Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity
This study focuses on the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a moving force under the influence of three types of motions, including accelerating, decelerating and constant velocity types of motion, respectively. The beam’s nonli...
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irk-123456789-1771292021-02-11T01:28:35Z Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity Mamandi, A. Kargarnovin, M.H. Farsi, S. This study focuses on the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a moving force under the influence of three types of motions, including accelerating, decelerating and constant velocity types of motion, respectively. The beam’s nonlinear governing coupled partial differential equations (PDEs) of motion for the bending rotation of warped cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. To obtain the dynamic response of the beam under the action of a moving force, the derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic response is obtained using mode summation technique. Furthermore, the calculated results are verified with those obtained by finite element method (F.E.M.) analysis. In the next step a parametric study on the response of the beam is conducted by changing the magnitude of the traveling concentrated force, its velocity and beam’s boundary conditions and likewise their sensitivity on the beam’s dynamic response are studied, respectively. It is observed that the existence of quadratic-cubic nonlinearity in the governing coupled PDEs of motion renders hardening/softening behavior on the dynamic response of the beam. Moreover, it is noticed that any restriction on the beam mid-plane stretching will introduce nonlinear behavior in the beam’s PDEs of motion. Вивчено нелiнiйну динамiчну реакцiю нахиленої балки Тимошенка з рiзними граничними умовами пiд дiєю рухомої сили, включаючи вплив рухiв трьох типiв, зокрема руху з прискоренням, уповiльненням та сталою швидкiстю. З допомогою принципу Гамiльтона отримано нелiнiйнi зв’язанi рiвняння з частинними похiдними для вигину обертання деформованого перетину, поздовжнього та поперечного зсувiв. Для встановлення динамiчної реакцiї балки пiд дiєю рухомої сили було розв’язано отриманi нелiнiйнi зв’язанi рiвняння з використанням методу Гальоркiна. Далi динамiчну реакцiю балки було одержано з використанням технiки модального пiдсумовування. Встановленi результати було перевiрено за допомогою методу скiнченних елементiв. На наступному кроцi було проведено параметричний аналiз реакцiї балки при змiнi величини рухомої концентрованої сили, її швидкостi та граничних умов, а також чутливостi реакцiї балки на цi параметри. Було помiчено, що наявнiсть квадратичної або кубiчної нелiнiйностi у зв’язаних рiвняннях з частиними похiдними робить динамiчну реакцiю балки бiльш твердою або м’якою, а будь-якi обмеження на розтягування середньої площини вводять нелiнiйнiсть у рiвняння руху балки. 2013 Article Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity / A. Mamandi, M.H. Kargarnovin, S. Farsi // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 385-407. — Бібліогр.: 21 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177129 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
This study focuses on the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a moving force under the influence of three types of motions, including accelerating, decelerating and constant velocity types of motion, respectively. The beam’s nonlinear governing coupled partial differential equations (PDEs) of motion for the bending rotation of warped cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. To obtain the dynamic response of the beam under the action of a moving force, the derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic response is obtained using mode summation technique. Furthermore, the calculated results are verified with those obtained by finite element method (F.E.M.) analysis. In the next step a parametric study on the response of the beam is conducted by changing the magnitude of the traveling concentrated force, its velocity and beam’s boundary conditions and likewise their sensitivity on the beam’s dynamic response are studied, respectively. It is observed that the existence of quadratic-cubic nonlinearity in the governing coupled PDEs of motion renders hardening/softening behavior on the dynamic response of the beam. Moreover, it is noticed that any restriction on the beam mid-plane stretching will introduce nonlinear behavior in the beam’s PDEs of motion. |
| format |
Article |
| author |
Mamandi, A. Kargarnovin, M.H. Farsi, S. |
| spellingShingle |
Mamandi, A. Kargarnovin, M.H. Farsi, S. Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity Нелінійні коливання |
| author_facet |
Mamandi, A. Kargarnovin, M.H. Farsi, S. |
| author_sort |
Mamandi, A. |
| title |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity |
| title_short |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity |
| title_full |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity |
| title_fullStr |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity |
| title_full_unstemmed |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity |
| title_sort |
nonlinear vibration solution of an inclined timoshenko beam under the action of a moving force with constant/non-constant velocity |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177129 |
| citation_txt |
Nonlinear vibration solution of an inclined Timoshenko beam under the action of a moving force with constant/non-constant velocity / A. Mamandi, M.H. Kargarnovin, S. Farsi // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 385-407. — Бібліогр.: 21 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
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| first_indexed |
2025-07-15T15:09:09Z |
| last_indexed |
2025-07-15T15:09:09Z |
| _version_ |
1837726066446172160 |
| fulltext |
UDC 517.9
NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO
BEAM UNDER THE ACTION OF A MOVING FORCE
WITH CONSTANT/NON-CONSTANT VELOCITY
НЕЛIНIЙНI ВIБРАЦIЙНI РОЗВ’ЯЗКИ НАХИЛЕНОЇ
БАЛКИ ТИМОШЕНКА ПIД ДIЄЮ РУХОМОЇ СИЛИ
ЗI СТАЛОЮ АБО ЗМIННОЮ ШВИДКIСТЮ
A. Mamandi
Parand Branch, Islamic Azad Univ., Tehran, Iran
e-mail: am_2001h@yahoo.com
M. H. Kargarnovin
Sharif Univ. Technology, Tehran, Iran
S. Farsi
Tarbiat Modares Univ., Tehran, Iran
This study focuses on the nonlinear dynamic response of an inclined Timoshenko beam with different
boundary conditions subjected to a moving force under the influence of three types of motions, inclu-
ding accelerating, decelerating and constant velocity types of motion, respectively. The beam’s nonlinear
governing coupled partial differential equations (PDEs) of motion for the bending rotation of warped
cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. To obtain
the dynamic response of the beam under the action of a moving force, the derived nonlinear coupled
PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic response is obtai-
ned using mode summation technique. Furthermore, the calculated results are verified with those obtained
by finite element method (F.E.M.) analysis. In the next step a parametric study on the response of the
beam is conducted by changing the magnitude of the traveling concentrated force, its velocity and beam’s
boundary conditions and likewise their sensitivity on the beam’s dynamic response are studied, respecti-
vely. It is observed that the existence of quadratic-cubic nonlinearity in the governing coupled PDEs of
motion renders hardening/softening behavior on the dynamic response of the beam. Moreover, it is noti-
ced that any restriction on the beam mid-plane stretching will introduce nonlinear behavior in the beam’s
PDEs of motion.
Вивчено нелiнiйну динамiчну реакцiю нахиленої балки Тимошенка з рiзними граничними умова-
ми пiд дiєю рухомої сили, включаючи вплив рухiв трьох типiв, зокрема руху з прискоренням,
уповiльненням та сталою швидкiстю. З допомогою принципу Гамiльтона отримано нелiнiй-
нi зв’язанi рiвняння з частинними похiдними для вигину обертання деформованого перетину,
поздовжнього та поперечного зсувiв. Для встановлення динамiчної реакцiї балки пiд дiєю рухо-
мої сили було розв’язано отриманi нелiнiйнi зв’язанi рiвняння з використанням методу Гальор-
кiна. Далi динамiчну реакцiю балки було одержано з використанням технiки модального пiдсу-
мовування. Встановленi результати було перевiрено за допомогою методу скiнченних елемен-
тiв. На наступному кроцi було проведено параметричний аналiз реакцiї балки при змiнi величи-
ни рухомої концентрованої сили, її швидкостi та граничних умов, а також чутливостi реакцiї
балки на цi параметри. Було помiчено, що наявнiсть квадратичної або кубiчної нелiнiйностi
у зв’язаних рiвняннях з частиними похiдними робить динамiчну реакцiю балки бiльш твердою
c© A. Mamandi, M. H. Kargarnovin, S. Farsi, 2013
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3 385
386 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
або м’якою, а будь-якi обмеження на розтягування середньої площини вводять нелiнiйнiсть у
рiвняння руху балки.
1. Introduction. Under actual operating conditions, the linear and nonlinear vibration analysis
of structural elements, such as strings, rods, beams, plates and shells traveling by a moving
mass/force is of considerable practical importance to the structural and railway engineers. For
over a century many analytical and numerical methods have been proposed to investigate the
dynamic behavior of different engineering structures. However, until now almost no attention
has been paid to the study of nonlinear dynamic analysis for the coupled bending rotation of
the warped cross-section, longitudinal and transverse deflections of an inclined Timoshenko
beam subjected to a moving force. Some practical examples for such behavior are a bridge
when traveled by moving vehicles or trains, an overhead traveling crane moving on its girder,
a beam subjected to pressure waves, simulation of high axial speed machining operations and
internal two-phase flow in piping systems. Further application on the subject of vibrations of
the inclined beams can be addressed in the aerospace and armed force industries such as rocket
launcher systems and cannon tubes.
It should be emphasized that from a mechanical component design point of view, inclusion
of acceleration/deceleration character of moving mass/force certainly plays a significant role
in the final results. This becomes more critical when one deals, for example, with take-off and
landing phase of aircrafts on runways/angle decks (flight decks) of warship aircraft carriers,
automobiles and locomotives at take off or sudden brake on roadways/highway bridges and
on rails/railway bridges, respectively. Furthermore, frequent braking and accelerating of rail-
guided cranes play important role in design steps of these types of structures.
Related to the vibration analysis of Euler – Bernoulli beam either under motion of traveling
force or traveling mass numerous works are reported [1 – 12]. Similarly, one can find number
of different studies on the dynamical behavior of Timoshenko beams subjected to motion of
moving load and mass [7, 10, 11, 13 – 18]. Nonlinear dynamic analysis of Euler – Bernoulli and
Timoshenko beams under the action of either moving force or mass are specifically investigated
in [4, 8, 9, 12, 14, 16 – 18].
From the experimental point of view, it appears that as the amplitude of vibration increases,
nonlinear effects come into play; therefore by considering that the source of nonlinearity may
be either inertial, geometric or material in nature, the influence of such terms on the beam
dynamic behavior should also be included (see Refs. [8, 14, 19]). In this paper, attention is paid
to geometric nonlinearity which may be caused by large curvatures and nonlinear stretching
of the mid-plane of an inclined Timoshenko beam. In general, due to existence of nonlinear
terms, the exact analytical (closed form) solutions for governing equations of motion are not
available.
In this study three nonlinear governing coupled PDEs of motion for the bending rotati-
on of warped cross-section, transverse and longitudinal vibrations of an inclined Timoshenko
beam under the action of a moving force are derived using Hamilton’s principle. Then by apply-
ing Galerkin’s method three obtained nonlinear second order ordinary differential equations
(ODEs) governing modal equations can be solved numerically using the Adams – Bashforth –
Moulton integration method via MATLAB solver package. It should be noted that in the
present study the nonlinear effects of axial strain, bending curvature and shear strain on the
dynamic responses of the inclined Timoshenko beam are all considered using the von-Karman
strain-displacement relation in conjunction with the moderately large deflection theory.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 387
Fig. 1. (a) A lateral force traveling on an inclined pinned-pinned beam. (b) Acting forces
on the elastic beam at the contact point in the equivalent moving force model.
In extending the issue of moving mass further to a more applicable study we believe that the
same problem but under motion of the moving force has its own importance in this filed. Based
on this postulate and on the same line to the other studies by the same authors [18] this study
is initiated. It should be mentioned that there is not going to be any novelty on the solution
technique in this paper with respect to the some previous works but what makes this work new
is related to the very important outcome results from the application point of view. Briefly, it
should be pointed out that the main contribution and significant technical advantages of this
paper is to present some tangible results which have not been reported in the earlier published
papers.
2. Mathematical modeling. An inclined Timoshenko beam with length l and inclination
angleϕ traveled by a concentrated forceF with velocity v and constant acceleration/deceleration
a is considered (see Fig. 1(a)). The longitudinal and lateral components of the force with which
the traveling force acts on the beam are F cosϕ and F (sinϕ+µ cosϕ), respectively, as shown in
Fig. 1(b). In our upcoming analysis, when the load enters the left end of the beam, zero initial
conditions for the beam are assumed, i.e., the beam is at rest at time t = 0. It is further assumed
that the external damping is not negligible and the damping behavior follows the viscous nature
[2, 8, 10, 11]. Moreover, the beam deforms within the linear elastic range and therefore Hooke’s
laws are prevailing. In this study the von-Karman’s moderately large strain-displacement relati-
ons are used as [14, 18]:
εxx = u,x +
1
2
w2
,x, w,x = ψ + γ and κ = ψ,x (1)
in which u = u(x, t) is the axial longitudinal time-dependent in-plane displacement, w =
= w(x, t) is the time-dependent transverse deflection of the beam measured upward from its
equilibrium position when unloaded and ψ = ψ(x, t) is the time-dependent rotation of the
warped cross-section of the beam due to the bending. The subscripts (, t) and (, x) stand for the
derivative with respect to the time (t) and spatial coordinate (x) to the related order, respecti-
vely. In addition, ε, γ and κ are the longitudinal (or normal) strain, shear strain and curvature
at the center line of the Timoshenko beam, respectively. To obtain the governing differenti-
al equations of motion by applying Hamilton’s principle the kinetic energy, K, and the strain
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
388 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
energy, U, of the beam are:
K =
1
2
l∫
0
ρAw2
,t dx+
l∫
0
ρAu2,t dx+
l∫
0
ρIdψ,t
2 dx
, (2a)
U =
1
2
l∫
0
EAε2xx dx+
l∫
0
EIdκ
2 dx+
l∫
0
kGAγ2 dx
. (2b)
The total external virtual work done by the traveling force with variable velocity, frictional force
and external viscous damping forces acting on the beam is:
δWe = −
l∫
0
(F (sinϕ+ µ cosϕ)δu+ F cosϕδw + c1ψ,tδψ + c2w,tδw + c3u,tδu)|x=ζ(t) dx (3)
in which c1ψ,t, c2w,t and c3u,t are regarded as the external viscous damping forces applied on
the beam [2, 8 – 11]. Now, we can establish the Lagrangian function of the system as: L =
= K − (U −We). Applying Hamilton’s principle on L as: δ
∫ t2
t1
Ldt = 0 or
δ
t2∫
t1
(U −K) dt =
t2∫
t1
δWe dt. (4)
By doing some mathematics, one would get the nonlinear governing coupled PDEs of motion
(EOMs) as well as the boundary conditions (BCs) for the problem at hand as follows:
the moment relation in ψ direction,
ρIdψ,tt − EIdψ,xx − kGA(w,x − ψ) + c1ψ,t = 0, (5)
the force relation in z direction,
ρAw,tt + kGA
(
ψ,x − w,xx)− EA(u,xxw,x + u,xw,xx +
3
2
w,xxw
2
,x
)
+
+ c2w,t = −F cosϕδ(x− ζ(t))χ(t), (6)
the force relation in x direction:
ρAu,tt − EA(u,xx + w,xw,xx) + c3u,t = −F (sinϕ+ µ cosϕ)δ (x− ζ(t))χ(t), (7)
and the boundary conditions at both ends for the Timoshenko beam are
i- either M = EIdψ,x or ψ prescribed; (8a)
ii- either Q = kGA(w,x − ψ) + EA
(
u,xw,x +
1
2
w3
,x
)
or w prescribed; (8b)
iii- either N = EA
(
u,x +
1
2
w2
,x
)
or u prescribed, (8c)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 389
where ρ is the beam density, A is the cross-sectional area of the beam, Id is the beam’s cross-
sectional second moment of inertia, E is Young’s modulus of elasticity, G is the shear modulus,
k is the shear correction factor, EId is the beam’s flexural rigidity, ρA is the beam’s mass per
unit length, F is the magnitude of the traveling force, µ is the kinetic frictional coefficient,
c1, c2 and c3 are external damping constants related to viscous damping of the beam, namely
η, also M, Q and N are the bending (flexural) moment, shear force and axial (longitudinal)
tension/compression force of the beam, respectively. Furthermore, δ(x − ζ(t)) is Dirac’s delta
function in which ζ(t) is the instantaneous position of the moving force with the velocity v and
constant acceleration/deceleration a on the beam such that ζ(t) = x0 + vt or ζ(t) = x0 + vt+
+1/2at2 for describing the constant velocity or accelerating/decelerating types of motion of the
traveling force, respectively, where x0 is the initial point of application of the force on the beam,
v is the traveling force entrance/exit velocity and a is the constant acceleration/deceleration of
traveling force on the beam, χ(t) is the pulse function which is equal to one while the force is
traveling on the beam, and zero when the traveling force is outside the beam span; for example
in the case of constant velocity type of motion that is described by χ(t) = u(t) − u(t − l/v),
in which u(t) represents the unit step function. The derivation of nonlinear governing coupled
PDEs of motion is rather lengthy and for brevity its details will not be given here.
3. Solution method. In this study Galerkin’s method is chosen as a powerful computational
tool to analyze the vibrations of an inclined Timoshenko beam. Based on the separation of
variables technique, the response of Timoshenko beam in terms of the linear free-oscillation
modes can be assumed as follows [18, 21]:
w(x, t) =
n∑
j=1
φj(x)pj(t) = ΦT (x)P(t), (9)
ψ(x, t) =
n∑
j=1
τj(x)qj(t) = ΓT (x)Q(t), (10)
u(x, t) =
n∑
j=1
θj(x)rj(t) = ΘT (x)R(t), (11)
where P(t), Q(t) and R(t) are vectors of order n listing the generalized coordinate pj(t), qj(t)
and rj(t), respectively, and Φ(x), Γ(x) and Θ(x) are some vector functions collecting the first n
mode shapes (eigen-functions) of φj(x), τj(x) and θj(x), respectively.
By substituting Eqs. (9), (10) and (11) into Eqs. (5), (6) and (7), pre-multiplying both sides
of Eqs. (5), (6) and (7) by ΓT (x), ΦT (x) and ΘT (x), respectively, integrating over the interval
(0, l) and imposing the properties of the Dirac’s delta function, the resulting nonlinear coupled
modal equations of motion in matrix form are as follows:
ρId
n∑
j=1
Sij q̈j(t) + 2ρIdηωi
n∑
j=1
Sij q̇j(t)−
n∑
j=1
[EIdKij − kGASij ] qj(t)−
− kGA
n∑
j=1
Eijpj(t) = 0, (12)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
390 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
ρA
n∑
j=1
Mij p̈j(t) + 2ρAηωi
n∑
j=1
Mij ṗj(t)− kGA
n∑
j=1
Hijpj(t) + kGA
n∑
j=1
Fijqj(t)−
− EA
n∑
j=1
n∑
k=1
rj(t)Gijkpk(t)− EA
n∑
j=1
n∑
k=1
pj(t)Tijkrk(t)−
− 3
2
EA
n∑
j=1
n∑
k=1
pj(t)Iijkpk(t)
2 = −F cosϕχ(t)bi(t), (13)
and
ρA
n∑
j=1
Jij r̈j(t) + 2ρAηλli
n∑
j=1
Jij ṙj(t)− EA
n∑
j=1
Nijrj(t)−
− EA
n∑
j=1
n∑
k=1
pj(t)Lijkpk(t) = −F [sinϕ+ µ cosϕ]χ(t)di(t) (14)
for i = 1, 2, . . . , n, in which the matrices S, K, L, M, N, I, J, T, E, F, G and H are defined as:
(M)ij =
l∫
0
φi(x)φj(x) dx, (H)ij =
l∫
0
φi(x)φ′′j (x) dx,
(F)ij =
l∫
0
φi(x)τ ′j(x) dx, (S)ij =
l∫
0
τi(x)τj(x) dx,
(K)ij =
l∫
0
τi(x)τ ′′j (x) dx, (E)ij =
l∫
0
τi(x)φ′j(x) dx,
(J)ij =
l∫
0
θi(x)θj(x) dx, (N)ij =
l∫
0
θi(x)θ′′j (x) dx,
(G)ijk =
l∫
0
φi(x)θ′′j (x)φ′k(x) dx, (I)ijk =
l∫
0
φi(x)φ′′j (x)φ′k
2
(x) dx,
(L)ijk =
l∫
0
θi(x)φ′′j (x)φ′k(x) dx, (T)ijk =
l∫
0
φi(x)φ′′j (x)θ′k(x) dx, (15)
where i, j, k = 1, 2, 3, . . . , n.
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 391
Prime and dot marks over any parameter indicate the derivative with respect to the position
(x) and time (t), respectively. Furthermore, the n× 1 column vectors of b and d are defined as
(b)i = φi (x = ζ(t)) and (d)i = θi (x = ζ(t)) . (16)
It is clear that Eqs. (12), (13) and (14) are three nonlinear coupled second-order ordinary dif-
ferential equations (ODEs). The boundary conditions for a pinned-pinned Timoshenko beam
with fixed end supports are [18, 20, 21]:
essential BCs :
{
u(0, t) = u(l, t) = 0 ⇒ θj(x) = 0 at x = 0 and l,
w(0, t) = w(l, t) = 0 ⇒ φj(x) = 0 at x = 0 and l;
(17)
natural BCs : M(0, t) = M(l, t) = 0 ⇒ τj,x(x) = 0 at x = 0 and l.
The boundary conditions for a clamped-pinned Timoshenko beam with immovable end supports
are [18, 20, 21]:
essential BCs :
u(0, t) = u(l, t) = 0 ⇒ θj(x) = 0 at x = 0 and l,
w(0, t) = w(l, t) = 0 ⇒ φj(x) = 0 at x = 0 and l,
ψ(0, t) = 0 ⇒ τj(x) = 0 at x = 0;
(18)
natural BCs : M(l, t) = 0 ⇒ τj,x(x) = 0 at x = l.
And the boundary conditions for a clamped-free Timoshenko beam are as follows [18, 20, 21]:
essential BCs :
u(0, t) = 0 ⇒ θj(x) = 0 at x = 0,
w(0, t) = 0 ⇒ φj(x) = 0 at x = 0,
ψ(0, t) = 0 ⇒ τj(x) = 0 at x = 0;
(19)
natural BCs :
M(l, t) = 0 ⇒ τj,x(x) = 0 at x = l,
Q(l, t) = 0 ⇒ kGA(w,x − ψ) = 0 at x = l,
N(l, t) = 0 ⇒ EA(u,x) = 0 at x = l.
Moreover, Initial Conditions (ICs) for the Timoshenko beam are
ICs : u(x, 0) = u,t(x, 0) = w(x, 0) = w,t(x, 0) = ψ(x, 0) = ψ,t(x, 0) = 0. (20)
In Eq. (14), λli denotes the natural angular frequency (rad/s) of longitudinal vibration of the
beam related to its type of boundary condition. For the either pinned-pinned or clamped-
pinned beam it is in the form of
iπ
l
√
E/ρ [18, 20, 21], and for a cantilever beam is given by
(2i− 1)π
2l
√
E/ρ [18, 20, 21], where i = 1, 2, 3, . . . , n.
To solve Eqs. (12), (13) and (14), all entries in the matrices in Eqs. (15) and (16) should
be calculated. Herein, by inspection it can be seen that the following functions (mode shapes)
for φj(x), τj(x) and θj(x) will satisfy both the linearized equations of motion of the beam and
different types of boundary conditions as the following [13, 18, 20, 21]:
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392 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
(i) For a pinned-pinned Timoshenko beam, the normal modes are expressed as follows:
φj(x) = D sin(bBξ),
τj(x) = H cos(bBξ), (21a)
and θj(x) = sin(jπξ), ξ =
x
l
, with j = 1, 2, 3, . . . , n,
in which the frequency equation is
sin bB = 0, (21b)
where, in this case, sin bB = nπ, with n = 1, 2, 3, . . . .
(ii) For a clamped-pinned Timoshenko beam, the normal modes are
φj(x) = D[cosh bAξ − coth bA sinh bAξ − cos bBξ + cot bB sin bBξ],
τj(x) = H
[
cosh bAξ +
σ
λZ
sinh bAξ − cos bBξ + σ sin bBξ
]
, (22)
and θj(x) = sin(jπξ), ξ =
x
l
, with j = 1, 2, 3, . . . , n,
in which the frequency equation is
λZ tanh bA− tan bB = 0. (23)
(iii) For a clamped-free Timoshenko beam, the normal modes are
φj(x) = D[cos bAξ − λZ∆ sinh bAξ − cos bBξ + ∆ sin bBξ],
τj(x) = H
[
cosh bAξ +
σ
λZ
sinh bAξ − cos bBξ + σ sin bBξ
]
, (24)
and θj(x) = sin[(j − 1/2)πξ], ξ =
x
l
, with j = 1, 2, 3, . . . , n,
in which the frequency equation is given by
2 + [b2(r2 − s2)2 + 2] cosh bA cos bB − b(r2 + s2)
(1− b2r2s2)1/2
sinh bA sin bB = 0, (25)
where, in Eqs. (21) – (25),
b2 =
ρAl4
EId
ω2, r2 =
Id
Al2
, s2 =
EId
kGAl2
, Z =
B2 − s2
A2 + s2
, λ =
A
B
,
A,B =
1√
2
{
∓(r2 + s2) + [(r2 − s2)2 +
4
b2
]
1
2
} 1
2
, (26)
σ = − λ sinh bA+ sin bB
1
Z cosh bA+ cos bB
, ∆ =
1
λ sinh bA− sin bB
Z cosh bA+ cos bB
,
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 393
in which Di and Hi, i = 1, 2, 3, . . . , n, are normal modal amplitudes which depend on natural
frequencies of the Timoshenko beam related to the type of BCs. Moreover, it should be poi-
nted out that ωi is the natural angular frequency (rad/s) of transversal/bending slope warping
vibration of the beam which depends on the type of boundary condition and can be obtained
from Eq. (26) [18, 20].
By inserting the corresponding normal modes for any type of boundary condition of the
beam into the Eqs. (15) and (16) all entries in all matrices can be calculated. In the next
step these evaluated matrices will be inserted in the Eqs. (12), (13) and (14) and the later
set of equations can be solved numerically using the Adams – Bashforth – Moulton integrati-
on method via MATLAB solver package out of which the values of pn(t), qn(t) and rn(t) can
be obtained. By back substitution of pn(t), qn(t) and rn(t) in Eqs. (9), (10) and (11), u(x, t),
w(x, t) and ψ(x, t) are established, respectively [18, 21].
In the next step based on the obtained values for u(x, t), w(x, t) and ψ(x, t), the dynamic
response of an inclined Timoshenko beam having three different types of boundary conditions
including pinned-pinned, clamped-pinned and clamped-free under influence of three types of
force motions of a) accelerating, b) decelerating and c) uniform velocity motion are obtained.
The obtained results for the beam response under each of those three types of force motions
and boundary conditions are shown separately in the Figs. 4 to 14. The detailed kinematical
discussions of the above different motions are described below [18].
a) In the case of constant accelerating type of motion (ζ(t) = x0+v0t+1/2at2, a = const >
> 0), it is assumed that the beam is at rest when the force F enters the beam at x0 = 0 and
t0 = 0 with initial velocity v0 = 0 and it arrives to the other end of the beam, i.e., x = l with
final velocity v. The total traveling time in the beam span, t1, and force exit velocity v will be:
t1 =
2l
v
, (27a)
v =
√
2al. (27b)
b) For constant decelerating type of motion (ζ(t) = x0 + v0t+ 1/2at2, a = const < 0), it is
also assumed that the beam is at rest when the force F enters the beam at x0 = 0 and t0 = 0
with entrance velocity v0 (non-zero initial velocity) and it stops (v = 0) at the other end of the
beam, i.e., x = l. The total traveling time in the beam span, t2, and force entrance velocity v0
will be:
t2 =
2l
v0
, (28a)
v0 =
√
2l|a|. (28b)
c) For uniform velocity type of motion (ζ(t) = x0 + v0t, v0 = const > 0), it is also assumed
that the beam is at rest when the force F enters the beam at x0 = 0 and t0 = 0 with force
constant velocity v0 and it reaches to the other end of the beam, i.e., x = l at instant t3. The
total traveling time in the beam span, t3 will be:
t3 =
l
v0
. (29)
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394 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
Fig. 2. Instantaneous normalized vertical displacement under a moving force
of F, (—) linear analysis using present study, (– · –) linear analysis: Ref [13].
4. Verification of results and case studies. As mentioned in the introduction at the moment
no specific results exist for the problem under consideration in the literature. Therefore, to
verify the validity of the obtained results in this study we have to consider some special cases
of our study to be compared with those existing in the literatures.
4.1. Verification of results in the linear case. In the first attempt we set the higher order
terms, i.e.,
(
u,xxw,x + u,xw,xx +
3
2
w,xxw,x
2
)
and (u,xx +w,xw,xx) of the left-hand sides of Eqs.
(6) and (7), respectively, equal to zero for a constant velocity type of motion (a = 0). Further-
more, referring to Eqs. (12) to (14) we set: c1 = c2 = c3 = 0, i.e., η = 0, µ = 0 and ϕ = 0.
This led us to a set of relations for u(x, t), w(x, t) and ψ(x, t) referring to the case known for
linear analysis for a horizontal undamped Timoshenko beam. To establish verifications of our
analysis, we consider the data given in [13] as: l = 1m, E = 207 × 109N/m2, G = 77.6 ×
×109N/m2, k = 0.9, ρ = 7700 kg/m3, Id = 6.236 × 10−5m4, A = 0.02736m2, F = 0.2ρg ×
×Al(N), α = 0.11 and β0 = 0.15, in which β0 = πr0/l is Rayleigh’s slenderness coefficient with
r0 taken as the radius of gyration of the beam. Based on the above data, the computer code was
run for the linear case, and the vertical displacement (w) of the instantaneous positions of the
moving force is calculated and the dimensionless outcome results are depicted and compared
with other existing results in Fig. 2. The normalization factor for the vertical displacement is
wst = Fl3/48EId, which is a mid point deflection of a simply supported beam under mid-span
concentrated load of F. A close inspection of the curves in the Fig. 2 indicates good agreements
between the two results.
4.2. Verification of results in the nonlinear case. As described earlier, in this study to extend
the validity of our obtained results we prepared appropriate APDL (ANSYS Parametric Design
Language) routine in the environment of the ANSYS software to simulate the response of a
moving force on an inclined Timoshenko beam. Then, the linear and nonlinear FEM solutions
were compared with those obtained by the linear and nonlinear analytical solutions applying
the mode summation technique. In the modeling of the Timoshenko beam we used BEAM-188
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 395
element defined in this software, which is suitable for analyzing beam structures. This element
is a 2-D 2-noded second order beam element having 6 (or 7) DOFs with 3-translational DOF
and 3-rotational DOF in each node. Now, to establish our calculations we consider an inclined
undamped Timoshenko beam with geometry and mechanical properties listed in Table 1.
Table 1. Geometric and material properties of considered inclined beam [18]
Parameter Symbol Value
Beam’s cross-sectional area (m2) A (b× h) 5× 10−3 (0.05× 0.1)
Beam’s length (m) l 6
Young’s modulus (N/m2) E 207× 109
Shear modulus (N/m2) G 77.6× 109
Beam’s density (kg/m3) ρ 7850
Poisson’s coefficient ν 0.25
Shear correction factor k 0.85
Gravitational acceleration (m/s2) g 9.81
Damping coefficient η 0.033
Kinetic frictional coefficient µ 0.2
Fig. 3 illustrates the variation of the mid-point deflection w (m) of an inclined undamped
pinned-pinned Timoshenko beam with ϕ = 36◦ vs. non-dimensional time vt/l at velocity ratio
α = 0.25 for the traveling force F = 2ρgAl (N) under influence of constant velocity motion
using FEM analysis and analytical analysis, respectively. From this figure, one can conclude
that the results for the beam’s mid-point lateral dynamic displacement obtained by FEM and
analytical solutions in either of the nonlinear or the linear analysis are almost the same, which
shows very good agreement between these analytical results obtained via mode summation
technique and FEM analysis. The suitable number of elements which has been used for the
beam to converge the linear/nonlinear results is 80 elements.
4.3. Results and discussions. In the all following case studies data given in Ref. [18] as well
as those beam listed in Table 1 are taken into consideration.
To clarify the results and in order to have a better insight on interpreting the variation
of the obtained results we tried to present the results in dimensionless forms. So we begin
with defining the dimensionless dynamic deflection w(xmax, t)/w0 and the dimensionless time
parameter t/ti, i = 1, 2, and 3 where w0 and xmax (= x|wmax
) denote the maximum static
deflection and the point on the beam that corresponds to this deflection, respectively. It should
be noticed that w(xmax, t) is obtained from dynamic analysis of the governing equations of
motion at xmax. Table 2 shows the values of w0 and position of xmax due to the lateral force, i.e.,
F, for three different types of the beam’s boundary conditions.
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396 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
Fig. 3. Variation of mid-point deflection w (m) vs. normalized time vt/l for an inclined
pinned-pinned Timoshenko beam with ϕ = 36◦ affected by a moving force
F = 2ρgAl(N) under constant velocity motion at α = 0.25 using analytical
and FEM analyses for linear and nonlinear solutions: (—) analytical linear
solution, (– –) analytical nonlinear solution, (—◦— ) ANSYS linear solution
(80 elements used), (—O— ) ANSYS nonlinear solution (80 elements used).
Table 2. Values of xmax (= x|wmax
) and w0 due to the applied lateral force F for different boundary conditions
of a beam [2, 18]; (ϕ is the beam’s inclination angle)
Beam geometry B.C.s
Position of applied
lateral force F ;
xmax(= x|wmax
)
Absolute value
of w0
pinned-pinned x = 0.5l w0 = Fl3 cosϕ
48EId
clamped-pinned x = 0.55l w0 = Fl3 cosϕ
48
√
5EId
clamped-free x = l w0 = Fl3 cosϕ
3EId
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 397
Fig. 4. Variation of dimensionless dynamic lateral deflection (wnl(xmax, t)/w0) at xmax = 0.5l vs. normalized
time (t/ti) for an inclined pinned-pinned Timoshenko beam (ϕ = 30◦) traversed by a moving force
of F = ρgAl(N) with different velocity ratios (α = 0.25, 0.5, 0.75 and 1) under influence of three
types of motion; (a – c) dimensionless results for nonlinear analysis: wnl(xmax, t)/w0, (d – f) dimension-
less difference between nonlinear and linear analysis in percent: (wnl − wl)/w0.
Moreover, let’s define the velocity ratio as α = v/vcr. For a Timoshenko beam, vcr is
the critical velocity of a concentrated moving force on this beam defined in general form as
(vcr)Timo = ω1l/π [10, 11, 18], where ωi is the any natural angular frequency (rad/s) of trans-
versal/bending slope warping vibration of this beam given as: (ωi)Timo =
bi
l2
√
EId
ρA
[18, 20]
with i = 1, 2, . . . , n (see Eq. (26); also please see [10, 11, 18]. Note that bi depends on the
type of boundary condition and for example for a simply supported Timoshenko beam, b1 =
= π/B1, bi = iπ/Bi, i = 1, 2, . . . , n; consequently the critical velocity of example first mode
is (vcr)Timo = (1/lB1)
√
EId/ρA [10, 11, 18]. On the other hand, for only a simply supported
kind of boundary condition a modified formula for (vcr)Timo is given in [15, 18]. For this case
of boundary condition, the difference between (vcr)Euler = (π/l)
√
EId/ρA [8, 10, 13, 18] and
(vcr)Timo [10, 11, 15, 18] is about 0.03%. However, in general, we prefer to use (vcr)Timo [10, 11,
15, 18] in our up-coming calculations for each type of BC.
It should be mentioned that based on the previous analysis and obtained results which
reveal that the friction force is very small [6], we neglect the effect of friction in the follo-
wing case studies. Figs. 4 – 6 illustrate the variation of dimensionless dynamic lateral deflection
(w(xmax, t)/w0) vs. dimensionless time t/ti, i = 1, 2 and 3, at reference point xmax (= x|wmax
)
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398 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
Fig. 5. Variation of dimensionless dynamic lateral deflection (wnl(xmax, t)/w0) at xmax = 0.55l vs. normalized
time (t/ti) for an inclined clamped-pinned Timoshenko beam (ϕ = 30◦) traversed by a moving force
of F = ρgAl(N) with different velocity ratios (α = 0.25, 0.5, 0.75 and 1) under influence of three
types of motion; (a – c) dimensionless results for nonlinear analysis: wnl(xmax, t)/w0, (d – f) dimensi-
onless ifference between nonlinear and linear analysis in percent: (wnl − wl)/w0.
for pinned-pinned, clamped-pinned and clamped-free Timoshenko beam, respectively, with
inclination angle ϕ = 30◦ traversed by a moving force F = ρgAl (N) with different velocity
ratios (α = 0.25, 0.5, 0.75 and 1) under influence of three types of motion. In the aforementi-
oned figures, the depicted results on the left, middle and right columns are related to the cases
of accelerating, decelerating and constant velocity motion, respectively. Moreover, in those fi-
gures, the first and the second rows show the dimensionless results obtained from the nonlinear
analysis (wnl(xmax, t)/w0) and percentage of dimensionless difference between the nonlinear
and the linear analysis ((wnl − wl)/w0), respectively.
As can be seen from Figs. 4(a – c) in the decelerating and uniform velocity types of moti-
on, by increasing the velocity ratio α up to α = 0.75 and α = 0.5, respectively, the value of
maximum dynamic deflection increases and decreases afterwards, respectively, whereas in the
accelerating type of motion an increasing trend can be seen for the maximum dynamic deflecti-
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 399
Fig. 6. Variation of dimensionless dynamic lateral deflection (wnl(xmax, t)/w0) at xmax = l vs. normalized
time (t/ti) for an inclined clamped-free Timoshenko beam (ϕ = 30◦) traversed by a moving force
of F = ρgAl(N) with different velocity ratios (α = 0.25, 0.5, 0.75 and 1) under influence of three
types of motion; (a – c) dimensionless results for nonlinear analysis: wnl(xmax, t)/w0, (d – f) dimensi-
onless difference between nonlinear and linear analysis in percent: (wnl − wl)/w0.
on of the beam’s mid-span. Moreover, in the decelerating type of motion the range of variation
of the maximum dynamic deflection is larger with respect to other two types of motion. From
Figs. 4(d – f), it is concluded that the maximum difference for the beam mid-span deflection
between nonlinear and linear analysis happens primarily in the decelerating motion and with
smaller difference in the case of uniform velocity and then in the accelerating type of motion,
respectively.
It can be observed from Figs. 5(a – c) that in the accelerating type of motion by increasi-
ng the velocity ratio α the values of maximum dynamic downward deflection always increase,
whereas in the decelerating and uniform velocity types of motion the value of maximum dynamic
downward deflection increases up to α = 0.5 and α = 0.25, respectively, and decreases after
these points. Also, it is seen from Fig. 5(a – c) that in the decelerating type of motion when
the moving force is at the right end of the beam, i.e., x = l, the maximum dynamic upward
(positive) deflection at xmax = 0.55 l which happens at α = 1 is the greatest value with respect
to the other two types of motion. Moreover, it can be observed from Figs. 5(d – f) that the
maximum difference at xmax = 0.55l for the beam deflection between nonlinear and linear
analysis happens primarily in the decelerating motion and with smaller difference in the case of
accelerating and then in the uniform velocity type of motion, respectively.
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400 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
(a) (b)
(c) (d)
Fig. 7. Variation of the dynamic response of a point at x = 0.5 l of an inclined pinned-pinned Timoshenko
beamvs. different values of the moving force for different velocity ratios in the uniform
velocity motion; (a) α = 0.25, (b) α = 0.5, (c) α = 0.75, (d) α = 1; (—) nonlinear solu-
tion, (– · –) linear solution.
It can be observed from Figs. 6(a – c) that in the decelerating type of motion by increasing
the velocity ratio α the value of maximum dynamic downward deflection at x = l on the beam
almost increases, whereas in the accelerating type of motion the reduction trend is always seen
for the maximum dynamic downward deflection. Furthermore, in the uniform velocity type
of motion by increasing the velocity ratio α up to α = 0.5, the value of maximum dynamic
downward deflection increases and vice versa afterwards. In addition, it can be observed from
Figs. 6(d – f) that the maximum difference between nonlinear and linear analysis happens pri-
marily in the decelerating motion and with smaller difference in the case of accelerating and
then in the uniform velocity type of motion, respectively. Moreover, in this type of boundary
condition, the differences are much lower than the one in the other two types of boundary
conditions.
In Figs. 7 and 8 the absolute values of maximum dynamic response (wmax(x)) variation of
a point at x = 0.5 l and 0.55l, respectively, on an inclined pinned-pinned and clamped-pinned
Timoshenko beam with ϕ = 30◦ vs. different values of the moving force is shown for vari-
ous velocity ratios of α = 0.25, 0.5, 0.75 and 1, respectively, using both linear and nonlinear
solutions in the uniform velocity type of motion, respectively. It can be seen from Fig. 7 that
the maximum dynamic deflection of the nonlinear analysis is always lower than the one obtai-
ned from the linear solution. The hardening behavior is seen in this case as reported in other
works [16, 18]. Also, the dynamic mid-point displacements of such beam using linear and nonli-
near solutions are almost the same for the value of F ≤ 0.5ρgAl(N). However, after this point,
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 401
(a) (b)
(c) (d)
Fig. 8. Variation of the dynamic response of a point at x = 0.55 l of an inclined clamped-pinned Timoshenko
beam vs. different values of the moving force for different velocity ratios in the uniform
velocity motion; (a) α = 0.25, (b) α = 0.5, (c) α = 0.75, (d) α = 1; (—) nonlinear solu-
tion, (– · –) linear solution.
the magnitude of the wmax of the nonlinear and the linear solutions differs gradually and the
difference grows rapidly as the value of F increases. Furthermore, from Fig. 7, the difference
between the linear and the nonlinear solutions has an increasing trend up to the load velocity
ratio of α = 0.5 and a reverse trend afterwards. The maximum difference between linear and
nonlinear solutions for all cases in this figure occurs at F = 3ρgAl(N) at α = 0.5 (see Fig. 7(b)).
Also, the variation of the linear solution mathematically follows a linear trend in this figure [8,
16].
It can be observed from Fig. 8 that both the hardening (stiffening) behavior in the lower
velocity ratio (α = 0.25) and the softening behavior in the higher velocity ratios (α = 0.5, 0.75
and 1) can be predicted in this type of boundary condition of the beam under the action of a
moving force. Also, the absolute values of maximum dynamic deflection (wmax(x = 0.55l))
of a point at xmax = 0.55 l of an inclined clamped-pinned Timoshenko beam using linear and
nonlinear solutions are almost the same no matter what values of F might be. In addition, the
difference between the linear and the nonlinear solutions has a bit increasing trend up to the
load velocity ratio of α = 0.75 and a reverse trend afterwards. In other words, the comparison
of results indicates that the trend of difference reduction between linear and nonlinear solutions
in the clamped-pinned type is much faster than the pinned-pinned type of BC (compare Figs.
7 and 8). Besides, it should be noted that the maximum difference between linear and nonlinear
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402 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
(a) (b)
(c) (d)
(e)
Fig. 9. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an inclined pinned-pinned Timo-
shenko beam with ϕ = 30◦ at different velocity ratios for different moving force value of F in the
constant velocity motion; (a) F = 0.5ρgAl(N), (b) F = ρgAl(N), (c) F = 2ρgAl(N), (d) F =
= 3ρgAl(N), (e) F = 4ρgAl(N); (—) nonlinear solution, (– · –) linear solution.
solutions for all cases in this figure occurs at F = 4ρgAl(N) at α = 0.75 (see Fig. 8(c)). The
variation of the linear solution mathematically also follows a linear trend in this type of BC.
It should be mentioned that for a clamped-free Timoshenko beam there is similar softening
behavior (for brevity results are not illustrated here) with much less intensity compared to a
clamped-pinned Timoshenko beam. It is believed that more pronounced reduction of softeni-
ng/hardening behavior can be seen when supports are allowed to slide or to be free.
Using nonlinear and linear analysis in the case of constant velocity type of motion the vari-
ation of normalized lateral dynamic displacement (w/w0) vs. x/l of an inclined pinned-pinned,
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 403
(a) (b)
(c) (d)
Fig. 10. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an inclined pinned-pinned Timo-
shenko beam with ϕ = 30◦ at different moving force size of F for different velocity ratios in the
constant velocity motion; (a) α = 0.25, (b) α = 0.5, (c) α = 0.75, (d) α = 1; (—) nonlinear solution,
(– · –) linear solution.
clamped-pinned and clamped-free Timoshenko beam by changing velocity ratios, α, and movi-
ng force value, F , are shown respectively, in Figs. 9 – 10, 11 – 12 and 13 – 14.
It can be seen from Fig. 9 that the nonlinear analysis results of w/w0 are almost lower
than the one obtained from the linear analysis. Also, when α increases the difference between
linear and nonlinear analysis decreases. This difference increases when the value of F increases.
The maximum difference between linear and nonlinear analysis happens at α = 0.25 when
F = 4ρgAl (N). The linear solutions always predict a unique value ofw/w0 in the same velocity
ratio of α no matter what values of F might be.
It can be seen from Fig. 10 that that for all velocity ratios α the linear analysis always has
an identical value for w/w0 no matter what the values of F are. Moreover, it is seen that the
point corresponding to the (wmax)linear is always in right side of the similar point on the nonli-
near analysis. And where the magnitude of moving force is small, i.e., F = 0.5 ρgAl(N) the
nonlinear and the linear solutions are almost the same no matter what the values of α might
be. However, after this point (F > 0.5ρgAl(N)) the difference between linear and nonlinear
solutions becomes more pronounced, and consequently the difference between the linear and
the nonlinear solution of maximum value of deflection increases accordingly. Furthermore, by
increasing the velocity ratio up to α = 0.5 the maximum dynamic deflection of linear and nonli-
near solutions increases and the reverse trend prevails afterwards so that the maximum dynamic
deflection of the linear and the nonlinear solutions occurs at F = 4ρgAl(N) at α = 0.5.
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404 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
Fig. 11. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an
inclined clamped-pinned Timoshenko beam with ϕ = 30◦ at different
velocity ratios for moving force value of F = 4ρgAl(N) in the constant
velocity motion; (—) nonlinear solution, (– · –) linear solution.
Fig. 12. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an
inclined clamped-pinned Timoshenko beam with ϕ = 30◦ at different
moving force of F for velocity ratio α = 0.25 in the constant velocity
motion; (—) nonlinear solution, (– · –) linear solution.
It can be observed from Fig. 11 that the instantaneous dynamic deflection calculated from
the nonlinear analysis is almost greater than the one obtained from the linear analysis. Also,
Fig. 12 shows that the linear analysis always presents a unique value for w/w0 no matter what
values of F might be. Moreover, for small magnitude of moving force F , i.e., F = 0.5ρgAl(N),
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NONLINEAR VIBRATION SOLUTION OF AN INCLINED TIMOSHENKO BEAM . . . 405
Fig. 13. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an
inclined clamped-free Timoshenko beam with ϕ = 30◦ at different
velocity ratios for moving force value of F = 4ρgAl(N) in the constant
velocity motion; (—) nonlinear solution, (– · –) linear solution.
Fig. 14. Variation of dimensionless dynamic deflection (w/w0) vs. (x/l) for an
inclined clamped-pinned Timoshenko beam with ϕ = 30◦ at different
moving force of F for velocity ratio α = 0.25 in the con-
stant velocity motion; (—) nonlinear solution, (– · –) linear solution.
the nonlinear and the linear solutions are almost the same. After this point (F > 0.5ρgAl(N))
the difference between the linear and the nonlinear solution increases slightly. It can be observed
from Figs. 13 and 14 that for the clamped-free beam the difference between the linear and the
nonlinear analysis can be negligible graphically. However, the maximum difference between the
linear and the nonlinear analysis occurs at F = 4ρgAl(N) at α = 0.25.
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406 A. MAMANDI, M. H. KARGARNOVIN, S. FARSI
5. Conclusions. Three nonlinear coupled partial differential equations of motion are solved
for the rotation of warped cross section, longitudinal and transversal displacements of an incli-
ned Timoshenko beam with different boundary conditions subjected to a moving force under
influence of three types of motions including accelerating, decelerating and constant velocity
motion, and the outcome results are the followings:
In the pinned-pinned and the clamped-pinned types of boundary condition under influence
of accelerating type of motion, the maximum dynamic deflection is reached at much later time
than the other two cases. Moreover, in the decelerating type of motion the range of variation of
maximum dynamic deflection is greater than the one obtained from other two types of motion
for all types of boundary conditions of the beam.
It is concluded that the maximum difference between nonlinear and linear analysis for the
beam deflection at a reference point, i.e., xmax (= x|wmax
) almost happens primarily in the
decelerating motion and with smaller difference in the case of accelerating and then in the
uniform velocity types of motion, respectively.
The maximum dynamic displacements of a moving force problem using linear and nonlinear
solutions are almost the same for lower values of F . However, for larger values of F , the magni-
tude of the beam deflection of the nonlinear and the linear solutions differs gradually and the
difference grows rapidly as the value of F increases. Also, by increasing the velocity ratio the
difference between linear and nonlinear solution of maximum dynamic deflection of the beam
becomes negligible. In addition, for pinned-pinned, clamped-pinned and clamped-free types of
boundary condition the variation of the linear solution mathematically follows a linear trend in
a moving force problem.
From the nonlinear analysis point of view, due to the existence of the quadratic-cubic nonli-
nearity nature of the governing coupled PDEs of motion, in a pinned-pinned Timoshenko beam
the system behaves like a hard spring. That is, by increasing the magnitude of the moving force,
the dynamic deflections become smaller than those from solution of linear system whereas in
the clamped-pinned and clamped-free types of boundary conditions, the system behaves like
a hard/soft and soft spring, respectively. For a soft system by increasing the magnitude of the
moving force, the nonlinear dynamic deflection becomes greater than those obtained from the
linear solution.
When α increases, the value of maximum instantaneous dynamic deflection decreases so
that for higher velocity ratios, i.e., α = 0.75 and 1, the values of linear and nonlinear soluti-
ons are almost the same no matter what values of F might be. Furthermore, by increasing the
magnitude of moving force F the difference between maximum value of the linear and the
nonlinear solution increases.
It can be observed that the linear solution of the instantaneous dynamic deflection under a
moving force always shows a unique value in any identical value of the velocity ratio α no matter
what values of F might be, whereas this behavior can not be seen in the nonlinear analysis.
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Received 16.12.11,
after revision — 10.07.12
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