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     http://pme.sagepub.com/ Engineers

    Proceedings of the Institution of Mechanical

     http://pme.sagepub.com/content/168/1/371The online version of this article can be found at:

     DOI: 10.1243/PIME_PROC_1954_168_040_02

     1954 168: 371Proceedings of the Institution of Mechanical Engineers G. B. Warburton

    The Vibration of Rectangular Plates 

    Published by:

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    Institution of Mechanical Engineers

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    371

    The Vibration of Rectangular Plates

    By G.

    B.

    Warburton, M.A.,

    Ph.D.*

    In

    this

    paper are considered the free transverse vibrations of rectangular plates with all possible

    boundary conditions obtained by combining free, freely-supported, and fixed edges. The Rayleigh

    method, assuming waveforms similar to those of beams, is used to derive a simple approximate

    frequency expression for

    all

    modes of vibration. The terms

    in this

    expression depend on the nodal

    pattern and the boundary conditions; they are tabulated for fifteen boundary con di ti on s4 four

    edges free, freely-supported, or fixed and the twelve cases in which some of the edges have one

    condition and the rest another. The expression can

    also

    be used to obtain frequencies for a plate

    which has a combination of all three boundary conditions. The effect on frequency of an edge being

    supported and partially restrained is discussed.

    For some boundary conditions it is possible to compare derived frequencies with those obtained

    by various methods of accurate analysis and by experiment; except for a few cases, the results from

    the approximate expression and accurate analysis agree closely.

    For rectangular plates and for most boundary conditions and modes of square plates, the nodal

    pattern consists of lines approximately parallel to the sides of the plate. T he exceptions are discussed

    and the gradual transition from these non-parallel patterns characteristic of a square plate to those

    of a rectangular plate is traced.

    Th e frequencies of extensional vibrations of rectangular plates are derived for two boundary con-

    ditions.

    I N T R O D U C T I O N

    In

    this

    paper are considered the frequencies of free transverse

    vibrations of rectangular plates for all combinations of free,

    freely-supported, and fixed edges. In transverse vibration, points

    in the plate undergo small displacements in the direction per-

    pendicular to the plane of the plate.

    All

    four edges of the plate

    may be free, freely-supported, or

    fixed;

    here are twelve mixed

    boundary conditions

    in

    which some of the edges have one con-

    dition and the rest another. The author has obtained an approxi-

    mate formula, expressing frequency

    in

    terms of the boundary

    conditions, the nodal pattern, the dimensions of the plate, and

    the constants of the material;

    this

    formula is applicable to all

    fifteen boundary conditions. For

    six

    of these the author has not

    found any reference giving frequencies; or only one- al l edges

    freely-supported-is there a well-known frequency expression.

    For the remaining conditions, previous investigators, whose

    work is referred to, have obtained frequencies for some modes

    of vibration for specific values of the ratio of the sides of the

    rectangle. For two of these eight conditions, Eschler (1950)t

    has

    obtained frequency equations similar to those found by the

    author and including a l l modes of vibration. The author also

    outlines a method of obtaining frequencies for a plate which has

    a combination of all three boundary conditions, for example,

    two free, one freely-supported, and one fixed edge.

    The approximate frequency formula is derived by applying

    the Rayleigh method (Rayleigh 1894, p. 109 et seq. ) and by

    assuming that the waveforms of vibrating plates and beams are

    similar;

    thus for a plate with all its edges fixed, the waveform

    assumed is the product of the characteristic functions for two

    beams with fixed ends. In effect, the Rayleigh method imposes

    additional constraints on a system, so that the resulting fre-

    quencies are higher than those given by an exact analysis.

    Whenever possible, the approximate frequencies are compared

    with the results of exact or more accurate analyses. For most

    boundary conditions, the errors are very small, but for a few

    they are appreciable. To use

    this

    method the nodal patterns must

    consist of lines approximately parallel to the sides of the plate.

    Usually

    t h i s

    condition is satisfied; the exceptions, and their

    effect on

    requency,

    are

    discussed.

    The

    MS.

    f

    t h i s

    paper

    was

    first received at the

    Institutionon.

    21st

    July 1953, and in its revised form, as accepted by the C o u n d for

    publication,

    on

    24th November 1953.

    * Lecturer, former1

    I.C.I.

    Research Fellow,Department of Engi-

    neering,

    University o?Edinburgh.

    t An

    alphabetical list of reference8

    is given

    in Appendix

    111.

    In practice, one edge of

    a

    panel may be supported and

    partially restrained-a condition between that of freely-

    supported and fixed. For any mode the increase of frequency,

    which is caused by the substitution of a fixed for a freely-

    supported edge, may be obtained from the approximate formula.

    Thus limits are imposed on the required frequency and, from a

    knowledge of the degree of restraint, an estimate of the frequency

    can

    be made. Some of the boundary conditions investigated

    appear to have little practical significance, but those containing

    various numbers of freely-supported edges are required in order

    to

    give lower limits for the frequencies of plates with partially

    restrained edges.

    Extensional vibrations can also

    occur in

    a plate; then motion

    is only

    in

    the plane of the plate. AS

    this

    type of vibration is

    associated with high frequencies and

    has

    ess practical significance

    than transverse vibrations, it is only considered briefly for two

    boundary conditions (Appendix I).

    Throughout the paper, the plates are assumed to be isotropic,

    elastic, free from applied loads and of uniform thickness, and

    the analysis is based on the ordinary theory of th n plates.

    Also

    the thickness is assumed to be small in comparison with the

    wavelength.

    Notation.

    Lengths of sides of rectangular plate. ,

    Ratios

    in

    expressions for displacement.

    Frequency.

    Acceleration due to gravity.

    Thickness of plate.

    Number of nodal lines in X and

    Y

    directions

    respectively.

    b2h

    d ( n - 1)2

    Y

    directions.

    rz =

    ime.

    Component displacements of a point in the

    X

    and

    Transverse disulacement of a Doint.

    x, Y

    A , B, C

    Component amplitudes.

    Co-orciinate distances in p h i f plate.

    Eh3

    D = 12 1

    -u2)'

    E Youne's

    modulus.

    Gx,

    Hx, Jx

    Fun zon s of

    m in

    frequency expression.

    Gy H,, Jy

    Functions

    of n

    n

    frequency expression.

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    T H E V I B R A T I O N

    O F

    R E C T A N G U L A R P L A T E S

    Ria integral.

    Kinetic energy.

    Strain

    energy.

    w =

    wsinwt.

    Indices.

    Factors in amplitude expressions defining nodal

    pattern.

    form.

    Functions

    in

    frequency expressions.

    Functions of

    x

    and

    y

    respectively, defining wave-

    2rf circular

    frequency.

    G E N E R A L

    METHOD

    FOR T R A N S V E R S E

    V I B R A T I O N S

    Before considering the various boundary conditions, the basic

    equations and methods of solution

    will

    be given. For a rectangular

    plate of sides of lengths a and b, the vibration form must satisfy

    Fig.

    1.

    Details of Rectangular Plate

    the boundary conditions at the edges OA,

    AB,

    BC,

    CO

    (Fig.

    1);

    also it must satisfy the plate equation (Love

    1927)

    :-

    where w, the displacement at any point

    x, y)

    at time

    t,

    is given

    by

    w

    =

    wsin

    wt

    = Ae(x)+Cy) inwt

    . 2)

    In general, it is not possible to find a form for

    w

    to satisfy

    equation 1) together with the boundary conditions. For these

    cases, an infinite series

    can

    be assumed for

    W;

    each term of the

    series satisfies equation

    1)

    and some of the boundary con-

    ditions and, by taking suitable values of the coefficients

    A,

    the

    remaining conditions are satisfied.

    Usually this

    procedure leads

    to an infinite determinant being zero, each term in the deter-

    minant being a different function of the frequency. The roots

    of the determinant give the natural frequencies for the modes.

    Normally, the roots obtained by taking

    two,

    three, four and

    five rows and columns of the determinant converge to a

    limit.

    Accurate values of frequency

    can

    also be obtained by con-

    siderations

    of

    energy. For a rectangular plate, the potential

    energy of bending

    U

    s given (Timoshenko

    1937)

    by

    and the kinetic energy

    T

    by

    The maximum values of potential and kinetic energy are given

    by

    Eh3

    T m U = t g

    h2 1:

    W2 dy . .

    .

    .

    .

    (6)

    whencc

    By the Rayleigh principle (Rayleigh 1894, p. 109 et

    seq.), if

    a

    suitable waveform W is assumed, satisfying approximately the

    boundary conditions, the resulting frequency value is very near

    to, but higher

    than,

    the true value, because the assumption of

    an incorrect waveform is equivalent to the introduction of

    constraints to the system. The characteristic beam functions in

    the directions

    OX

    and

    OY

    are normally assumed. By this

    method the approximate frequency expressions in

    this

    paper are

    obtained.

    The energy method gives more accurate values

    if

    the

    Rayleigh-Ritz method*

    is

    used. In

    ths

    a

    series is assumed for

    W, of the form

    where

    (x) and

    are the

    various

    characteristic

    bcam

    functions corresponding to different modes. The

    coefl icients

    Aab have values, which make w2, given by equation (7),a

    minimum.

    For any mode of vibration the nodal pattern is dciined by

    m

    and

    n,

    the

    number of nodal lines in the

    X

    nd

    Y

    directions

    respectively. This number includes the edge

    if it

    is supported;

    thus, for

    two

    parallel

    edges

    supported,

    m

    (or

    n) can

    be

    any

    integer given by

    m> 2.

    For a plate with one edge supported and

    the parallel edge free,

    m =

    1 is an additional value, and for

    two

    parallel

    edges

    free,

    m

    = 0 and m =

    1 are

    possible. The

    method of denoting the mode of vibration by the numbers of

    nodal lines is suitable, if the

    nodal

    lines are approximately

    parallel to the sides of the rectangle. However, when the plate

    is square and

    similarly

    supported in the

    X

    nd

    Y

    irections, the

    nodal lines

    may

    not be parallel to the sides. Thus, for a square

    plate with

    all

    edges fixed (or free) or with

    two

    neighbouring

    edges lixed and

    two

    free (or freely-supported), the patterns

    may

    not be

    similar

    to those for rectangles; the case of a square

    plate with

    a l l

    edges freely-supported is an exception-the nodal

    lines are always parallel to the edges. These non-parallel

    patterns were first observed experimentally for a square plate

    with free edges by

    Chladni 1802

    and

    1817).

    For

    this

    plate, the

    normal

    modes of vibration are of the type

    m/n

    (that is,

    m nodal

    lines

    in

    the

    X

    direction and n nodal lines in the

    Y

    direction)

    only when m = n or

    m

    s odd and n is even, or

    v i c e

    versa. When

    m-n = f2, f4, f6 he normal modes are of the types

    m/n+n/m

    and

    m/ n- n / m.

    These patterns

    do

    not consist of lines

    parallel to the sides of the plate; the amplitude of any point is

    the sum of the amplitudes of that’point for the component

    vibrations

    m/n

    and

    n/ m.

    Thus

    two

    new nodal patterns are

    obtained; the plus and minus

    signs

    corresponding to

    m/n

    and

    n/m

    being in phase and out of phase respectively. However, the

    normal modes are

    m/nfn/m

    and,

    in

    these

    cases,

    the modes

    m/n, n / m

    cannot exist separately. Simple examples of modes of

    the

    types

    m/nfn/m

    are given in Fig.

    2

    (first

    column)

    for

    a / b

    = 1 for various boundary conditions. The

    two

    types have

    discrete frequencies as well

    as

    discrete

    nodal

    patterns. The

    gradual transition from

    normal

    modes of the types

    m/nfn/m

    to normal modes of the types m/n and n/m,

    as

    a / b deviates

    from

    unity,

    will

    be considered later in

    this

    paper.

    When

    m

    is even and

    n

    is odd (or

    vim

    versa)

    for

    a free square

    plate, the normal modes of vibration have nodal

    lines

    parallel

    This

    method was first applied by Rim

    1909)

    to the transverse

    vibration

    of free plates; the

    method

    and its ap

    licatian

    to

    various

    plate problana

    have

    been described

    in

    detpil

    by

    &instock

    1952).

    W=z%A

    O(x)[email protected])

    .

    .

    .

    8)

    ab

    ab a

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    T H E V I B R A T I O N O F RECTANGULAR PLATES

    373

    are only approximate for free edges. The appropriate expressions

    for e ( x ) are given below; corresponding expressions for

    $(y) can

    be obtained by substituting

    y b, E,

    and

    c

    for

    x ,

    a,

    y,

    and

    k

    respectively.

    to the sides, and the modes m/n and

    n/m

    have, in theory,

    identical frequencies. The

    tw o

    modes may vibrate together and

    an

    infinite number of nodal patterns

    is

    theoretically possible,

    corresponding to combinations of the modes m/n and n/m,

    \Iv’th

    varying amplitude ratios and phase angles.

    Such

    combinauons (1) Freely-suppfled

    at 0

    and

    = a.

    are given by Chladni (1802) and are discussed

    in

    detail by

    Waller (1939). In practice, the frequencies of modes mln and

    n/m may not be identical, owing to small imperfections in the

    plate, and the

    nodal

    pattern obtained

    d

    epend

    on

    he method

    of

    excitation.

    Characteristic Beam Functions.

    Using the

    Rayleigh

    method,

    the characteristic

    beam

    functions are substituted for

    O(x) and

    for

    m

    = 5 4 ,

    6, . .

    (lOa)

    &)

    n equation

    2).

    These functions satisfy the

    boundary

    conditions for plates

    wt

    fixed or freely-supported edges, but

    m-

    >nx

    a

    (x) = sin

    or

    m = 2,3,4,. .

    (9)

    2)

    Fixed at

    = 0

    and

    =

    a.

    e x )

    =

    cos

    y( ; -

    j

    k

    cosh

    y z-+)

    X

    Sin

    fr

    s l n h

    4-Y

    here

    k

    nd

    tan b+tanh fr =

    0.

    L a d

    Bl

    H

    lo

    w

    p o p

    2//1+52

    C /I-l/2

    i/;

    ]

    h - -DO

    alb-o.98

    X

    e( )

    =

    sin yt ;-j) +K’

    inh

    yt(f-j)

    for m

    = 3a5a

    7,

    (lob)

    and tan w tanh w

    0.

    in

    fr’

    S i n h

    r’

    here k’ =

    --

    e(x) =

    1 form

    =

    0 . . . .’ .

    (lla)

    e(x)

    =

    1--

    form

    = 1 . . . . .

    (llb)

    d x ) = cosy( ; -+ ) + k

    cosh

    y(f-+)

    for

    m

    = 2,4,

    6,

    (11~)

    and

    tan fr+tanhfr = 0.

    here

    k

    =

    --

    3) Free

    at x

    =

    0 and

    x

    = a.

    2

    a

    x

    Sin

    4-Y

    w r

    €3

    a l b -

    0.95

    . .

    (lld)

    En

    I

     

    [email protected]

    Fig.

    2.

    Variation of

    Nodal

    Pattern with u / b

    a All

    CS

    fixed, m =

    4,

    n = 2.

    b

    All

    edges

    free, m = 2, n = 0.

    c

    T w o

    neighbouring edges fixed, two Erce,m = 2,

    n

    = 1.

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    374 THE

    VIBRATION

    O F RECTANGULAR PLATES

    (4)

    Fixed at

    x =

    0,

    free

    at

    x = u.

    e(x )

    =

    cosF-cosh$+k(sin$-sinhy)

    for

    m

    =

    1 ,2 ,3 , . .

    . . . .

    12)

    ( 5 )

    Fixed at x

    =

    0, freely-supported at

    x

    =

    a.

    e( )

    = sin

    yfZh-+)

    + A ’

    sinh

    y t ( - + )

    for

    m = 2,3,4,. . . . . . 13)

    where y’ and

    k’ are

    defined in equation

    (lob).

    e( ) = I-;

    for m =

    1 . . . . . . . . . .

    14a)

    (6) Free at

    x

    = 0, freely-supported at

    x

    = a.

    x

    e x>

    = siny‘ ~--f)+k‘sinhy‘ ~-f)

    form

    =

    Z, 3,4,

    . . .

    14b)

    where y’ and k‘ are defined in equation (lld).

    A P P R O X I M A T E F R E Q U E N C Y E X P R E S S I O N

    For any boundary condition, expressions for

    O(x)

    and

    +(y)

    are obtained from equations

    (9)-(14),

    and are substituted

    in

    equations (2) and (7) to give an expression for frequency. In

    each case a non-dimensional frequency factor

    A

    proportional to

    frequency, is derived, where

    pU4(2&Z12(1- 02)

    . . .

    15)

    ltoEh2g

    2 =

    It is found that for

    all

    possible boundary conditions consisting

    of any combination of My-supported,

    free,

    and

    fixed edges,

    frequencies

    can

    be determined from the expression

    where the coefficients

    Gxa Gya Hxa Hya

    Jxa

    and

    Jy

    depend

    on

    he

    nodal pattern and the boundary conditions. Values of these

    coefficients are given in Table

    1

    for the three pure boundary

    conditions,

    in

    which

    all

    the plate edges have the same condition,

    and for the twelve cases

    in

    which some of the plate edges have

    one condition and the rest another.

    Suppose the frequency

    of

    a rectangular plate with three edges

    fixed and one freely-supported (parallel to the Y-axis) is required

    for the modem

    = 2, n

    =

    3.

    Then from Table

    1

    and equation

    (16)

    +(l--0). 1*252(1-mT) ,2 52

    d

    UZ

    =

    2*44+39.0%+ lO45p

    Thus the frequency factor

    h

    is obtained for any ratio a / b ; the

    frequency is given by

    f

    In

    this

    case,

    A

    is independent

    of

    Poisson’s ratio

    u,

    because

    H x

    =

    Jx

    and

    H, = J,; this

    is true for any plate with no

    free

    edges.

    Equation (1 6) and Table

    1

    can

    also

    be used to give frequencies

    of plates with boundaries including

    all

    three edge conditions-

    free, freely-supported, and fixed. For instance,

    if

    a rectangular

    plate has the side

    x = 0

    fixed,

    y =

    0 freely-supported, and

    x = a

    and

    y = b

    free, the values of G,,

    Hx,

    and

    Jx

    are idenucal

    with those of cases

    12, 13,

    and

    15

    in Table

    1

    ; hat is, only the

    boundary

    conditions for the edges

    x

    = 0 and

    x = a and

    the

    number of nodes

    in

    the X-direction affect

    the

    values of

    G,, Hx,

    and

    Jx.

    Similarly, the edge conditions for

    y = 0

    and

    y = b

    and

    the value of

    n

    control the values of

    Gy, H,,,

    and Jy, and the latter

    are identical

    in this

    case with those of

    9 m

    Table

    1.

    Thus for

    the fundamental mode 111 the frequency factor

    h

    is given by

    2a2 3

    h2 =

    05974+0+~[u(-0*0870). O+(l-u)O.471

    a]

    a2

    = O.l27+0*287g(l-~)

    The last column of Table

    1

    gives the conditions for which

    modes

    of

    the typesm/ n &n / m

    occur

    or square plates. Sometimes

    the effect of these modes on requency is considerable;

    this w i l l

    be discussed in the

    next

    seaion, where the necessary modified

    frequency expressions are given. In that section the frequencies

    for modes

    of

    he

    typem/n,

    given by equation

    (16),

    are compared

    with those obtained by previous investigators. However,

    it

    may

    be noted that the errors in the frequencies obtained from

    equation (16) for plates with no free edges are less

    than

    0.6 per

    cent.

    A C C U RA C Y O F A P P R O X I M A T E F R E Q U E N C Y E X P R E S S IO N

    A N D D I S C U S S I O N O F M O D E S m / n f n / m

    (1) Four Edges Freely-suppo lted.

    This is the only boundary

    condition for which h

    can

    be expressed exactly by a simple

    formula. The frequency equation derived by the Rayleigh

    method, equation (16), is identical with the well-known exact

    expression, Timoshenko

    (1937).

    For square or rectangular plates

    the

    nodal

    lines are always parallel to the sides.

    (2)

    Four Edges Ftked.

    Sezawa (1931) obtained frequencies for

    several modes, but,

    owing

    to his assumptions, his values are

    inaccurate by 3-5 per cent. Kato (1932),by progressive approxi-

    mation, obtained a value

    h =

    3.641 for the fundamental mode

    for a square plate. A more accurate value for

    this

    mode,

    h

    = 3.646, has been obtained by Merent methods by Tomotika

    (1935 and 1936) and Iguchi (1938). The latter also obtained

    frequencies for various higher modes of square plates and for

    the

    fundamental for rectangular plates. By using a modified

    minimal

    problem, Weinstein

    (1937

    and

    1951)

    obtained upper

    and lower bounds for the frequency of ten modes for a square

    plate. Hopkins

    (1945)

    extended the theory to include transverse

    load, the influence of stiffeners and aeolotropy of the material;

    the only calculated frequency

    in

    his paper agrees with those of

    Tomotika and Iguchi for the fundamental mode for a square

    plate. The methods of Tomotika

    (1936), Iguchi (1938),

    and

    Hopkins

    (1945)

    were all

    similar

    to that previously described,

    whereW onsists of an infinite series considerable computation

    is required to obtain a single frequency by these methods.

    Young (1950) applied the Rayleigh-Ritz method to square

    plates, and obtained frequencies agreeing with those of Iguchi

    (to within

    0.1

    per cent). Eschler

    (1950)

    used the Rayleigh method

    to obtain an approximate frequency expression,

    similar

    to but

    more complicated

    than

    equation (16). Comparing values of

    h

    given by approximate and exact analyses, the maximum err01

    is 0.5 per cent.

    Existence

    of Modes

    m/nfn/m. When a clamped plate is

    square or almost square, the nodal patterns

    no

    onger consist of

    lines parallel

    to

    the sides.

    Also

    a square plate does not have two

    normal modes

    m/n, n/m

    with identical frequencies, but

    has

    normal modes represented by

    m/n n/m

    with two discrete

    frequencies. The nodal patterns of the modes 4/2&2/4 are

    shown in Fig. 2u. The literature contains no record of the

    change-over

    in

    pattern from a rectangle to a square. Young

    (1950)

    obtained discrete frequenciesfor the

    two

    modes

    4/2&2/4,

    but Iguchi (1938) obtained a frequency only for the mode

    4/2+2/4. In order to

    check

    the existence of these two modes

    with discrete frequencies, the author exteaded the analysis of

    Tomotika (1936) to these modes, and found that they exist

    separately with discrete frequencies.

    The author investigated the transition of patterns as

    a / h l

    by using the Rayleigh-Ritz method, but

    limiting

    the analysis

    to the two principal terms. For a rectangular plate

    in

    which

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    T H E V I B R A TI O N OF R E C T A N G U L A R P L A T E S

    TABU

    . COEFFICIENTS

    N FREQUENCY

    QUATION

    16)

    Freely-supported edge,

    m

    ixed

    edge.

    - - - - - -

    375

    Free edge.

    ~

    Boundary

    conditioi

    a

    1

     

    2,3,4,. .

    2

    3,4,5, .

    0

    1

    2

    3,4,5,

    .

    2,3,4,

    1,3,4,..

    2

    3,4,5, .

    ,

    .4, .

    1

    , .4,

    1

    t,

    3.4,.

    0

    1

    2

    I,4,5,

    .,

    1

    I 3,4,. .

    1

    2

    1,4,5,

    I

    2

    I4.5,.

    . I

    2

    ,4,5,. .

    1

    2

    ,4.5,. .

    2,3,4, .

    2

    I, 4.5,. .

    0

    1

    2

    I,

    4,5,

    2

    1,4,5,

    3,4,

    .

    I 3.4, .

    r 3 r

    4,. .

    0

    1

    2

    3 4,5,

    1

    8 3.4,. .

    1

    3.4, .

    B3.4,. .

    2

    I

    4,5,

    .

    1

    2

    4,5,. .

    0

    1

    2

    4,5,

    0

    1

    2

    4,5,

    .

    -

    GY

    -

    n--.

    1-50

    n-

    0

    0

    193

    n-1

    160(

    n-4

    n-4

    n-I

    n-1

    0

    0

    1*50(

    n-4

    0

    n-f

    n - 1

    0-1

    L.506

    *-f

    1.597

    ,494

    1-+

    0

    0

    -506

    I-+

    0

    0

    -506

    1-f

    -

    HY

    (n-l)z

    0

    0

    2

    1-+)2 c y n - * ) n

    1.248

    I w1

    -An

    (n-l)a

    (n-l)1

    (n-1)*

    -0.0870

    1.347

    2

    0

    0

    ~

    JY

    1.248

    2

    0

    (n-l)2

    (n- 111

    0.471

    3.284

    2

    Modes

    m/n&nlm

    exist

    for

    a -

    if,

    None

    m-n= f2,4,6... .

    r-n = i2.4.6, ...

    None

    None

    None

    None

    None

    None

    None

    m i t n

    None

    None

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    376 T H E V I B R A T IO N O F R E C T A N G U L A R P L A T E S

    there are an even number of nodes in both the

    X

    nd

    Y

    directions,

    the displacement may be expressed as

    W

    = A [ . . s y ( ~ - ~ ) + k r o s h y ~ - , ) ] [ 0 0 ~ € ( f - f )

    +c cosh c(f-+)]

    +B[cosc~--f)+ccosh~~--))][cosY(%--f)

    + k coshy($-t)]

    where the amplitude ratioA/B s chosen for each mode to make

    the frequency a minimum for that mode. This is expressed

    mathematically by

    aR

    aR

    A = 0 , a T = O

    where

    R

    is the Riu integral given by

    a2w a2w

    ]&dy .

    (17)

    azw

    2

    12(1-u2)pw2~

    Eh2g

    The evaluation of

    t h i s

    integral is simplified by noting that

    provided that W = 0 at the four edges. This w a s proved by

    Leggett (1941) in a paper on the buckling of plates.

    Hence

    aR

    9 4 2y2czr) phw2

    o = - -

    -

    g + p + x - x ) A + & j (18)

    . . (19)

    where

    2 2

    [1- -

    ‘I -;

    sin y ) ] [1-

    -

    l)n(cz-; sin

    € I

    and

    Eh3

    12(1-

    2 )

    D=-

    Owing to the existence of the small term T]’he two modes

    m/ n

    and n / m are never completely independent

    ;

    heir effect on each

    other

    wll

    be investigated for particular values of

    m

    and n. If the

    modes 412 and 214 (or the associated modes 4/2f2/4) be con-

    sidered, then substitutingy = 3.57~

    nd

    e = 1.506~n equations

    (18) and (19), two frequency factorshl and

    X2,

    with corresponding

    amplitude ratios

    @/A ) ,

    and (B/A)2, are obtained for any value

    of

    a / b . It

    should be noted that (B/A)l

    =

    -(A/&.

    In Fig. 3a,

    hl and h2 are plotted against a/b for these modes; the dotted

    lines represent the values of X 1 and hz obtained by applying

    equation (16) to these modes. I t

    wil l

    be observed that the

    resulting difference in frequency, if a dotted line is used instead

    of its associated full line, is negligible. Thus f it is erroneously

    0:6 0 9

    a/b

    b

    All edges free, m = 2, n = 0.

    16

    u

    All

    edges

    fixed, m = 4, n = 2.

    a 6

    c

    Two

    neighbouringedges

    fixed,

    two free, m

    = 2, n =

    1.

    Fig. 3. Variation of Frequency Factor X with a /b

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    T H E V I B R A T I O N OF R E C T A N G U L A R P L A T E S

    377

    corrected the work of Goldmann and showed theo re ti dy and

    experimentally that for

    alb

    = 1.5, the nodal patterns were

    approximately parallel to the sides, conesponding to modes of

    the type mln. By taking three plates for which alb = 1,1.04, and

    1.07,

    Pavlik showed theoretically and experimentally the transi-

    tion from the characteristic patterns of square plates to those of

    rectangular plates. Waller's papers (1939 and 1949), which

    include a large number of experimental patterns for square and

    r e ~ . 3 u l a rree plates, showed the same transition.

    Equation (16), with Table 1, gives frequencies for all modes of

    the type

    mln;

    that

    is,

    it

    gives all frequencies for rectangular

    plates and frequencies for square (or almost square) plates when

    m =

    n

    or m - n is an odd number. When the plate is square and

    m - n 1s

    an even number, the modes are of the types m/nfn/m.

    Th e range of values of

    a / b ,

    in which the transition from modes

    m / n to modes m l n f n l m occurs, is considerably greater for free

    plates than for fixed plates.

    Also,

    the percentage difference in

    frequency between two associated modes m/n+n/m and

    mln-nlm is much greater for free plates than for fixed plates.

    Thus it is necessary to modify the frequency expressions

    previously given.

    Modes

    m10fOlmfor

    Square Plates Where

    m

    s Even. Assuming

    vibrauon forms

    LV

    =

    A[cosy(~-+)+k coshy(:-+)]

    assumed that the normal modes for a square plate with fixed

    edges are

    412

    and

    214,

    instead of

    4/2f2/4,

    the frequency error

    will be very small. The amplitude ratio ( B / A ) l is plotted in

    Fig. 4 (curve

    I);

    these values have been used to construct the

    0

    fA[cos y e - i ) +k a s h y e - + )

    Fig. 4. Variation of Amplitude Ratio with

    a / b

    I

    All edges fixed, m =

    4,

    n = 2.

    I1

    All edges free,

    m

    =

    2,

    n =

    0.

    I11

    TWO

    eighbouring

    edges

    fixed,

    two

    free, m =

    2,

    n =

    1.

    nodal patterns for a / b = 1 and a / b = 0.98, shown in Fig. 2 ~ .

    Fig. 2 shows the rapid change from patterns characteristic of

    square plates to those for rectangular plates, as

    a / b

    deviates from W = A[sin

    y E-+)

    +k' sinh

    f(:-f)](l-?)

    it is found that

    A2

    =

    (m-+)4f2o(m-+)2;;-,8 . . (20)

    Modes m l l lm for Square Plates Where m = 3 ,

    5,

    7 . .

    Assuming vibration forms

    ..n; .r

    y . Y L , .

    Similar results are obtained for modes

    with

    odd numbers of

    nodes in the

    X

    and Y directions, provided m

    n.

    When

    m

    is

    odd and

    n

    even, or vice versa, there is no term in

    AB

    in the

    integral

    R.

    Hence the two equations, corresponding to equations

    (18) and (19), are independent. Thus, in this case, m l n and n l m

    are normal modes for

    all

    values of

    a / b

    and there is only one

    frequency for the square plate, as the two equations are identical

    for

    a / b = 1.

    Only one pair of modes has been investigated in detail, but

    these modes have the lowest frequencies at which this pheno-

    menon occurs. From the nature of

    7

    [equation (19a)],

    it

    can

    be seen that the percentage difference between the frequencies

    for modes

    m/n+n/m

    and

    m/n-n/m

    decreases as

    m

    and

    n

    increase. Thus the author has not attempted to allow for these

    modes in his approximate frequency equation (16). When the

    plate is square, and m and n are unequal but both even (or both

    odd), there will be two frequencies, both nearly equal to that

    given by the approximate formula. The nodal patterns for these

    modes

    can

    be determined by addition

    and

    subtraction of the

    relevant patterns for modes

    m / n

    and n/m.

    Chladni

    (1802

    and

    1817)

    determined

    experimentally the nodal patterns of square plates. The

    theoretical determination of frequencies and nodal patterns of

    free square plates was the first application by Ritz (1909) of

    his method, which has since been applied to many other

    problems. Lemke (1928) repeated the work of Ritz with greater

    accuracy (in effect this means taking more terms in the series

    for displacement) and she also obtained experimental frequencies

    and nodal patterns; for the graver tones there was good agree-

    ment between theory and experiment. Ritz showed that for

    square plates the normal modes were of the type mln with nodal

    lines approximately parallel to the sides, if m

    =

    n or if m was

    odd and n even (or vice versa). For m - n

    =

    5 2 , f4, f6, .

    the normal modes were of the types

    m/nfn/m.

    Goldmann (1918) extended the work of Ritz to rectanguh

    plates, but erroneously assumed that modes of the types

    m/nfn/m existed in rectangular plates. Pavlik (1936 and 1937)

    (3)

    Four

    Edges

    Free.

    it

    is

    found that

    Modes

    mlnfn lm

    for Square Plates Where m and n are Like

    Quantities but Unequal ( m > l ,

    n>l). Assuming vibraaon

    forms

    W

    =

    A[cos

    y ( z - 4 )

    +k cosh

    y z-+)] [ m s c ~ - + )

    +c coshce;'-+)]

    & A [ c o s c ~ - + ) + c o s h ~ ( ~ - f f ) ] [ c o s y ~ - f )

    +k coshy(:-+)]

    it is found that

    A2 = (m-. ) )4+(n-+)4

    6 6

    160 1

    m - n

    ,2 (m

    -

    )4{-m

    + n -

    1+ m- )2+ (n- )*

    m+n-1 2

    . . (22)

    1

    m - n

    32(1-

    )

    1

    2

    (m-

    +)2(n-+)2[

    m+n--l+ (m -

    )2

    +

    n- 121

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    378 THE V I B R A T I O N OF R E C T A N G U L A R P L A T E S

    It

    will be seen that for

    all

    modes

    m/n&n/m,

    the expression for

    A2 consists of the corresponding expression for the non-existent

    mode

    m / n

    plus (or minus) a ‘correction factor’. These factors are

    not negligible; the difference in frequency for the modes

    2/0+0/2

    nd

    210-0/2

    is approximately

    25

    per cent for

    I

    =

    3.

    However, in the correction factor

    of

    equations

    (20)

    nd

    (22),

    t

    is sufficiently accurate to use the integral value m

    = 2 or n =

    2)

    with no modification; the usual modifications

    are

    made in the

    remainder of the expressions for hz Frequencies have been

    calculated from these equations for various modes for square

    plates and for rectangles, where

    a / b = 1.5,

    and have been

    compared with the accurate values of

    Ritz (1909),

    Lemke

    (1928),

    and Pavlik

    (1936).

    For square plates, frequencies given by the

    approximate formulae are

    5

    per cent high for modes

    111,

    211,

    and 2/2,but for most other modes the error is less than 1 per

    cent. For the rectangular plate the error for the fundamental

    mode

    1/1

    is

    6

    per cent, but for other modes the errors in fre-

    quency are about

    1

    per cent.

    The author has investigated the transition as a/ l from the

    modes

    210

    and 0/2 f rectangular plates to the modes 2/0+0/2

    and

    2/0-0/2

    of square plates. Th e method

    has

    been described

    for plates with fixed edges. Th e waveform assumed is

    :-

    W

    =

    A[cos

    y(:-+)

    + k

    ~~shy(:-+)]

    In Fig.

    3b

    the frequency factors,

    A 1

    and

    h2,

    for the

    two

    modes

    are plotted against

    a / b .

    The frequency factors that would be

    obtained by erroneously using equation

    (16)

    are shown by

    dotted lines. Over the range

    0.8< a / b

    <

    1.25, if

    frequencies are

    obtained from equation (16) or these modes, they are inaccurate.

    In Fig.

    4

    (curve 11) the variation of the amplitude ratio

    with

    a / b

    is shown;

    this has

    been used to plot the nodal patterns

    for the two modes for various values of

    a / b

    n Fig. 2

    t

    will be

    seen

    from Figs.

    2

    and

    4

    hat

    the transition from modes charac-

    teristic of rectangular plates to those of square plates

    is

    much

    more gradual for free plates than for fixed plates. T he amplitude

    ratios agree with those obtained theoretically and experimentally

    by Pavlik

    (1936

    nd

    1937)

    his results show that for other modes

    ( B / A ) l

    decreases as the difference between m and

    n

    increases,

    for a particular value of

    a / b .

    Thus

    it

    is safe to conclude that for

    all modes for which

    ( m - n )

    is an even number, equation (16) s

    inaccurate only over the range

    0.8

    < a / b

    <

    1.25.

    (4,

    Mixed Boundmy Conditions.

    For

    six

    of the twelve mixed

    boundary conditions listed

    in

    Table 1 accurate values of

    frequency have been obtained for some modes by other investi-

    gators. For each of the conditions, in which

    two

    parallel edges

    are freely-supported (conditions

    6,7, 10,

    and 1

    ,

    there exists a

    waveform, which satisfies the plate equation

    (1)

    and

    all

    the

    boundary conditions. The resulting frequency equations are

    transcendental and can be solved numerically only for particular

    values of (n-l)a/b. For a plate with two parallel edges fixed

    and

    two

    freely-supported, Sezawa (1931) and Kato (1932)

    obtained the fundamental frequency, and Iguchi (1938) btained

    frequencies for several modes. Eschler

    (1950)

    olved the problem

    with the addition of loads applied in the plane of the plate and

    he also obtained an approximate frequency expression similar

    to equation

    (16).

    Voigt

    (1893)

    irst derived the transcendental

    equations for plates with

    two

    parallel edges free, two freely-

    supported; following

    this

    analysis Zeissig

    (1898)

    calculated

    frequencies for various modes and performed experimental

    investigations, which gave frequencies and nodal patterns

    agreeing with

    his

    calculations. Eschler (1950) epeated Voigt’s

    analysis, adding the effect on frequency of loads applied to the

    plate in its plane, and he also obtained an approximate frequency

    expression, but it included only modes with two or more nodal

    lines parallel to the free edges. Iguchi (1938) calculated fre-

    quencies for several modes for a plate with one edge fixed, three

    edges freely-supported. For the last of these four conditions-

    one edge free, three edges freely-supported-the author has

    derived the transcendental frequency equation (Appendix

    11).

    Young

    (1950)

    used the Rayleigh-Rim method to calculate

    frequencies of the five gravest modes of vibration of square

    plates with

    two

    neighbouring edges free and

    two

    fixed, and with

    one edge fixed and three edges free. For the latter condition

    Barton (1951) performed similar calculations for rectangular

    plates and obtained experimental frequencies for these modes.

    Eschler (1950) btained an approximate solution for plates with

    two parallel edges free and two fixed, but his solution considered

    only modes in which there were two or more nodal lines parallel

    to the free edges; thus the

    two

    gravest families of modes-with

    no nodal line and one nodal line parallel to the free edges-were

    ignored.

    The accuracy of the frequencies obtained from equation (16)

    depends on the boundary conditions.

    As

    has been mentioned,

    the assumed waveforms do not satisfy exactly the conditions for

    a free edge

    ;

    ence the frequencies are not as accurate for plates

    with one or more free edges. When there are no free edges, the

    frequencies given by equation

    (16)

    and by exact analysis agree

    to within 0.6 per cent. When free edges exist, the approximate

    frequencies are usually within

    1

    per cent of the exact values,

    but they are sometimes

    5

    per cent too high when the charac-

    teristic functions for free-free or fixed-free beams are used in

    the analysis. The greatest errors occur for cantilever plates

    (condition 15) or the modes I and 2/1; or a square plate the

    approximate frequencies are

    11

    and 3 per cent too high and for

    a / b

    = 5

    they are 27 and 12 per cent too high for these modes.

    For both these modes the waveform

    in

    the Y-direction is

    I

    2

    3

    4

    5

    6

    7

    n t , NUMBER OF NODA L LINES

    IN

    S

    DIRECTION

    Fig.

    5.

    Frequencies of Cantilever Plate

    a/b = 1.855.

    o

    Grinsted’s experimental points.

    n

    Number

    of nodal

    lines

    in

    Y

    direction.

    assumed to be linear. This assumption, which cannot be

    expected to give good results, is also associated with the less

    accurate frequencies obtained for free plates. I n Fig.

    5

    the

    frequencies of a rectangular cantilever plate

    ( a / b = 1.855),

    obtained from equation (16) and Table 1, have been plotted

    against the number of nodal lines

    in

    the X-direction, m, each

    curve representing a different number

    of

    nodes in the Y-direc-

    tion, n. As

    m

    and n can only be integers, the curves connecting

    the frequencies have no physical significance, but merely indicate

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    379

    HE VIBRATION OF RECTANGULAR PLATES

    the different families of modes. Grinsted (1952) obtained experi-

    mental frequencies for

    24

    modes for a plate of these dimensions

    and his experimental points are shown in Fig. 5. Apart from the

    mode

    111

    the greatest difference between experimental and

    theoretical frequencies is 6.4 per cent and the mean error is

    2-5 per cent. The diagram shows that, in general, the frequency

    error decreases as m and

    n

    increase.

    For three of the mixed boundary conditions of Table

    1

    (5, 9, and 13) modes of the types

    m/nfn/m

    exist for square

    plates, when

    m

    and n are unequal. When the plate has two

    neighbouring edges fixed and two freely-supported (condition

    5),

    the difference in frequency between associated modes

    is

    negligible, and equation (16) with Table 1 can be used to give

    approximate frequencies; the rapid transition from nodal

    patterns characteristic of square plates to those of rectangular

    plates is similar to that described for a plate with

    all

    its edges

    fixed. A square plate with two neighbouring edges free and two

    freely-supported (condition

    9)

    behaves

    in

    a manner similar to

    that of a plate with free edges. Thus the percentage difference in

    frequency for associated modes, m(n+n m .and mln-nlm, may

    be considerable. Frequency expressions slrmlar to equations (21)

    and

    (22) can

    be derived, and for modes

    m / l

    /m,

    here

    m> ,

    The

    behaviour of the modes

    211

    and

    1/2

    for a plate with two

    neighbouring edges lixed and two free

    (condition 13)

    has been

    investigated by the Rayleigh-Ritz method, assuming

    W =

    A [ c o s ~ - c o s h ~ + k ( s i n ~ - s i n h ~ ) ] [ c o s $

    -

    osh

    ?+ c(sin - sinh f ) ]

    €X

    w

    ( :

    a

    +B[cos%-cosh EX

    +c s i n --sinh -)][ax 7

    a

    - c o s h ~ + k ( s i n ~ - s i n h p ) ] . .

    ,

    (24)

    In Fig.

    4,

    curve I11 gives the

    variation of the

    amplitude ratio

    A/B with a/b. Fig. 3c shows the variation of frequency with a/b

    for the two modes; the full lines represent the frequencies

    calculated by the Rayleigh-Ritz method and the dotted lines

    represent those calculated from equation

    (16)

    and Table

    1 .

    The points marked ere obtained for a square plate by

    Young, using several terms

    in

    his series expression for

    W,

    nd

    show that the use of two terms in the expression for W nstead

    of the approximate frequency expression (16) may not, in fact,

    increase the accuracy. The author considers that approximate

    frequencies for all modes

    can

    be obtained from equation (16),

    particularly as the percentage difference in frequency between

    associated modes m/ n+ n/ m and m / n - n / m decreases as

    m

    and n

    increase; the errors for modes

    m/n&tn/mwill

    not be greater than

    those existing for modes m / n for

    this

    boundary condition. If

    exact values are required for the higher modes, a series of

    several terms must be used for

    W.

    he variation of nodal pattern

    with

    a/b

    is shown

    in

    Fig.

    2c

    for the modes

    2/1

    and

    1/2.

    E F F F C T O F A P A R T I A L L Y R E S T R A I N E D E D G E

    For a plate which has three edges fixed and the other supported

    and partially restrained frequency limits can be found for any

    mode by considering the fourth side to be (a) fixed and (b) freely-

    supported. The greatest percentage difference between these

    limits will occur for the fundamental mode. For

    t h i s

    mode for

    a square plate, taking values from Table

    1,

    with four edges fixed

    h,2

    =

    1.5064+1.5064+2. 1.2482

    = 13.41

    A, = 3.66

    with three edges fixed and one freely-supported

    hb2 = 1.254+1*5064+2.

    1.252 1--=-)1*248

    = 10.49

    & = 3.24

    Therefore 3*24

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    380

    THE VIBRATION O F RECTANGULAR PLATES

    where

    A,

    B, and

    C

    are component amplitudes. From the ampli-

    tude ratios for cylinders [Arnold and Warburton

    1949,

    equations

    (17)

    and

    (re)]

    it follows that for plates,

    for

    h1

    A = B = O

    for

    h2

    (m

    -

    )b

    (n- 1)a

    =O a n d - A =B

    (n- 1)a

    (m

    -

    )b

    =Oand --A=B

    nd for

    h3

    Thus, the vibrations associated with hl are transverse vibra-

    tions, and the frequency expression for hl agrees with that pre-

    viously given [equation

    (16)].

    The other roots give much higher

    frequencies, as

    h / a

    must be small for a

    th n

    plate, and are asso-

    ciated with extensional vibrations; that is, the motion occurs

    only in the plane of the plate.

    Love

    (1927,

    p.

    497)

    gives equations for extensional vibrations

    of plates; by substituting

    u

    and w from equations

    (26)

    in his

    equations, the expressions for

    X2

    and

    h3,

    given in equations

    (25b

    and

    c),

    can be

    obtained. For transverse vibrations the

    boundary conditions considered were freely-supported, free,

    and fixed; for extensional vibrations new definitions are neces-

    sary. In general, there are

    two

    conditions for the edge

    OY;

    these

    conditions depend on whether that edge is free to move in the

    X-direction. These conditions

    will

    be referred to as free and

    restrained; in this section

    a l l

    edges of the plate are free. From

    equation (26) it

    can

    be seen that there are m equally-spaced lines.

    parallel to the Y-axis, including the edges, for which

    u = 0;

    between these lines there are

    (m-1)

    lines for which

    u

    = 0.

    Similarly, parallel to the X-axis there are

    n

    lines, including the

    edges, for which

    u = 0

    and

    n-1

    lines for which u

    = 0.

    Thus

    the only real nodes are a finite number of points, given

    by

    the

    intersection of these two series of lines. Further investigation

    shows the difference between the two types of mode associated

    with

    h2

    and

    A 3

    respectively. For the former type the plate edges,

    x = 0, x

    =

    a,y =

    0,

    and

    y

    = b, must be free from direct stress,

    but a shear

    stress

    must

    be

    applied to the edges for the equili-

    brium equation to be satisfied. For the latter type no stresses

    are

    applied to the edges. Thus equation

    (25c)

    gives the frequency of

    extensional vibrations of rectangular plates with perfectly free

    edges.

    Plates with

    Two

    Parallel

    Sides

    Fixed

    and

    Two Freely-supported.

    From the results for cylinders with fixed ends

    [Arnold

    and

    Warburton

    1953,

    equation

    (23)]

    a frequency equation is obtained

    for plates of this type; the roots of the equation are

    d

    2aZ

    b4 b2

    h12

    = (m-+)4+(n-11)4-+-(m-+)2(n-l)2p

    .

    +4(nr-f)z(n-l)$(1+~)2p] t . (27b

    and c)

    for m

    = 3, 4, 5, . .

    here

    p

    =

    l

    (m-

    T

    2

    ; = 0.550

    nd

    m-+

    is replaced by 1.506 for m = 2.

    The lowest mot agrees with that given by equation

    (16)

    for

    transverse vibrations; this is to be expected, as the theories for

    cylinders and

    beams

    are both based on assuming the beam

    vibration form between the fixed ends. The two higher mots

    give the frequencies of extensional vibrations for plates with two

    parallel edges restrained and

    two

    free. As the waveform for a

    cylinder with fixed ends is only approximate, the frequencies

    given by equation (27) are also approximate, whilst those given

    by equations

    (25)

    are exact, because the assumed waveform

    [equation

    (26)]

    satisfies the plate equations.

    A P P E N D I X

    I 1

    T R A N SV E R SE V I B R A T I O N OF R E C T A N G U L A R P L A T E W I T H

    O N E E D G E F R EE A N D T H R E E E D G E S F R E E L Y - S U P P O R T E D

    ConsideringOC (Fig.

    1) to

    be free and the other edges fieely-

    supported the waveform which satisfies the plate equation is

    w

    =

    sinotsin- n(n-l)y[

    Asrn

    (n(n;1)2u rZ-l)+($+)}

    From the boundary conditions

    and the frequencies

    are

    given by

    2fr2+l)f

    coth

    [ ~ ( r z + l ) + ]

    ( -l>a

    =

    0.

    (30)

    This equation

    has

    to be solved numerically for particular values

    of

    (n-l)a/b

    and

    U.

    A P P E N D I X

    I 1 1

    R E F E R E N C E S

    ARNOLD,

    R. N.,

    and

    WARBURTON,

    .

    .

    1949 P m .

    OY.SOC.

    A$ vol.

    197,

    p.

    238,

    ‘Flexural Vibrations of the Walls of

    Thin

    Cylindrical Shells Having Freely-supported Ends’.

    1953

    Proc. 1.Mech.E. (A), vol.

    167,

    p.

    62,

    ‘The Flexural

    Vibrations of

    Thin

    Cylinders’.

    BARTON,

    M.

    V. 1951

    Jl. of Applied Mechanics,

    Trans.

    A.S.M.E., vol.

    73,

    p.

    129,

    ‘Vibration of Rectangular and

    Skew Cantilever Plates’.

    CHADN

    E.

    F.

    F.

    1802

    ‘Die

    Akustik‘.

    ESCHLBR,

    H.

    1950

    Ingenieur-Archiv, vol.

    18, p. 330,

    ‘Zur

    Ermittlung der Eigenschwingungmhlen der

    in

    ihrer

    Mittelebene belasteten Rechteckplatte’.

    GOLDMA”,E. 1918 ‘Anwendung der Ritzschen Methode auf

    die Theorie der Tr an sv er sa ls ch wi ng n freischwingen-

    der Platten von rechteckiger, rhombischer, dreieckiger

    und elliptischer B e g r m g ’ (Dissertation, Breslau).

    GRINSTID,

    B.

    1952

    Proc. 1.Mech.E.

    (A),

    vol.

    166,

    p.

    309,

    ‘Nodal Pattern Analysis’.

    HM

    R. F. S.

    1952

    JI. of Applied Mechanics, Trans.

    A.S.M.E., vol.

    74,

    p.

    402,

    ‘The Frequency of Vibration of

    Rectangular Isotropic Plates’.

    1945

    Aeronautical Research Council, Reports

    and Memoranda No. 2234, ‘The Solution of

    Small

    Displacement, Stability or Vibration Problems Con-

    cerning a Flat Rectangular Plate when the Edges are

    either Clamped or Simply-supported’.

    IGUCHI, . 1938

    Memoirs of the Faculty of Engineering,

    Hokkaido Imperial University, ‘Die Eigenwertprobleme

    f i r die elastische rechteckige Platte’.

    1817

    ‘Neue Bietriige

    zur

    Akustik‘.

    HOPKINS,. .

    1937

    Ingedeur-Archiv,

    vol. 8,

    p.

    11.

    1938a Ingenieur-Archiv, vol. 9, p.

    1.

    KATO,

    H.

    1932 J1. of the Society of Naval Architects, vol. 50,

    p.

    209.

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    THE VIBRATION

    OF

    RECTANGULAR PLATES 381

    LECGBTT,

    . M. A. 1941 Aeronautical Research Council,

    TOMOTIKA,.

    1935 Report of the Aeronautical Research

    Reports and Memoranda No. 1991, ‘The Buckling of a Institute, Tokyo Imperial University, No. 129, ‘Trans-

    Square Panel Under Shear when One Pair of Edges is verse Vibration of a Square Plate with Four Clamped

    Clamped and the Other Pair is Simply-supported’.

    Edges’.

    LEMKE,LICE

    1928 Annalen der Physik, Leipzig, fourth 1936 Phil. Mag., seventh series, vol. 21, p. 745, ‘The

    series, vol.

    86,

    p.

    717,

    ‘Experimentelle Untersuchungen Transverse Vibration of a Square Plate Clamped at Four

    zur W. Ritzschen Theorie der Transversalschwingungen Edges’.

    quadratischer Platten’. VOIGT,W. 1893 Nachrichten, Giittingen, p. 225, ‘Bemerkung

    LOVE, A. E. H. 1927 ‘Mathematical Theory of Elasticity’, zu dem Problem der transversalen Schwingungen

    fourth edition (Cambridge University Press).

    rechteckiger Platten’.

    PAVLIK,B.

    1936

    Annalen der Physik, Leipzig,

    fifth

    series,

    WALLER,Mary

    D.

    1939 Proc. Phys. SOC., vol. 51, p. 831,

    vol. 27, p. 532, ‘Beitrag

    zur

    theoretischen und experi- ‘Vibrations of Free Square Plates’.

    mentellen Untersuchung der Biegungsschwingungen bei 1949 Proc. Phys. SOC. B), vol. 62, p. 277, ‘Vibrations of Free

    rechteckigen Platten mit freien Riindern’. Rectangular Plates’.

    1937 Annalen der Physik, Leipzig,

    fifth

    series, vol. 28,

    1937

    Memorial des Sciences Mathematiques,

    p. 632, ‘Beitrag zur Untersuchung des Zusammenhanges No. 88, ‘Etude des spectres des equations aux dtrivkes

    der bei Biegungsschwingungen an rechteckigen und partielles de la thb rie des plaques elastiques’.

    quadratischen Platten beobachteten Staubfiguren’. 1951 Jl. of Applied Mech. Trans. A.S.M.E., vol. 73, p. 229,

    RAYLEIGH,ord 1894 ‘Theory of Sound’, vol. 1, second Contributions to the Discussion of paper ‘Vibration of

    edition (Macmillan and

    Co.,

    London). Rectangular Plates by the Ritz Method’.

    RITZ, W. 1909 Annalen der Physik, fourth series, vol. 28, 1952 ‘Calculus of Variations with Applications

    p. 737, ‘Theorie der Transversalschwingungen einer to Physics and Engineering’ (McGraw-Hill, New York

    quadratischen Platte

    mit

    freien Riindern’.

    and London).

    SEZAWA,

    K 1931 Report of the Aeronautical Research YOUNG,

    D.

    1950 Jl. of Applied Mechanics, Trans. A.S.M.E.,

    Institute, Tokyo Imperial University, No. 70, ‘On the vol. 72, p. 448, ‘Vibration of Rectangular Plates by the

    Lateral Vibration of a Rectangular Plate Clamped at

    Ritz

    Method’.

    Four Edges’. ZEISSIG,

    C.

    1898 Annalen der Physik, Leipzig (Wiedemann),

    1937

    ‘Vibration Problems in Engineering’, vol.

    64,

    p.

    360,

    ‘Ein einfacher

    Fall

    der transversalen

    second edition (Van Nostrand). Schwingung einer rechteckigen elastischen Platte’.

    WEINSTEIN, .

    WEINSTOCK,.

    TIMOSHENKO,

    .

    Communications

    Mr. R.

    F.

    S.

    HEARMON

    Princes Risborough) wrote that the

    author had shown that the systematic application of beam

    functions not only simplified the approximate derivation of the

    transverse frequencies of rectangular isotropic plates under all

    possible boundary conditions, but also completely unified the

    presentation

    of

    the results.

    It was worth discussing briefly the possibility of applying

    beam functions to plates which differed in specified ways from

    those dealt with by the author and, in particular, with plates

    (1) which were isotropic, but oblique;

    2)

    which were rectangular but anisotropic.

    (1) Barton (1951) had calculated the frequencies of the first

    two modes of oblique (or skew or swept) cantilever plates, having

    all sides the same length. He had used a series formed from

    suitable beam functions to represent the deflexion

    W

    nd, in his

    treatment, had limited the series to eighteen terms, but even

    so,

    he had been unable to avoid somewhat complex calculations.

    Bereuter (1946)* had obtained estimates for the fundamental

    frequency of oblique plates with all edges either clamped or

    supported and of the clamped oblique plate with a concentrated

    mass at its mid-point. He had assumed straightforward trigono-

    metric or polynomial functions for

    W

    and,,by applying the

    *BERBUTER,. 1946 ‘Theoretische Untersuchungen iiber die

    Eigenfrequenz Parallelogramftrmiger Platten’. Ecole Polytechnique

    Fedkrale Publications du Laboratoire de PhotdlasticitC No.

    3.

    Edition S.

    A.

    Leeman frkres et Cie, Stockerstrasse

    64,

    Zurich.

    (Acknowledgement is made to Professor B. G. Johnston, University

    of

    Michigan, or drawing attention

    to t h i s

    reference.)

    Rayleigh-Ritz method, had been able to derive for the funda-

    mental frequency of the clamped oblique plate a closed formula

    which was substantially ndependent of the form assumed for W.

    In the case of the supported oblique plate, Bereuter had been

    unable to find any simple expression for W which satisfied the

    boundary conditions, and, as a result, the Rayleigh-Ritz method

    proved unreliable. Bereuter, however, developed a finite-dif-

    ference method applicable to the supported plate, but the solution

    was only slowly convergent with respect to the fineness of the net

    into which the plate was divided, and thus the accuracy was not

    high. The problem of buckling was closely allied to that of

    frequency, and Wittrick (1953)t had studied the buckling of the

    clamped oblique plate. It was evident that there w a s some

    interest at the present time in oblique plates, and it seemed that

    the application of the author’s beam function method might

    yield results

    of

    considerable value.

    (2) The differential equation and the potential energy

    equation had been derived for a generally anisotropic plate (for

    example

    see

    Hearmon 1948) , but the only case which had

    so

    far proved amenable to treatment was the ‘specially orthotropic’

    plate (Hearmon and Adams 1952)9, that is, a plate made from a

    t

    WIITRICX, .

    H. 1953

    Aeronautical Quarterly,

    vol.

    4, p. 151,

    ‘Budding

    of

    Oblique Plates with Clamped Edges Under Uniform

    Compression’.

    *

    HFARMON,. F. S.,and

    ADAMS,

    E.

    H. 1952

    British Jl. of Applied

    Phys., vol. 3, p. 150, ‘The Bending and Twisting of Anisotropic

    Plates’.

    HEARMON,

    . F. S. 1946

    Proc. Phys.

    SOC.

    Lond.,

    vol. 58, p. 78,

    ‘ T h e

    Fundamental Frequency of Vibration

    of

    Rectangular Wood and

    Plywood

    Plates’.

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    382

    C O M M U N I C A T I O N S O N T H E V I B R A T I O N O F R EC TA N GU LA R PLA TES

    material possessing three perpendicular axes of elastic symmetry,

    two

    of which lay in the plane of the plate,

    in

    directions parallel

    and perpendicular to the plate edges. Expressions for the fre-

    quency of vibration had been derived for all the modes of the

    simply-supported orthotropic plate, but

    only

    for the funda-

    mental mode of the clamped plate (Hearmon 1946)*. However,

    using the appropriate beam functions (equations (1Oa) and (lob))

    in conjunction with the potential energy expression from

    reference Hearmon (1948) or Hearmon (1946), the frequencies of

    a l l

    the modes, except the fundamental of

    a

    clamped orthotropic

    plate, were obtained as :

    the

    difiicult

    question of ‘single’ vibrations which had no separate

    physical existence and of true ‘combined’ modes.

    The nodal systems for a square plate with fixed edges shown

    in Fig. 2u were 3/1+ 1/3and 3/1-

    1

    3 systems respectively, not

    as stated, 4/2&2/4 systems. That might be made

    clear

    by con-

    sidering the analogous (and theoretically simpler) case of a

    where

    y

    = (m-+>rr

    for

    m =

    3a4a 5

    c = (n-+>rr

    for n

    =

    3,4, 5

    . . .

    Dl, z,

    nd

    D 3

    were the

    flexural

    and torsional rigidities defined

    in Hearmon

    (1948)

    and Hearmon (1946) and the other symbols

    were defined as in the author’s paper, exceptp, which represented

    mass per

    unit

    volume. For an isotropic material

    and equation

    31)

    reduced to the equation given by the author

    for the clamped isotropic plate.

    The fundamental corresponded to

    m = n

    =

    2; in

    the case in

    question

    D,= D~=

    o3

    ~h3/12(1--*)

    where the numerical factors were given to

    three

    significant

    f igures.

    The previous estimate(Hearmon1946) had been obtained by

    assuming

    and gave for the fundamental mode :

    W = K x 3 2 ( u - x ) 2 ( b - y ) 2 sin w t

    Evidently, thetwo estimates from equations (32) and (33) were in

    good agreement, and M ered appreciably only in the num eri d

    factor of the third term; the effect of that difference on the

    calculated frequencies was, however, small.

    The above example showed that it was possible to apply the

    beam functions to calculating the frequency of vibration of

    anisotropic plates. There was no reason to expect difficulties

    with boundary conditions other

    than

    clamped, and

    it

    was thus

    possible

    in

    principle to construct a table similar to Table 1, but

    valid for anisotropic instead of isotropic plates.

    I 2 3

    m /n -

    /m

    .-

    Fig.

    7.

    Square Membrane (Theory)

    Compare plate

    wt

    fixed edges.

    0

    I 2

    c

    t

    I-----

    Imlm

    L

    .

    Fig. 8. Free Square Plate (Experimental)

    Dr.

    MARY

    . WALLERLondon) wrote that it had been stated

    in the paper that, ‘the analysis is based on the ordinary theory of

    thin plates’. That assumed that the principle

    of

    superposition

    was valid, namely, that a stationary wave

    in

    one direction was

    unaffected in frequency

    or

    amplitude by the superposition of a

    second stationary wave of the same frequency at right angles to

    it. It then followed that, as the author had remarked, ‘When

    m

    is odd and

    n

    s even (or

    wice

    wersa)

    for a free square plate

    wo

    modes

    may

    vibrate together and an infinite number of nodal

    patterns is theoretically possible, corresponding to combinations

    of the modes m / n and n/m, with varying amplitude ratios and

    phase angles’. This

    was orthodox

    theory. She had, however,

    shown (Waller 1952)t that Such additionalmodes were impossible

    and that the displacements of free-vibrating surfaces of geo-

    metrical shapes must

    be

    ‘mechanically-balanced’ about one

    or

    more lines

    of

    w e t r y . She had also discussed (W der 1953)$

    *

    HEARMON,. F.

    S. 948

    Forest Products Research Laboratory,

    Stationery ice, London).

    t WALLER,MARY.

    1952 Proc.

    ROY.

    OC.

    A),

    V O ~ . 211, p. 265,

    ‘Vibrations

    of

    Free Plates :Line Symmetry: Corresponding Modes’.

    w m m , y >.

    1953 ACU(tjCa, ~01.

    ,

    P. 370, ‘Concerning

    Combined and Degenerate Vibrations

    of

    Plates’.

    vibrating membrane some of whose nodal systems were

    shown

    in Fig.

    7.

    They corresponded

    to

    the formula

    Special Re

    rt

    No. 7, Elasticity of

    Wood and Plywood’

    (H.M. m X

    mx

    z = s i n - s i n ~ & s i n - s i n ~ = O

    .

    (34)

    U

    U

    a

    where, for the case in question m = 3 and n = 1 and wice verso.

    Fig.

    7

    might be compared with Fig.

    8

    (which was part of a

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    A U T H O R ’ S

    R E P L Y

    O N T H E V I B R A T I O N O F R E C T A N G U L A R

    P L A T E S

    383

    diagram in Waller

    1939)

    which related to the free square plate.

    The nodal systems of a hypothetical free membrane could be

    similarly arranged and their equation would read

    e = c o s - c o s ~ f c o s n ~ c o s ~ = Ox . . .

    (35)

    a U

    givingm =

    2, n

    =

    0

    r erice oersa, for the square nodal systems

    so marked in Fig.

    2b.

    A U T H O R ’ S R E P L Y

    Dr. G. B. WARBURTONrote,

    in

    reply to the communications,

    that the possibility of applying beam functions to plates, other

    than rectangular isotropic plates, as suggested by

    Mr.

    Hearmon,

    was very interesting. He agreed with

    Mr.

    Hearmon that the

    application of the beam function method might give valuable

    results for some boundary conditions for oblique isotropic

    plates. Probably an oblique cantilever plate was an exception;

    there were discrepancies between the theoretical and experi-

    mental results of Barton

    (1951)

    or that case, although he had

    used a series of eighteen terms to represent

    W.

    arton had

    concluded that the Rayleigh-Ritz method, using beam deflexion

    forms, was suitable for that type of plate only

    if

    the angles of the

    plate were between

    60

    and

    120

    deg. Thus, the beam function

    method was likely to be inaccurate for the oblique cantilever

    plate. However, frequencies obtained for oblique plates

    with

    other boundary conditions might be accurate, as the work on

    rectangular plates had shown that it was difficult to apply his

    own method to a cantilever plate.

    It seemed that his o w n method could be expected to give

    frequencies for anisotropic rectangular plates with a fair degree

    of accuracy. For the ‘specially orthotropic’ clamped plate,

    mentioned by Mr. Hearmon, values of the fundamental natural

    frequency had been given in a paper by Kanazawa and Kawai

    (1952)*. The method was

    similar

    to that of Tomotika (1935),

    successive approximations being obtained for M Y required

    frequency. Unfortunately, for anisotropic plates, only the first

    approximations for the frequency factor h had been given for

    various values of DI/D and 0 2 / 0 3 between

    +

    and

    3.

    The author

    had compared the values of

    h

    given by Kanazawa and Kawai with

    those obtained from equations

    (32)

    and

    (33)

    for a square plate.

    Over the above range of values of

    D11D3

    and

    D2/D3

    equation

    (32),

    based on he beam function method, always gave the lowest

    value of

    A;

    the corresponding values of frequency given by

    equation

    (33)

    were up to 1 per cent higher, and those of Kanazawa

    and Kawai were up to

    1.5

    per cent higher. As the Rayleigh

    method always gave frequencies higher than the exact values,

    the frequencies obtained from equation

    (32)

    had the smallest

    error. The fundamental frequencies of ‘specially orthotropic’

    square plates with other boundary conditions (combinations of

    fixed and freely-supported edges), had been given by Kanazawa

    and Kawai.

    They had also tabulated frequencies for two or three modes of

    isotropic rectangular plates with one or more edges clamped and

    the rest freely-supported. In some cases they had given only a

    first approximation, but in others they had included better

    approximations

    ;

    hey had compared their values of frequency

    with those of other Japanese workers (for example, Iguchi

    (1938),

    Sezawa

    (1931)a

    Kato

    (1932)a

    and Tomotika

    (1935)),

    where

    available. His own values of frequency from equation (16)were

    always lower than the first approximations of Kanazawa and

    Kawai and sometimes lower than their second approximations.

    Comparing the errors in the Mere nt methods applied to iso-

    tropic and orthotropic plates, it appeared that frequencies

    obtained for orthotropic plates with combinations of fixed and

    freely-supported edges would be accurate, if the appropriate

    beam functions, equations

    (9)a

    (lo), and

    (13),

    were used for the

    plate displacement.

    Congress for Applied Mechanics, p. 333, ‘On he Lateral Vibration of

    Anisotropic RectPngular Plates (Studed by the Integral Equation)’.

    K A N A u W A a

    Tea

    and

    KAWAIa

    T. 952 pro^. Second

    Japan

    National

    Dr.

    Waller had misinterpreted

    his

    notation, when she had

    stated that the nodal patterns for a square plate with fixed edges

    shown in Fig.

    2a

    were not

    4/2f2/4.

    The author had defined

    ma

    n

    as the number of nodal lines in the

    X

    and

    Y

    directions

    respectively; if an edge were freely-supported or fixed, it was

    included in the number of nodal ines in the relevant direction.

    Thus

    or a plate with all its edges fixed, there were at least

    two

    nodes in each direction, and patterns such as

    3/1+1/3

    nd

    311

    -

    13

    were impossible. Tha t notation could be applied to a l l

    boundary conditions without any inconsistency occurring, .and

    was convenient because the nodal patterns consisted of h e s

    approximatelyparallel to the sides of the plate for most boundary

    conditions. Dr. Waller had consideredm and n to be the number

    of half waves in the X and Y directions respectively; that was

    shown by the form of equations

    (34)

    and

    (35).

    Orthodox theory predicted that when

    m

    was odd and tz was

    even (or vice aersa) for

    a

    free square plate,

    two

    modes of equal

    frequency might vibrate together and an infinite number of

    nodal patterns was possible. Dr. Waller had pointed out that

    such patterns were impossible to obtain experimentally; she

    called them ‘degenerate modes’ (Waller

    1953).

    However, in that

    paper she had admitted that for particular values of m and

    n

    two

    patterns might be obtained, similar o those of two degenerate

    modes, owing to faulty experimentation or lack of uniformity

    of the plate. That result was t o be expected from a considera-

    tion of the effects of small imperfectionson he vibrations of a

    plate.

    In the papers quoted

    in

    her contribution to the discussion

    (Waller

    1952

    and

    1953)

    Dr. Waller had considered the classes of

    symmetry of the nodal patterns for vibrations of free plates of

    various forms. For a free square plate

    six

    classes of symmetry

    had been given

    ; wo

    of them included

    all

    the patterns for which

    (m-n)was an odd number, and the remaining four classes those

    patterns for which

    (m-n)

    was an even number and m and n

    were equal. The four latter classes had been defined in terms of

    their lines of symmetry, the two diagonals and the

    two

    medians

    of the square, as given below :-

    (a) both diagonals and both medians antinodal (for

    example, patterns for modes

    2/0+0/2

    nd

    212,

    Fig.

    8);

    (b)

    both diagonals and both medians nodal;

    (c) both diagonals antinodal and both medians nodal (for

    example, pattern for mode

    111,

    Fig.

    8)

    ( d ) both diagonals nodal and both medians antinodal (for

    example, pattern for mode

    210

    -

    12,

    Fig.

    8).

    Dr. Waller had stated that if the patterns of a free square plate

    were tabulated as in Fig.

    8,

    which could be extended indefinitely

    downwards and to the right, the classes of symmetry repeated

    themselvesin every four columns and every two ows. Thus the

    class of symmetry for modes

    2/0-0/2, 610-016..

    .a

    212,

    4/2+2/4,6/2+2/6

    . .,should be the same. The nodal patterns

    of a free square plate had been obtained experimentally and

    analytically, and had been given for many modes by Chladni

    (1802),Ritz (1909), Lemke (1928)a Waller (1939), and Grinsted

    (1952).A

    study of any of those papers showed that the classes of

    symmetry, as defined above, did not repeat themselves

    in

    every

    four columns and every

    two

    rows. For instance, it was obvious

    from Fig.

    8

    hat the patterns for modes

    210-012

    and

    212

    did

    not belong to the same class of symmetry.

    He did not find any law governing the repetition of classes of

    symmetry; M erent laws were applicable, depending on whether

    the mode was to the left or to the right of the diagonal line,

    representing modes for which

    m

    = n, in Fig.

    8.

    However, the

    various modes could be divided amongst the above four classes

    of symmetry as given below :-

    Class a) modes

    m / n + n / m

    when both

    m

    and n were even; also

    (b) modes

    mln-nlm

    when

    m

    and n were both odd, but

    (c) modes

    m / n + n / m

    when both

    m

    and n were odd; also

    ( d ) modes mln-n lm when

    m

    and

    n

    were both even, but

    modes 2/29 414,616 .

    ;

    unequal;

    modes 111,

    313,515

    . .;

    unequal.

    at IOWA STATE UNIV on May 10, 2014pme.sagepub.comDownloaded from 

    http://pme.sagepub.com/http://pme.sagepub.com/http://pme.sagepub.com/

  • 8/17/2019 Ritz Function

    15/15

    384

    He had stated that the value of the frequency factor, A,

    depended on Poisson’s ratio u f one or more edges of the plate

    were free. Th e value of

    u

    also affected the nodal patterns obtained

    for modes of the types

    m/nfn/m.

    Those effects were small, but

    it should be noted that a value of u

    =

    0.3 had been assumed in

    A U T HO R ’S R E P L Y O N

    THE

    V I B R A T I O N O F R E C T A N G U L A R P L A T E S

    obtaining the patterns of Fig. 2 b and c), the frequency factors

    of Fig. 3 b and c), and curves I1 and I11 in Fig.

    4.

    However, any

    other practical value of u would have little effect on the results

    plotted

    in

    Figs. 2

    and

    4; the effect on frequency of varying values

    of

    u

    was

    usually

    s m a l l and could be obtained from equation (16).


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