Aerotecnica M&S 100 Years Ago: A Study on Aircraft Longitudinal Stability

Aerotecnica Missili & Spazio https://doi.org/10.1007/s42496-022-00111-y PREFACE Aerotecnica M&S 100 Years Ago: A Study on Aircraft Longitudinal Stability Preface to Issue 1, Volume 101 Aldo Frediani1 · Vittorio Cipolla1 · Sergio De Rosa2 · Paolo Gasbarri3 Accepted: 7 February 2022 © The Author(s) under exclusive licence to AIDAA Associazione Italiana di Aeronautica e Astronautica 2022 1 On the Longitudinal Stability of Airplanes1 1.1 Relationship Between Static and Dynamic Stability in the Hypothesis of no Oscillations of the CG: Relative Measurement Method. Approximate Study of the Jolt 1.1.1 F. Burzio, Politecnico di Torino The general treatment of the longitudinal stability problem (which is the study of a rigid body in the plane of motion, composed of the motion of the Centre of Gravity (or “jolt” in the following) in a fixed vertical plane and of the rotation of the system (pitch) around an axis normal to the plane with the origin in the Centre of Gravity) does not reveal in a simple way the relationship existing between what is usually called static longitudinal stability, or initial, or intrinsic, and the dynamic phenomenon (dynamic stability, measured by certain parameters, which we will see later on in the paper) which is the consequence of statics and other factors (moment of inertia). Also with the obtained methods of measurement, we determine independently the static stability2and the dynamic one,3 without the possibility to compare them. A simple relationship between the two quantities can be established with the assumption that the oscillation around the barycentric axis occurs without giving rise to alterations in the motion of the centre of gravity, that is, with the study of simple pitch, supposed the null jolt. The resulting difference is discussed in the following. * Aldo Frediani 1 Università di Pisa, Pisa, Italy 2 Università Federico II Napoli, Naples, Italy 3 Università La Sapienza Roma, Rome, Italy The general procedure leads, for small oscillations4 to a system of three differential equations in the variables: q = dθ/dt. (angular pitch speed), u, v (velocity components of the CG, in the direction of motion and in the normal one, respectively): ⎧ du + v0 q = −g 𝜃 cos𝜃0 + uXu + vXv + qXq ⎪ dtdv ⎨ dt − u0 q = −g 𝜃 sin𝜃0 + uZu + vZv + qZq ⎪ I dq = uMu + vMv + qMq , ⎩ dt (1) where I is the moment of inertia of the system around the pitch axis, Xu Xv Zu … M q are the incremental ratios (resistance derivatives in the original test) of the components X and Z of the thrust and of its moment M relevant ot the CG with respect to u, v and q. The integration of this system can be carried out by the Routh's method whose determinant gives, for the general integrals of the form 𝜃 = 𝜃1 e𝜆1 t + 𝜃2 e𝜆2 t + 𝜃3 e𝜆3 t + 𝜃4 e𝜆4 t , … , a 4th degree equation in λ: 𝜆4 + A1 𝜆3 + B1 𝜆2 + C1 𝜆 + D1 = 0 where the coefficients are functions of Xu … Mq , and of the known quantities. The 4 roots of the characteristic equation in λ, in the most interesting case for stability, are of the type ) )( ( 𝜆 + a1 ± ib1 𝜆 + a2 ± ib2 = 0, 1 Presented on July 15th, 1922 in Turin. A. Rota, “Sulla stabilità longitudinale dei velivoli”, Rendiconti I.S.A., Anno IX, Serie 2a.“Statical longitudinal stability of airplanes”, NACA-TR-96, National Advisory Committee for Aeronautics, 1921. 3 Reports of National Advisory Committee for Aeronautics, p. 330 and following, 1917. 4 Bairstow, L., “Applied Aerodynamics”, p. 457 and following. 2 13 Vol.:(0123456789) A. Frediani et al. i.e.: complex conjugate roots corresponding to two damped oscillations of the form ) ( 2𝜋 t , 𝜃 = 𝜃0 e−𝜇t sen T where the parameters μ and T are functions of a 1 and b 1 in the 1st oscillation, of a2 and b2 in the 2nd. These, or rather the roots of two equations of the 2nd degree that are derived from the approximate decomposition of the equation of 4th degree into λ, (Bryan, Bairstow) are physically interpreted as two coexisting pitch oscillations: the 1st (short oscillation) occurs at small T and great μ, the 2nd (phugoid oscillation) generally occurs at large T and small μ: the latter, at high speeds, could become an aperiodic motion. If we make the hypothesis of null jolt (u = v = 0), the system (1) is reduced to the 3rd equation, modified as follows: I d2 𝜃 = qMq = 𝛿M, dt2 (2) and we find5 (5) 𝛿M = A d𝜃 + B𝜃, dt where A is proportional to the damping torque (or rotational, or dynamic stabilizer), B to the static stabilizer torque.6 Equation (2) gives rise (in the case, more interesting for stability, of negative roots of the characteristic equation, and taking the initial condition θ = 0 for t = 0 into account), to the general solution: 𝜃 = 𝜃0 e−𝜇t sen(ht), where: 𝜇= A 2 B ; h = − 𝜇2 , 2I I that is, 𝜇 and h are real part and modulus of the imaginary part of the roots of the characteristic equation, respectively. Therefore, neglecting the jolt is the same as assuming pitch consisting of a single damped oscillation instead of the two present in the general solution: and it will be interesting to study in what relationship is that with these. Now, we observe that, because (as is well known) the period of oscillation is: 5 See, for example, Verduzio “Teoria del volo dell’aeroplano” (Theory of the flight of aircraft), p. 230 and following or De Villers, “La dynamique de l’avion”, p. 215 and following. 6 The relationship with the longitudinal stability index,c, given by A. Rota, is B = QV2δc, where Q is the aircraft weight, V the speed and δ the density. 13 T= 2𝜋 2𝜋 =√ , h B∕I − h2 it results: B 4𝜋 2 = 2 + 𝜇2 , I T (3) This is, therefore, the simple relationship sought between static and dynamic stability, the parameters of the latter being 𝜇 and T. Equation (3) tells us that a great dynamic stability (large 𝜇, small T) can be obtained with a large static stabilizing moment, and with a small moment of inertia, and indeed, that the dynamic stability is the ratio of static stability and the moment of inertia. This result is achieved in the hypothesis of simple pitching, that is, of zero jolt. It will therefore be important to study, in second approximations, this motion of the jolt, which has been neglected so far. This is also because, in experimental research on stability into the wind tunnel, it is possible to obtain only the simple pitch, the speed of the center of gravity (the wind speed) remaining, evidently, unchanged. Equation (3) also offers a way for the experimental determination of static stability as a function of the dynamics one. The relevant apparatuses (aerodynamic torsion pendulums) are used for tunnel tests on models and can be of various types. Let's see the base theory. In these devices, in addition to the damping moment A' and the static stabilizer moment of the model, (having the meaning of their corresponding A and B defined above for the actual aircraft), two other active moments have to be considered: • a first one proportional to the angular displacement, and (...truncated)