Aerospace Engineering - Airplane performance and dynamics

Completed notes of the course

Complete course

Politecnico di Milano - Master in Aeronautical engineering Notes of the course of: 055736 - Airplane Performance and Dynamics of the academic year 2023/24, held by Professor Carlo RIBOLDI Disclaimer: professor Riboldi is not the author of these notes. The numbers in pencil that appear in the notes refer to the number of the page of professor’s notes available on WeBeep. These notes only cover the theoretical lectures (exercise sessions are not included). I Flight Dynamics Introduction - Airplane modelling −→different models for different analysis • Steady/quasi-steady analysis–Aircraft−→Point in 3D space [good for the initial design] –Input (?)−→Basic *weight *Aerodynamic polar (Lift and drag coefficients) *Thrust / power installed –Suitability−→Flight performances for most legs of the mission (which are performed in steady conditions) • Dynamic analysis [good for later stages of the design]–Aircraft is modelled as *rigid body *assembly of rigid bodies *flexible body (all of them in 3D space) –input−→more detailed, and it will depend on geometry, structural choices and configuration –Stability *Studying dynamic manoeuvres *Assessing flying qualities *Setting up control systems / testing control systems *Building up a simulator We are going to start from the modelling of an aircraft as a rigid body in 3D space−→We will have to simplify the model to the point they are suitable for static / quasi static analysis. II Reference Systems Why do we need reference systems? • They allow to write simple expressions of physical quantities–kinematics-related *body position/orientation in 3D space *velocity/acceleration –dynamics-related *inertial properties *forces (gravity, aerodynamic forces ...) Airplane performance and dynamics, lectures 1,2,3 • They allow to bring signals into computations Structure of reference systems:• Origin • 3 mutually orthogonal axis (cartesian frame) Notion: Inertial reference system• We need it since Newton’s 2nd law applies to quantities taken with respect to an inertial reference system. • Inertial reference = unaccelerated reference–No linear acceleration –no rotation−→notation always associated to centripetal acceleration. –uniformly translated –standing still • Hard to visualize: an example of visualization is fixed-stars system • Several non-inertial (strictly) systems can be approximately considered inertial. Earth Inertial System - italics "I"I •Origin: center of EarthO i •Axes: ⃗ i1,⃗ i2−→ define equatorial plane. ⃗ i3Earth’s rotation axis, from the center to the north pole. •Notes: –It is parallel to fixed-stars –It is fixed with respect to daily rotation of the Earth –moves around the Sun with the planet –It is not strictly inertial Earth-fixed SystemE •Origin: center of EarthO e≡ O i •Axes: ⃗e1, ⃗e 2−→ equatorial plane. ⃗e3≡⃗ i3Earth’s rotation axis. •Notes: –e 1, e 2are attached to the Earth −→rotational speed:Ω e=2 π24 ·3600r ads –It is mostly employed for travel motion trajectories (GPS relying to ...) ???? –It is not strictly inertial2 Airplane performance and dynamics, lectures 1,2,3 Navigational System ( ˜ N) •Origin: A point on the surface of the Earth, origin for trajectory (typically) •Axes: ⃗˜ n 1, ⃗ ˜ n 2−→ they define a plane (local horizon plane), locally tangent to the surface of the Earth. (⃗ ˜ n 1is to the North, ⃗ ˜ n 2is to the East) ⃗˜ n 3−→ to the center of Earth •Notes: –frame is attached to the Earth –It is very useful for defining gravity Local vertical reference system, orlocal horizontal frame, orNorth-East-Down (aka the acronimous NED)(N) •Origin: Aircraft center of gravityG •Axes: ⃗n1, ⃗n 2−→ they define a plane (local horizon planewhich moves with the aircraft) parallel to the one tangent to the surface of Earth. (⃗n 1is to the North, ⃗n 2is to the East, ⃗n3to the center of Earth). •Notes: –Reference is moving with the aircraft (it is not attached) –it is rarely inertial (depends on the type of flight) –It is useful for the instantaneous trajectory of the a/c *e.g. inclination with respect to the horizon of the a/c *e.g. inclination with respect to the horizon of velocity Body System(B) •Origin: Arbitrary point on a/c –typically, the center of gravityG –Alternatively, a material point on the a/c •Axes: Also arbitrary, but it is typically (for our description goals): –⃗ b1,⃗ b3define a plane of vertical symmetry of a/c –⃗ b1is aligned with the longitudinal direction of the a/c. –⃗ b2points to the right. ⃗n1, ⃗n 2−→ they define a plane (local horizon plane which moves with the a/cparallel to the one tangent to the surface of Earth. (⃗n 1is to the North, ⃗n 2is to the East, ⃗n 3to the center of Earth. •Notes:3 Airplane performance and dynamics, lectures 1,2,3 – Generally not inertial –Onboard measurements refer to a body reference –easy definition of inertial properties –easy definition of thrust forces –it is often employed for the expression of the operations of motion of the a/c Stability System(S) Defined starting from body reference •Origin: Arbitrary (same asB) •Axes: –⃗s 1, ⃗s 3define a plane of vertical symmetry of a/c –⃗s 1points the projection of the velocity vector on the symmetry plane. –⃗s 2points to the right. •Notes: –employed for eigenanalysis –the angle of attack is, by definition, the angle between⃗ b1and ⃗s 1 Wind System(W) It is defined starting from body system •Origin: Arbitrary (same asB) •Axes: –⃗w 1is aligned with airspeed vector –⃗w 3is in the vertical plane of symmetry –⃗w 2comes as consequence of the previouses •Notes: –easy definition of "aerodynamic angles": angle of attack and sideslip –very easy definition of aerodynamic force components *lift: ⃗w 3 *drag: ⃗w 1 *lateral force: ⃗w 24 Airplane performance and dynamics, lectures 1,2,3 Rotations Why do we need rotations? Different quantities are easily expressed in different systems. (eg: inertia for a constant mass rigid body is constant (so, easier to express) in B; gravity is easy to define inN; aerodynamic forces are easy to define inW). Equations of motion are written in a single reference. Interfacing "signals" (or measurements) into that single reference requires some form of rotation. There are several ways to express3D rotations. In aerospace, the following are used: • Euler angles (also calledTait-Bryansequence) • Direction cosine matrix • Quaternions Note that for one, single rotation, many descriptions are possible. Euler Angles consider two reference systems:• TheBaseline I(with origin O and axes⃗ i1,⃗ i2and⃗ i3); • TheRotated system K(with origin O and axes⃗ k1,⃗ k2and⃗ k3); We would like to linkItokby means of a precise sequence of three planar rotations. Each planar rotation is defined by an axis and an intensity (so we have a rotation vector⃗σ). Many sequences are possible, yielding: the same final results, different axis or intensities and different planar rotations. We concentrate on one specific sequence: theTait-Bryansequence: 1. rotation aroundi 3of intensity σ 1. (Therefore⃗ i3≡⃗ i′ 3. We get the intermediate system I’ (⃗ i′ 1,⃗ i′ 2,⃗ i′ 3) ). 2. rotation aroundi′ 2s of intensity σ 2. (Therefore⃗ i2≡⃗ i′ 2. We get the intermediate system I”(⃗ i′′ 1,⃗ i′′ 2,⃗ i′′ 3) ). 3. rotation aroundi′′ 1s of intensity σ 3. (Therefore⃗ i′′ 1≡ k 1. We get the final system (⃗ k1,⃗ k2, ⃗ k3) There are several intensities which provide by combination the same result. Limitations are put onσ 1, σ 2, σ 3: −π≤σ 1≤ +π −π2 ≤ σ 2≤ +π2 −π≤σ 3≤ +π; Let’s see the mathematical expressions corresponding to the Tait-Bryan sequence. We have two operations available, which are very similar from a mathematical point of view, but totally different in meaning: theRotation of a vectorand theChange of reference5 II.1 Rotation of a vector Airplane performance and dynamics, lectures 1,2,3II.1 Rotation of a vector Say we want to rotate the vector vector ⃗vinto vector⃗w. A rotation is an application which, once applied to a vector, preserves the modules (intensity), and changes the direction. ⃗w=R⃗v Lets see howRis get. Suppose we are rotating a vector along axis x. (Rotation ofσ 1) vI =        1 0 0         ; wI =        cos σ 1 sinσ 1 0         What is the matrix that transfersvI intowI ? wI =        cos σ 1 sinσ 1 0         =         cos σ 1∗ ∗ sinσ 1∗ ∗ 0∗ ∗                  1 0 0         ; And we call         cos σ 1∗ ∗ sinσ 1∗ ∗ 0∗ ∗         = RI |I′ Let’s now see how we have to set the second and third columns: v=i 2=        0 1 0         ; w=i′ 2=        cos σ 1 sinσ 1 0         So, the R matrix is: wI =        − sinσ 1 cosσ 1 0         =         ∗ − sinσ 1∗ ∗cosσ 1∗ ∗0∗                  0 1 0         ; v=i 2=        0 0 1         ; w=i′ 3=        0 0 1         So, the R matrix is: wI =        0 0 1         =         ∗ ∗ 0 ∗ ∗0 ∗ ∗1                  0 0 1         ; SinceRI |I is the same for all three cases (choices of v and w),RI |I′ =         cos σ 1− sinσ 10 sinσ 1cos σ 10 0 0 1        Is true that RI =RI ′ =RI |I′ Let’s take the same I, I’ of previous example, write components of vectors (same choices) in I’ v=i 1, w=i′ 1, components in I’ : vI ′ =        cos σ 1 −sinσ 1 0         , wI ′ =        1 0 0         wI ′ =        1 0 0         =         cos σ 1− sinσ 1∗ sinσ 1cos σ 1∗ 0 0∗                  cos σ 1 −sinσ 1 0         v=i 2, w=i′ 2, components in I’ : vI ′ =        sin σ 1 cosσ 1 0         , wI ′ =        0 1 0         wI ′ =        0 1 0         =         cos σ 1− sinσ 1∗ sinσ 1cos σ 1∗ 0 0∗                  sin σ 1 cosσ 1 0         v=i 3, w=i′ 3, components in I’ : vI ′ =        0 0 1         , wI ′ =        0 0 1         6 II.2 Change of reference Airplane performance and dynamics, lectures 1,2,3w I ′ =        0 0 1         =         cos σ 1− sinσ 10 sinσ 1cos σ 10 0 0 1                  0 0 1                 cos σ 1− sinσ 10 sinσ 1cos σ 10 0 0 1         = RI ′ By comparingwI ′ =RI ′ vI ′ andwI =RI vI we notice thatRI ′ =RI Remark: despite it had been shown with planar notations, this has general validity. Remark: It is possible to associate a notation to two references, a baseline and a rotated one, which are connected by the rotation: this is reflected by this notation:R j−→k II.2 Change of reference Take a vectorv, define its components in a first reference I,vI , obtain the components of the same vector v into another reference I’, rotated with respect to I. We call thisvI ′ . Its mathematical expression is:vI ′ =QI |I′ vItake I, I’; take v=i 1. Its components are vI =        1 0 0         ; vI ′ =        cos σ 1 −sinσ 1 0         sovI ′ =        cos σ 1 −sinσ 1 0         =         cos σ 1∗ ∗ −sinσ 1∗ ∗ 0∗ ∗                  1 0 0         Other coefficients comprehend the other two cases: v=i 2: vI ′ =        sin σ 1 cosσ 1 0         =         ∗ sinσ 1∗ ∗cosσ 1∗ ∗0∗                  0 1 0         v=i 3: vI ′ =        0 0 1         =         ∗ ∗ 0 ∗ ∗0 ∗ ∗1                  0 0 1         By collecting info for all cases:QI |I′ =        cos σ 1sin σ 10 −sinσ 1cos σ 10 0 0 1         Note that this is(RI |I′ )T , that meansQI |I′ = (RI |I′ )T Note that this is thre in general (not only for planar notations. II.3 Summarizing Rotation of a vector:w=Rv, where v is the original vector, w is the rotated vector, R is the rotation tensor, and R is associated to a baseline and a rotational frame. Baseline system is I, rotated system is k. The notation isR I−→k w=R I−→kv7 II.3 Summarizing Airplane performance and dynamics, lectures 1,2,3Representation of rotation of a vector: pick a reference system, e.g.: I: wI =RI I−→kvI e.g.: k:wk =Rk I−→kvk Since for rotationsRI I−→k= Rk I−→k, we can introduce this notation: Rk |I I−→k Properties of rotations: Orthogonality:RT I−→k= R− 1 I−→kProof: take a vector v and a rotational tensor R I−→k Definew=R I−→kv Compute the transpose:wT =vT RT I−→k Recall a rotation preserves the modules:wT w=vT v Let’s substitute:wT w= (vT RT I−→k) w=vT RT I−→k= R I−→kv =vT v RT I−→kR I−→k= I−→RT I−→k= R− 1 I−→k Similarly, it can be shown thatR I−→kRT I−→k= I So,R I−→kRT I−→k= RT I−→kR I−→k= I From this property: w=R I−→kv , andv=RT I−→kw vk =RI |k/ T I−→kvI , andvI =RI |k I−→kvk change of reference by tensors a generic tensor is an operator, changing a vector into another vector Rotation is an example of tensor Other examples include theinertia tensor Consider a generic tensorT Consider two reference systems I,k , linked byR I−→k assign two vectorsw=T v Express this both in I and k : I:wI =TI vI k:wk =Tk vk Operate a change of reference from k to I ofvandw wI =RI |k I−→kwk vI =RI |k I−→kvk We want to writeTk wI =RI |k I−→kwk =TI vI =TI RI |k I−→kvk wk =RI |k−T I−→kTI RI |k I−→kvk Tk =RI |k−T I−→kTI RI |k I−→k By exploiting orthogonality:TI =RI |k I−→kTk RI |k−T I−→kNote (example): consider a rotational tensor T=R I−→k TI =Tk =RI |k−T I−→k8 II.3 Summarizing Airplane performance and dynamics, lectures 1,2,3According to the first property: Tk =RI |k I−→k= ( RI |k−T I−→kRI |k I−→k) RI |k I−→k, the terms in the parenthesis are the Identity matrix. The property is verified. According to the second property:TI =RI |k I−→k= RI |k I−→k( RI |k I−→k) RI |k−T I−→k) , the terms in the parenthesis are the Identity matrix. The property is verified. Composition of rotations Consider three vectors: v, w, q such thatw=R I−→kv , andq=R k−→Fw Clearly:q=R k−→F( R I−→kv ) =R k−→F( R I−→k) v RI−→F= R k−→FR I−→k( R I−→k) v Representation of composed rotations: Consider the scenario just introduced (previous point): Write expressions in referenceI:qI =RI I−→FvI =RI k−→FRI I−→kvI We haveRI k−→F, which is non-trivial to express. We apply properties for relaying the reference of tensors, to get an easier expression. According to the properties we introduced: RI k−→F= RI |k I−→kRk |F k−→FRI |k−T I−→k Substituting:qI =RI |k I−→kRk |F k−→FRI |k−T I−→kRI I−→kvI =RI |k I−→kRk |F k−→FvI Write expressions in F: qF =RF I−→FvF −→qF =Rk |F k−→FRF I−→kvF −→RF I−→F= Rk |F−T k−→FRI |k I−→kRk |I k−→I; By substitution, qF =Rk |I k−→FRk |F−T k−→FRI |k I−→kRk |F k−→FvF =RI |k I−→kRk |F k−→FvF (We usedRk |I k−→FRk |F−T k−→F= I). Note: it should not be surprising that expressions ofqI andqF are the same, since they are RI I−→F= RF I−→F= RI |F I−→F Change of reference of a vector trough composed rotations: Consider the scenario previously introduced (two rotations, composed), and consider vector v ComponentsvF =RI |F−T I−→FvI ExpressingRI |F−T I−→F= ( RI |k−T I−→kRk |F−T k−→F)T =Rk |F−T K−→FRI |k−T I−→k Therefore, substituting:vF =Rk |F−T K−→FRI |k−T I−→kvI Let’s go back to the Tait-Bryan sequence: It is a composition of three planar rotations Let us start from I and generate the intermediate and target systems one by one: 1. Rotation aroundi 3, intensity σ 1, from ItoI′ RI |I′ I−→I′ =         cos σ 1− sinσ 10 sinσ 1cos σ 10 0 0 1         9 II.3 Summarizing Airplane performance and dynamics, lectures 1,2,32. Rotation around i′ 2, intensity σ 2, from I′ toI′′ RI ′ |I′′ I′ −→I′′ =         cos σ 20 sin σ 2 0 1 0 −sinσ 00 cos σ 2        3. Rotation around i′′ 1, intensity σ 3, from I′′ tok RI ′′ |k I′′ −→k=         1 0 0 0 cos σ 3− sinσ 3 0 sinσ 3cos σ 3        We can compose three planar rotations to set an arbitrary 3D rotation. In terms of tensors: R I−→k=????? In terms of representation:RI |k I−→k= RI |I′ I−→I′ =RI ′ |I′′ I′ −→I′′ =RI ′′ |k I′′ −→k′ Note that: RI |I′ I−→I′ =RI |I′ I−→I′ (σ 1) RI ′ |I′′ I′ −→I′′ =RI ′ |I′′ I′ −→I′′ (σ 2) RI ′′ |k I′′ −→k= RI ′′ |k I′′ −→k( σ 3)10