NumberDB$^\beta$ Advanced search Tags Tables Help Login
Roots of the frequency equations of the Euler-Bernoulli beam
edit · history · discussion · files · long url · physics eigenvalue
Numbers
boundary conditions
$n$ 
$z_n=\beta_nL$
clamped-free
1:
1.875104068711961166445308241078214162570111733531069988245413713105679952840428638526566550581886037
comment: Fundamental bending mode of the cantilever.
clamped-free
2:
4.694091132974174576436391778019812049389896737545766828972803277849077936801052508003588502781554273
clamped-free
3:
7.854757438237612564861008582764570457848541929230046694423281448826561421408653528234986678939998005
clamped-free
4:
10.99554073487546699066734910785470293961297277465158688750572876853268790287767665585379428140017424
clamped-free
5:
14.13716839104647058091704681255177206860307679297466257660390221060862813796997317553720514490403464
clamped-free
6:
17.27875953208823633354392841437582208593451963555020518019583409820517182765950506238397180374363584
clamped-free
7:
20.42035225104125099441581194794783704613728889454422146971007757693218464227461742599605706277847926
clamped-free
8:
23.56194490180644350152025324019807551703126599005089175895107824631683682878780174034732903185349800
clamped-free
9:
26.70353755551829880544547873808817400922749669144352092021072147715785917311705187846408706782389604
clamped-free
10:
29.84513020910281726378873090743040506336991605023740363191556084701360702365066041922847460719876666
clamped-free
11:
32.98672286269283844616831418385368300036703604164605498080877644930525722952757744250611964695088177
clamped-free
12:
36.12831551628262183428116220464844632308953935250206079305040271746198340987786876751288080901841924
clamped-clamped
1:
4.730040744862704026024048100833884819898341800706848689096355845369136669852013182181954893927320917
clamped-clamped
2:
7.853204624095837556477066687254049790322304173990674614841337308510559417819292849483886760031243884
clamped-clamped
3:
10.99560783800167090666903251910589241754280303057636199707655590265692282890028712037617876202353498
clamped-clamped
4:
14.13716549125746417710591785509333075287529510012847128811887550869002686113217387908169505230447322
clamped-clamped
5:
17.27875965739948143809107397576864401359534661339648225684683709882290021988498478152723922533455527
clamped-clamped
6:
20.42035224562606109093641118931307964990611597910278384669642225163506215031582802152786071719153648
clamped-clamped
7:
23.56194490204045507539201680064149781898303074333166351028307806420424760504760268178030665237038191
clamped-clamped
8:
26.70353755550818624841940764575857225221680788710419590862536843386243219815220731747457268598935630
clamped-clamped
9:
29.84513020910325426700149327939424575405883331015441626550676962449415820763576070822505510988914263
clamped-clamped
10:
32.98672286269281956154719686483756311040117568283227892102769304821772658683437565346748656460046447
clamped-clamped
11:
36.12831551628262265035963561077978702140797839221643609157497495284133067338051355136760079948602132
clamped-clamped
12:
39.26990816987241546315006806769548805305544022385809493455057688640137636476300530392228256148741510
clamped-pinned
1:
3.926602312047918778238533343627024895161152086995337307420668654255279708909646424741943380015621942
clamped-pinned
2:
7.068582745628732088552958927546665376437647550064235644059437754345013430566086939540847526152236608
clamped-pinned
3:
10.21017612281303054546820559465653982495305798955139192334821112581753107515791401076393035859576824
clamped-pinned
4:
13.35176877775409312420970382287928612610840394355209795431268421693121609907610365873728634299467815
clamped-pinned
5:
16.49336143134640978077359843241878155520058784141613946051384652410886329341718782673374328230023223
clamped-pinned
6:
19.63495408493620773157503403384774402652772011192904746727528844320068818238150265196114128074370755
clamped-pinned
7:
22.77654673852600097883770024400054954773980032060890364716625043238064842798445394275638116884362865
clamped-pinned
8:
25.91813939211579421731677716594696136984281203404615521578811119093050438959281423535513484554656121
clamped-pinned
9:
29.05973204570558745577945123791880401532786929640276597151665652827996403292422309138096835453697206
clamped-pinned
10:
32.20132469929538069424209467850768235503488894320356731189052823438851621440335891973354237922559313
clamped-pinned
11:
35.34291735288517393270473806189420721588517442310660915038597253083677604667361510116041247790769340
clamped-pinned
12:
38.48451000647496717116738144517390995749477770617716469370115023971556670079232835847403947426126966
pinned-pinned
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
pinned-pinned
2:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
pinned-pinned
3:
9.424777960769379715387930149838508652591508198125317462924833776923449218858626995884104476026351204
pinned-pinned
4:
12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827
pinned-pinned
5:
15.70796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
pinned-pinned
6:
18.84955592153875943077586029967701730518301639625063492584966755384689843771725399176820895205270241
pinned-pinned
7:
21.99114857512855266923850368295652018938018579562574074682461214615471484400346299039624377739481948
pinned-pinned
8:
25.13274122871834590770114706623602307357735519500084656779955673846253125028967198902427860273693654
pinned-pinned
9:
28.27433388230813914616379044951552595777452459437595238877450133077034765657588098765231342807905361
pinned-pinned
10:
31.41592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
pinned-pinned
11:
34.55751918948772562308907721607453172616886339312616403072439051538598046914829898490838307876328775
pinned-pinned
12:
37.69911184307751886155172059935403461036603279250126985169933510769379687543450798353641790410540482
free-free
1:
0
comment: Rigid translation, represented by $X(x)=1$.
free-free
2:
0
comment: Rigid rotation, represented by $X(x)=x-L/2$.
free-free
3:
4.730040744862704026024048100833884819898341800706848689096355845369136669852013182181954893927320917
equals: clamped-clamped,1
free-free
4:
7.853204624095837556477066687254049790322304173990674614841337308510559417819292849483886760031243884
equals: clamped-clamped,2
free-free
5:
10.99560783800167090666903251910589241754280303057636199707655590265692282890028712037617876202353498
equals: clamped-clamped,3
free-free
6:
14.13716549125746417710591785509333075287529510012847128811887550869002686113217387908169505230447322
equals: clamped-clamped,4
free-free
7:
17.27875965739948143809107397576864401359534661339648225684683709882290021988498478152723922533455527
equals: clamped-clamped,5
free-free
8:
20.42035224562606109093641118931307964990611597910278384669642225163506215031582802152786071719153648
equals: clamped-clamped,6
free-free
9:
23.56194490204045507539201680064149781898303074333166351028307806420424760504760268178030665237038191
equals: clamped-clamped,7
free-free
10:
26.70353755550818624841940764575857225221680788710419590862536843386243219815220731747457268598935630
equals: clamped-clamped,8
free-free
11:
29.84513020910325426700149327939424575405883331015441626550676962449415820763576070822505510988914263
equals: clamped-clamped,9
free-free
12:
32.98672286269281956154719686483756311040117568283227892102769304821772658683437565346748656460046447
equals: clamped-clamped,10
pinned-free
1:
0
comment: Rigid rotation about the pinned end, represented by $X(x)=x$.
pinned-free
2:
3.926602312047918778238533343627024895161152086995337307420668654255279708909646424741943380015621942
equals: clamped-pinned,1
pinned-free
3:
7.068582745628732088552958927546665376437647550064235644059437754345013430566086939540847526152236608
equals: clamped-pinned,2
pinned-free
4:
10.21017612281303054546820559465653982495305798955139192334821112581753107515791401076393035859576824
equals: clamped-pinned,3
pinned-free
5:
13.35176877775409312420970382287928612610840394355209795431268421693121609907610365873728634299467815
equals: clamped-pinned,4
pinned-free
6:
16.49336143134640978077359843241878155520058784141613946051384652410886329341718782673374328230023223
equals: clamped-pinned,5
pinned-free
7:
19.63495408493620773157503403384774402652772011192904746727528844320068818238150265196114128074370755
equals: clamped-pinned,6
pinned-free
8:
22.77654673852600097883770024400054954773980032060890364716625043238064842798445394275638116884362865
equals: clamped-pinned,7
pinned-free
9:
25.91813939211579421731677716594696136984281203404615521578811119093050438959281423535513484554656121
equals: clamped-pinned,8
pinned-free
10:
29.05973204570558745577945123791880401532786929640276597151665652827996403292422309138096835453697206
equals: clamped-pinned,9
pinned-free
11:
32.20132469929538069424209467850768235503488894320356731189052823438851621440335891973354237922559313
equals: clamped-pinned,10
pinned-free
12:
35.34291735288517393270473806189420721588517442310660915038597253083677604667361510116041247790769340
equals: clamped-pinned,11
Definition
The dimensionless frequency roots $z_n=\beta_nL$ of a uniform Euler-Bernoulli beam of length $L>0$ are the nonnegative values for which $X^{(4)}=\beta_n^4X$ on $0\leq x\leq L$ has a nonzero transverse mode $X$ satisfying the selected endpoint conditions [1].
Parameters
boundary conditions
—   endpoint conditions (The endpoints have one of the six listed pairs of clamped, pinned or free conditions.)
$n$
—   mode index ($n\geq1$, with modes ordered by nondecreasing $z_n$ and repeated according to multiplicity.)
Formulas
(1)
For $z>0$, the frequency equations are $\cos z\cosh z=-1$ for clamped-free conditions, $\cos z\cosh z=1$ for clamped-clamped and free-free conditions, and $\sin z\cosh z-\cos z\sinh z=0$ for clamped-pinned and pinned-free conditions [1]. For pinned-pinned conditions, $z_n=n\pi$.
(2)
Writing the endpoint pair as a superscript, $z_{n+2}^{\mathrm{free\text{-}free}}=z_n^{\mathrm{clamped\text{-}clamped}}$ and $z_{n+1}^{\mathrm{pinned\text{-}free}}=z_n^{\mathrm{clamped\text{-}pinned}}$ for $n\geq1$.
(3)
The eigenvalue of $d^4/dx^4$ is $\beta_n^4=z_n^4/L^4$. With $E$ the Young modulus, $I$ the second moment of cross-sectional area, $\rho$ the mass density and $A$ the cross-sectional area, the angular frequency is $\omega_n=(z_n^2/L^2)\sqrt{EI/(\rho A)}$, and the frequency in cycles per unit time is $f_n=\omega_n/(2\pi)$.
Comments
(4)
In each pair the first condition is imposed at $x=0$ and the second at $x=L$. A clamped end satisfies $X=X'=0$, a pinned end satisfies $X=X''=0$, and a free end satisfies $X''=X'''=0$. Primes denote derivatives with respect to $x$. Reversing the endpoints gives the same spectrum.
(5)
The beam has constant bending stiffness and mass per unit length, with no axial load, added masses or elastic supports. The model describes one transverse displacement component and neglects shear deformation and rotary inertia. Clamped is also called fixed or built-in, pinned is also called hinged or simply supported, and clamped-free is the cantilever.
(6)
The index $n$ includes rigid-body modes. For free-free conditions $z_1=z_2=0$, with mode space spanned by $1$ and $x-L/2$; its first positive root is $z_3$. For pinned-free conditions $z_1=0$, with mode $X(x)=x$; its first positive root is $z_2$. The other four pairs have no zero mode. Positive roots are simple. A formal zero of a characteristic equation is included only when it satisfies the original boundary value problem.
(7)
[1], Tables 2 and 5, gives the endpoint conditions and characteristic equations. The authors provide implementations and finite-element comparisons in [2].
Programs
(P1)
Python
import mpmath as mp

mp.mp.dps = 120

def beam_root(boundary, n):
    """Return beta_n*L, counting rigid-body modes with multiplicity."""
    if n < 1:
        raise ValueError('n must be positive')
    if boundary == 'free-free':
        if n <= 2:
            return 0
        boundary, n = 'clamped-clamped', n-2
    elif boundary == 'pinned-free':
        if n == 1:
            return 0
        boundary, n = 'clamped-pinned', n-1
    if boundary == 'pinned-pinned':
        return n*mp.pi
    if boundary == 'clamped-free':
        f = lambda z: mp.cos(z)+mp.sech(z)
        a, b = (n-mp.mpf('0.75'))*mp.pi, (n-mp.mpf('0.25'))*mp.pi
    elif boundary == 'clamped-clamped':
        f = lambda z: mp.cos(z)-mp.sech(z)
        a, b = (n+mp.mpf('0.25'))*mp.pi, (n+mp.mpf('0.75'))*mp.pi
    elif boundary == 'clamped-pinned':
        f = lambda z: mp.sin(z)-mp.cos(z)*mp.tanh(z)
        a, b = n*mp.pi, (n+mp.mpf('0.25'))*mp.pi
    else:
        raise ValueError('unknown endpoint conditions')
    return mp.findroot(f, (a,b), solver='bisect', maxsteps=500)

print(mp.nstr(beam_root('clamped-free', 1), 100))
References
[1]
Firas A. Khasawneh and Daniel Segalman, Exact and numerically stable expressions for Euler-Bernoulli and Timoshenko beam modes, Applied Acoustics 151 (2019), 215-228. (arXiv) (doi)
Links
Similar tables
Dirichlet eigenvalues of the classical planar domains —   the second-order membrane analogue with displacement fixed at the boundary
Neumann eigenvalues of the classical planar domains —   the second-order membrane analogue with zero normal derivative at the boundary
Zeros of Bessel functions of the first kind $J_\alpha$ —   Bessel roots determine the vibration modes of a circular membrane
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds modes $1\leq n\leq12$ for each of the six listed endpoint pairs, giving a compact reference for the lowest modes; the free-free and pinned-free ranges include their two and one rigid-body modes, respectively)
How they were obtained:

The positive roots are recorded to 100 significant decimal digits; the zero modes are exact. The positive transcendental roots are isolated by rational bisection. Arb evaluates the scaled characteristic functions and their derivatives at 429 bits. Each initial bracket has opposite endpoint signs and a derivative enclosure excluding zero. Bisection continues to absolute width below $10^{-112}$. The measured relative-accuracy lower bound is 374 bits, exceeding 112 decimal digits. Pinned-pinned roots are enclosed as $n\pi$ in Arb. The three zero entries follow exactly from the polynomial solutions of $X^{(4)}=0$.

more

The attached checks construct the four-by-four endpoint matrix directly from the fundamental solutions and evaluate its determinant by permutation expansion, independently of the scalar equations in the generator. At 768 bits they certify opposite determinant signs at both exact endpoints of every stored nonzero decimal interval. Exact row reduction verifies the zero-mode multiplicities. Independent determinant roots computed with mpmath at 150 and 200 decimal digits are compared to all entries; ordering and the isospectral equalities are also checked. The Programs example supplies approximations; the attached generator carries the certified enclosures.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.