VIBRATION ANALYSIS OF WINGS WITH TIP-MOUNTED ENGINE

Transcription

1 VIBRATION ANALYSIS OF WINGS WITH TIP-MOUNTED ENGINE by Sabrina Chowdhury Undergraduate Student, Widener University, Chester, Pennsylvania. AIAA Student Member Abstract Vibration analysis was being conducted on a cantilevered beam wing with a motor attached at the tip. The analysis calculated the natural frequencies and mode shapes of the beam taking into account the inertial as well as the gyroscopic forces induced by the motor. The partial differential equation for each mode of vibration is uncoupled. The gyroscopic forces couple the torsional, longitudinal bending, and transverse bending vibrations of the beam. The coupling enters the problem through the application of boundary conditions. The theory studied leads to information about essential vibrational characteristics. A model of a wing with an attached rotating motor will be setup in order to test the theory derived. Nomenclature A = cross-sectional area of beam E = modulus of elasticity of beam material G = modulus of rigidity of beam material h = distance between the mass center and elastic axis H = angular momentum I = moment of inertia of beam I = moment of inertia of the engine J = cross-sectional polar moment of inertia L = length of Beam m = mass of the engine t = temporal variable (time) w = bending deflection (positive down) x = spatial variable along the length of the beam θ = twist (positive clockwise) ρ = density of beam material Superscripts ( ) = derivative w.r.t. beam axis ( ) = time derivative

2 Motivation The motivation behind this research project stems from the V- Osprey. The gyroscopic forces on the wings of the Osprey change as the engine nacelles are rotated converting the aircraft from a helicopter to an airplane. The effect of the rotating engine propellers on the wings corresponds to the effect of the rotating motor fixed to the cantilevered beam in the research. Analysis A cantilevered beam is a continuous system. A continuous system has infinite degrees-of-freedom. The analysis is aimed at calculating the first few modes of free vibration of the beam. To calculate the free vibration characteristics of the beam one needs to solve the dynamic equations of motion of the beam. These dynamics of the beam are represented by partial differential equations. The partial differential equations are separated and turned into ordinary differential equations. Boundary conditions are applied to the homogeneous ordinary differential equations. This application of boundary conditions leads to equations that can be solved to obtain the nontrivial solutions. The partial differential equation for the dynamics of the beam in bending is given by, w w ρ A + = (Eq.) t x and, the partial differential equation for the dynamic of the beam in torsion is given by, θ θ ρ J = (Eq.) t x The bending deflection and the twist are functions of independent variables x and t. Assuming that each function can be represented by a product of separate functions of space and time, we have, w ( x, = W ( x) T ( (Eq.) θ ( x, = Θ( x) T( (Eq.) Substituting Eq. in Eq., we have, rearranging, () ρ AW T & + T = T&& W = T ρa W () = λ

3 The left hand side of the above equation is only a function of t, while the right hand side is only a function of x. Since x and t are independent the above equality is only true if both sides are equal to a constant, here this constant is denoted by λ. Thus, T&& λt = The solution to the above ordinary differential equation can be written as, where, T = Asin ωt + B cosωt ω = λ Since the system is undamped, the solution for T( is harmonic as expected. This means that the natural frequencies and mode shapes are related only to the space component. The fourth order ordinary differential equation for the bending is given by, where, W () + β W = λρ A = β The solution to this fourth order differential equation is given by, W = C cos βx + C sin βx + C cosh βx + C sinh βx (Eq.) where, C, C, C, C, are constants to be determined. Similarly, the separation of variables technique is applied to the torsion equation. Here the Eq. is substitute into Eq., to get, ρj ΘT& Θ T = Rearranging so that once again the independent variables are separated on either side of the equation and then equating each to the constant λ, we have, T & T = ρ J Θ Θ = λ

4 Then the second order ordinary differential equation for torsion results as, Θ γ Θ = where, λg γ = ρ The solution to this ordinary differential equation is, Θ = C cosγx + C6 sin γx (Eq.6) where, C, C 6, are constants to be determined. Now Eq. and Eq.6 denote the possible solutions to the problem. The constants are determined by application of the boundary conditions. The boundary conditions can be derived as, w(, = w (, = w ( L, = & θh = iωθh w ( L, = mw&& mh && θ = mω w + θ θ (, = θ ( L, = ( I + mh ) && θ + mhw&& + w& H = ( I + mh ) ω θ w + iωw H The application of the boundary conditions to the bending equations yields, W () = C + C W () = βc + βc W ( x) = β C cos βl β C sin βl + β C cosh βx + β C sinh βl = C cosγl C6 sin γl W ( x) = β C sin βl β C cos βl + β sinh βl + β C cosh βl mω mω = C cos βl C sin βl mω mω C cosh βl C sinh βl + C cosγl + C6 sin γl

5 Similarly the application of boundary conditions to the torsion equations results in, θ () = C = θ ( L) = γc sin γl + γc cos γl β = C[ cos βl + sin βl] β C [ sin βl cos βl] β C[ cosh βl sinh βl] β C [ sinh βl + cosh βl] + ( I + mh ) ω + C[ cos γl] ( I + mh ) ω + C 6[ sin γl] The following non-dimensional parameters are applied to the boundary conditions, 6 I = I + mh ρjl m = m ρ AL h h = L A = AL J ωh = β L H ω m = mβ ( βl) = γ ( γl)mah

6 Thus the equations can be written in nondimensional matrix form as, cos βl sin βl + mβl cos βl γlmah cos βl + iγl AH sin βl sin βl cos βl + mβl sin βl γlmah c sinh βl iγl AH cosh βl coshβl sinhβl + mβl coshβl γlmahcoshβl iγlah sinhβl sinhβl coshβl + mβl sinhβl γlmahsinhβl iγlah coshβl where, + i H cos γl m β L h cos γl sin γl γl I cos γl C C + i H sin γl C = m β L h sin γl C C cos γl γl I sin γl C 6 γ L = ( βl) = ( βl) L GA A non-trivial solution exists only if the determinant is singular, i.e., the equations are not linearly independent. One can now find the frequencies of the system by finding the βl where the determinant goes to zero. For any given βl one can find five of the constants in terms of the sixth one. Thus, the mode shape can be calculated.

7 Results The possible values for βl in the six by six matrix were solved for using MATLAB. Graphs were produced plotting the determinant against the square root of the frequency. First the mass of the tip was varied in order to find the effect mass has on the frequency. Change in Mass Determinant BetaL The points at which the graph crosses the zero axis is the frequency at which the determinant is equal to zero. The first and third crossings correspond to bending and the second crossing is due to torsion. Since the mass has no torsional effect on the frequency, the variants of mass all cross at the same frequency during the second crossing. The following graph shows the change in frequency at each of the three modes due to the change in mass. Change in Mass BENDING Frequency TORSION BENDING Mass

8 Next the effect of inertia was studied. The graph of the determinant against the frequency shows a reverse effect in comparison to the mass. Inertia leaves no impression on the bending and therefore the graph crosses at one point at the bending modes. Only a change in frequency may be noted at the torsional mode. Change in Inertia Determinant BetaL Again the frequency change was plotted. The graph shows that no change occurred for the two bending modes, and a decrease in frequency resulted in the torsion mode due to the change in inertia. Change in Inertia Frequency BENDING TORSION BENDING Inertia

9 As the mass was offset from the center of the beam the frequency changed. The determinant versus frequency graph shows what happens at each mode. The first mode does not experience much frequency change. All three modes are a mixture of bending and torsion, so the frequency effects due to either bending or torsion individually are not known. Their combined effect on the frequency as the mass is moved further from the center is graphed by MATLAB. The change in frequency shown as the distance from the center grows larger reflects the determinant graph MATLAB produced. As expected the change in frequency due to the first mode is negligible, and the second two combination modes show an increase in frequency. Change in Distance from Center Frequency BENDING TORSION BENDING Distance

10 Changing the angular momentum was the final effect on frequency studied. Much like the mass offset trials, angular momentum couples bending and torsion. The effect however is more evident in the second mode, while the first and third modes have a minimal effect on the frequency. Below the graph shows the frequency change as angular momentum is increased. The second mode of coupled torsion and bending has a slight effect on the frequency, while the first and third modes show no change. Change in Angular Momentum BENDING Frequency TORSION BENDING Angular Momentum

11 Conclusions The equations for the vibration of a wing with a tip engine were developed and MATLAB was used in order to solve the six by six matrix for the possible values of the variable βl. Once βl is known one can find the frequencies and the mode shapes of the system. Tip mass reduces the bending frequencies and tip inertia reduces the torsional frequencies. Both the tip mass offset and tip angular momentum couples the bending and torsional modes resulting in an increase in the frequency. Possible future work includes analyzing the cumulative effect of mass, inertia, mass offset, and angular momentum. Further analysis of the mode shapes particularly in the mass offset and angular momentum cases. An experimental setup is also needed to test the theory derived. A model of a wing with an attached rotating motor will be setup in order to test the theory derived. References [] Shabana, A.A., Theory of Vibration; Volume I: An Introduction, Springer-Verlag, New York, 99. [] Graham, Kelly, S. Fundamentals of Mechanical Vibrations, McGraw-Hill Higher Education, nd ed. Boston,.