Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction
A spectral problem describing small transversal vibrations of an elastic rod with a concentrated mass (bead) at the right end subject to viscous friction is considered. The left end is hinge-jointed. The location of the spectrum of this problem is described and the asymptotic formula of the eigenval...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2009
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| Цитувати: | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction / I.V. Gorokhova // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 375-385. — Бібліогр.: 9 назв. — англ. |
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| citation_txt | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction / I.V. Gorokhova // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 375-385. — Бібліогр.: 9 назв. — англ. |
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| description | A spectral problem describing small transversal vibrations of an elastic rod with a concentrated mass (bead) at the right end subject to viscous friction is considered. The left end is hinge-jointed. The location of the spectrum of this problem is described and the asymptotic formula of the eigenvalues is obtained.
|
| first_indexed | 2025-12-07T15:40:53Z |
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| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2009, vol. 5, No. 4, pp. 375–385
Small Transversal Vibrations of Elastic Rod with Point
Mass at One End Subject to Viscous Friction
I.V. Gorokhova
K.D. Ushinskii South-Ukrainian State Pedagogical University
26 Staroportofrankovskaya Str., Odessa, 65020, Ukraine
E-mail:I.Gorochova@rambler.ru
Received December 1, 2008
A spectral problem describing small transversal vibrations of an elastic
rod with a concentrated mass (bead) at the right end subject to viscous
friction is considered. The left end is hinge-jointed. The location of the
spectrum of this problem is described and the asymptotic formula of the
eigenvalues is obtained.
Key words: eigenvalues, operator pencil, algebraic multiplicity, boundary
conditions.
Mathematics Subject Classification 2000: 47A56, 74K10.
1. Introduction
Small transversal vibrations of a homogeneous rod of density ρ = 1, stretched
by a distributed longitudinal force proportional to g(x) > 0, g ∈ C1[0, l] subject
to homogeneous viscous friction of the coefficient k > 0, can be described by the
equation
∂4u
∂x4
+
∂2u
∂t2
+ k
∂u
∂t
− ∂
∂x
(
g(x)
∂u
∂x
)
= 0. (1)
Here x is the longitudinal coordinate measured from the left end of the rod, t is
the time, u(x, t) is the transversal displacement of a point lying at the distance x
from the left end of the rod at the time t. The left end of the rod is hinge-jointed
without damping while at the right end there is a massive ring with mass m > 0
that is able to move in the direction orthogonal to the equilibrium position of the
rod. The hinge connection of the left end is described by the boundary conditions
u(0, t) = 0, (2)
c© I.V. Gorokhova, 2009
I.V. Gorokhova
∂2u
∂x2
∣∣∣∣
x=0
= 0. (3)
The boundary conditions at the right end are
∂2u
∂x2
∣∣∣∣
x=l
= 0, (4)
− ∂3u
∂x3
∣∣∣∣
x=l
+ m
∂2u
∂t2
∣∣∣∣
x=l
+ g(x)
∂u
∂x
∣∣∣∣
x=l
+ β
∂u
∂t
∣∣∣∣
x=l
= 0, (5)
where l > 0 is the length of the rod, β > 0 is the coefficient of viscous friction
(damping) of the ring. Condition (4) shows that the rod is hinge-connected to
the ring while condition (5) means that the ring can move with damping in the
direction orthogonal to the equilibrium position of the rod. Various boundary
conditions for undamped rods were considered in [1]. Condition (5) is physically
motivated for the case of damping (see, e.g., [1–4]. It should be mentioned that
the results similar to the ones obtained in this paper can be found in [5] for
another type of boundary conditions. Excluding the time by usual transformation
u(x, t) = eiλty(λ, x), we arrive at the following spectral problem:
y(4) − λ2y − (gy′)′ + ikλy = 0, (6)
y(λ, 0) = 0, (7)
y(2)(λ, 0) = 0, (8)
y(2)(λ, l) = 0, (9)
−y(3)(l)−mλ2y(l) + g(l)y′(l) + iλβy(l) = 0. (10)
2. Operator Theory Approach
To consider the operator-theoretical approach to problem (6)–(10) let
us introduce the operators acting in the Hilbert space L2(0, l)⊕ C according to
D(A) =
{(
y(x)
c
)
:
y(x) ∈ W 4
2 (0, l),
c = y(l), y(0) = y(2)(0) = 0, y(2)(l) = 0
}
,
A
(
y(x)
c
)
=
(
y(4)
−y(3)(l)
)
, G
(
y
c
)
=
( −(gy′)′
g(l)y′(l)
)
,
M =
(
I 0
0 m
)
, K =
(
k 0
0 β
)
376 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Small Transversal Vibrations of Elastic Rod with Point Mass at One End
and introduce the quadratic operator pencil
L(λ) = (A− λ2M + G + iλK)
with the domain D(L) = D(A) independent of the spectral parameter λ by defi-
nition. The spectrum of L(λ) can be identified with the spectrum of problem
(6)–(10) because the components of the vectorial equation L(λ)Y = 0 are nothing
but (6) and (10) while conditions (7)–(8) can be found in the description of D(A).
The coefficients in the equations of problem (6)–(10) are entire functions of λ.
Therefore (see [6, p. 27]), the spectrum of problem (6)–(10), i.e., the spectrum of
the pencil L, consists of normal eigenvalues accumulating at infinity. We use the
following definitions.
Definition 1. The set of values of λ such that the inverse operator L(λ)−1
exists as a bounded closed operator is said to be the resolvent set of L(λ) and the
compliment is said to be the spectrum of L(λ).
Definition 2. A number λ0 ∈ C is said to be an eigenvalue ([7]) of the
operator pencil L(λ) if there exists a vector y0 ∈ D(A) called eigenvector such
that y0 6= 0 and L(λ0)y0 = 0. The vectors y1, . . . , yp−1 are said to be the chain of
associated vectors for y0 if
k∑
s=0
1
s!
dsL(λ)
dλs
∣∣∣
λ=λ0
yk−s = 0, k = 1, p− 1.
The number p is said to be the length of the chain composed by the eigen-
and associated vectors. The geometric multiplicity of an eigenvalue is defined to
be the number of the corresponding linearly independent eigenvectors. The al-
gebraic multiplicity of an eigenvalue is defined to be the greatest value of the
sum of the lengths of chains corresponding to linearly independent eigenvectors.
An eigenvalue is said to be isolated if it has some deleted neighborhood contained
in the resolvent set. An isolated eigenvalue λ0 of finite algebraic multiplicity is
said to be normal if the image ImL(λ0) is closed.
Lemma 1. The operator A is selfadjoint and nonnegative.
P r o o f. Let Y =
(
y(x)
y(l)
)
∈ D(A) and Z =
(
z(x)
z(l)
)
, where z(x) ∈
W 4
2 (0, l), then taking into account y(0) = 0, y(2)(0) = 0, we integrate by parts
twice and obtain
(AY, Z) =
l∫
0
y(4)(x)z(x) dx− y(3)(l)z(l) (11)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 377
I.V. Gorokhova
= −y(3)(0)z(0) + y′(l)z(2)(l)− y′(0)z(2)(0)− y(l)z(3)(l) + y(0)z(3)(0) +
l∫
0
yz(4) dx.
It is clear that if we set
z(0) = z(2)(0) = z(2)(l) = 0, (12)
then we arrive at (AY, Z) = (Y,AZ) and D(A∗) = D(A). Let us show that A
is nonnegative. To do it we consider the inner product (AY, Y ). Setting Z = Y
into (11) and taking into account (12), we obtain
(AY, Y ) =
l∫
0
|y(2)|2 dx > 0.
Lemma 2. The operator G is symmetric and nonnegative.
P r o o f. Let Y ∈ D(A) = D(G), Z ∈ D(A) = D(G). Consider the inner
product
(GY, Z) = −
l∫
0
(g(x)y′)′z(x) dx + g(l)z(l)y′(l)
=
l∫
0
gy′z′ dx = g(l)z′(l)y(l)−
l∫
0
(gz′)′y dx = (Y, GZ). (13)
Thus G is symmetric. Let us show that it is nonnegative on the domain D(G) =
D(A). To do it we consider the inner product (GY, Y ). With account of definition
of G equation (13) implies
(GY, Y ) =
l∫
0
g(x)|y′|2 dx > 0.
Lemma 3. The spectrum of L(λ) is located in the closed upper half-plane.
This result is known (see [8]) for bounded operator pencils, however it remains
true for pencils of unbounded operators (see, e.g., [9]).
To obtain more detailed information about the spectrum of the pencil L(λ)
it is convenient to transform the spectral parameter
τ = λ− ik
2
. (14)
378 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Small Transversal Vibrations of Elastic Rod with Point Mass at One End
Then equation (6) takes the form
τ2y − y(4) +
k2
4
y + (gy′)′ = 0, (15)
while boundary condition (10) is
mτ2y(l) + y(3)(l) + (
kβ
2
− mk2
4
)y(l)− iτ(β −mk)y(l)− g(l)y′(l) = 0. (16)
Since problem (7)–(9), (15), (16) is deduced from problem (6)–(10) by spectral
parameter transformation (14), the spectrum of problem (7)–(9), (15), (16) also
consists of normal eigenvalues accumulating at infinity. We introduce new ope-
rators:
 = A + G +
(
−k2
4 I 0
0 (−mk2
4 + βk
2 )
)
, D(Â) = D(A),
M̂ = M, M̂ À 0, K̂ =
(
0 0
0 (β −mk)
)
and consider the following quadratic operator pencil with the domain D(L̂(τ)) =
D(L(λ)) = D(A):
L̂(τ) = τ2M̂ − iτK̂ − Â.
It is obvious that
I) if mk < β, then K̂ > 0;
II) if mk = β, then K̂ ≡ 0.
Theorem. I. Let mk < β, then:
1) eigenvalues of the pencil L̂ lie in the closed upper half-plane and on the
interval of the imaginary axis [−ik/2, 0);
2) the interval [−ik/2, 0) contains only finite number of eigenvalues which are
semisimple, i.e., the corresponding eigenvectors do not possess associated
vectors;
3) the geometric multiplicity of each eigenvalue of the pencil L̂ does
not exceed 2;
4) the set of eigenvalues {τk} can be arranged as the union of two subsequences
{τk} = {τ (1)
k } ∪ {τ (2)
k }
such that
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 379
I.V. Gorokhova
4a) the set {τ (2)
k } lies on the imaginary and on the real axes symmetrically with
respect to these axes, i.e., under proper enumeration τ
(2)
−k = −τ
(2)
k ;
4b) if 0 ∈ {τ (2)
k }, then 0 is a double element of the set {τ (2)
k } while all other
τ
(2)
k are simple;
4c) the elements of the set {τ (1)
k } lie in the open upper half-plane and on the
interval [−ik/2, 0] of the imaginary axis; the elements in the open upper
half-plane are located symmetrically with respect to the imaginary axis, i.e.,
τ
(1)
−k = −τk
(1) for each not pure imaginary τ
(1)
k , and symmetrically located
eigenvalues possess equal multiplicities. Denote the elements of {τ (1)
k },
which are located in the open lower half-plane (if any) by −iθ1, −iθ2, . . . ,
−iθκ, where
θ1 > θ2 · · · > θκ,
then
4d) iθj /∈ {τ (1)
k }, (j = 1, 2, . . . ,κ);
4e) the number of elements of the set {τ (1)
k } with account of multiplicities in
each of the intervals (iθj+1, iθj), j = 1, 2 . . .κ − 1 is odd;
4f) the number of elements of {τ (1)
k } in the interval (0, iθκ) is even, if 0 /∈ {τ (1)
k }
and is odd, if 0 ∈ {τ (1)
k }.
II. If mk = β, then the spectrum of the pencil L̂ lies on the real axis and on
the interval [− ik
2 , ik
2 ]. The spectrum of L̂ is symmetric with respect to the real
axis and to the imaginary axis. The number of pure imaginary eigenvalues is
finite. If 0 belongs to the spectrum, then its algebraic multiplicity is 2 or 4 and
its geometric multiplicity is 1 or 2, respectively. The algebraic multiplicity of any
nonzero eigenvalue coincides with its geometric multiplicity and does not exceed 2.
P r o o f.
I. As it was mentioned above, for mk < β the inequality K̂ > 0 is valid.
Therefore, it is possible to apply the results of [9] to the pencil L̂ and to obtain
1) the eigenvalues of L̂ lie in the closed upper half-plane and on the imaginary
axis (Lemma 2.2 in [9]). Using Lemma 1, we conclude that the eigenvalues
in the open lower half-plane are located only on the interval [−ik/2, 0];
2) applying Lemma 2.3 (Part 1) from [9] to the pencil L̂ we obtain that
the eigenvalues located on the intervals [−ik/2, 0), (∞, 0) and (0,∞) are
semisimple. The operator A > 0 implies that  is bounded from below and
380 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Small Transversal Vibrations of Elastic Rod with Point Mass at One End
then it follows by Theorem 2.3 from [9] that the number of eigenvalues on
the interval [−ik/2, 0) is finite;
3) the canonical fundamental system of solutions yk(λ, x), (k = 1, 2, 3, 4) of
equation (6) ([6, p. 25]) is defined by the boundary conditions: y
(n−1)
k (λ, 0) =
δkn, k, n = 1, 2, 3, 4, where δkn is the Kronecker symbol. Two of these solu-
tions, y2(λ, x) and y4(λ, x), satisfy conditions (7) and (8). Consequently, the
geometric multiplicity of each eigenvalue of the pencil L̂ does not exceed 2;
4) it is evident that M̂ +K̂ is a strictly positive operator for mk < β, therefore
the operators involved in L̂ satisfy the conditions of Corollary 3.1 and of
Theorem 3.1 in [9]. Thus statements 4a) and 4f) follow from Corollary 3.1,
while statements 4b)–4e) from Theorem 3.1.
R e m a r k.
In the case of mk > β the problem can be reduced to the case of mk < β by
transformation τ → −τ .
Let us consider again the pencil L(λ).
Corollary.
I. If mk < β, then:
1) the eigenvalues of the pencil L, i.e., the eigenvalues of problem (6)–(10) lie
in the closed half-plane Im λ ≥ k/2 and on the interval [0, ik/2);
2) the interval [0, ik/2) contains only finite number of eigenvalues which are
semisimple;
3) the geometric multiplicity of each eigenvalue of L does not exceed 2;
4) the set of eigenvalues {λk} can be arranged as a union of two subsets
{λk} = {λ(1)
k } ∪ {λ(2)
k },
such that:
4a) the set {λ(2)
k } lies on the imaginary axis and on the line Imλ = k/2 and
is located symmetrically with respect to the imaginary axis and to the line
Imλ = k/2;
4b) if ik/2 ∈ {λ(2)
k }, then ik/2 is a double element of the set {λ(2)
k }; all other
λ
(2)
k are simple;
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 381
I.V. Gorokhova
4c) the elements of the set {λ(1)
k } lie in the open half-plane Imλ > k/2 and on
the interval [0, ik/2]; the elements which lie in the half-plane Imλ > k/2 are
located symmetrically with respect to the imaginary axis, i.e., λ
(1)
−k = −λk
(1)
for each not pure imaginary λ
(1)
k and symmetrically located eigenvalues are
of the same multiplicity. Denote by iγ1, iγ2, . . . , iγκ the elements of the set
{λ(1)
k }, which lie on the interval [0, ik/2), indexed such that
γ1 > γ2 · · · > γκ,
then
4d) i(k − γj) /∈ {λ(1)
k }, j = 1, 2 . . .κ;
4e) the number of elements of the set {λ(1)
k } located on each of the intervals
(i(k−γj), i(k−γj+1)), j = 1, 2 . . .κ− 1, counted with multiplicities is odd;
4f) the interval (ik/2, i(k − γ1)) contains even number of elements of {λ(1)
k }, if
ik/2 /∈ {λ(1)
k }, and odd number if ik/2 ∈ {λ(1)
k }.
II. If mk = β, then the spectrum of L lies on the line Imλ = k/2 and on
the interval [0, ik]. In this case the spectrum of L is symmetric with respect to
the line Imλ = k/2 and to the imaginary axis. The number of pure imaginary
eigenvalues is finite. If 0 belongs to the spectrum, then its algebraic multiplicity
is even and does not exceed 4.
III. If mk > β, then
1) The eigenvalues of the pencil L, i.e., the eigenvalues of problem (6)–(10)
lie in the strip 0 < Imλ ≤ k/2 and on the interval on the imaginary axis
[ik/2, ik];
2) the interval (ik/2, ik] contains only finite number of eigenvalues which are
semisimple;
3) the geometric multiplicity of each of the eigenvalues of the pencil L does
not exceed 2;
4) the spectrum {λk} can be given as a union of two subsequences
{λk} = {λ(1)
k } ∪ {λ(2)
k }
such that
4a) the set {λ(2)
k } is located on the imaginary axis and on the horizontal line
Imλ = k/2 symmetrically with respect to the imaginary axis and to the
line Imλ = k/2;
382 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4
Small Transversal Vibrations of Elastic Rod with Point Mass at One End
4b) if ik/2 ∈ {λ(2)
k }, then ik/2 is a double element of the set {λ(2)
k }; all other
λ
(2)
k are simple elements of the set;
4c) all elements of {λ(1)
k } lie in the strip 0 < Imλ < k/2 and on the in-
terval [ik/2, ik]; the elements of this subsequence which lie in the strip
0 < Imλ < k/2 are located symmetrically with respect to the imaginary
axis, i.e., λ
(1)
−k = −λk
(1) for each not pure imaginary λ
(1)
k and the multiplici-
ties of symmetrically located elements coincide. Let us denote the elements
of {λ(1)
k } which are located in the interval (ik/2, ik] by iζ1, iζ2, . . . , iζκ,
where
ζ1 > ζ2 · · · > ζκ,
then
4d) i(k − ζj) /∈ {λ(1)
k }, (j = 1, 2 . . .κ);
4e) the number of elements of {λ(1)
k } with account of multiplicities is odd in
each of the intervals (i(k − ζj), i(k − ζj+1)), (j = 1, 2 . . .κ − 1);
4f) the number of elements of {λ(1)
k } in the interval (i(k − ζκ), ik/2) is even if
ik/2 /∈ {λ(1)
k } and is odd if ik/2 ∈ {λ(1)
k }.
3. Eigenvalue Asymptotics
In this section we assume g = const > 0. Direct calculation gives:
y2(λ, x) = − z2
2sh(z1x)
z1(z2
1 − z2
2)
+
z2
1sh(z2x)
z2(z2
1 − z2
2)
, (17)
y4(λ, x) =
sh(z1x)
z1(z2
1 − z2
2)
− sh(z2x)
z2(z2
1 − z2
2)
, (18)
where
z1 =
√
g +
√
g2 − 4(−λ2 + ikλ)
2
,
z2 =
√
g −
√
g2 − 4(−λ2 + ikλ)
2
.
The solution of (6), which satisfies conditions (7)–(9), is
y = y
(2)
4 (λ, l) · y2(λ, x)− y
(2)
2 (λ, l) · y4(λ, x). (19)
Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 383
I.V. Gorokhova
Substituting (19) into boundary condition (10), we obtain
y
(2)
4 (λ, l) · (−y
(3)
2 (λ, l)−mλ2y2(λ, l) + gy
(1)
2 (λ, l) + iλβy2(λ, l))
−y
(2)
2 (λ, l) · (−y
(3)
4 (λ, l)−mλ2y4(λ, l) + gy
(1)
4 (λ, l) + iλβy4(λ, l)) = 0,
what with account of (17) and (18) gives
−z3
2z
2
1sh(z1l)ch(z2l)− z2
1(mλ2 − iλβ) · sh(z1l)sh(z2l)
+g(l)z2
1z2sh(z1l)ch(z2l) + z3
1z
2
2sh(z2l)ch(z1l)
+z2
2(mλ2 − iλβ)sh(z1l)sh(z2l)− g(l)z1z
2
2sh(z2l)ch(z1l) = 0. (20)
To find asymptotics of the roots of equation (20) we substitute
√
λ =
πn
l
+
A
n
+
B
n2
+
C
n3
+ O(
1
n4
). (21)
Then the coefficients before the powers of 1/n must vanish, what implies
λn =
π2n2
l2
+
ik
2
+
g
2
+
1
lm
− 1
2πnm2
− 1
4π2m2n2
+
lg
2π2mn2
− l2g2
8π2n2
− l2k2
8π2n2
+
l
6π2m3n2
+
il(β −mk)
π2m2n2
+ O
(
1
n3
)
.
This shows how to find the parameters l, k, g, m, β using the spectrum of problem
(6)–(10), i.e to solve the inverse problem for the case of g = const.
The author is grateful to V. Pivovarchik for setting the problem.
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[9] V.N. Pivovarchik, On Spectra of a Certain Class of Quadratic Operator Pencils
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Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 4 385
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| id | nasplib_isofts_kiev_ua-123456789-106549 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1812-9471 |
| language | English |
| last_indexed | 2025-12-07T15:40:53Z |
| publishDate | 2009 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Gorokhova, I.V. 2016-09-30T08:23:40Z 2016-09-30T08:23:40Z 2009 Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction / I.V. Gorokhova // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 4. — С. 375-385. — Бібліогр.: 9 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106549 A spectral problem describing small transversal vibrations of an elastic rod with a concentrated mass (bead) at the right end subject to viscous friction is considered. The left end is hinge-jointed. The location of the spectrum of this problem is described and the asymptotic formula of the eigenvalues is obtained. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction Article published earlier |
| spellingShingle | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction Gorokhova, I.V. |
| title | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction |
| title_full | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction |
| title_fullStr | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction |
| title_full_unstemmed | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction |
| title_short | Small Transversal Vibrations of Elastic Rod with Point Mass at One End Subject to Viscous Friction |
| title_sort | small transversal vibrations of elastic rod with point mass at one end subject to viscous friction |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/106549 |
| work_keys_str_mv | AT gorokhovaiv smalltransversalvibrationsofelasticrodwithpointmassatoneendsubjecttoviscousfriction |