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Closed-form solution of large deflection of a spring-hinged beam subjected to non-conservative force and tip end moment
Accepted Manuscript Closed-Form Solution of Large Deflection of a Spring-Hinged Beam Subjected to Non-Conservative Force and Tip End Moment Ibrahim Abu-Alshaikh , Riyad Abu-Mallouh , Osama Ghazal , Hashem Alkhaldi PII:
S0997-7538(14)00038-2
DOI:
10.1016/j.euromechsol.2014.02.019
Reference:
EJMSOL 3044
To appear in:
European Journal of Mechanics / A Solids
Received Date: 26 October 2013 Accepted Date: 28 February 2014
Please cite this article as: Abu-Alshaikh, I., Abu-Mallouh, R., Ghazal, O., Alkhaldi, H., ClosedForm Solution of Large Deflection of a Spring-Hinged Beam Subjected to Non-Conservative Force and Tip End Moment, European Journal of Mechanics / A Solids (2014), doi: 10.1016/ j.euromechsol.2014.02.019. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
A novel analytical approach to obtain closed-form solutions of large
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deflection of beams subjected to inclined non-conservative load and tipend moment. The beams are free at one end and spring-hinged at the other
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end, the spring at the hinged support is assumed to be linear or nonlinear. The elliptic integral approach is applied to solve the governing non-linear differential equation which based on Euler–Bernoulli nonlinear beam
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theory.
The analytical results were extensively analyzed and compared with
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existing results based on various numerical methods in the literature.
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Closed-Form Solution of Large Deflection of a Spring-Hinged Beam Subjected to Non-Conservative Force and Tip End Moment Ibrahim Abu-Alshaikh1, Riyad Abu-Mallouh2, Osama Ghazal2 and Hashem Alkhaldi1 (1) Mechanical Engineering Department, The University of Jordan, Amman-11942-Jordan (2) Mechanical and Industrial Engineering Department, Applied Sciences University, Amman-11931-Jordan
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( [email protected], [email protected] , [email protected], [email protected])
Abstract
The thrust of this paper is to present a novel approach to obtain the closed-form solutions of large deflection of beams subjected to inclined non-conservative load and tip-end moment. The beams are
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free at one end and spring-hinged at the other end, the spring at the hinged support is assumed to be nonlinear. The mathematical formulation based on Euler–Bernoulli nonlinear beam theory is presented, and the governing differential equation for the problem is established. The nonlinear governing
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differential equation was solved using the elliptic integral approach. The results were extensively analyzed and compared with existing solutions based on various numerical methods in the literature. The analytical solution presented shows very good agreements with those from literature with more adequateness, efficiency, and less time effort.
Keywords large deflection, follower load, tip moment, nonlinear spring, elliptic integrals 1. Introduction
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Nowadays, deflection of a cantilever beam has been the subject of numerous engineering problems
which have very attractive civilian applications, e.g.,
shipbuilding, forestry, roofed structures, cranes, heavy bridges, flexible manipulator, etc. Compliant mechanisms are also composed of elastic links whose deformations
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are utilized to produce the desired output motion for a given input actuation. In comparison to a rigid body mechanism, a compliant mechanism yields smaller
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workspace (Banerjee et al., 2009; Tolou and Herder, 2009). Furthermore, large deformation of its links enforces the consideration of geometric and material nonlinearity, thus making its design and synthesis a challenging task. The large deflection of a cantilever beam subjected to inclined conservative
load using the Elliptic integral formulation has been investigated by many researchers (Bisshopp and Drucker, 1945; Timoshenko and Gere, 1963; Frisch-Fay, 1962). Closed-form solutions for the large deflections of a constant curvature cantilever beam subjected to combined non-follower force and moment is presented by (Lau, 1993). Lee investigated the large deflection of cantilever beam under a combined loading consisting of a uniformly distributed load and one vertical concentrated load
ACCEPTED MANUSCRIPT at the free end. He proposed a simple numerical integration scheme using Butcher’s fifth order Runge–Kutta method along with one parameter (curvature) shooting technique and solved the problem (Lee, 2002). A new technique for large deflection analysis of non-prismatic cantilever beam based on the integrated least square error of the non-linear governing differential equation in which the angle of rotation is
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represented by a polynomial has recently been presented (Dado and AL-Sadder, 2005).
The large deflection of an elastic beam with a follower load has been investigated by many authors. The problem of the instability of a cantilevered beam
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which is fixed from one side and free from the other and subjected to a compressive end follower force has been studied numerically using static and dynamic analysis (Leipholz, 1970). The problem of an overhanging beam with intermediate support and
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restrained against rotation subjected to a follower force using Newton’s method has been investigated (Wang et al., 1998). Large deflection of a cantilever beam subjected to a tip-concentrated load is studied by utilizing a fourth-order Runge–Kutta method (Mutyalarao et al., 2010) where the inclination to the deformed axis of the beam is assumed as constant. Large deflection behavior of a cantilever beam subjected to a tip
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concentrated load which rotates in relation with the tip-rotation of the beam is analyzed (B.N. Rao and G.V. Rao, 1987 and 1988). They applied the static and dynamic stability criteria to uniform and non-uniform cantilever beams and solved the problem using Runge–Kutta method with iterative shootings to reduce the
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computational effort. It was shown that a linear spring-hinged uniform cantilever subjected to a follower compressive force applied to its free end can exhibit only
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flutter instability (Kounadis, 1980). An iterative shooting method has been used to analyze the large deflections of curved prismatic cantilever beam with uniform curvature subjected to a follower load at the tip (Nallathambi et al., 2010). The finite element method has been investigated by (Wang and Shahinpoor, 1996; Argyris, Symeonids, 1981). The large deflection of curved beam under follower loads using the finite difference method has been solved (Srpcic and Saje, 1986). The direct numerical method for the large deflection problem of a non-uniform straight cantilever under a tip-concentrated follower force with constant inclined angle is proposed (Shvartsman, 2007, 2009, 2013), this problem is solved by using a simple Runge–Kutta method without shooting any kind of iterations. In the paper (Shvartsman, 2009) these results were generalized for the case of the cantilever under
ACCEPTED MANUSCRIPT two concentrated follower forces. The purpose of paper (Shvartsman, 2013) was to assess the validity of the non-iterative direct method for the static analysis of the flexible curved cantilever subjected to tip- concentrated follower force. The analog equation method for evaluation critical load based on the eigenvalue sensitivity is used to optimize the critical load of an Euler–Bernoulli cantilever beam with constant
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volume subjected to a tangential compressive tip load (Katsikadelis and Tsiatas, 2007).
A literature review reveals that the non-conservative (follower load) deflection of elastic beams was usually solved by various numerical methods, Runge–
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Kutta method, iterative shooting method, the finite difference method, the finite element method, or the direct non-iterative numerical method. In this paper, a successful attempt is made to present an exact closed form solution for the large
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deflection of a spring-hinged cantilever beam under a tip moment and a concentrated follower force (non-conservative) acting at the free end of the beam for different force inclined angle. To the author’s best knowledge it is noted that closed-form solutions for such a problem has not been proposed in the past. The results obtained by the proposed method are in good agreement with those determined by different numerical
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method in the literature, but the present method takes less computational time and it is more efficient than other methods.
2. Problem Formulation
Consider an inextensible slender cantilever beam of length L and a flexural
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stiffness EI subjected to both a concentrated follower right-end load P inclined with an angle α and a tip end moment. These loading conditions are represented in the
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global ( x, y ) coordinate-system, where the curved coordinate along the deflected axis of the beam measured from the free end is denoted by the arc length s , Fig.1. The angle of inclination of the concentrated end load P with respect to the deformed axis of the beam remains unchanged after deformation. Considering the free body diagram of the segments of the beam where the length of the right hand segment is s and the left hand segment becomes ( L − s ) . Hence, the beam weight is assumed to be neglected, the horizontal and vertical static equilibrium equations lead that the force components are independent of the arc length s . However, the internal bending moment is a function of the arc length s . Noting that the angle φ ( s ) represents the rotation of the beam with respect to the positive x -axis and ds denotes the length of
ACCEPTED MANUSCRIPT infinitesimal element of the beam. Hence, the shear force balance relation leads to (Shvartsman, 2007) d 2φ ( s ) = − β 2 sin (φ ( s ) + α − φ (0) ) ds 2
(1)
where P EI
(2)
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β2 =
The nonlinear ordinary differential equation, Eq. (1), governs the behavior of the cantilever beam. The right hand side of the beam is free of supports while the left hand side is supported by a frictionless hinge and a nonlinear spring of torsional and kt 2 . The other end is free and subjected to both a
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stiffness coefficients kt1
concentrated follower force and a tip end moment M . The boundary conditions
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associated with these supports are governed by the relations (Shvartsman, 2007)
dφ ( s ) M − = 0, ds s =0 EI EI
dφ ( s ) + kt 1 φ ( L ) + kt 2 φ 3 ( L ) = 0 ds s = L
(3)
(4)
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Once Eq. (1) has been solved for the slope φ ( s ) , the Cartesian coordinates ( x, y ) of the beam are readily determined by integrating the following relations along the interval from s to L
dx = cos(φ ( s )) ds
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dy = sin(φ ( s )), ds
(5)
We now introduce the new variable
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z ( s ) = φ ( s ) + α − φ (0)
(6)
where
z (0) = α
(7)
The Euler-Bernoulli equation under the transformation presented in Eq. (6) becomes d 2 z (s) = − β 2 sin ( z ( s ) ) 2 ds
(8)
Although Equation (8) is straightforward in appearance, it is in fact rather difficult to be solved analytically because of the nonlinearity term sin ( z ( s ) ) and because it is
ACCEPTED MANUSCRIPT difficult to predict the slope at the free end, that is φ (0) . In order to obtain the exact solution of Eq. (8), this equation is multiplied by dz ( s ) / ds , so that it becomes d 2 z ( s ) dz ( s ) dz ( s ) + β 2 sin ( z ( s ) ) =0 2 ds ds ds
(9)
Equation (9) can be further written as
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2 d 1 dz ( s ) 2 − β cos ( z ( s ) ) = 0 ds 2 ds
(10)
Equation (10) corresponds to the conservation of mechanical energy, it can be immediately integrated and rearranged, by equating the term inside the square
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brackets by a constant, as 2
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dz ( s ) 2 = 2C + 2 β cos ( z ( s ) ) ds
(11)
where C is an integrating constant that should satisfy the boundary conditions associated
with
the
beam.
Introducing
the
trigonometric
relation;
cos ( z ( s ) ) = 1 − 2sin 2 ( z ( s ) / 2 ) , into equation (11), after making some rearrangements and taking the square root of both sides, becomes
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dz ( s ) 1 z (s) = 2β − sin 2 2 ds ζ 2 where
2β 2 C +β2
(13)
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ζ2 =
(12)
From Eqs. (3), (6) and (12), ζ at s = 0 can be expressed as 2
M 2 α = + sin 2 ζ 2 2 β EI
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1
(14)
To obtain the possible ranges of the slope φ (0) at the free end, related to a particular load inclined by an angle α , the expression corresponds to the curvature of the beam presented in Eq. (12) is examined. Noting that the sign of the term under the square root in Eq. (12) must be positive or zero at all points on the beam including both ends of the beam; this is due to the fact that the curvature is always a real number. However, to ensure that this term is positive at the whole points of the beam, the following condition should be satisfied for every value of the arc length s
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M 2 α 2 z ( s) = + sin ≥ sin 2 ζ 2 2 2 β EI 1
(15)
Replacing the sine term in Eq. (12) by χ , where z (s) , 2
χ = sin
(16) d χ / ds as a function of dz ( s ) / ds . Thus,
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it is easy to obtain the value of
differentiating Eq. (16) with respect to s and squaring both sides, leads that
dχ 1 2 dz ( s ) = (1 − χ ) 4 ds ds 2
2
(17)
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Substituting Eq. (12) into (17) and rearranging the terms of the resultant equation gives
ζ dχ
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ds =
β (1 − χ 2 )(1 − ζ 2 χ 2 )
(18)
Equation (18) is a separable first order ordinary differential equation which can be solved directly by integrating both sides as
F ( χ ;ζ ) =
βs + C1 ζ
(19)
α C1 = F sin( ); ζ 2
, thus Eq. (19) becomes
βs α + F sin( ); ζ ζ 2
(20)
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F ( χ ;ζ ) =
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In Eq. (19) C1 is a constant of integration that can be found at s = 0 , where
Equation (20) can be rewritten in terms the Jacobi elliptic function sn[u; m] as
β s
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z (0) + F sin( ); ζ ; ζ 2 ζ
χ = sn
(21)
From Eqs. (6), (16) and (21), the slope angle can be written as
β s
α + F sin( ); ζ 2 ζ
φ ( s) = φ (0) − α + 2 arcsin sn
;ζ
(22)
To carryout numerical computations, Eq. (22) will be written with respect to dimensionless quantities by introducing the following non-dimensional parameters
s=
s L
(23)
ACCEPTED MANUSCRIPT β = βL
(24)
ML EI
(25)
P=
PL2 EI
(26)
kt 1 =
kt1 L3 EI
(27)
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M=
The slope angle appears in Eq. (22) can be rewritten again in terms of the above dimensionless quantities as β s
α + F sin( ); ζ 2 ζ
;ζ
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φ ( s ) = φ (0) − α + 2 arcsin sn
(28)
Differentiating both sides of Eq. (28) with respect to s
where
;ζ
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β s dφ ( s ) 2 β α = dn + F sin( ); ζ ds ζ 2 ζ
(29)
dn[u; m] is the well-known type of Jacobi elliptic functions. The
quantity β represents the non-dimensional resultant force applied at the free end of the beam. From Eqs. (2), (24), (14) and (26) β and ζ can be expressed as
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β= P 2
M 2 α = + sin 2 ζ 2 2β 1
(30) (31)
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Equation (28) with the value of β and ζ given in Eqs. (30-31) represents the exact solution of the problem, where the slope at the free end φ (0) can be determined from
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the non-dimensional form of the condition presented in Eq. (4) which leads that
β 2β α dn + F sin( ); ζ kt1ζ 2 ζ
kt 2 3 (ψ ) = 0 ; ζ + (ψ ) + kt 1
(32)
where
β
α + F sin( ); ζ 2 ζ
ψ = 2 arcsin sn
; ζ − α + φ (0)
(33)
The numerical value for φ (0) when kt 2 ≠ 0 can be determined by Newton-Raphson
method from Eq. (32) and Eq. (33). Otherwise, the constant φ (0) can be found
ACCEPTED MANUSCRIPT exactly for both linear spring-hinged and cantilever beams. The following Section is utilized to handle different problems.
3. Results and Discussion In this section many case studies are utilized; in the first one, the beam is assumed to be cantilever beam, i.e., the stiffness coefficients of the spring are
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assumed to be very large, in the second case the spring at the hinge support is considered to be linear while in the third case the spring is assumed to be nonlinear. In these three cases the problem is solved with and without applying tip moments. 3.1 Verification Problems
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3.1.1 Cantilever beam subjected to concentrated follower force
As the spring stiffness coefficients kt 2 = 0 and kt1 approaches infinity the
ϕ (0) as
β
α + F sin( ); ζ 2 ζ
φ (0) = − 2 arcsin sn
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left hand support of the beam becomes fixed, in this case Eq. (32) is solved exactly for
;ζ + α
(34)
The three curves shown in Fig. 2 are drawn using Eq. (34), these curves show the load variation with the tip-angle for three different values of load-inclination
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angle α = −π / 2 , α = −π / 3 and α = −π / 6 . The curves corresponding to the positive values of load-inclination angle are symmetric with the curves of Fig. 2 about the load axis. For the specified load and load-inclination angle, only one tip-angle can be
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observed. However, for the specified tip-angle and load-inclination angle, one can find many loads. Many loads for a tip-angle show different deformed configurations of the beam, this implies uniqueness on the solution of non-linear differential equations for
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the present cantilever beam problem (Shvartsman, 2007; Mutyalarao et al., 2010). Table 1 shows some numerical values for the tip-angle φ (0) and their corresponding loads at α = −π / 2 . However, for each tip-angle there is infinitely many corresponding loads, in Table 1 only loads less than five hundred ( P < 500 ) are mentioned. To achieve a high accuracy for both of the tip angle and the associated boundary conditions, these loads are evaluated correctly for 54 digits using the Maple software. These high-precision numerical results which are directly found from Eq. (34) and presented in Table 1, fit the approximate values presented in (Shvartsman, 2007; Mutyalarao et al., 2010). It can be noted from Fig. 2 that the tip-angle φ (0) of the
ACCEPTED MANUSCRIPT cantilever beam does not exceed double the value of the inclination angle of the load. For this case study Eq. (28) combined with Eq. (34), represents the exact solution for the cantilever beam which can be written as
β s
α + F sin( ); ζ 2 ζ
β α 2 arcsin sn + F sin( ); ζ 2 ζ
;ζ −
;ζ
(35)
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φ ( s ) = 2 arcsin sn
Equation (35) with β and ζ given in Eqs. (30-31) represents the exact solution for the cantilever beam which is subjected to both the end-tip moment M and
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concentrated follower force P inclined by angle α . The curves presented Fig. 3, show deformed configurations of the beams with different values of the inclined forces
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P selected from in Table 1. These elastica curves are drawn directly using the closedform formula presented in Eq. (35) for five different values of the applied nondimensional follower force with M = 0 , kt 2 = 0 , α = π / 2 and
kt1 = 1× 1030 . The
curves of Fig. 3 which represent the deformed shape of the cantilever beam are exactly the same as those obtained numerically by (Shvartsman, 2007; Mutyalarao et al.,
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2010).
3.1.2 Cantilever beam subjected to pure bending moment The second verification problem is related to a cantilever beam subjected to only a pure bending moment. In the above formulation, when the beam is subjected
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to a pure bending moment the value of β approaches zero, accordingly from Eq. (14)
ζ approaches zero. However, the ratio β / ζ approaches the value M / 2 and Eq.
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(28) becomes
φ ( s ) = M ( s − 1)
(36)
In the Eq. (36) M is the non-dimensional tip moment applied at the free end of the beam. Substituting Eq. (36) directly into Eq. (5) leads to the following relations
dx = cos( M ( s − 1)) → x = sin( M ( s − 1)) / M + C1 , ds
(37)
dy = sin( M ( s − 1)) → y = − cos( M ( s − 1)) / M + C2 . ds
(38)
ACCEPTED MANUSCRIPT The homogeneous boundary conditions at the fixed support ( s = 1 ) lead that C1 = 0 and C2 = 1 / M . Equations (37) and (38) represent a circle of radius (1 / M ) centered at
the point ( 0,1/ M ) , which can be written as 2
2
1 1 x + y− = . M M 2
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(39)
Equation (39) states that the slope of elastica curves of the beam is simply related linearly to the non-dimensional arc length and it is deflected as a circle of radius (1 / M ) . However, the deflected shapes of the beam yield to; an arc of a circular
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shape when ( M < 2π ) , a complete circle of a unity circumference when ( M = 2π ) , Fig. 4. However, when M > 0 the beam is deflected upward and when M < 0 the
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beam is deflected downward. The analytical solution achieved through the Elliptic formulation stated above in Eq. (39) agrees with the recent results presented in (Katsikadelis and Tsiatas, 2007; Tari, 2013). The curves presented Fig. 4 show the deformed shape of the cantilever beam for different values of the applied moment ( M ) with β = 1× 10 −30 , kt 2 = 0 and kt1 = 1× 1030 . These curves reveal that as
(M )
increases, the angle of rotation at the hinge support
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the applied moment
increases. Furthermore, since the beam is fixed at the supported end, the slope for all curves is zero at the origin. It can be also noted that the deflected shape of the beam
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makes a semi-circular arc for M = π and a complete circle when M = 2π . Table 2, show some selected numerical results for the curves obtained from Eq. (35) and
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presented in Figs. 2-4, these numerical values agree with the results of Fig. 2 and Table 1 and show perfect agreement with the numerical results given in (Shvartsman, 2007; Mutyalarao et al., 2010; Katsikadelis and Tsiatas, 2007; Tari, 2013).
3.2 Linear spring hinged beam For this special case, where kt 2 = 0 , Eq. (32) is solved exactly for the value of
φ (0) as φ (0) =
β −2 β α dn + F sin( ); ζ kt1ζ 2 ζ
;ζ −
β α 2 arcsin sn + F sin( ); ζ 2 ζ
;ζ + α
(40)
ACCEPTED MANUSCRIPT Thus Eq. (28) becomes β −2 β α dn + F sin( ); ζ ; ζ − kt1ζ 2 ζ β β s α α + F sin( ); ζ 2 arcsin sn + F sin( ); ζ ; ζ + 2 arcsin sn 2 2 ζ ζ
φ (s ) =
;ζ
(41)
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Equation (41) with the value of β and ζ given in Eqs. (30-31) represents the exact solution for the problem when the spring is linear ( kt 2 = 0 ). The general closed form solution presented in Eq. (41), is applied easily for the remaining part of this Subsection. The curves presented Fig. 5 show the deformed shape of the beam for
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different values of the spring constant ( kt1 ) with α = −π / 2 , M = 0 and kt 2 = 0 . For this special case it is noted that ζ always converges to the numerical values ( ± 2 ).
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These curves illustrated in Fig. 5 reveal that as the spring stiffness kt1 increases the angle of rotation at the hinge support decreases. Furthermore, as the spring stiffness kt1 approaches infinity the beam becomes fixed at the supported end. Accordingly, the slope for all curves is non-zero at the origin except that corresponds to the cantilever beam represented by the solid line, where kt1 goes to infinity.
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The above problem is resolved but with an applied moment instead of the applied load. Fig. 6 shows the deformed shape of the beam for different values of the spring constant kt1 with M = π , kt 2 = 0 and both P and α approach zero. These
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curves reveal that as the spring stiffness kt1 increases the angle of rotation at the hinge support decreases. Furthermore, as the spring stiffness kt1 approaches infinity the
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beam becomes fixed at the supported end. Accordingly, the slope for all curves is non-zero at the origin except that corresponds to the solid line, where kt1 goes to infinity. Table 3, show some selected numerical results for the curves obtained from Eq. (41) and presented in Figs. 5 and 6.
3.3 Nonlinear spring hinged beam As the spring stiffness coefficients kt 2 and kt1 become nonzero, Eq. (32) becomes a nonlinear third order algebraic equation with respect to ψ which can be solved numerically by Newton-Raphson method with a high accuracy since in all
ACCEPTED MANUSCRIPT cases it has one real root. Then, from Eq. (33) φ (0) can be computed, and from Eq. (28) the slope can found exactly for this case. The curves presented Figs. 7, 8 and 9 show the deformed configurations of the beams with different values of kt 2 when P = 2π and M = π for three different values of load-inclination angle α = −π / 6 , α = −π / 2 and α = −2π / 3 , respectively. The
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curves presented Fig. 10 show the deformed shapes of the beams for different values of α with spring constants kt1 = kt 2 = 1 , P = 2π and M = π . The curves presented Fig.
11 show the deformed shapes of the beams for different values of the nonwhen kt1 = kt 2 = 1 , α = −7π / 12 and M = π . The curves
P
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dimensional load
illustrated Figs. 7, 8 and 9 reveal that as stiffness becomes smaller the slope angle of the deflected beam the hinged support becomes larger (the beam rotates more).
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Furthermore, these curves show that when the x-component of the force is compressive ( α = −π / 6 ) at the free end of the beam the radius of curvature of the deflected beam is more than the case when the x-component of the force is tensile ( α = −2π / 3 ) at the free end. The deformed configurations of the beams in Figs. 7, 8 and 9 for a specified value of the load inclination angle α with various spring
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stiffness coefficients kt 2 , the radius of curvature of the deflected beam is constant, i. e., the slope angle of the deflected beam at the hinged support increases as the stiffness coefficients kt 2 decreases, however the curvature is unchanged for the same loading conditions at a constant inclination angle α . The curves illustrated Figs. 7, 8
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and 9 reveal further when the x-component of the force is compressive ( α = −π / 6 ) at the free end of the beam the slope angle of the deflected beam at the hinged support
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increases from φ (0) = 0.0 at kt 2 = ∞ to φ (0) = 271.6044324 degrees at kt 2 = 0.1 . However, when the x-component of the force is tensile ( α = −2π / 3 ) at the free end, the slope angle of the deflected beam at the hinged support increases from φ (0) = 0.0 at kt 2 = ∞ to φ (0) = 89.01331333 degrees at kt 2 = 0.1 . Figure 10, shows the deformed configurations of the beams with different load inclination angle α , these configurations show that as the load inclination angle α increases the discrepancy between the elastic curves decreases. These curves further illustrate that the slope angle of the deflected beam at the hinged support is directly proportional to the load inclination angle α . On the other hand, the radius of curvature is inversely
ACCEPTED MANUSCRIPT proportional to the load inclination angle α .
Fig. 11 shows the deformed
configurations of the beams with different load intensity when with kt1 = kt 2 = 1 ,
α = −7π / 12 and M = π . Table 4, shows some selected numerical results for the curves presented in Figs. 7-11.
4. Conclusions
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Based on elliptic integral approach, the closed-form solutions of large deflection of beams subjected to a concentrated non-conservative inclined force and end-tip moment at the free are found. The beams are free at one end and springhinged at the other end are found, when the spring at the hinged support is non-linear
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the closed-form solution exists, however the tip-angle φ (0) which is indeed found by the Newton-Raphson method. The obtained results are compared with existing
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numerical solutions in the literature; they show very good agreements with more adequateness, efficiency, and less time calculations.
The closed-form solutions for cantilever beams subjected to either a concentrated non-conservative inclined force or pure end moment are found as special case. Furthermore, it shown that the solution of the governing non-linear differential equation for a specified loading conditions is unique because for the specified load
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and load-inclination angle only one tip-angle can be observed and many loads for a tip-angle show different deformed configurations of the beams. It also noted that the tip-angle of the cantilever beam does not exceed double the value of the inclination angle of the load. It is further noted that, the deformed configurations of the spring-
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hinged beams for specified loading conditions and various spring stiffness coefficients effects only the slope angle of the deflected beam at the hinged support, but not the
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radius of curvature of the deflected beam. Thus, for a specified end load and tip moment, the slope angle of the deflected beam at the hinged support is directly proportional to the load inclination angle and the radius of curvature is inversely proportional to the load inclination angle.
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5. References Argyris, J.H., Symeonidis, Sp., 1981. Non-linear finite element analysis of elastic
problems. Comput. Meth. Appl. Mech. Eng. 26, 75–123.
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systems under non-conservative loading-natural formulation. Part 1, quasi- static
Banerjee, A., Bhattacharya, B., Mallik, A.K., 2009. Forward and inverse analyses of smart compliant mechanisms for path generation. Mechanism and Machine Theory.
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44, 369–381.
Appl. Math. 3, 272–275.
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Bisshopp, K.E., Drucker, D.C., 1945. Large deflections of cantilever beams. Quart.
Dado, M., AL-Sadder, S., 2005. A new technique for large deflection analysis of nonprismatic cantilever beams. Mech. Res. Commun. 32, 692-703.
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Frisch-Fay, R., 1962. Flexible Bars. Butterworths, London.
Katsikadelis, J.T., Tsiatas, G.C., 2007. Optimum design of structures subjected to
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follower forces. Int. J. Mech. Sci. 49, 1204–1212.
Kounadis, A.N., 1980. On the static stability analysis of elastically restrained
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structures under follower forces. AIAA J. 18, 473–476.
Lau, J.H., 1993. Closed-form solutions for the large deflections of curved optical glass fibers under combined loads. J. Electron. Packag. 115, 337–339.
Lee, K., 2002. Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Non-Linear Mech. 37, 439–443.
Leipholz, H.H.E., 1970. Stability Theory, Academic Press, New York.
ACCEPTED MANUSCRIPT Mutyalarao, M., Bharathi, D., Nageswara Rao, B., 2010. Large deflections of a cantilever beam under an inclined end load. Appl. Math. Comput. 217, 3607– 3613.
Nallathambi, A.K., Rao, C.L., Srinivasan, S.M., 2010. Large deflection of constant
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curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445.
Rao, B.N., Rao, G.V., 1987. Applicability of static and dynamic criterion for the stability of a cantilever column under a tip-concentrated sub-tangential follower force.
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J. Sound Vib. 120, 197–200.
Rao, B.N., Rao, G.V., 1988. Applicability of static or dynamic criterion for the stability of a non-uniform cantilever column subjected to a tip-concentrated sub-
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tangential follower force. J. Sound Vib. 122, 188–191.
Shvartsman, B.S., 2007. Large deflections of a cantilever beam subjected to a follower force. J. Sound Vib. 304, 969–973.
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Shvartsman, B.S., 2009. Direct method for analysis of flexible cantilever beam subjected to two follower forces. Int. J. Non-Linear Mech. 44, 249–252.
Shvartsman, B.S., 2013. Analysis of large deflections of a curved cantilever subjected
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to a tip-concentrated follower force. Int. J. Non-Linear Mech. 50, 75–80.
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Srpčič, S., Saje, M., 1986. Large deformations of thin curved plane beam of constant initial curvature. Int. J. Mech. Sci. 28, 275–287.
Tari, H., 2013. On the parametric large deflection study of Euler–Bernoulli cantilever beams subjected to combined tip point loading. Int. J. Non-Linear Mech. 49, 90-99.
Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, McGraw-Hill, New York.
ACCEPTED MANUSCRIPT Tolou, N., Herder, J.L. 2009. A semi analytical approach to large deflections in compliant beams under point load. J. Math. Prob.
Eng., Volume 2009, art. no.
910896, DOI: 10.1155/2009/910896.
Wang, C.M., Lam, K.Y., He, X.Q., 1998. Instability of variable arc-length elastica
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under follower force. Mech. Res. Comm. 25, 189–194.
Wang, G., Shahinpoor, M., 1997. Design prototyping and computer simulations of a novel large bending actuator made with a shape memory alloy contractile wire. Smart
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Mater. Struct. 6, 214–221.
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Captions of Figures Figure1 Spring-hinge beam subjected to end-tip moment and concentrated load inclined with an angle .
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Figure 2 Variation of load P with tip-angle (0) of the cantilever beam for three different values of load-inclination angle . Figure 3 The deformed configurations of cantilever beams for some selected nondimensional follower forces from Table 1.
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Figure 4 The deformed configurations of the cantilever beams with different values of the pure tip-moment M .
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Figure 5 The deformed configurations of beams for different values of the spring constant kt1 when kt 2 0 , P 2 and M 0 . Figure 6 The deformed configurations of beams for different values of the spring constant kt1 when kt 2 0 , P 0 and M .
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Figure 7 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , / 6 and M . Figure 8 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , / 2 and M .
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Figure 9 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , 2 / 3 and M .
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Figure 10 The deformed configurations of beams for different values of the angle with kt1 kt 2 1, P 2 and M . Figure 11 The deformed configurations of beams for different values of the applied follower force with kt1 kt 2 1, 7 /12 and M .
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Figure1
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Figure2
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Figure3
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Figure4
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Figure5
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Figure6
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Figure7
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Figure8
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Figure9
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Figure10
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Figure11
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Captions of Tables Table 1 Tip-angle φ (0) and corresponding loads for α = − π / 2 .
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Table 2 Some numerical results related to Figs. 2- 4 where α = π / 2 kt 2 = 0 and kt1 → ∞ .
Table 3 Some numerical results related to Figs. 5-6 where α = − π / 2 and kt 2 = 0 .
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M =π .
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Table 4 Some numerical results related to Figs. 7-11 where kt1 = 1 and
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Table 1 Tip-angle, φ (0)
Loads, P 55.00148654416298262
φ (0) = 0
343.7592909010186414 2.175452548796267936 35.29972500248758509 79.05415318343091602
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φ (0) = −π / 3
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220.0059461766519305
178.4269705445048675
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265.9358269063915293 431.5571891748481151 3.437592909010186414
φ (0) = −π / 2
30.93833618109167772 85.93982272525466034
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168.4420525414991343 278.4450256298250995 415.9487419902325561 4.987217139601180858
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φ (0) = −2π / 3
26.86443123007284632 93.11297613745548063 158.7446184088704770 291.2417082236357456 400.6277786759940730 13.75037163604074565
φ (0) = −π
123.7533447243667109 495.0133788974668436
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Table 2
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y(0)
0 0 0 0 0
0.49119033446049566 0.45695449640313449 0.31243412284471124 0.4422792576101×10-5 -1.392735918945×10-6
0.0095801609592501 0.254235109367029 0.436176358355093 -0.45694860213681 -0.456947217475544
-120.0000000000000000 -90.00000000000000000 -60.00000000000000000 0.000000000000000000 -180.0000000000000000
π /5 3π / 5 π 6π / 5
0.9354892837891453 0.5045511524492214 6.283185157323×10-20 -0.1559148807407946
0.3039588939179082 0.6944550841840032 0.6366197725829090 0.4798566616565867
35.99999999999999999 107.9999999999999999 179.9999999999999999 215.9999999999999999
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x(0)
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0 0 0 0
φ (0)
M
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P From Table1 26.864… 30.938… 35.299… 55.001… 123.753…
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Table 3
M
kt 1
x(0)
y(0)
φ (0)
2.0
0.0
0.6
0.87872918541161417
0.3820615414123622
402.18544842064809
2.0
0.0
0.8
0.43234908443340605
-0.855108339282307
315.50808555174462
2.0
0.0
1
-0.4077510192431759
-0.867107322669288
2.0
0.0
2
-0.7425709865762506
0.6055776850377852
2.0
0.0
3
-0.2662241416897635
0.9204676578251394
2.0
0.0
5
0.19276108002505243
0.9386049055636373
97.081131122107889
2.0
0.0
10
0.51353938912903221
0.8089581562315380
76.278564033571057
2.0
0.0
20
0.65115087489107233
0.7029498850196507
65.877280489302641
2.0
0.0
∞
0.76736219151321488
0.5738390625921045
55.475996945034225
0.0
π
1.75
0.43539034251927788
0.4644566551191410
496.85009029047167
0.0
π
2
0.63153861484008493
0.0802727401387400
457.24382900416271
0.0
π
5.0
-0.5947428235638634
-0.227080841690351
290.89753160166508
0.0
π
6.0
-0.6360545305421878
-0.026821055539853
272.41460966805424
0.0
π
8.0
-0.5955651898651948
0.2249151828231019
249.31095725104068
0.0
π
10.0
-0.5243323796211528
0.3610544149076897
235.44876580083254
0.0
π
20.0
-0.2961674824612977
0.5635331021120691
207.72438290041627
0.0
π
∞
6.2831851573230×10-20
0.6366197725829090
179.99999999999999
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P
159.48883238771838 124.81788724015700
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263.50166783040254
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Table 4
α
x(0)
y(0)
5
−π / 6
-0.102839903899515
-0.1842192968507671
362.632496383311601
2π
10
−π / 6
-0.155804212491188
-0.1422597713208244
6.00747568916105861
2π
5
−π / 2
-0.405829660587963
0.17425595085192099
277.534385424785132
2π
10
−π / 2
-0.357524355379033
0.25930519688401004
344.203002516374278
2π
5
− 2π / 3
-0.111182032376387
0.62465712985116463
201.980993311099055
2π
10
− 2π / 3
-0.037749028662611
0.63335060198303137
195.299584061016238
2π
1
− π / 12
0.1722639544756102
-0.1341074389484171
434.978237537209998
2π
1
− 3π / 1 2
0.0382915098318173
-0.2318870955144558
407.651993623329616
2π
1
− 4π / 1 2
-0.076139204473931
-0.2749592717986707
384.491097647097952
2π
1
− 8π / 12
-0.315892564488034
0.55024527430811466
221.748549270323836
4π
1
− 7 π / 12
0.0964086125687454
-0.5670556023774596
41.6109211005907538
5π
1
− 7 π / 12
-0.453197224138077
-0.4144163826091202
-15.585384191692458
2 8π
1
− 7 π / 12
0.4631952732930986
-0.4057339001088209
388.421828248114791
5 7π
1
− 7 π / 12
0.5903938574885914
0.16641598734819502
-264.18274471860111
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φ (0)
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2π
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kt 2
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S0997-7538(14)00038-2
DOI:
10.1016/j.euromechsol.2014.02.019
Reference:
EJMSOL 3044
To appear in:
European Journal of Mechanics / A Solids
Received Date: 26 October 2013 Accepted Date: 28 February 2014
Please cite this article as: Abu-Alshaikh, I., Abu-Mallouh, R., Ghazal, O., Alkhaldi, H., ClosedForm Solution of Large Deflection of a Spring-Hinged Beam Subjected to Non-Conservative Force and Tip End Moment, European Journal of Mechanics / A Solids (2014), doi: 10.1016/ j.euromechsol.2014.02.019. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Highlights
A novel analytical approach to obtain closed-form solutions of large
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deflection of beams subjected to inclined non-conservative load and tipend moment. The beams are free at one end and spring-hinged at the other
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end, the spring at the hinged support is assumed to be linear or nonlinear. The elliptic integral approach is applied to solve the governing non-linear differential equation which based on Euler–Bernoulli nonlinear beam
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theory.
The analytical results were extensively analyzed and compared with
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existing results based on various numerical methods in the literature.
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Closed-Form Solution of Large Deflection of a Spring-Hinged Beam Subjected to Non-Conservative Force and Tip End Moment Ibrahim Abu-Alshaikh1, Riyad Abu-Mallouh2, Osama Ghazal2 and Hashem Alkhaldi1 (1) Mechanical Engineering Department, The University of Jordan, Amman-11942-Jordan (2) Mechanical and Industrial Engineering Department, Applied Sciences University, Amman-11931-Jordan
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( [email protected], [email protected] , [email protected], [email protected])
Abstract
The thrust of this paper is to present a novel approach to obtain the closed-form solutions of large deflection of beams subjected to inclined non-conservative load and tip-end moment. The beams are
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free at one end and spring-hinged at the other end, the spring at the hinged support is assumed to be nonlinear. The mathematical formulation based on Euler–Bernoulli nonlinear beam theory is presented, and the governing differential equation for the problem is established. The nonlinear governing
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differential equation was solved using the elliptic integral approach. The results were extensively analyzed and compared with existing solutions based on various numerical methods in the literature. The analytical solution presented shows very good agreements with those from literature with more adequateness, efficiency, and less time effort.
Keywords large deflection, follower load, tip moment, nonlinear spring, elliptic integrals 1. Introduction
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Nowadays, deflection of a cantilever beam has been the subject of numerous engineering problems
which have very attractive civilian applications, e.g.,
shipbuilding, forestry, roofed structures, cranes, heavy bridges, flexible manipulator, etc. Compliant mechanisms are also composed of elastic links whose deformations
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are utilized to produce the desired output motion for a given input actuation. In comparison to a rigid body mechanism, a compliant mechanism yields smaller
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workspace (Banerjee et al., 2009; Tolou and Herder, 2009). Furthermore, large deformation of its links enforces the consideration of geometric and material nonlinearity, thus making its design and synthesis a challenging task. The large deflection of a cantilever beam subjected to inclined conservative
load using the Elliptic integral formulation has been investigated by many researchers (Bisshopp and Drucker, 1945; Timoshenko and Gere, 1963; Frisch-Fay, 1962). Closed-form solutions for the large deflections of a constant curvature cantilever beam subjected to combined non-follower force and moment is presented by (Lau, 1993). Lee investigated the large deflection of cantilever beam under a combined loading consisting of a uniformly distributed load and one vertical concentrated load
ACCEPTED MANUSCRIPT at the free end. He proposed a simple numerical integration scheme using Butcher’s fifth order Runge–Kutta method along with one parameter (curvature) shooting technique and solved the problem (Lee, 2002). A new technique for large deflection analysis of non-prismatic cantilever beam based on the integrated least square error of the non-linear governing differential equation in which the angle of rotation is
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represented by a polynomial has recently been presented (Dado and AL-Sadder, 2005).
The large deflection of an elastic beam with a follower load has been investigated by many authors. The problem of the instability of a cantilevered beam
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which is fixed from one side and free from the other and subjected to a compressive end follower force has been studied numerically using static and dynamic analysis (Leipholz, 1970). The problem of an overhanging beam with intermediate support and
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restrained against rotation subjected to a follower force using Newton’s method has been investigated (Wang et al., 1998). Large deflection of a cantilever beam subjected to a tip-concentrated load is studied by utilizing a fourth-order Runge–Kutta method (Mutyalarao et al., 2010) where the inclination to the deformed axis of the beam is assumed as constant. Large deflection behavior of a cantilever beam subjected to a tip
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concentrated load which rotates in relation with the tip-rotation of the beam is analyzed (B.N. Rao and G.V. Rao, 1987 and 1988). They applied the static and dynamic stability criteria to uniform and non-uniform cantilever beams and solved the problem using Runge–Kutta method with iterative shootings to reduce the
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computational effort. It was shown that a linear spring-hinged uniform cantilever subjected to a follower compressive force applied to its free end can exhibit only
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flutter instability (Kounadis, 1980). An iterative shooting method has been used to analyze the large deflections of curved prismatic cantilever beam with uniform curvature subjected to a follower load at the tip (Nallathambi et al., 2010). The finite element method has been investigated by (Wang and Shahinpoor, 1996; Argyris, Symeonids, 1981). The large deflection of curved beam under follower loads using the finite difference method has been solved (Srpcic and Saje, 1986). The direct numerical method for the large deflection problem of a non-uniform straight cantilever under a tip-concentrated follower force with constant inclined angle is proposed (Shvartsman, 2007, 2009, 2013), this problem is solved by using a simple Runge–Kutta method without shooting any kind of iterations. In the paper (Shvartsman, 2009) these results were generalized for the case of the cantilever under
ACCEPTED MANUSCRIPT two concentrated follower forces. The purpose of paper (Shvartsman, 2013) was to assess the validity of the non-iterative direct method for the static analysis of the flexible curved cantilever subjected to tip- concentrated follower force. The analog equation method for evaluation critical load based on the eigenvalue sensitivity is used to optimize the critical load of an Euler–Bernoulli cantilever beam with constant
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volume subjected to a tangential compressive tip load (Katsikadelis and Tsiatas, 2007).
A literature review reveals that the non-conservative (follower load) deflection of elastic beams was usually solved by various numerical methods, Runge–
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Kutta method, iterative shooting method, the finite difference method, the finite element method, or the direct non-iterative numerical method. In this paper, a successful attempt is made to present an exact closed form solution for the large
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deflection of a spring-hinged cantilever beam under a tip moment and a concentrated follower force (non-conservative) acting at the free end of the beam for different force inclined angle. To the author’s best knowledge it is noted that closed-form solutions for such a problem has not been proposed in the past. The results obtained by the proposed method are in good agreement with those determined by different numerical
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method in the literature, but the present method takes less computational time and it is more efficient than other methods.
2. Problem Formulation
Consider an inextensible slender cantilever beam of length L and a flexural
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stiffness EI subjected to both a concentrated follower right-end load P inclined with an angle α and a tip end moment. These loading conditions are represented in the
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global ( x, y ) coordinate-system, where the curved coordinate along the deflected axis of the beam measured from the free end is denoted by the arc length s , Fig.1. The angle of inclination of the concentrated end load P with respect to the deformed axis of the beam remains unchanged after deformation. Considering the free body diagram of the segments of the beam where the length of the right hand segment is s and the left hand segment becomes ( L − s ) . Hence, the beam weight is assumed to be neglected, the horizontal and vertical static equilibrium equations lead that the force components are independent of the arc length s . However, the internal bending moment is a function of the arc length s . Noting that the angle φ ( s ) represents the rotation of the beam with respect to the positive x -axis and ds denotes the length of
ACCEPTED MANUSCRIPT infinitesimal element of the beam. Hence, the shear force balance relation leads to (Shvartsman, 2007) d 2φ ( s ) = − β 2 sin (φ ( s ) + α − φ (0) ) ds 2
(1)
where P EI
(2)
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β2 =
The nonlinear ordinary differential equation, Eq. (1), governs the behavior of the cantilever beam. The right hand side of the beam is free of supports while the left hand side is supported by a frictionless hinge and a nonlinear spring of torsional and kt 2 . The other end is free and subjected to both a
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stiffness coefficients kt1
concentrated follower force and a tip end moment M . The boundary conditions
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associated with these supports are governed by the relations (Shvartsman, 2007)
dφ ( s ) M − = 0, ds s =0 EI EI
dφ ( s ) + kt 1 φ ( L ) + kt 2 φ 3 ( L ) = 0 ds s = L
(3)
(4)
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Once Eq. (1) has been solved for the slope φ ( s ) , the Cartesian coordinates ( x, y ) of the beam are readily determined by integrating the following relations along the interval from s to L
dx = cos(φ ( s )) ds
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dy = sin(φ ( s )), ds
(5)
We now introduce the new variable
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z ( s ) = φ ( s ) + α − φ (0)
(6)
where
z (0) = α
(7)
The Euler-Bernoulli equation under the transformation presented in Eq. (6) becomes d 2 z (s) = − β 2 sin ( z ( s ) ) 2 ds
(8)
Although Equation (8) is straightforward in appearance, it is in fact rather difficult to be solved analytically because of the nonlinearity term sin ( z ( s ) ) and because it is
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(9)
Equation (9) can be further written as
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2 d 1 dz ( s ) 2 − β cos ( z ( s ) ) = 0 ds 2 ds
(10)
Equation (10) corresponds to the conservation of mechanical energy, it can be immediately integrated and rearranged, by equating the term inside the square
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brackets by a constant, as 2
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dz ( s ) 2 = 2C + 2 β cos ( z ( s ) ) ds
(11)
where C is an integrating constant that should satisfy the boundary conditions associated
with
the
beam.
Introducing
the
trigonometric
relation;
cos ( z ( s ) ) = 1 − 2sin 2 ( z ( s ) / 2 ) , into equation (11), after making some rearrangements and taking the square root of both sides, becomes
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dz ( s ) 1 z (s) = 2β − sin 2 2 ds ζ 2 where
2β 2 C +β2
(13)
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ζ2 =
(12)
From Eqs. (3), (6) and (12), ζ at s = 0 can be expressed as 2
M 2 α = + sin 2 ζ 2 2 β EI
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1
(14)
To obtain the possible ranges of the slope φ (0) at the free end, related to a particular load inclined by an angle α , the expression corresponds to the curvature of the beam presented in Eq. (12) is examined. Noting that the sign of the term under the square root in Eq. (12) must be positive or zero at all points on the beam including both ends of the beam; this is due to the fact that the curvature is always a real number. However, to ensure that this term is positive at the whole points of the beam, the following condition should be satisfied for every value of the arc length s
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M 2 α 2 z ( s) = + sin ≥ sin 2 ζ 2 2 2 β EI 1
(15)
Replacing the sine term in Eq. (12) by χ , where z (s) , 2
χ = sin
(16) d χ / ds as a function of dz ( s ) / ds . Thus,
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it is easy to obtain the value of
differentiating Eq. (16) with respect to s and squaring both sides, leads that
dχ 1 2 dz ( s ) = (1 − χ ) 4 ds ds 2
2
(17)
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Substituting Eq. (12) into (17) and rearranging the terms of the resultant equation gives
ζ dχ
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ds =
β (1 − χ 2 )(1 − ζ 2 χ 2 )
(18)
Equation (18) is a separable first order ordinary differential equation which can be solved directly by integrating both sides as
F ( χ ;ζ ) =
βs + C1 ζ
(19)
α C1 = F sin( ); ζ 2
, thus Eq. (19) becomes
βs α + F sin( ); ζ ζ 2
(20)
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F ( χ ;ζ ) =
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In Eq. (19) C1 is a constant of integration that can be found at s = 0 , where
Equation (20) can be rewritten in terms the Jacobi elliptic function sn[u; m] as
β s
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z (0) + F sin( ); ζ ; ζ 2 ζ
χ = sn
(21)
From Eqs. (6), (16) and (21), the slope angle can be written as
β s
α + F sin( ); ζ 2 ζ
φ ( s) = φ (0) − α + 2 arcsin sn
;ζ
(22)
To carryout numerical computations, Eq. (22) will be written with respect to dimensionless quantities by introducing the following non-dimensional parameters
s=
s L
(23)
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(24)
ML EI
(25)
P=
PL2 EI
(26)
kt 1 =
kt1 L3 EI
(27)
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M=
The slope angle appears in Eq. (22) can be rewritten again in terms of the above dimensionless quantities as β s
α + F sin( ); ζ 2 ζ
;ζ
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φ ( s ) = φ (0) − α + 2 arcsin sn
(28)
Differentiating both sides of Eq. (28) with respect to s
where
;ζ
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β s dφ ( s ) 2 β α = dn + F sin( ); ζ ds ζ 2 ζ
(29)
dn[u; m] is the well-known type of Jacobi elliptic functions. The
quantity β represents the non-dimensional resultant force applied at the free end of the beam. From Eqs. (2), (24), (14) and (26) β and ζ can be expressed as
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β= P 2
M 2 α = + sin 2 ζ 2 2β 1
(30) (31)
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Equation (28) with the value of β and ζ given in Eqs. (30-31) represents the exact solution of the problem, where the slope at the free end φ (0) can be determined from
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the non-dimensional form of the condition presented in Eq. (4) which leads that
β 2β α dn + F sin( ); ζ kt1ζ 2 ζ
kt 2 3 (ψ ) = 0 ; ζ + (ψ ) + kt 1
(32)
where
β
α + F sin( ); ζ 2 ζ
ψ = 2 arcsin sn
; ζ − α + φ (0)
(33)
The numerical value for φ (0) when kt 2 ≠ 0 can be determined by Newton-Raphson
method from Eq. (32) and Eq. (33). Otherwise, the constant φ (0) can be found
ACCEPTED MANUSCRIPT exactly for both linear spring-hinged and cantilever beams. The following Section is utilized to handle different problems.
3. Results and Discussion In this section many case studies are utilized; in the first one, the beam is assumed to be cantilever beam, i.e., the stiffness coefficients of the spring are
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assumed to be very large, in the second case the spring at the hinge support is considered to be linear while in the third case the spring is assumed to be nonlinear. In these three cases the problem is solved with and without applying tip moments. 3.1 Verification Problems
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3.1.1 Cantilever beam subjected to concentrated follower force
As the spring stiffness coefficients kt 2 = 0 and kt1 approaches infinity the
ϕ (0) as
β
α + F sin( ); ζ 2 ζ
φ (0) = − 2 arcsin sn
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left hand support of the beam becomes fixed, in this case Eq. (32) is solved exactly for
;ζ + α
(34)
The three curves shown in Fig. 2 are drawn using Eq. (34), these curves show the load variation with the tip-angle for three different values of load-inclination
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angle α = −π / 2 , α = −π / 3 and α = −π / 6 . The curves corresponding to the positive values of load-inclination angle are symmetric with the curves of Fig. 2 about the load axis. For the specified load and load-inclination angle, only one tip-angle can be
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observed. However, for the specified tip-angle and load-inclination angle, one can find many loads. Many loads for a tip-angle show different deformed configurations of the beam, this implies uniqueness on the solution of non-linear differential equations for
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the present cantilever beam problem (Shvartsman, 2007; Mutyalarao et al., 2010). Table 1 shows some numerical values for the tip-angle φ (0) and their corresponding loads at α = −π / 2 . However, for each tip-angle there is infinitely many corresponding loads, in Table 1 only loads less than five hundred ( P < 500 ) are mentioned. To achieve a high accuracy for both of the tip angle and the associated boundary conditions, these loads are evaluated correctly for 54 digits using the Maple software. These high-precision numerical results which are directly found from Eq. (34) and presented in Table 1, fit the approximate values presented in (Shvartsman, 2007; Mutyalarao et al., 2010). It can be noted from Fig. 2 that the tip-angle φ (0) of the
ACCEPTED MANUSCRIPT cantilever beam does not exceed double the value of the inclination angle of the load. For this case study Eq. (28) combined with Eq. (34), represents the exact solution for the cantilever beam which can be written as
β s
α + F sin( ); ζ 2 ζ
β α 2 arcsin sn + F sin( ); ζ 2 ζ
;ζ −
;ζ
(35)
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φ ( s ) = 2 arcsin sn
Equation (35) with β and ζ given in Eqs. (30-31) represents the exact solution for the cantilever beam which is subjected to both the end-tip moment M and
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concentrated follower force P inclined by angle α . The curves presented Fig. 3, show deformed configurations of the beams with different values of the inclined forces
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P selected from in Table 1. These elastica curves are drawn directly using the closedform formula presented in Eq. (35) for five different values of the applied nondimensional follower force with M = 0 , kt 2 = 0 , α = π / 2 and
kt1 = 1× 1030 . The
curves of Fig. 3 which represent the deformed shape of the cantilever beam are exactly the same as those obtained numerically by (Shvartsman, 2007; Mutyalarao et al.,
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2010).
3.1.2 Cantilever beam subjected to pure bending moment The second verification problem is related to a cantilever beam subjected to only a pure bending moment. In the above formulation, when the beam is subjected
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to a pure bending moment the value of β approaches zero, accordingly from Eq. (14)
ζ approaches zero. However, the ratio β / ζ approaches the value M / 2 and Eq.
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(28) becomes
φ ( s ) = M ( s − 1)
(36)
In the Eq. (36) M is the non-dimensional tip moment applied at the free end of the beam. Substituting Eq. (36) directly into Eq. (5) leads to the following relations
dx = cos( M ( s − 1)) → x = sin( M ( s − 1)) / M + C1 , ds
(37)
dy = sin( M ( s − 1)) → y = − cos( M ( s − 1)) / M + C2 . ds
(38)
ACCEPTED MANUSCRIPT The homogeneous boundary conditions at the fixed support ( s = 1 ) lead that C1 = 0 and C2 = 1 / M . Equations (37) and (38) represent a circle of radius (1 / M ) centered at
the point ( 0,1/ M ) , which can be written as 2
2
1 1 x + y− = . M M 2
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(39)
Equation (39) states that the slope of elastica curves of the beam is simply related linearly to the non-dimensional arc length and it is deflected as a circle of radius (1 / M ) . However, the deflected shapes of the beam yield to; an arc of a circular
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shape when ( M < 2π ) , a complete circle of a unity circumference when ( M = 2π ) , Fig. 4. However, when M > 0 the beam is deflected upward and when M < 0 the
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beam is deflected downward. The analytical solution achieved through the Elliptic formulation stated above in Eq. (39) agrees with the recent results presented in (Katsikadelis and Tsiatas, 2007; Tari, 2013). The curves presented Fig. 4 show the deformed shape of the cantilever beam for different values of the applied moment ( M ) with β = 1× 10 −30 , kt 2 = 0 and kt1 = 1× 1030 . These curves reveal that as
(M )
increases, the angle of rotation at the hinge support
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the applied moment
increases. Furthermore, since the beam is fixed at the supported end, the slope for all curves is zero at the origin. It can be also noted that the deflected shape of the beam
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makes a semi-circular arc for M = π and a complete circle when M = 2π . Table 2, show some selected numerical results for the curves obtained from Eq. (35) and
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presented in Figs. 2-4, these numerical values agree with the results of Fig. 2 and Table 1 and show perfect agreement with the numerical results given in (Shvartsman, 2007; Mutyalarao et al., 2010; Katsikadelis and Tsiatas, 2007; Tari, 2013).
3.2 Linear spring hinged beam For this special case, where kt 2 = 0 , Eq. (32) is solved exactly for the value of
φ (0) as φ (0) =
β −2 β α dn + F sin( ); ζ kt1ζ 2 ζ
;ζ −
β α 2 arcsin sn + F sin( ); ζ 2 ζ
;ζ + α
(40)
ACCEPTED MANUSCRIPT Thus Eq. (28) becomes β −2 β α dn + F sin( ); ζ ; ζ − kt1ζ 2 ζ β β s α α + F sin( ); ζ 2 arcsin sn + F sin( ); ζ ; ζ + 2 arcsin sn 2 2 ζ ζ
φ (s ) =
;ζ
(41)
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Equation (41) with the value of β and ζ given in Eqs. (30-31) represents the exact solution for the problem when the spring is linear ( kt 2 = 0 ). The general closed form solution presented in Eq. (41), is applied easily for the remaining part of this Subsection. The curves presented Fig. 5 show the deformed shape of the beam for
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different values of the spring constant ( kt1 ) with α = −π / 2 , M = 0 and kt 2 = 0 . For this special case it is noted that ζ always converges to the numerical values ( ± 2 ).
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These curves illustrated in Fig. 5 reveal that as the spring stiffness kt1 increases the angle of rotation at the hinge support decreases. Furthermore, as the spring stiffness kt1 approaches infinity the beam becomes fixed at the supported end. Accordingly, the slope for all curves is non-zero at the origin except that corresponds to the cantilever beam represented by the solid line, where kt1 goes to infinity.
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The above problem is resolved but with an applied moment instead of the applied load. Fig. 6 shows the deformed shape of the beam for different values of the spring constant kt1 with M = π , kt 2 = 0 and both P and α approach zero. These
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curves reveal that as the spring stiffness kt1 increases the angle of rotation at the hinge support decreases. Furthermore, as the spring stiffness kt1 approaches infinity the
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beam becomes fixed at the supported end. Accordingly, the slope for all curves is non-zero at the origin except that corresponds to the solid line, where kt1 goes to infinity. Table 3, show some selected numerical results for the curves obtained from Eq. (41) and presented in Figs. 5 and 6.
3.3 Nonlinear spring hinged beam As the spring stiffness coefficients kt 2 and kt1 become nonzero, Eq. (32) becomes a nonlinear third order algebraic equation with respect to ψ which can be solved numerically by Newton-Raphson method with a high accuracy since in all
ACCEPTED MANUSCRIPT cases it has one real root. Then, from Eq. (33) φ (0) can be computed, and from Eq. (28) the slope can found exactly for this case. The curves presented Figs. 7, 8 and 9 show the deformed configurations of the beams with different values of kt 2 when P = 2π and M = π for three different values of load-inclination angle α = −π / 6 , α = −π / 2 and α = −2π / 3 , respectively. The
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curves presented Fig. 10 show the deformed shapes of the beams for different values of α with spring constants kt1 = kt 2 = 1 , P = 2π and M = π . The curves presented Fig.
11 show the deformed shapes of the beams for different values of the nonwhen kt1 = kt 2 = 1 , α = −7π / 12 and M = π . The curves
P
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dimensional load
illustrated Figs. 7, 8 and 9 reveal that as stiffness becomes smaller the slope angle of the deflected beam the hinged support becomes larger (the beam rotates more).
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Furthermore, these curves show that when the x-component of the force is compressive ( α = −π / 6 ) at the free end of the beam the radius of curvature of the deflected beam is more than the case when the x-component of the force is tensile ( α = −2π / 3 ) at the free end. The deformed configurations of the beams in Figs. 7, 8 and 9 for a specified value of the load inclination angle α with various spring
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stiffness coefficients kt 2 , the radius of curvature of the deflected beam is constant, i. e., the slope angle of the deflected beam at the hinged support increases as the stiffness coefficients kt 2 decreases, however the curvature is unchanged for the same loading conditions at a constant inclination angle α . The curves illustrated Figs. 7, 8
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and 9 reveal further when the x-component of the force is compressive ( α = −π / 6 ) at the free end of the beam the slope angle of the deflected beam at the hinged support
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increases from φ (0) = 0.0 at kt 2 = ∞ to φ (0) = 271.6044324 degrees at kt 2 = 0.1 . However, when the x-component of the force is tensile ( α = −2π / 3 ) at the free end, the slope angle of the deflected beam at the hinged support increases from φ (0) = 0.0 at kt 2 = ∞ to φ (0) = 89.01331333 degrees at kt 2 = 0.1 . Figure 10, shows the deformed configurations of the beams with different load inclination angle α , these configurations show that as the load inclination angle α increases the discrepancy between the elastic curves decreases. These curves further illustrate that the slope angle of the deflected beam at the hinged support is directly proportional to the load inclination angle α . On the other hand, the radius of curvature is inversely
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Fig. 11 shows the deformed
configurations of the beams with different load intensity when with kt1 = kt 2 = 1 ,
α = −7π / 12 and M = π . Table 4, shows some selected numerical results for the curves presented in Figs. 7-11.
4. Conclusions
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Based on elliptic integral approach, the closed-form solutions of large deflection of beams subjected to a concentrated non-conservative inclined force and end-tip moment at the free are found. The beams are free at one end and springhinged at the other end are found, when the spring at the hinged support is non-linear
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the closed-form solution exists, however the tip-angle φ (0) which is indeed found by the Newton-Raphson method. The obtained results are compared with existing
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numerical solutions in the literature; they show very good agreements with more adequateness, efficiency, and less time calculations.
The closed-form solutions for cantilever beams subjected to either a concentrated non-conservative inclined force or pure end moment are found as special case. Furthermore, it shown that the solution of the governing non-linear differential equation for a specified loading conditions is unique because for the specified load
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and load-inclination angle only one tip-angle can be observed and many loads for a tip-angle show different deformed configurations of the beams. It also noted that the tip-angle of the cantilever beam does not exceed double the value of the inclination angle of the load. It is further noted that, the deformed configurations of the spring-
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hinged beams for specified loading conditions and various spring stiffness coefficients effects only the slope angle of the deflected beam at the hinged support, but not the
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radius of curvature of the deflected beam. Thus, for a specified end load and tip moment, the slope angle of the deflected beam at the hinged support is directly proportional to the load inclination angle and the radius of curvature is inversely proportional to the load inclination angle.
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5. References Argyris, J.H., Symeonidis, Sp., 1981. Non-linear finite element analysis of elastic
problems. Comput. Meth. Appl. Mech. Eng. 26, 75–123.
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systems under non-conservative loading-natural formulation. Part 1, quasi- static
Banerjee, A., Bhattacharya, B., Mallik, A.K., 2009. Forward and inverse analyses of smart compliant mechanisms for path generation. Mechanism and Machine Theory.
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44, 369–381.
Appl. Math. 3, 272–275.
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Bisshopp, K.E., Drucker, D.C., 1945. Large deflections of cantilever beams. Quart.
Dado, M., AL-Sadder, S., 2005. A new technique for large deflection analysis of nonprismatic cantilever beams. Mech. Res. Commun. 32, 692-703.
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Frisch-Fay, R., 1962. Flexible Bars. Butterworths, London.
Katsikadelis, J.T., Tsiatas, G.C., 2007. Optimum design of structures subjected to
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follower forces. Int. J. Mech. Sci. 49, 1204–1212.
Kounadis, A.N., 1980. On the static stability analysis of elastically restrained
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structures under follower forces. AIAA J. 18, 473–476.
Lau, J.H., 1993. Closed-form solutions for the large deflections of curved optical glass fibers under combined loads. J. Electron. Packag. 115, 337–339.
Lee, K., 2002. Large deflections of cantilever beams of non-linear elastic material under a combined loading. Int. J. Non-Linear Mech. 37, 439–443.
Leipholz, H.H.E., 1970. Stability Theory, Academic Press, New York.
ACCEPTED MANUSCRIPT Mutyalarao, M., Bharathi, D., Nageswara Rao, B., 2010. Large deflections of a cantilever beam under an inclined end load. Appl. Math. Comput. 217, 3607– 3613.
Nallathambi, A.K., Rao, C.L., Srinivasan, S.M., 2010. Large deflection of constant
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curvature cantilever beam under follower load. Int. J. Mech. Sci. 52, 440–445.
Rao, B.N., Rao, G.V., 1987. Applicability of static and dynamic criterion for the stability of a cantilever column under a tip-concentrated sub-tangential follower force.
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J. Sound Vib. 120, 197–200.
Rao, B.N., Rao, G.V., 1988. Applicability of static or dynamic criterion for the stability of a non-uniform cantilever column subjected to a tip-concentrated sub-
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tangential follower force. J. Sound Vib. 122, 188–191.
Shvartsman, B.S., 2007. Large deflections of a cantilever beam subjected to a follower force. J. Sound Vib. 304, 969–973.
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Shvartsman, B.S., 2009. Direct method for analysis of flexible cantilever beam subjected to two follower forces. Int. J. Non-Linear Mech. 44, 249–252.
Shvartsman, B.S., 2013. Analysis of large deflections of a curved cantilever subjected
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to a tip-concentrated follower force. Int. J. Non-Linear Mech. 50, 75–80.
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Srpčič, S., Saje, M., 1986. Large deformations of thin curved plane beam of constant initial curvature. Int. J. Mech. Sci. 28, 275–287.
Tari, H., 2013. On the parametric large deflection study of Euler–Bernoulli cantilever beams subjected to combined tip point loading. Int. J. Non-Linear Mech. 49, 90-99.
Timoshenko, S.P., Gere, J.M., 1961. Theory of Elastic Stability, McGraw-Hill, New York.
ACCEPTED MANUSCRIPT Tolou, N., Herder, J.L. 2009. A semi analytical approach to large deflections in compliant beams under point load. J. Math. Prob.
Eng., Volume 2009, art. no.
910896, DOI: 10.1155/2009/910896.
Wang, C.M., Lam, K.Y., He, X.Q., 1998. Instability of variable arc-length elastica
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under follower force. Mech. Res. Comm. 25, 189–194.
Wang, G., Shahinpoor, M., 1997. Design prototyping and computer simulations of a novel large bending actuator made with a shape memory alloy contractile wire. Smart
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Mater. Struct. 6, 214–221.
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Captions of Figures Figure1 Spring-hinge beam subjected to end-tip moment and concentrated load inclined with an angle .
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Figure 2 Variation of load P with tip-angle (0) of the cantilever beam for three different values of load-inclination angle . Figure 3 The deformed configurations of cantilever beams for some selected nondimensional follower forces from Table 1.
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Figure 4 The deformed configurations of the cantilever beams with different values of the pure tip-moment M .
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Figure 5 The deformed configurations of beams for different values of the spring constant kt1 when kt 2 0 , P 2 and M 0 . Figure 6 The deformed configurations of beams for different values of the spring constant kt1 when kt 2 0 , P 0 and M .
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Figure 7 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , / 6 and M . Figure 8 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , / 2 and M .
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Figure 9 The deformed configurations of beams for different values of the spring constant kt 2 with kt1 1, P 2 , 2 / 3 and M .
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Figure 10 The deformed configurations of beams for different values of the angle with kt1 kt 2 1, P 2 and M . Figure 11 The deformed configurations of beams for different values of the applied follower force with kt1 kt 2 1, 7 /12 and M .
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Figure1
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EP
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Figure2
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EP
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Figure3
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EP
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Figure4
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EP
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Figure5
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EP
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Figure6
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EP
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Figure7
AC C
EP
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Figure8
AC C
EP
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Figure9
AC C
EP
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Figure10
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EP
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Figure11
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Captions of Tables Table 1 Tip-angle φ (0) and corresponding loads for α = − π / 2 .
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Table 2 Some numerical results related to Figs. 2- 4 where α = π / 2 kt 2 = 0 and kt1 → ∞ .
Table 3 Some numerical results related to Figs. 5-6 where α = − π / 2 and kt 2 = 0 .
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M =π .
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Table 4 Some numerical results related to Figs. 7-11 where kt1 = 1 and
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Table 1 Tip-angle, φ (0)
Loads, P 55.00148654416298262
φ (0) = 0
343.7592909010186414 2.175452548796267936 35.29972500248758509 79.05415318343091602
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φ (0) = −π / 3
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220.0059461766519305
178.4269705445048675
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265.9358269063915293 431.5571891748481151 3.437592909010186414
φ (0) = −π / 2
30.93833618109167772 85.93982272525466034
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168.4420525414991343 278.4450256298250995 415.9487419902325561 4.987217139601180858
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EP
φ (0) = −2π / 3
26.86443123007284632 93.11297613745548063 158.7446184088704770 291.2417082236357456 400.6277786759940730 13.75037163604074565
φ (0) = −π
123.7533447243667109 495.0133788974668436
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Table 2
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y(0)
0 0 0 0 0
0.49119033446049566 0.45695449640313449 0.31243412284471124 0.4422792576101×10-5 -1.392735918945×10-6
0.0095801609592501 0.254235109367029 0.436176358355093 -0.45694860213681 -0.456947217475544
-120.0000000000000000 -90.00000000000000000 -60.00000000000000000 0.000000000000000000 -180.0000000000000000
π /5 3π / 5 π 6π / 5
0.9354892837891453 0.5045511524492214 6.283185157323×10-20 -0.1559148807407946
0.3039588939179082 0.6944550841840032 0.6366197725829090 0.4798566616565867
35.99999999999999999 107.9999999999999999 179.9999999999999999 215.9999999999999999
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x(0)
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EP
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0 0 0 0
φ (0)
M
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P From Table1 26.864… 30.938… 35.299… 55.001… 123.753…
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Table 3
M
kt 1
x(0)
y(0)
φ (0)
2.0
0.0
0.6
0.87872918541161417
0.3820615414123622
402.18544842064809
2.0
0.0
0.8
0.43234908443340605
-0.855108339282307
315.50808555174462
2.0
0.0
1
-0.4077510192431759
-0.867107322669288
2.0
0.0
2
-0.7425709865762506
0.6055776850377852
2.0
0.0
3
-0.2662241416897635
0.9204676578251394
2.0
0.0
5
0.19276108002505243
0.9386049055636373
97.081131122107889
2.0
0.0
10
0.51353938912903221
0.8089581562315380
76.278564033571057
2.0
0.0
20
0.65115087489107233
0.7029498850196507
65.877280489302641
2.0
0.0
∞
0.76736219151321488
0.5738390625921045
55.475996945034225
0.0
π
1.75
0.43539034251927788
0.4644566551191410
496.85009029047167
0.0
π
2
0.63153861484008493
0.0802727401387400
457.24382900416271
0.0
π
5.0
-0.5947428235638634
-0.227080841690351
290.89753160166508
0.0
π
6.0
-0.6360545305421878
-0.026821055539853
272.41460966805424
0.0
π
8.0
-0.5955651898651948
0.2249151828231019
249.31095725104068
0.0
π
10.0
-0.5243323796211528
0.3610544149076897
235.44876580083254
0.0
π
20.0
-0.2961674824612977
0.5635331021120691
207.72438290041627
0.0
π
∞
6.2831851573230×10-20
0.6366197725829090
179.99999999999999
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P
159.48883238771838 124.81788724015700
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EP
AC C
263.50166783040254
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Table 4
α
x(0)
y(0)
5
−π / 6
-0.102839903899515
-0.1842192968507671
362.632496383311601
2π
10
−π / 6
-0.155804212491188
-0.1422597713208244
6.00747568916105861
2π
5
−π / 2
-0.405829660587963
0.17425595085192099
277.534385424785132
2π
10
−π / 2
-0.357524355379033
0.25930519688401004
344.203002516374278
2π
5
− 2π / 3
-0.111182032376387
0.62465712985116463
201.980993311099055
2π
10
− 2π / 3
-0.037749028662611
0.63335060198303137
195.299584061016238
2π
1
− π / 12
0.1722639544756102
-0.1341074389484171
434.978237537209998
2π
1
− 3π / 1 2
0.0382915098318173
-0.2318870955144558
407.651993623329616
2π
1
− 4π / 1 2
-0.076139204473931
-0.2749592717986707
384.491097647097952
2π
1
− 8π / 12
-0.315892564488034
0.55024527430811466
221.748549270323836
4π
1
− 7 π / 12
0.0964086125687454
-0.5670556023774596
41.6109211005907538
5π
1
− 7 π / 12
-0.453197224138077
-0.4144163826091202
-15.585384191692458
2 8π
1
− 7 π / 12
0.4631952732930986
-0.4057339001088209
388.421828248114791
5 7π
1
− 7 π / 12
0.5903938574885914
0.16641598734819502
-264.18274471860111
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φ (0)
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2π
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kt 2
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P