Symmetry exploitation in the natural vibrations of rod systems Текст научной статьи по специальности «Математика»

DOI: 10.18698/0236-3941-2017-4-28-41

SYMMETRY EXPLOITATION IN THE NATURAL VIBRATIONS OF ROD SYSTEMS

A.M. Pavlov A.N. Temnov

[email protected] [email protected]

Bauman Moscow State Technical University, Moscow, Russian Federation

Abstract

The purpose of this work was to study spectral and Cau-chy problem for the mechanical system consisting of three rods, two of them being identical and connected with the third one by linear elastic elements. We stated the corresponding spectral problem and studied its spectrum. Findings of the research show that eigenfunctions of the considered spectral problem are classified according to the irreducible representations of the finite group of transformations despite the fact that the initial equations system admits continuous (Lie) transformation groups. We considered the weak solution of Cauchy problem and revealed its simplification in case of special "symmetrical" form of initial conditions and right-hand side of the corresponding operator equation system

Keywords

Rod system, symmetry group, irreducible representation, spectral problem, eigenfunction classification, projection operator

Received 26.12.2016 © BMSTU, 2017

Introduction. In the natural vibrations problem symmetry of mechanical system plays an important role. The representation theory of symmetry groups is an approach which allows recognizing and exploiting the influence of the system symmetry on the corresponding spectral problem. The main result of the representation theory with respect to spectral problems can be formulated in the following theorem [1].

Theorem. Let linear operator A commutes (permutable) with the representation operators of symmetry group G and has a discrete spectrum of eigenvalues with finite multiplicity, then its eigenfunctions are basis functions of group G irreducible representations.

As a result, with help of projection operators on subspaces of irreducible representations, spectral problem can be solved in these subspaces [2] which usually have lower dimension than initial space [3].

This approach has been successfully applied to mechanical systems with a finite number of degrees of freedom [4] (molecular vibrations [5], mass-spring models of mechanical systems [6, 7], finite element and finite difference models [8-10] etc.). In this case the symmetry group is a finite group, usually represented by spatial symmetry of the system, and the irreducible representations and their respective projectors can be easily found with the tables of characters of the finite groups irreducible representations [11].

If the mechanical system includes a continuous medium, it is much more difficult to determine the symmetry group only out of the spatial symmetry, since the displacements of the system are described by differential equations in partial derivatives, which generally have the continuous symmetry groups [12].

The complete set of differential equation system symmetry groups can be obtained by universal algorithms [13]. And it is also possible to write irreducible representations and corresponding projectors for continuous groups [14]. But the irreducible representations of continuous groups were found only for some particular cases [15]. For an arbitrary continuous group, finding them is quite a challenge. Also for the boundary value problems, this approach requires the invariance of the manifold defined by boundary conditions, which is not always possible to follow [16].

But there are special cases when obtained continuous symmetry groups illustrate linearity of the operator, corresponding to the original differential equations system, and in addition includes some finite group of permutations. Then, taking into account that eigenfunctions set is not invariant under linear transformations, eigenfunc-tions of this operator can be classified according by the irreducible representations of obtained permutations group or its subgroup, with respect to which the boundary conditions are invariant.

In this paper one of such cases was considered. Using an approach based on the theory of both finite and continuous groups we have obtained and then verified symmetry classification of eigenfunctions of the operator, generated by the natural vibrations problem of the rod system.

The problem statement. We will consider a mechanical system consisting of three linear elastic rods of length l, two of which have the same inertia and stiffness characteristics and are connected by linear-elastic connections with a third rod different from the first two, in cross-section with the coordinate x = l. We denote the stiffness and distributed mass of the rods as pj (x) and mj (x), j = 0,1,2 respectively for central and lateral rods. Functions pj (x)e C1 [0,l], mj (x)e C0 [0,l], j = 0,1,2 are bounded positive functions of coordinate x.

Displacements of the rods cross-sections uj (x, t), j = 0,1,2 along the axis X are defined by the equations

Pl(x) -

m(x)

¿4

l-L*1-

p0(x) -m0(x)

лГ-

p2(pc) -m2(x)

«1»

«0»

«2»

Fig. 1. Rod system of three rods

m0 (x)

d2u ( x, t ) d

dt2

дх

Po (x)

duo ( x, t )

dx

= qo (x, t);

(1)

mi {x )

d2ui { x, t ) d

dt2

dx

pi {x)

dui { x, t )

dx

= qi {x, t);

m2

. . d2u2 {x, t) d f , 4du2 {x, t) , ч

{x[p2 {x) dl l = q2 {x, t).

dt2

dx

(i)

Where qj ( x), j = 0,1,2 are bounded functions of coordinate x which determine the external loads.

Boundary conditions for cross sections with coordinates x = 0 and x = l are given by

u0(0, t) = uj(0, t) = u2(0, t) = 0;

pi (l)ui(l, t) + k(ui (l, t)-uo (l,t)) = 0;

p2 (l )u2 (l, t) + k (u2 ((, t)-U0 (l, t )) = 0;

p0 (l)u0 ((, t) + k (0 ((, t)- u ((, t)- u2 (l, t)) = 0.

To state evolutionary problem we will add to (1), (2) the initial conditions

uj (x,0) = y) (*);

(2)

duj

~st

{x,0) = v) {x), j = 0, i,2.

(3)

Operator statement. We assume Hilbert space

( 2

H =

X©L2j{[0, l], mj{x)) e{1}т,

J=0

vector functions U(t) = {uj (t)}T, where Uj (t)eC2,2 (L2j ([0, l], mj (x)), [0, <x>)). Vector F(t) = {{j (t)}T, where fj (t)eC([0,œ),L2j ([0, l],mj (x))) and f (x,t) =

= qj (t )

mj (x)

The inner product in H is defined by

2 l

{V,W )= X I mj {x )vj Wj dx.

j = 00

(4)

Let the matrix operator A = diag

i

V mj

{x) dx

d

pj{x

dx

\

has domain

D(A): Q.U(t) = 0, where Q is an operator of the boundary conditions (2).

From (i)-(3), we obtain the nonhomogenious Cauchy problem for the hyperbolic operator equation:

U (t) + AU (t) = F (t); (5)

U(0) = ^0, U(0) = ^1.

Spectral problem for the operator A. Let us assume F(t) = 0, U (t) = Uemt,

where U is the eigenfunction, ® is the natural frequency of the rod system. We obtain the spectral problem for the operator A

AU-XU = 0, (6)

where X = ra2.

Lemma 1. The operator A is unbounded, self-adjoint and uniformly positive in H operator. The squared norm in the energetic space of the operator A is given by

||u|IA = i Jpj (x1 dx + k I ( (l)-uj (l))2 = 2n > 0.

j=0 0 V ax J j=1

Proof. Uboundedness of operator A directly follows from that every matrix element of A is ubounded. Direct evaluation of (AU,V), (U,AV) and (AU,U) shows that A is self-adjoint and uniformly positive in H.

Similarly to work [17], we formulate a theorem on the spectrum of the operator A. Theorem 1. The operator A has a positive real discrete spectrum of eigenvalues

0<X1 <X2 <...<Xk <..., Xk

and the system of complete and orthogonal in spaces H and HA eigenfunctions {Uk }™= 1, with the following orthogonality conditions

2 l

(Uk, Us )= I Jmj (x)ukj usj dx = 5ks;

j = 00

([№>a vljpjwVMf-)^ (7)

2

+ k I ( (l)-ukj (l))(^0 (l)-usj (l)) = X5ks.

j =1

Proof. Positive and real properties of the spectrum follow from Lemma 1. Discreteness of spectrum follows from Sobolev's embedding theorem which is proved for this problem by comparison of norms in space HA and Sobolev space Wj.

Lie symmetries analysis of the spectral problem. We will consider that inertial-mass characteristics of the rods do not depend on the coordinate x

p1 (x ) = p2 (x ) = ps; p0(x) = p0; (8) m1 (x) = m2 (x) = ms; m0 (x) = m0.

Problem (6) corresponds to a system of differential equations

* + Xu0 (x) = 0;

m0 dx2

Ps d2U\ (x )

ms dx2

Ps d2u2(x)

(9)

ms dx2

Let the M c XxU is open subset of space XxU, where X :={x},

U :={wo, ui, U2 }T. We define the space X xU x U «xU (2) , which is a second prolonga-

T

d'u0 d'u d'u2

tion of space XxU. Here U(i) :=j-

, i = 1,2. In the space

dx' dx1 dx'

X x U x U(1) x U(2) the subset M(2) corresponds to a subset M.

Mapping A = {Ai, A 2, A 3j defined by left-hand side of system (9) translates M2) into three-dimensional Euclidean space: A : M(2) ^ M3. System (9) defines a subvarie-ty pA) = {l e M(2) : A (|) = 0 j consisting of its solutions.

Definition 1 [13]. A local group of transformations G acting on an open subset M c X x U, is the symmetry group of the system of differential equations (9), if its prolongation leaves the subvariety pA2) invariant, i.e. for V|aepA2) it follows that pr(2 )g | e pA2 ), where pr(2 )g is a second prolongation of the group G.

We will obtain the Lie algebra of infinitesimal generators v that meet the criteria of the infinitesimal invariance of the system (9) using known methods [13, 16]:

pr(2)yi (Av(^)) = 0, v = 1,2,3,

(10)

where p-ep^).

The Lie algebra of infinitesimal generators of the system (9) symmetry groups is determined by 12 vector fields

5 5 5 5 5 5

vi = —; v2 = Uo —; v3 = ui —; v4 = U2—; v5 = ui —; v6 = U2—;

5x u0 u1 u1 u2 u2

f

v 7 = Sin

Xm0 Po

Л

f

-; v8 = cos

f

v10 = cos

Xms

U0

— ; vn = sin

u1

'Xm0 Л 5

x >

P0 U0

IXms 5

■ x —

V Ps U2

f

v 9 = sin

\Xm,

_5 u1

(11)

Xmj,

U2

One-parameter groups G generated by the vector fields (11) v, act as follows:

Gi: (x + s,uo,ui,U2); G2 : (x,eeUo,ui,u); G3: (x,uo,esui,U2); (x,u0,u1 + su2,u2); G5: (x,u0,u1, esu2); G6: (x,u0,u1,u2 +su1);

G4 G7

f f x, uo +s sin

x, uo, u, +ssin

(

Gil

x, uo, U,, U2 +ssin

Xmo ^

--x , u,

po v

IXms Л

x

Vv ps v

IXms

U2

f ( x, uo +scos

v

(

Xmo

V

po

Л

Jio

/ ЛЛ

x, uo, u, +scos

Xms

Ui, U2

U2

V

■•x

V

x, uo, u,, u2 +scos

v

(Xm,

ЛЛ

V

(12)

The group Q corresponds to invariance of solutions under translations along the X axis. For the boundary value problems, this group is not a symmetry group. The groups G2,G3,G5 illustrate the linearity of the operator A. The groups G4 and G represents the ability of permutation of the functions ui (x ) and u2 ( x ) in the system (9). The invariance of the solutions on the summation with the fundamental solutions of equations (9) is shown by the groups G _Q2.

Symmetry classification of the operator A eigenfunctions. Lie symmetry analysis allows us to formulate following result.

Theorem 2. Eigenfunctions of the operator A are the basis functions of the irreducible representations of the group S2.

Proof. It was shown above that besides the linearity of operator A functions ui (x), u2 (x) are permutable. Set of these permutations can be described as a finite

group, isomorphic (by Cayley's theorem) to the group S2.

Since invariance of equality AU-XU = 0 with respect to permutations has been shown in symmetry analysis of the system (9), then the invariance of problem (6) requires the invariance of the set D ( A ) under group actions.

The group S2 consists of two elements: the identity element gI and permutation gi-2. In the space H, representation of the group T ( S2 ) consists of two matrix operators T ( gI ) and T ( gi-2 ), corresponding to elements of the group.

( i 0 0 ^ ( i 0 0 ^

T (gj ) =

o 1 o o o 1

T (gl-2 ) =

o o 1 o 1 o

(13)

Substitution of vector components T (gs )U and T (g1_2 )U in equations (2) shows that the set D (A) is invariant under the representation T (S2). Then for the problem (6) it follows that AT (S2) Uk _ A,kT (S2) Uk = 0, where Uk is an eigenelement of operator A corresponding to its eigenvalue

As a result operator A is commutative with the operators of the representation T (S2), and in this case, from the Wigner theorem [18] and Theorem 1 on the spectrum of the operator A, it follows that eigenfunctions of operator A are basis elements of the irreducible representations of the group S2.

The characters x(a' P) of the irreducible representations a and P of the group S2, and the characters x(x) of regular representation x of the group S2 in the space H are shown in Table 1.

Table 1

Characters of representations of the group S2

Representation gi g1-2

а 1 1

ß 1 -1

1 3 1

Projectors on the subspaces of irreducible representations in space H are given

by

P (a)= 2 (T (gf) + T (g1_2));

2 (14)

P (P)= 2 (T (g! )-T (g1_2 )).

For the eigenfunction Uk that belongs to the subspace of i-th irreducible representation, take place following equations [19].

p(i )Uk = Uk. (15)

It follows that the eigenfunctions of operator A that belong to the subspace of irreducible representation a have the coordinates ju0, us, us }T; the elements that belong to the subspace of irreducible representation P have coordinates {0, us, -us }T. It is also necessary to note that, by virtue of the theorem on the orthogonality of characters of irreducible representations [18], the spaces of irreducible representations are orthogonal. The subspaces of irreducible representations are found as follows:

H = H (a)0 H(P);

H (a) = p (a)H;

H (P) = P (P)H. Solution of the spectral problem. Let us assume that

rnfr =a, fr =b, fr =c

' P0 \Psm0 k k

Taking into account the boundary conditions (2) for coordinate x = 0, the solutions are given by

u0 ( x ) = Co cos ( A • x ) ;

u1 (x ) = C1cos (aA^ x ); (16)

u2 (x ) = C2 cos (aA^ x ).

For the subspace H(a), we have U0 (x ) = Qcos (Ax ); U1 (x ) = U2 (x ) =

= C1 cos ( aAx ). Due to calculate eigenvalues X we obtain characteristic matrix La) by substituting the functions (16) to the boundary conditions of (2) for the coordinate x = l :

( - cos (Л/) -caA sin (aAl) + cos (aAl

-bA sin (Al ) + 2cos (Al ) -2cos (aAl )

(17)

The eigenvalues X corresponding to the elements of this subspace are the roots of the determinant of the matrix L a):

det ( L a)) = bA sin (Л/ )( cos ( aAl ) - caA sin ( aAl )) + +2cos( Al )caA sin( aAl ) = 0.

Let us assume that C1 = 1. Eigenfunctions of the subspace H(a) are given by

Ы0 ( X) = - (caAsin(aA^)-C°S(aA1 )) cos(Ax); , Ч

cos (Al ) (19)

u1 (x ) = u2 (x ) = cos (aAx ).

Highlighting of the part of the coefficient for u0 ( x ) in (19) was made for convenience of further statement.

For the subspace we have u0 ( x ) = 0, u1 ( x) = -u2 ( x ) = Q cos ( aAx ). Similar to subspace hH we obtain for subspace HP) when C1 = 1:

L(P) =(-caA sin ( aAl ) + cos ( aAl )) ;

det ( L(p) ) = - ( саЛзт ( аЛ1 ) - cos ( аЛ1 ) ) = 0;

U0 (x) = 0; u1 (x) = -u2 (x) = cos (aAl).

(20)

Now let us will consider the solution of the spectral problem (6) in the initial space H. By substituting the functions (16) into the boundary conditions of (2) for the coordinate x = l, we write characteristic matrix for the calculation of the eigenvalues X:

' - cos (A/) -сяЛ sin (яЛ/) + cos (яЛ/) 0 ^

L = - cos (Л/) 0 -caA sin (яЛ/) + cos (яЛ/)

^-ЬЛ sin (Л/) + 2cos (Л/) - cos (яЛ/) - cos (яЛ/)

(21)

Eigenvalues X are the roots of the matrix L determinant:

det ( L ) = ( саЛзт ( аЛ1 )- cos ( аЛ1 ))( ЬЛ sin (Л/ )(cos (яЛ/ - caA sin ( яЛ/ +

+2cos(A/ )сяЛ sin^AZ )) = 0. (22)

Let us assume that C2 = 1, then the eigenfunctions (16) are written as follows:

( саЛшг( aAl)- cos ( аЛ )) £os (л, );

K ' cos (A/ ) V '

щ (x) =

(ЬЛ sin (A/)- cos (A/))(caЛsin (аЛ1 )- cos (аЛ1 ))- cos (Л/)(сяЛ sin (яЛ/)) (23)

cos(Л/)cos(яЛ/ )

x cos ( яЛх ) ; u2 (x) = cos (яЛх).

The expression (22) for the determinant of matrix L consists of two multipliers, the first of them is highlighted in bold and equal to det (l(p)), up to sign. If this multiplier is equal to zero, then found eigenvalues will coincide with the eigenvalues corresponding to the eigenvalues of the subspace H1^). In this case, in (23) the function u0 ( x) will be identically zero, and щ (x) will be equal to -u2 (x).

The second multiplier in (22) coincides with det (L(a))and the eigenfunctions

corresponding to zeroes of this multiplier will belong to The coefficient of cos (яЛх ) in (23) becomes equal to one.

In addition to the decomposition of the initial space to subspaces of smaller dimension, we can divide the whole spectrum œ(A) of the operator A to corresponding subspectra

a(A ) = aH(A )ua^( A ) (24)

defining equations of which (18), (20) have simpler form than initial one (22).

In summary the group-theoretic approach shown above has allowed us to simplify the solution of the spectral problem.

Numerical example. Let us assume that a = 1,3, b = 2,5, c = 0,7, l = 1. The diagrams det (L(a)) and det (L(p)) are shown at the figures 2a and 2b, respectively. Diagrams for det(L(a)), det(L(p)) and det (L) are shown at figure 2c.

We will give the values A corresponding to the first 4 eigenvalues of the operator A (see Table 2) and diagrams of eigenfunctions corresponding to them (Fig. 3a-d). Here eigenfunctions were normalized by their maximum value in [0, l] .

Fig. 2a. Diagram of det (L(a)} Fig. 2b. Diagram of det (L(p)}

с

Fig. 2c. Combined diagram of det(L(a)), det(L(p)), det (L)

Table 2

Eigenvalues of the operator A

No. Л Subspectrum

1 0,75 a(ß)(A)

2 0,95 аН( А)

3 2,62 а(а)( А)

4 2,71 ст(|3) (А)

0,4 0,6 d

Fig. 3. (a — Л = 0,75; б — Л = 0,95; с — Л = 2,62; d — Л = 2,71)

Summary of the results obtained above is written in the form of the following theorem.

Theorem 3. Spectrum of the operator A consists of two subspectra: ct( A ) = CT(")( A )UCT(P)( A ).

Eigenfunctions of the operator belong to the subspaces H W and H (P) respectively, so that H = H(a)© H(P), H (a)= P(a) H, H(P) = P(P) H where P(a), P(P) are orthogonal projectors onto subspaces of irreducible representations of the symmetric group S2.

Solution of the Cauchy problem. If the initial conditions and the right side of the Cauchy problem considered above, transforms under irreducible representations of group S2 (e. g. belongs to spaces of these representation) results of symmetry classification of the corresponding spectral problem eigenvalues are useful in following solution and analysis of the problem (5).

We will consider a weak solution of the problem (5). In the metric of space H = H © {1}, operator A eigenfunctions decomposition of the solution U (t ) is given by:

U (t )=£q>k (t )Uk, 9k (t ) = (U (t ), Uk )

k=0

where the coefficients 9k ( t ) are solutions of the Cauchy problems d 29k ( t )

(25)

dt2

-Xk9k (t) = fk (t), k = 0...<x>;

(26)

9k (0) = (0,Uk), cpk (0) = ( Uk ), fk (t) = (F(t),Uk ).

Here U0 = {1}, = 0 is the solution of the system (9), orthogonally complementing space H. It is easy to find that U0 transforms according to the irreducible representation a of the group S2.

Taking into account (25) and (26), weak solution of problem (5) is written in the following form [20]

u(t) = £f(°, Uk)cos((ktui^^lsin((kt) +

k=01 V^k

V

sin (( (t - 5 ))(F (t ), Uk )ds ^

Uk-

yfXk I

J J (F (s ), Uo ) dsds + ( Uo )t + (0, Uq )|Uq. (27)

,00

We will decompose F(t), ^ on the basis of the group S2 irreducible representations

F(t)= £ P(v)F(t)= £ F(v) (t);

v = a,p v = a,p

= £ p (v)^ ° = £ ^ °(v); (28) v = a,p v = a,p

^ = £ p= £ ^l(v). v = a, P v = a, P

Since the subspaces H(a) and H(p) are orthogonal, the scalar products where non-zero projections of vectors F(t), ¥°, ^1on one side and the eigenfunction Uk on the other side, belong to different subspaces, will be zeroed out. If in (28), projections of the vectors F(t), on any of the subspaces will be simultaneously equal to zero then in the weak solution of (27), we can neglect the subspectra and corresponding eigenfunctions that belong to subspaces where projections of vectors F(t), ^1are both equal to zero.

Conclusion. The group-theoretic approach considered in this work is useful in solution and analysis of the more complex spectral and evolutionary problems generated by vibrations of symmetrical rod and beams systems, presented, for example, in [21].

Symmetry classification of eigenfunctions of the boundary value problem operator allows to simplify eigenvalues calculation process via solving the problem in subspaces of lower dimension than the initial space. Hence, the spectrum is divided into subspectra corresponding to the elements of these subspaces. As a benefit asymptotic behavior for each of the subspectra can be obtained.

In the weak solution of the evolutionary problem, orthogonality of the obtained subspaces admits not to use eigenfunctions that belong to those subspaces where

projections of the initial conditions and the right-hand side of the equation are equal to zero at the same time. In case of limitations on the number of eigenfunctions used, this approach increases convergence of the solution to the exact one.

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DOI: 10.7463/1014.0728802 Available at: http://technomag.edu.ru/jour/article/view/702

Pavlov A.M. — post-graduate student of Spacecraft and Launch Vehicles Department, Bau-man Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).

Temnov A.N. — Cand. Sc. (Phys.-Math.), Assoc. Professor of Spacecraft and Launch Vehicles Department, Bauman Moscow State Technical University (2-ya Baumanskaya ul. 5, str. 1, Moscow, 105005 Russian Federation).

Please cite this article in English as:

Pavlov A.M., Temnov A.N. Symmetry Exploitation in the Natural Vibrations of Rod Systems. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Mashinostr. [Herald of the Bauman Moscow State Tech. Univ., Mech. Eng.], 2017, no. 4, pp. 28-41. DOI: 10.18698/0236-3941-2017-4-28-41