AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016)

Transcription

1 AA 242B/ ME 242B: Mechanical Vibrations (Spring 2016) Homework #2 Due April 17, 2016 This homework focuses on developing a simplified analytical model of the longitudinal dynamics of an aircraft during the rollout phase of landing. Model Components The analytical longitudinal model to be developed contains several parts corresponding to: The aircraft body subsystem. A nose landing gear subsystem and two identical main landing gear subsystems. A nose wheel/tire subsystem and two identical main wheel/tire subsystems. The assembly of the subsystems outlined above is graphically depicted in Figure 1 where two coordinate systems are shown: one attached to the ground (X, Y, and Z axes), and one to the aircraft body (x, y, and z axes). Both frames have their origin located at the center of gravity (CG) of the aircraft. Aircraft Body Subsystem The aircraft body has a mass, m, and a pitching moment of inertia around the Y axis, I y. It is assumed to be rigid and to undergo a motion in the X-Z plane (which is also the x-z plane). The engine thrust, T, is assumed to act through the CG of the aircraft and to be aligned with the body x-axis for simplicity. Figure 1 also shows the velocity vector of the aircraft, V. The angle between V and the body x-axis is the angle of attack α. This angle and oll other angles relevant to the dynamics of the aircraft during the rollout phase of landing are assumed to be small. Figure 1 does not show all forces acting on all of the aircraft subsystems outlined above. For convenience, this figure highlights however the thrust engine 1

2 vector, T, the angle of attack, α, the aerodynamic lift, L, whose direction is orthogonal to V, the aerodynamic drag, D, which is aligned with V but in the opposite direction, and the aerodynamic (pitching) moment, M, which is assumed to act through the CG of the aircraft in the Y direction (which is also the y direction). The lift, drag, and pitching moment are given by L = 1 2 ρv 2 SC L D = 1 2 ρv 2 SC D M = 1 2 ρv 2 ScC M where ρ denotes the free-stream air density, S denotes the aircraft planform area, and c is the wing mean aerodynamic chord. C L, C D and C M are the lift, drag, and pitching moment coefficients and are given by C L = C L0 +C Lα α C D = C D0 + C2 L πe 0 A R C M = C M0 +C Mα α+c MQ Q+C Mδ δ where C L0, C Lα, C D0, e 0, A R, C M0, C Mα, C MQ, and C Mδ are constant coefficients, Q denotes the pitch rate (angular velocity about the CG of the aircraft), and δ denotes the pitch control effectiveness. Wheel/Tire Subsystem Each wheel is assumed to behave as a lumped mass, m t, located at the hub centerline so that the wheel has no moment of inertia. The tire is considered to be massless. Its stiffness and damping characteristics are represented by a linear spring and a linear damper, respectively. These are assumed to act between the tire contact area on the runway and the wheel, through the hub centerline, in the vertical direction. The spring and damping coefficients are denoted by K t and C t, respectively, and the radius of the wheel/tire subsystem is denoted by r t (see Figure 2). Landing Gear Subsystems Each landing gear is assumed to behave essentially like a nonlinear, uniaxial damper (or shock absorber) with an oleo-pneumatic (oil-gas) mechanization (see Figure 3). It has a mechanical stop which limits its extension when it is down and locked before touchdown. The force in each landing gear, F g, is assumed to act at the wheel hub along the assumed centerline of the damper. Its value is given by F g = C 1 ż g + C 2 ( z g0 z g l u l l where C 1 and C 2 are two constant coefficients, k is the gas constant and assumed to be equal to 1.4, z g is the effective length of the landing gear measured along ) k 2

3 the assumed gear centerline from the aircraft x-body axis to the center of the wheel hub (it is also equal to the gas volume divided by the cylinder crosssectional area), z g0 is the effective length of the gas column when the landing gear is at its extension limit, and l u and l l are two constant distances shown in Figure 3. Note that a positive force F g is required to compress the landing gear and move it off its stop. As shown in Figure 1, the nose and main landing gears are attached at distances a and b from the CG of the aircraft along the x-body axis. Questions Adopt the free-body diagrams shown in Figure 1 and Figure 2. The forces displayed in these diagrams are explained in Table 2. Treat the following variables as control variables: δ, T, X gn, X gm where the subscripts n and m designate the nose and main landing gears, respectively. Then, answer the following questions: 1. Write all dependencies between any subsets of the kinematic variables given in Table Using Newton s laws of motion, derive the dynamic equations of motion of the entire multi-degree-of-freedom system that is, derive these equations from equilibrium. For this purpose, assume that all time-dependent angles remain small. Write these equations in the first-order form ξ = F (ξ, c), where ξ is a state vector (vector of degrees of freedom) and c is the vector of control variables described above (note that the set of kinematic variables defined in Table 1 was chosen to facilitate this specific task). State which of these equations can be interpreted as a constraint and state its type. 3. Inspire yourself from Table 1 and the dependencies derived in the response to the first question above, and propose a minimal set of generalized degrees of freedom q i describing the entire mechanical system. 4. For each chosen generalized degree of freedom q i, specify the corresponding non-conservative external generalized force Q i. Present your result in a table with a column for q i and a column for Q i. 5. Using the Lagrange equations, derive the equivalent dynamic equations of equilibrium of the entire multi-degree-of-freedom system in second-order form. Note. In both cases, simplify the dynamic equations of equilibrium of the wheel/tire subsystems by assuming that in the horizontal direction, and only in that direction, the effects of the inertial force and landing gear force components Z gn and Z gm can be neglected. 3

4 Figure 1: Planar view of the free-body diagram of the aircraft body subsystem during the rollout phase of landing: forces, applied moments, and kinematic variables. 4

5 Figure 2: Free-body diagram of the nose wheel/tire subsystem. Figure 3: Landing gear subsystem. 5

6 U W Q θ X cg H cg z gm z gn h m h n X m X n velocity in the x direction of the aircraft velocity in the z direction of the aircraft angular velocity about the CG of the aircraft pitch angle about the CG of the aircraft longitudinal position/displacement in the horizontal direction of the CG of the aircraft vertical position of the CG of the aircraft measured from the ground in the vertical direction perpendicular distance between the main wheel hub center and the x axis perpendicular distance between the nose wheel hub center and the x axis main hub height relative to the ground nose hub height relative to the ground main hub displacement in the X direction nose hub displacement in the X direction Table 1: A set of kinematic variables describing the entire mechanical system. X gn free-body diagram force related to the action/reaction at the connection between the nose landing gear and its wheel hub and applied there in the x direction X gm free-body diagram force related to the action/reaction at the connection between the main landing gear and its wheel hub and applied there in the x direction Z gn in Figure 1, this is the reaction to the nose landing gear force F gn ; it is applied in the z direction at the hub of the nose landing gear Z gm in Figure 1, this is the reaction to the main landing gear force F gm ; it is applied in the z direction at the hub of the main landing gear T thrust force in the x direction L lifting force D drag force M a aerodynamic moment mg weight of the aircraft Table 2: Force definitions. 6