Vibration and Aeroelasticity
Flutter of plates
Consider a plate occupying a domain S on the x, y plane, bounded by the
contour Γ (hereafter, Γ is supposed to be piecewise-smooth). One side of
the plate is subjected to gas flow with the velocity vector v = {vx, vy} = {v
cos θ , v sin θ } , v = |v|. If, in addition to the unperturbed state w0 ≡ 0, we
consider a perturbation w = w(x, y, t), then the aerodynamic pressure Δp
caused by interaction with the perturbed flow appears. It will be shown in
the following analysis that Δp is a linear operator of w, which will allow us to
present the solution in the form
w = φ(x, y) exp(ω t), Δp = Δp0(x, y) exp(ω t)
in all cases except the flutter problem for a viscoelastic plate. The equation
for vibrations of a constant-thickness plate takes the form
DΔ2 w + ρ h𝜕2w 𝜕t2 = Δp,
where D = Eh3/(12(1− 𝜈2)) is the cylindrical rigidity, E, 𝜈, and ρ are Young’s
modulus, Poisson’s ratio, and density of plate material, and h is its
thickness. From the above considerations, we have Δp0 = L1(φ) + L2(φ,
ω), and therefore,can be rewritten in the form DΔ2 φ + L1(φ) + ρhw2 φ +
L2(φ; ω)=0.On the contour Γ, the deflection amplitude φ(x, y) satisfies the
boundary conditions x, y ∈ Γ, M1(φ)=0, M2(φ)=0, where the boundary
operators M1 and M2 are problem-specific and will be given in each
particular case. We assume hereafter that the plate is not subjected to any
loads in its median plane. The system of equations represents a
complicated eigenvalue problem with a non-self-adjoint operator; its
eigenvalues are denoted by ω. By definition, we take that the perturbed
,motion of the plate is stable if Re ω < 0, and unstable if Re ω > 0; the
critical parameters of the system (plate, flow) are determined by the
condition Re ω = 0. In the further analysis, we consider the following main
questions: determination of Δp, formulation of new problems; development
of an efficient analytical approach, and identification of new mechanical
effects.
Determination of aerodynamic pressure
Numerous studies on the vibrations and stability of plates in supersonic
high-speed flows are carried out on the basis of the piston theory for the
aerodynamic pressure Δp caused by the interaction of the flow and
vibrating plate. This formulation has become so common that it was applied
even in the cases where its validity is questionable. Here, we derive Δp in
the cases of “moderate” supersonic (M ∼ 1.5–2) and low subsonic
velocities. Consider an elastic strip occupying domain S : {0 ≤ x ≤ l, y = 0,
|z|<∞}. On the side y ≥ 0, the strip is placed in a gas flow with unperturbed
parameters (planar problem) v = {u0, 0}, p0, ρ0, a0 = (𝛾p0/ρ0) 1/2, so that
the unperturbed flow potential is φ0 = u0x. Small vibrations of the strip w(x,
t) (with w/ℓ ≪ 1) cause flow perturbation; the perturbed flow potential is
denoted by φ1 = φ0 + φ. Then we proceed in the usual way: from the
Cauchy–Lagrange integral, equations of motion, mass conservation, and
equation of state we obtain an equation for φ1 and linearize it with respect
to the perturbation φ to obtain 1 a2 0 𝜕2φ 𝜕t2 + (M2 − 1) 𝜕2φ 𝜕x2 + 2 M a0
𝜕2φ 𝜕x𝜕t − 𝜕2φ 𝜕y2 = 0, where M = u0/a0. The potential φ must vanish at
infinity and satisfy the impermeability condition on the line
y = 0: y = 0, 0 ≤ x ≤ l, 𝜕φ 𝜕y = 𝜕w 𝜕t + u0 𝜕w 𝜕x
,y = 0, x ≤ 0, x ≥ l, 𝜕φ 𝜕y = 0. (2.3) The overpressure in the flow is obtained
from p = −ρ0 (𝜕φ 𝜕t + u0 𝜕φ 𝜕x ) .We search for the solution in the class of
functions φ(x, y, t) = f(x, y) exp(ω t), w(x, t) = W(x) exp(ω t), and p(x, y, t) =
q(x, y) exp(ω t). Introduce now the nondimensional coordinates x/l and y/l,
retaining hereafter the previous notation. Also, introduce the
nondimensional frequency lω/a0 = Ω.
The system of equations is transformed to,
(M2 − 1) 𝜕2f 𝜕x2 + 2MΩ𝜕φ 𝜕x + Ω2 f − 𝜕2f 𝜕y2 = 0 (2.5) y = 0, 0 ≤ x ≤ 1, 𝜕f
𝜕y = a0 (ΩW + M 𝜕W 𝜕x ) (2.6) y = 0, x ≤ 0, x ≥ 1, 𝜕φ 𝜕y = 0
q = −ρ0a0 l (Ωf + M 𝜕f 𝜕x ) .
In what follows, it is necessary to distinguish the cases of M < 1 and M > 1;
we consider them one by one. For M > 1, perturbations are absent to the
left of the point x = 0; therefore, it is possible to apply the Laplace transform
along the x coordinate; condition is not relevant, and the function W(x) can
be prolonged to x ≥ 1 arbitrarily (as long as applicability conditions for the
Laplace transform are satisfied), and this will not affect the overpressure
q(x, 0) acting on the strip. From we obtain for the Laplace transform ̃f(s, y)
In what follows, it is necessary to distinguish the cases of M < 1 and M > 1;
we consider them one by one. For M > 1, perturbations are absent to the
left of the point x = 0; therefore, it is possible to apply the Laplace transform
along the x coordinate; condition is not relevant, and the function W(x) can
be prolonged to x ≥ 1 arbitrarily (as long as applicability conditions for the
Laplace transform are satisfied), and this will not affect the overpressure
q(x, 0) acting on the strip. From we obtain for the Laplace transform ̃f(s, y)
β 2 ̃f − 𝜕2 ̃f 𝜕y2 = 0, β 2 = (M2 − 1) s 2 + 2MΩs + Ω2 . A solution bound at
infinity is ̃f = c1e −β y. (2.9) From the boundary condition for Laplace
transform 𝜕 ̃f 𝜕y y=0 = −β c1 = a0(Ω + Ms)W̃ it is possible to determine the
, parameter c1, and therefore it follows from (2.9) that ̃f = −a0 Ω + Ms β Wẽ
−β y. (2.10) The overpressure (in terms of Laplace transforms) is now
obtained from equation
q̃ (s, 0) = Δp̃ (s) = ρ0a2 0 l (Ω + Ms) 2 β W̃(s). (2.11)
The inverse Laplace transform is found from tables and convolution
theorems. We first write β = √M2 − 1√(s + s1)(s + s2) ≡ √M2 − 1β0; s1 =
Ω/(M−1), s2 = Ω/(M + 1); (s1 + s2)/2 = MΩ/ (M2 − 1) ≡ α1; (s1 − s2)/2=Ω/
(M2 − 1) ≡ α2. We now have L(−1) ( 1 β0 ) = I0(α2x)e −α1x ≡ H(x), where
I0(z) is the modified Bessel function; therefore L(−1) (W̃ β0 ) = x ∫ 0 H(x −
τ)W(τ) dτ L(−1) (sW̃ β0 ) = x ∫ 0 H(x − τ) 𝜕W 𝜕τ dτ L(−1) (s 2W̃ β0 ) = 𝜕 𝜕x x ∫
0 H(x − τ) 𝜕W 𝜕τ dτ.
We perform the necessary calculations and substitute the results in
equation (2.11) to obtain finally,
Δp(x) = ρ0a2 0M l(M2 − 1) 1/2 [ [ M2 − 2 M2 − 1 ΩW + M 𝜕W 𝜕x + (M2 + 2)
Ω2 2M (M2 − 1) 2 x ∫ 0 e −α1(x−τ) I0 (α2(x − τ)) W(τ) dτ − 2Ω2 (M2 − 1) 2 x
∫ 0 e −α1(x−τ) I1 (α2(x − τ)) W(τ) dτ + MΩ2 2 (M2 − 1) 2 x ∫ 0 e −α1(x−τ) I2
(α2(x − τ)) W(τ) dτ] ] ,
where I𝜈(z), 𝜈 = 1, 2, are the modified Bessel functions. There are
important implications of equation The formula of the piston theory is
obtained in the limit M ≫ 1, and it is valid only for the calculation of a few
first eigenvalues Ωn such that |Ωn|/M2 ∼ 1 because I𝜈(z) grow exponentially
with increasing argument. This important point has not been taken into
account so far. 2. If |z| < 1, then I𝜈(z) ∼ (z/2)𝜈, therefore for “moderately”
supersonic velocities M2 > 2 the first few eigenvalues Ωn can be calculated
with the last two integral terms in equation (2.12) omitted and, also, with
Δp(x) taken in the form
Flutter of plates
Consider a plate occupying a domain S on the x, y plane, bounded by the
contour Γ (hereafter, Γ is supposed to be piecewise-smooth). One side of
the plate is subjected to gas flow with the velocity vector v = {vx, vy} = {v
cos θ , v sin θ } , v = |v|. If, in addition to the unperturbed state w0 ≡ 0, we
consider a perturbation w = w(x, y, t), then the aerodynamic pressure Δp
caused by interaction with the perturbed flow appears. It will be shown in
the following analysis that Δp is a linear operator of w, which will allow us to
present the solution in the form
w = φ(x, y) exp(ω t), Δp = Δp0(x, y) exp(ω t)
in all cases except the flutter problem for a viscoelastic plate. The equation
for vibrations of a constant-thickness plate takes the form
DΔ2 w + ρ h𝜕2w 𝜕t2 = Δp,
where D = Eh3/(12(1− 𝜈2)) is the cylindrical rigidity, E, 𝜈, and ρ are Young’s
modulus, Poisson’s ratio, and density of plate material, and h is its
thickness. From the above considerations, we have Δp0 = L1(φ) + L2(φ,
ω), and therefore,can be rewritten in the form DΔ2 φ + L1(φ) + ρhw2 φ +
L2(φ; ω)=0.On the contour Γ, the deflection amplitude φ(x, y) satisfies the
boundary conditions x, y ∈ Γ, M1(φ)=0, M2(φ)=0, where the boundary
operators M1 and M2 are problem-specific and will be given in each
particular case. We assume hereafter that the plate is not subjected to any
loads in its median plane. The system of equations represents a
complicated eigenvalue problem with a non-self-adjoint operator; its
eigenvalues are denoted by ω. By definition, we take that the perturbed
,motion of the plate is stable if Re ω < 0, and unstable if Re ω > 0; the
critical parameters of the system (plate, flow) are determined by the
condition Re ω = 0. In the further analysis, we consider the following main
questions: determination of Δp, formulation of new problems; development
of an efficient analytical approach, and identification of new mechanical
effects.
Determination of aerodynamic pressure
Numerous studies on the vibrations and stability of plates in supersonic
high-speed flows are carried out on the basis of the piston theory for the
aerodynamic pressure Δp caused by the interaction of the flow and
vibrating plate. This formulation has become so common that it was applied
even in the cases where its validity is questionable. Here, we derive Δp in
the cases of “moderate” supersonic (M ∼ 1.5–2) and low subsonic
velocities. Consider an elastic strip occupying domain S : {0 ≤ x ≤ l, y = 0,
|z|<∞}. On the side y ≥ 0, the strip is placed in a gas flow with unperturbed
parameters (planar problem) v = {u0, 0}, p0, ρ0, a0 = (𝛾p0/ρ0) 1/2, so that
the unperturbed flow potential is φ0 = u0x. Small vibrations of the strip w(x,
t) (with w/ℓ ≪ 1) cause flow perturbation; the perturbed flow potential is
denoted by φ1 = φ0 + φ. Then we proceed in the usual way: from the
Cauchy–Lagrange integral, equations of motion, mass conservation, and
equation of state we obtain an equation for φ1 and linearize it with respect
to the perturbation φ to obtain 1 a2 0 𝜕2φ 𝜕t2 + (M2 − 1) 𝜕2φ 𝜕x2 + 2 M a0
𝜕2φ 𝜕x𝜕t − 𝜕2φ 𝜕y2 = 0, where M = u0/a0. The potential φ must vanish at
infinity and satisfy the impermeability condition on the line
y = 0: y = 0, 0 ≤ x ≤ l, 𝜕φ 𝜕y = 𝜕w 𝜕t + u0 𝜕w 𝜕x
,y = 0, x ≤ 0, x ≥ l, 𝜕φ 𝜕y = 0. (2.3) The overpressure in the flow is obtained
from p = −ρ0 (𝜕φ 𝜕t + u0 𝜕φ 𝜕x ) .We search for the solution in the class of
functions φ(x, y, t) = f(x, y) exp(ω t), w(x, t) = W(x) exp(ω t), and p(x, y, t) =
q(x, y) exp(ω t). Introduce now the nondimensional coordinates x/l and y/l,
retaining hereafter the previous notation. Also, introduce the
nondimensional frequency lω/a0 = Ω.
The system of equations is transformed to,
(M2 − 1) 𝜕2f 𝜕x2 + 2MΩ𝜕φ 𝜕x + Ω2 f − 𝜕2f 𝜕y2 = 0 (2.5) y = 0, 0 ≤ x ≤ 1, 𝜕f
𝜕y = a0 (ΩW + M 𝜕W 𝜕x ) (2.6) y = 0, x ≤ 0, x ≥ 1, 𝜕φ 𝜕y = 0
q = −ρ0a0 l (Ωf + M 𝜕f 𝜕x ) .
In what follows, it is necessary to distinguish the cases of M < 1 and M > 1;
we consider them one by one. For M > 1, perturbations are absent to the
left of the point x = 0; therefore, it is possible to apply the Laplace transform
along the x coordinate; condition is not relevant, and the function W(x) can
be prolonged to x ≥ 1 arbitrarily (as long as applicability conditions for the
Laplace transform are satisfied), and this will not affect the overpressure
q(x, 0) acting on the strip. From we obtain for the Laplace transform ̃f(s, y)
In what follows, it is necessary to distinguish the cases of M < 1 and M > 1;
we consider them one by one. For M > 1, perturbations are absent to the
left of the point x = 0; therefore, it is possible to apply the Laplace transform
along the x coordinate; condition is not relevant, and the function W(x) can
be prolonged to x ≥ 1 arbitrarily (as long as applicability conditions for the
Laplace transform are satisfied), and this will not affect the overpressure
q(x, 0) acting on the strip. From we obtain for the Laplace transform ̃f(s, y)
β 2 ̃f − 𝜕2 ̃f 𝜕y2 = 0, β 2 = (M2 − 1) s 2 + 2MΩs + Ω2 . A solution bound at
infinity is ̃f = c1e −β y. (2.9) From the boundary condition for Laplace
transform 𝜕 ̃f 𝜕y y=0 = −β c1 = a0(Ω + Ms)W̃ it is possible to determine the
, parameter c1, and therefore it follows from (2.9) that ̃f = −a0 Ω + Ms β Wẽ
−β y. (2.10) The overpressure (in terms of Laplace transforms) is now
obtained from equation
q̃ (s, 0) = Δp̃ (s) = ρ0a2 0 l (Ω + Ms) 2 β W̃(s). (2.11)
The inverse Laplace transform is found from tables and convolution
theorems. We first write β = √M2 − 1√(s + s1)(s + s2) ≡ √M2 − 1β0; s1 =
Ω/(M−1), s2 = Ω/(M + 1); (s1 + s2)/2 = MΩ/ (M2 − 1) ≡ α1; (s1 − s2)/2=Ω/
(M2 − 1) ≡ α2. We now have L(−1) ( 1 β0 ) = I0(α2x)e −α1x ≡ H(x), where
I0(z) is the modified Bessel function; therefore L(−1) (W̃ β0 ) = x ∫ 0 H(x −
τ)W(τ) dτ L(−1) (sW̃ β0 ) = x ∫ 0 H(x − τ) 𝜕W 𝜕τ dτ L(−1) (s 2W̃ β0 ) = 𝜕 𝜕x x ∫
0 H(x − τ) 𝜕W 𝜕τ dτ.
We perform the necessary calculations and substitute the results in
equation (2.11) to obtain finally,
Δp(x) = ρ0a2 0M l(M2 − 1) 1/2 [ [ M2 − 2 M2 − 1 ΩW + M 𝜕W 𝜕x + (M2 + 2)
Ω2 2M (M2 − 1) 2 x ∫ 0 e −α1(x−τ) I0 (α2(x − τ)) W(τ) dτ − 2Ω2 (M2 − 1) 2 x
∫ 0 e −α1(x−τ) I1 (α2(x − τ)) W(τ) dτ + MΩ2 2 (M2 − 1) 2 x ∫ 0 e −α1(x−τ) I2
(α2(x − τ)) W(τ) dτ] ] ,
where I𝜈(z), 𝜈 = 1, 2, are the modified Bessel functions. There are
important implications of equation The formula of the piston theory is
obtained in the limit M ≫ 1, and it is valid only for the calculation of a few
first eigenvalues Ωn such that |Ωn|/M2 ∼ 1 because I𝜈(z) grow exponentially
with increasing argument. This important point has not been taken into
account so far. 2. If |z| < 1, then I𝜈(z) ∼ (z/2)𝜈, therefore for “moderately”
supersonic velocities M2 > 2 the first few eigenvalues Ωn can be calculated
with the last two integral terms in equation (2.12) omitted and, also, with
Δp(x) taken in the form
