability or parametric reverberation. The instability may happen underneath the critical load of the structure under compressive loads over wide scopes of excitation frequencies. Therefore the parametric reverberation attributes of overlaid composite twisted cantilever panels are of awesome significance for comprehension the frameworks under intermittent loads. II. MATHEMATICAL FORMULATION 2.1. Energy Equations: Bending moment: Bending energy density is defined as U b = M α βdk α β The energy densities when curvature reaches its maximum value k α. k α β = ηk αβ Considering the straight path, dk α β = k αβdη The bending density is expressed in terms of components as Stress equations: The total stresses at any layer are the sum of the initial stresses plus the stresses due to bending and shear deformation. The strain energy Uo due to initial in-plane stresses is written as the stresses are given as U o = 1 2 {εo } T {σ o }da {σ o } = {ε o }[D P All Rights Reserved 224

3 The strain energy is expressed as The potential energy is given as U o = 1 2 {qo } T [K P ]{q o } Π = U b σ o 2.2. Finite element equations: Every plate in finite element is divided into two types 1. Plane Stress 2. Plane Strain The given plate comes under plane stress conditions. Where the stress in one direction is zero because of less thickness. When compared to its lateral and longitudinal directions. The strains in plane stress is given as Using the principle of stationary potential energy, the plane stress is expressed as [KP]{q 0 }= {p 0 } Finite element involves separation of structure into finite element particles, finding individual properties i.e., Shape function and stiffness matrix. Later forming global stiffness matrix. The shape functions Ni are defined as for 8 noded elements. Where u, v, w and uo, vo, wo are components of displacements in the x, y, z directions respectively. θ x and θ y are the rotations of the mid surface normal about the x and y axes respectively. x = Ni xi, y = Ni yi U = Ni ui, V = Ni vi, W = Ni wi θ x = Ni θ xi, θ y = Ni θ yi ie., U=N1u1+N2u2+..N8u8 If we differentiate the equations with there respective axis and plane coordinates we get strain displacement relation. The linear strains can be expressed in terms of displacements as {ε} = [B]{d e } Where [B] is connectivity matrix As shape functions are in local coordinates, It is changed into global coordinates with help of Jacobian matrix. Finally stiffness matrix can be defined All Rights Reserved 225

4 [K] = [B] T [D][B]dv Where [B] is obtained from strain displacement relations And [D]= For plane stress condition, 2.3. Natural frequency:- The natural frequency for a twisted cantilever plate is given as fn : n-th frequency (Hz) of the plate, Xij 2 : non-dimensional frequency parameter L : blade length (m) E : Young modulus of elasticity (N/m 2 ) ρ : material density (kg/m 3 ) υ : Poisson s ratio. 3.0: Results f n = X ij 2 2πL 2 E t2 12ρ(1 υ 2 ) III. NATURAL FREQUENCY OF TWISTED CANTILEVER PLATE Given the material is elastic =0.33 Youngs modulus of elasticity, E= GPa Fig1. 8x8 shape of mesh Table 3.1: Natural frequency of twisted cantilever plate L//C =1 Mesh i j x Table 3.2: Natural frequency of twisted cantilever plate L//C =3 Mesh i j x All Rights Reserved 226

5 i=1 j=1 i=1 j=0 i=0 j=1 i=0 j= Fig: Comparison of natural frequency IV. CONCLUSION The effect of frequencies on varies length is studied as follow The frequency of element is directly proportion to t/c ratio. The values of t/c increases with frequency. Similarly the values of L is also directly proportional to frequency and frequency increases with increase in length The values increases drastically when the thickness increases as shown in fig 2 The corner elements have more frequency than inner elements REFERENCES 1. Kielb, R.E., Macbain, J.C., and Leissa, A.W. (1985 c ): Vibrations of twisted cantilevered plates experimental investigation, ASME Journal of Engineering for Gas Turbines and Power, Vol. 107, pp Kuang, Jao-Hwa and Hsu, Ming-Hung (2002): The effect of fiber angle on the natural frequencies of orthotropic composite pretwisted blades, Composite Structures, Vol. 58 (4), pp Kielb, R.E., Leissa, A.W., and Macbain, J.C. (1985 b ): Vibrations of twisted cantilever plates a comparison of theoretical results, International Journal of Numerical Methods in Engineering, Vol.21, pp All Rights Reserved 227