Lowest Natural Frequencies of Structures
With Rigid-Body Degrees of Freedom
By Ir. S. Hyiaridesi
To calculate the lowest natural frequencies, use can be made o f the power method, for which the flexibility matrix or the inverted stiffness matrix has to be available. Struc-tures with rigid-body degrees of freedom, however, have a stiffness matrix which is not invertible. In this paper three methods are described to derive from the singular stiffness matrix a regular one. These methods are based on the some principles, but differ in particulars. One method is recommended, since it involves less numerical o p -erations, while its accuracy is the same.
Nomenclature K = stiffness matrix M = mass matrix
n = mmiber of degrees of freedom
V = mimber of rigid-body degrees of freedom qy j ' - t h eigenvector
Q = matrix whose columns are formed by eigenvector q R matrix representing momentum relations
T = kinetic energy
Mi generalized displacement or coordinate of node i U = displacement or coordinate vector
y = strain energy = y-th eigenvalue
= circular frequencj^ of vibration Introduction
I N " M A T R I X structural analysis, f r u i t f u l use is made of the displacement method, which leads to the stiffness matrix of the structure. Vibration problems are solved by introducing inertia forces and assuming harmonic vibration. Similar equations result by applying La-grange's equations.
Several methods are available to calculate the natural frequencies f r o m the derived equations. The iterative procedure, called the power method,''''''' is very useful. Applied to the stiffness matrix, this iterative method first yields the highest natural frequency. The lower frequencies can only be calculated by using the sweeping procedure. For calculation of the lowest natural fre-quencies, of general interest only, the flexibility matrix
1 Netherlands Ship Model Basin, Haagsteeg 2, Wageningen,
The Netherlands.
2 C. B. Bienzeno and R. Crammel, Technische Dynamik, Band 1,
Springer-Verlag, Berlm, 1953.
Koch, "Enige Toepassingen Van de Leer der Eigenfuncties op Vraagstukken u i t de Toegepaste Mechanica," Thesis, Delft, 1929.
* S. Timoshenko and J. N . Goodier, Theory of Elasticity, Mc-Graw-Hill Book Co., Inc., New York, N . Y . , 1951.
Manuscript received at S N A M E Headquarters, August 21, 1967.
i n the iterative methods has to be used. This matrix can be obtained either directly by the flexibility method or indirectly by inverting the stiffness matrix.
Structures w i t h one or more ligid-body degrees of freedom—for example, a railroad carriage—have a stiffness matrix which is singular, whicli means t h a t its i n -verse does not exist. Physically this means that the t o t a l motion of vibration may be considered as superposition of rigid-body motion and the motion of deformation. I n the rigid-body motion the strain energy is zero. Considering the principle of Rayleigh, i t w i l l be clear t h a t the square of the natural frequency is proportional to the strain energy; so the rigid-body motion may be consid-ered as a vibration w i t h frequency zero.
The idea now is to sweep the rigid-body motions out of the t o t a l motion. Then the structure behaves as if i t has been constrained i n its nodes and, hence, the stiff-ness matrix of this system is not singular anymore, for the displacement configuration is now uniquely defined. The three methods described herein concern this sweep-ing procedure.
I n the calculation of the natural frequencies of ship's hulls, successful use can be made of this method. I n the calculation of the vertical vibrations, the ship can be considered as a free-free beam w i t h two rigid-body de-grees of freedom, a translation and a rotation i n its vertical plane. I n the calculation of the coupled horizontal and torsional vibrations, three rigid-body degrees must be taken into account.
Elimination of Cyclic Coordinates
When rigid-body motions are possible, we can dis-tinguish two kinds of coordinates:
(a) The coordinate vector t h a t describes tlie
rigidbody motion, composed by the socalled cychc or i g -norable coordinates.
(6) The coordinate vector that describes the defor-mation of the structure.
lead to a uniquely defined problem whose stifl'ness matrix w i l l be regular. I n general, i t is impossible to indicate these cyclic coordinates directly. Therefore, two meth-ods are described i n the following subsections by which i t is possible to eliminate these cychc coordinates i n -directly. The first method has been based upon the orthogonality of all the eigenvectors. The second method involves impulse-momentum considerations.
I n vibration problems, generally i t is a custom to take the components of the displacement of a point, measured f r o m its equilibrium position, as the coordinates. W i t h these coordinates, the deformed state of the structure is described. This is done also i n this paper and, hence, the displacement vector u represents also the coordinate vector of the structure.
Orthogonality Considerations
Consider a structure w i t h p cyclic coordinates. There are then p eigenvectors q i . . . qp corresponding to the p eigenvalues zero. Then obviously
K - Q i = 0 (1) where K is the stiffness matrix and Qi is the matrix of
which the columns are composed by the eigenvectors qi to q„.
I f the structure has n degrees of freedom {n > p), then {11 — p) vectors q ^ + i * . . . q„* can be determined which are M-orthogonal to the first p eigenvectors. Thus
Q i ^ M ' Q j * = 0
where M is the mass matrix of the structure and Q 2 * is a matrix given by the column vectors q^+i* to q,,*, which have t o be mutually independent. I n general, these {n — p) vectors q^+i* to q„* are not tbe required eigen-vectors, but linear functions of them. Therefore they are distinguished by an asterisk. The original coordi-nates u are then replaced by a new set:
such t h a t
[Ql Q2 I f u i * ' (3) As a consequence of the definition of Q i and Q 2 * , the vector U l * (of order p X I) represents the cyclic co-ordinates and U 2 * (of order {n — p) X I) the remaining coordinates.
Substitution of equation (3) i n the expression of the strain energy gives us
. Q a * ^ . K [ Q i Q2 .U2^ = [ i u i * ^ U 2 * ^ : • Q r ^ K . Q i Q / . K Q 2 * . Q 2 * ^ ' K - Q i Q 2 * ' ' - K - Q 2 * U i ^
Considering equation (1), taking into account the sym-metry of K , the equation of the strain energy reduces to
V 4u2*^-Q * ^ - K - Q 2 = ^ U 2 * = i U 2 * ^ - K * . U 2 * (4)
The noncyclic coordinates are represented by U 2 * and, hence, the reduced stiffness matrix 7^* is invertible.
Similarly we obtain for the kinetic energy:
T = i i i i * ^ - M - d i * + i t i 2 * ^ . M * - u , * (5)
where
M = Q i ' - M Q i
M * = Q 2 * ' ' M - Q 2 *
Use has been made of the fact t h a t M is a diagonal matrix. ^ Indeed the strain energy is now determined only by the vibratory deformation U 2 * , whereas the kinetic energy is the sum of the kinetic energies of the rigid-body motion U l * and of the periodical motion U a * separately. Thus the Lagrange equations f o r the latter motion lead to
M * - Ü 2 * + K * - U 2 * = 0 and the eigenvalue problem beconres
Xii = 0 (6)
where M * and K * are symmetrical and of rank and of order (n - p), and where X = l/co^. The reduced stiff-ness matrix K * has to be invertible, because i t refers only to the deformation mode U2*.
In the Appendix a simple mechanical system is given f r o m which the natural frequencies are derived using the method described.
(2) Impulse-Momentun) Considerations
I n free vibration the linear and angular momenta of a completely unconstrained structure are constant.= As we are only interested i n the periodical motions, we may put the momenta equal to zero, which means t h a t the rigid-body motions are stopped. This also holds for a partially constrained structure; hence we have to satisfy as many extra equations as the number p of the rigid-body motions.
I n matrix f o r m the equations of linear and angular momentum can be w r i t t e n
R-u (7)
where R is of order p X n. This equation is partitioned such t h a t p of the displacements u are expressed i n terms of the remaining (n — p) displacements, which are compiled i n the column matrix U2. So
[ R 1 R 2 U l
U2 = 0
where U i is of the order p X I. From this equation follows
U l = - R i - i - R 2 - U 2 (8)
The original coordinates u are then related to this reduced column matrix U2 by
5 W. C. H u r t y and M . F. Rubinstein, Dynamics of Structures,
u = - R i - i - R 2 •Ua = R * - U 2 (9) This means a transformation f r o m coordinates u to a reduced set of coordinates U2. Consequently the mass and stiffness matrices i n the new coordinates are
M * = R * ' ' - M . R * a n d K * = R * ' ' - K - R * (10) where M and K are the mass and stiffness matrices i n the original coordinates and, hence, of order n X n, whereas the reduced matrices are of order (n — p) X {n — p) and still symmetric. I n the reduced coordinates, the natural vibrations are given bj^
M * - Ü 2 + K * ' U 2 = 0
A t the same time, equation (7) is satisfied, which means t h a t the motions as a rigid-bodj^ have been eliminated. Thus we can consider the structure to be constrained i n its nodes. Obviously these constraints exert no forces on the structure, and due to them the stiffness matrix K * w i l l be regular. The eigenvalue problem then may be w r i t t e n
| K * - i . M * - X-II 0 (11) where \ = 1/co^.
I n the Appendix, this method too is illustrated i n a simple sj'stem.
Replacement of Dependent R o w s in Stiffness Matrix
From a s t r u c t u r é w i t h p rigid-body degrees of free-dom, the stiffness matrix K , calculated i n the displace-ment method, has nullitj^ p. I f we can indicate p rows, depending on the remaining, and replace them b y i n -dependent rows, the new matrix K , w i l l be regular-These independent rows are given bj^ the momentum relation R - u = 0. Correspondingly the mass matrix M has to be adapted.
The stiffness matrix also has to be inverted i n the cal-culation of static deformations of structures w i t h rigid-bodj^ degrees of freedom. A loading w i t h a resultant force w i l l lead to a continuous accelerated rigid-body motion. When the loading is i n equilibrium the struc-ture w i l l onl}' deform, and this deformation has been uniqueljr defined. Then we may constrain the structure i n such a waj^ t h a t rigid-body motions are prevented, but we do not exert forces to the structure.
I n case of six rigid-body degrees of freedom, this method of constraint is realized hy choosing three non-collihear points i n the following manner:
Point A i n three perpendicular directions.
Point B i n two perpendicular directions i n a plane perpendicular to the line connecting these two points.
Point C i n one direction perpendicular to the plane suspended by these noncoUinear points.
I n this way the structure has been constrained against rigid-body motions; however, no reaction forces w i l l occur i n these points, because only well-balanced loadings are considered. The deformation then has been uniquely
determined. Thus, hy striking out f r o m the complete matrix K the rows and columns corresponding to these zero displacements, we get a reduced matrix K * which is nonsrngular. Similar considerations hold for partially constrained structures. Hence, f r o m static considera-tions i t is easy to indicate p rows, such that the other rows are independent.
I n the dynaimc calculations we can take these same rows and replace them by the momentum relation. I n i t i a l l y we have
u = 0
OY
'Mn 0 1 K i i K12 " U i "
_0 M22_ Oj2 _K21 K22_ -U2_ = 0 (12) where the partitioning has been done such that U i contains the displacements of the points constrained i n the static calculations (for example M i is of order p X p ) . Use has been made of the fact t h a t the mass matrix is diagonal.
Replacing the first matrix row of equation (12) by the momentum relation R-u = [Rl R 2 ] U l .U2 = 0 we find 0 0 — X R l R 2 . 0 M 2 2 . _K21 K22_ .U2_ = 0 (13) Premultiplying equation (13) w i t h R l R 2 _K21 K22. leads to - X I i 0 [K21 [K22 — - K 2 2 - R 2 - ^ - R l ] - ^ - M 2 2 K 2 i - R i ^ ' - R 2 ] - ' - M 2 2 - X I 2 " U i " - U 2 . = 0 (14) where I i is a unit matrix of order p X p and I2 a u n i t matrix of order (?i — p) X {11 — p). Obviously this eigenvalue problem leads first to the p eigenvalues zero of X, corresponding to the rigid-body modes. Secondlj^, equation (14) leads to the eigenvalue problem
|[K22 - K 2 1 - R l ^ l - R 2 ] ^ l - M 2 2 - I2! = 0 (15) f r o m which we find (n — p) eigenvalues X^; 5^ 0, corre-sponding to the deformation modes. M a t r i x
K ~ [K22 — K 2 i - R i ~ ' - R 2 ]
can be compared w i t h the reduced stiffness matrices of the preceding methods. Obviously its derivation is much simpler and there is the added advantage t h a t M = M22 is directly given b y the original mass matrix.
The J*'' mode U 2 j can be calculated i n the usual Avay. Then U i j is given by
From the momentum relations we have deduced equa-tion (8):
U l = - R 1 - 1 . R 2 . U 2 - (8)
Substitution of U i i n equation (16) leads to the eigenvalue problem (15). Hence, equations (8) and (16) are equiva-lent. The derivation of equation (8) is more direct than that of equation (16) and is the obvious method. A n application is given i n the Appendix.
I t s advantage of fewer calculations, compared w i t h the two methods mentioned before, seems to be reduced by the loss of symmetry of the stiffness matrix. However, i n general, the product-matrix of two symmetric matrices is not symmetric. Thus the eigenvalue problems, given by the other methods, must also be treated as being unsymmetric. Then i n each of the three cases the computer capacity of the internal memory is to the same order the decisive factor of the magnitude of the problem. Using external memories to enlarge the order of the eigenvalue problem is not advisable, paying attention to their large access-time.
Conclusions
The determination of the lowest natural frequencies of structures, partially or totally unconstrained, has i n each of the three methods been based on the same prin-ciple; namely, preventing the rigid-body motion i n the total motion. This results in a reduced stiffness matrix whose order and number of deformation modes are equal. This matrix is nonsingular. The mass matrix has to be reduced accordingly.
From a mathematical aspect the first two methods are more elegant than the t h i r d one, but the latter has ad-vantages i n numerical application. For example, the calculation of the reduced stiffness matrix is much less laborious, and the reduced mass matrix is inmiediately determined f r o m the original matrix. The first method still needs some additional requirements for the calcu-lation of the independent vectors q ^ + i * to q „ * . Each method leads to an eigenvalue problem of the same order.
Acknowledgment
The theory described herein arose as a mathematical tool during an investigation of dynamic properties of plate constructions, sponsored by the Netherlands' Shipresearch Centre T.N.O. Thanks are due to A. D . de Pater, Professor at the Technological University of D e l f t , f r o m whom originated the idea of eliminating the cyclic coordinates, using orthogonality considerations.
Appendix
A very simple mechanical system, Fig. 1, is considered t o illustrate the outlined theories. Only axial displace-ments are permitted, so that the number of degrees of freedom ?i = 3 and the number of rigid-body degrees of freedom p = 1. For the masses and spring-stiffnesses the following values are chosen:
- W W m2 m 3 = 2 L 1 F i g . 1 = 1 nil = ms ?)l2 =
2
Cl = C2 =1
W i t h these values i t is easily seen that, apart f r o m the value 0)1 = 0 for the rigid-body motions,ithe natural frequencies of the sj^stem are
W2 = 1 and W3 The corresponding modes are
V2
qi = (rigid-body mode)
- r
- r
q2 =
0
and qs =- 1
_ - i j
_ 1_
(the two deformation modes)
The mass and stiffness matrices of the system are given by
- 1 0 0" - 1 - 1
0-M = 0
2
0 and K = - 12
- 1_0 0 1_ _ 0 - 1 1_
Method 1
The cyclic coordinates are given b y equation (1): K - Q i = 0
As there is only one cyclic coordinate, we can write
Ql = qi Xl
and, hence, equation (1) leads to
- 1 - 1 0 Xl
- 1
2
- 1 X2 = 0_ 0 - 1 1_ _0_
f r o m which follows as a possible solution
q i
The M-orthogonal vectors are given by equation (2):
Q i ^ - M - Q 2 * = 0
where
Q 2 * = [ q 2 * q 3 * ] =
yi Zl 2/2 Z2
For the determination of q2* we can write
-1 0 0"
"2/1[1 1 1] 0
20
2/2_0 0 1_
-2/3-a possible solution is - 2" q2* =- 1
_ 0.
Then equation (9): R* can be written as I R2 R* = 1113 Vli Vh1
0
0
1
Similarly, for qs can be found
r 0¬
1
- 2 .
(independent of q2*)
where we have chosen the partitioning of
The only restriction i n the solution of equation (2) is the mutual independence of the vectors q j * (Ï = p + 1,. . Then f r o m equations (4) and (5):
K * = Q 2 * ' ' - K - Q 2 * M * = Q 2 * ^ - M - Q 2 *
the reduced stiffness and mass matrices can be deduced
such that
R l = [mi] and R 2 = [m^ m{\
Then the reduced mass and stiffness matrices, given by equations (10): M * = R * ' ' - M - R * K * =
I
5 - 3 - 3 5 and a n d^ i f
3- r
' L - 1 3.So the reduced eigenvalue problem, given b y equation (6):
| K * - i . M * - XI| = 0 leads to the equation
K * = R * ^ . K - R * become f o r the given values of m i and Ci;
M * = 2 and K * •3 1 1 1 _ o 1
~ Ll 1.
4 — A z i 3 — >>= 0
Hence, the eigenvalue problem appears to be
- X
0
1 - X= 0
f r o m which follows: X2 = 1 ÜJ2 = 1 and X 3= I
W3 = -\/2 The corresponding modes are- 1-
- r
0
and ^ 3 =- 1
_ - l _ _ i _
which leads to the same solutions as i n the foregoing.
Method 3
Choosing the same partitioning as i n M e t h o d 2, which means t h a t the mass mi is constrained i n the static calculation, the reduced matrix M 2 is obtained b y re-moving the first row and the first colunm f r o m the orig-inal mass matrix; hence
M 2 2
0'
0 1
all i n agreement w i t h the solutions already found.
Method 2
The equation of linear and angular momentum (7): R - u = 0
leads to
The matrices R i and R2 are the same as i n Method 2, so the reduced stiffness matrix given by equation (15) becomes K22 — K21 - R l ^ • R2 — whose inverse is 4
0'
- 1 1 [mi mi mi] M l L M 3 J0
10'
1 4Then the eigenvahie problem becomes modes qz* and qa* i n Method 1. Tor, i f we choose
which again leads to the same solutions.
- Q- - r
fl * —
q2 — 1 and qa* = 0
_ - 2 _
^ For this very simple mechanical structure, the applica- the eigenvalue problem leads to the determinant t i o n of the last t w o methods leads to an unsymmetric
eigenvalue problem, whereas the first one gave a sym-metric eigenvalue problenr. However, i t appears that
tliis has been caused by the choice of the deformation the same as found by ]\fethods 2 and 3! i - X 0
