Studies on the natural modes of structures play a very important role in vibration theory and its applications. Besides the quantitative extractions of natural modes of different mechanical systems,qualitative analyses of modes are useful in checking the results of experiments and numerical calculations in structure designs and the solutions of inverse vibration problems^{[1, 2, 3, 4, 5, 6, 7, 8, 9]} .

For the modes of the single span one-dimensional continuums such as strings,bars,torsion bars,and Euler beams (the beams mentioned hereafter are single span Euler beams if not specified),Krein and Gantmacher^[1] summarized the following qualitative properties:

P₁ There is no repetitive natural frequency (except for a beam with two free ends whose first two frequencies are both zero);

P₂ The 1st mode shape has no unconstrained zero point;

P₃ The ith mode shape has exactly (i − 1) nodes;

P₄ The nodes of two contiguous mode shapes alternate;

P₅ Any superposition of the ith,(i + 1)th,· · · ,jth mode shapes reverses its sign no less than (i − 1) times and no more than (j − 1) times.

The ensemble of P₁-P₅ is known as the oscillation property (OP),from which many other qualitative properties can be obtained^{[2, 3, 4, 5, 6]} . The pioneering work on the OP can be traced back to Sturm,a French mathematician in the 19th century,who established the OP of analytic bars. The unified proof of the OP for analytic strings,bars,and beams was given by Gantmacher and Krein with their excellent oscillatory kernel/matrix theory^[1] . It has been proved that some constrained one-dimensional continuums possess the OP as well^{[7, 8]} .

Vibration modes of various discrete models of one-dimensional continuums are supposed to reflect the OP. Otherwise,the discrete models are not reasonable. The discrete form of the OP is given by replacing the mode shapes in P₂-P₅ with u-lines,which are defined as the polylines connecting the successive deflected nodes of the discrete models (see the dot-dash line in Fig. 1)^{[1, 6, 7]} .

Fig. 1. Simple supported beam loaded by three concentrated forces.

Figure options

In fact,commonly used discrete models possess the OP. Some finite difference (FD) bars and finite element (FE) bars are proved to possess the property for any arbitrary mesh^{[6, 10]}by stiffness matrix factorization. The stiffness matrices of 3-point FD bars and 2-node FE bars are both tri-diagonal,and thus can be uniformly factorized into the simple matrix as follows:

Click Browse formula

where L and are diagonal matrices,whose elements are grid sizes and nodal stiffness coeffi- cients,respectively,and E is known as the differential matrix expressed as

Click Browse formula

The factors of K can be easily identified as sign-oscillatory matrices. Therefore,the 3-point FD and the 2-node FE bars possess the OP unconditionally^{[6, 10]}. However,for discrete beams,the patterns of stiffness matrices differ from model to model. Therefore,the discussion based on matrix-factorization is model-specific and boundary-condition- specific. The OP of 5-point FD beams was proved^{[6, 7, 8, 9]} by factoring the stiffness matrix as follows:

Click Browse formula

in which the differential matrix E depends on the boundary conditions. The OP of 2-node FE beams via the potential energy principle (PE-FE beams) was first studied by Gladwell^[11]. For cantilever conditions,he provided

Click Browse formula

where

Click Browse formula

Click Browse formula

For other boundary conditions,the factor matrices are more complicated than the above. Even for cantilever beams,the factors (5) and (6) are too complicated to implement further discus- sion. Assuming that the bending stiffness (flexural rigidity) EI was element-wise constant, Gladwell^[12] simplified the matrices and established the OP of PE-FE beams.

In fact,if EI is element-wise constant,then the mode shapes of PE-FE beams and the analytic beams are both piecewise cubic polynomial between masses. According to the charac- teristics of the FE method (FEM),these PE-FE beams possess identical mode shapes at nodes to the analytic beams with concentrated masses only,and thus possess the OP^[12]. This idea also delivers that arbitrarily meshed 2-node HR-FE beams with the variable bending stiffness EI possess the OP^[12]. But if EI is not element-wise constant,the OP may fail for PE-FE beams,as shown in Ref. ^[12].

However,since the above two approaches,i.e.,the stiffness matrix factorization and the ana- log to analytic beam,are closely coupled to the discrete models,when more complicated discrete beam models are put into consideration,the existing proofs cannot be easily transplanted.

In this paper,a new approach is proposed to discuss the OP of beams,which makes unified discussion on the properties of more complicated discrete beam models possible. We start our discussion with a proposition about a static property of one-dimensional continuums closely related to the OP,which is hence named as the statics oscillation property (SOP).

Proposition 1 (A) If a single span one-dimensional continuum is under the action of one concentrated force (nodal force for discrete systems), then the deflection of any unconstrained point (unconstrained node for discrete models) is non-zero and has the same sign.

(B) If a single span one-dimensional continuum is under the action of k (k = 2,3,• • • ) arbitrary concentrated forces (nodal forces for discrete systems),then its deflection line (u-line for discrete systems) reverses the sign no more than (k − 1) times.

This proposition agrees with our common experiences. See Fig. 1 for example,when a simply supported beam is under the action of three concentrated forces,the deflection line reverses its sign no more than twice. Gantmacher and Krein^[1] proved that possessing the SOP is a sufficient condition for a one-dimensional continuum to possess the OP. Zheng et al.^{[12, 13]} extended the conclusion to discrete models with nodal concentrated masses only,and proved the equivalence between the OP and the SOP in discrete cases.

In this paper,the SOP is directly studied instead of the OP. The SOP of one-dimensional continuums can be recognized as a corollary of Rolle’s Theorem^[1].Similarly,the SOP of discrete strings,bars,and torsion bars can be proved by analyzing the monotonicity of the u-lines. Unfortunately,the proof of the SOP for discrete bar models cannot be directly used for discrete beam models. Up to now,direct study on the SOP of discrete beam models has not been found in literatures.

Based on the local criteria of the balance,the deformation,and the positivity of the strain energy obtained in an analytic beam model,the SOP of discrete beam models is proved con- sistently in this paper. The result covers the known conclusions that the SOP/OP exists in 5-point FD beam models^{[6, 7, 8]} and 2-node HR-FE beam models^[12]. A sufficient condition for the SOP/OP of PE-FE beam models is also delivered from the result,which includes the condition that the bending stiffness is element-wise constant,as discussed in Refs. ^{[11, 12]},and supports the conjecture in Ref. ^[11] that the OP exists in the PE-FE beams with element-wise linear bending stiffness. Besides,the SOP/OP of FE Timoshenko beams with a small slenderness ratio is guaranteed by the approach.

The proposed approach in this paper can also be transplanted to discuss the qualitative property of vibrating multibearing beams. The natural vibration of multibearing beams shows a qualitative property closely related to P₁-P₅,which is known as the OP of multibearing beams^[9]. The corresponding property of FD multibearing beams has also been proved by matrix factorization with an assumption that the sizes of the two grids clamping any inner support do not vary acutely^[9]. However,through the present approach,the OP of a few kinds of discrete multibearing beam models,such as FD and FE beams,can be obtained by performing a simple transformation,and the mesh size assumption is no longer necessary.

The entire discussion of the OP for discrete beam models with lumped masses will be given as follows. In Section 2,the preliminary knowledge is given. For easy understanding,a proof to the SOP of analytic beams presented by Gantmacher and Krein^[1] is shortly reviewed. Based on their concept,a unified sufficient condition for the SOP of discrete beams is proposed in Section 3. In Section 4,many practical models are shown to satisfy the proposed condition, such as 5-point FD beams,2-node HR-FE beams,most 2-node PE-FE beams,and most 2-node Timoshenko FE beams. In Section 5,the OP of multibearing beams is briefly introduced and discussed by the proposed approach,and other discrete multibearing beam models are found to possess the property besides the FD ones discussed in literatures. In Section 6,the main conclusions of this paper are reviewed. 2 Preliminary

Before discussion,we first review a few lemmas and the establishment of the SOP/OP of analytic beams in Ref. ^[1].

Lemma 1^{[1, 10]} One-dimensional continuums possess the OP for any mass distribution (continuous and/or lumped) possessing the SOP. The discrete models of one-dimensional con- tinuums with any lumped mass possessing the SOP possess the OP.

For analytic beams,the SOP can be derived by analyzing the governing equation with the following result.

Lemma 2 If a continuous function f (x) has k roots in an interval (a,b),then its integral

Click Browse formula

has at most (k + 1) roots in [a,b] for any arbitrary constant C.

Take a simply supported beam for example,whose governing equation is

Click Browse formula

where q(x) is the distributed force,and EI(x) is the bending stiffness. Integrating (7) twice yields

Click Browse formula

in which M (x) is the bending moment distribution in physics.

Suppose that the beam is under the action of k concentrated forces. Because the shear force varies only at the place where there is a concentrated force,the bending moment M (x) is a polyline with (k + 1) segments,and thus has at most (k + 1) roots. Since two of these roots are at the ends according to the boundary condition,there are (k − 1) roots in (0,L). According to Lemma 2,there are at most k roots of w '(x) in [0,L],and there are (k + 1) roots of w(x) in the close interval. Since two of the roots of w(x) are at the ends because of the boundary condition,w(x) reverses its sign no more than (k − 1) times. Figure 2 shows the case k = 3,in which the sign reversion numbers of M (x),w '(x),and w(x) are 2,3,and 2,respectively.

Fig. 2. Beam under action of three concentrated forces.

Figure options

The beams under other boundary conditions can also be discussed similarly. As shown in Table 1,for the beams with the last two boundary conditions,the conclusions are obtained by analyzing their conjugated beams^{[7, 8]} whose corresponding boundary conditions are given as fixed-fixed and fixed-hinged,respectively.

Table 1. Maximum number of isolated roots under k concentrated forces

Table options

Although the analytic beams possess the SOP,it is not obvious that their discrete coun- terparts reflect the property,especially for the FE models which involve a weighted residual procedure. The deduction for analytic beams does not work directly for discrete beams,which is not governed by differential equations. A counter example is given to demonstrate that in Ref. ^[12]. 3 Sufficient condition for SOP/OP of discrete beams

Before the main theorem of this paper is given,let us review a few local qualitative properties of a beam segment [a,b] subject to the concentrated moments M_a and M_b and the concentrated forces f_a and f_b at its two ends,as shown in Fig. 3. The equilibrium delivers

Click Browse formula

Fig. 3. Static balanced beam segment under action of moments and forces.

Figure options

If M_a> 0 and M_b> 0,then M (x) > 0 in [a,b]. Integrating (8) yields

Click Browse formula

in which θ_a and θ_bare the slopes at a and b,respectively. This inequality indicates that if M_a and M_b are both positive,the average slope defined as satisfies

Click Browse formula

Clearly,if M_a and M_b are both negative in turn,the inequality (11) reverses its direction.

Besides,the semi-positivity of the strain energy reserved in the segment leads to

Click Browse formula

The three conclusions based on the beam theory,i.e.,the balance (9),the deformation (11),

Theorem 1 Suppose a discrete beam with N nodes {x_i}^N₁(0 = x₁< x₂< • • • < x_N= L) is under the action of any given set of nodal forces. The discrete beam possesses the SOP if the corresponding nodal sequences of the deflection {w_i}^N₁, the rotation angle {θ_i}^N₁, and the bending moment {M_i}^N₁ satisfy the following three local static properties:

(i) If the bending moments at any two successive nodes M_i and M_i+1 are both non-negative, then

Click Browse formula

where both equalities in (13) hold simultaneously only when M_i and M_i+1 both vanish; if M_i and M_i+1 are both non-positive,the inequality signs in (13) reverse (see Fig. 4 for example); for FD models,in which the nodal rotation θ_i is not given explicitly,the nodal rotation can be defined as

Click Browse formula

Fig. 4. Deformation of beam segment under action of two identical-sign nodal moments.

Figure options

(ii) The strain energy of any beam segment [x_i,x_i+1] is non-negative,i.e.,

Click Browse formula

(iii) The internal shear forces Q(x⁺_i ),Q(x⁻_i+1 ) and bending moments M_i,M_i+1 are statically balanced in (x⁺_i ,x⁻_i+1 ).

Furthermore,if the concentrated mass matrix is employed,it possesses the OP.

As shown before,all assumptions are derived from an analytic beam segment. The proof generally follows the concept in Section 2,where the monotonicity of w(x) is indicated by the slope θ(x). However,the slope sequence {θ_i}^N₁is no longer directly related to the deflection sequence {w_i}^N₁for discrete beams. Instead of {θ_i}^N₁,the average slope sequence {θ_i,i+1}}^N−1₁ plays the role of θ(x) in turn. Therefore,a bridge between {θ_i}^N₁and {θ_i,i+1}}^N−1₁ is the key of the proof.

Proof For convenience,simply supported beams are considered again. Beams with other boundary conditions can be discussed similarly.

Assume that a simply supported discrete beam is under the action of k concentrated forces. Because of the equilibrium condition (iii) in Theorem 1,there are at most (k+1) monotonic sub- sequences of {M_i}^N₁. Except the two ending ones,in each monotonic sub-sequence of {M_i}^N₁ , there is at most one sign reversion. Thus,the sequence {M_i}^N₁can be divided into p (p≤k) sub-sequences,i.e.,{M_i}_nr^n_r+1−1(r = 1,2,3,• • • ,p),such that each sub-sequence {M_i}_nr^n_r+1−1is sign-definite (see Fig. 2(b),where the three sign-definite sub-sequences are separated by vertical dash lines).

If a sub-sequence {M_i}_nr^n_r+1−1 is non-negative,Assumption (i) in Theorem 1 yields θ_nr≤ θ_nr+1 ≤• • • ≤θ_nr+1−1, i.e.,the sub-sequence {θ_i}_nr^n_r+1−1 increases. Similarly,if the sequence {M_i}_nr^n_r+1−1 is non-positive,the sub-sequence {θ_i}_nr^n_r+1−1 decreases. Particularly,if M_j−1M_j<0 and M_jM_j+1<0 appear simultaneously for a node j,we say that M_j forms a sign-definite sub-sequence of {M_i}^N₁ . In that case,we define θ_j as a monotonic sub-sequence of {θ_i}^N₁ (see Fig. 5(b)). Anyhow,the sequence {θ_i}^N₁ can be partitioned into exactly p monotonic sub- sequences corresponding to the p sign-definite sub-sequences of {M_i}^N₁ ,and the monotonicity of the sub-sequences alternates (see Figs. 2(b) and 2(c),in which the monotonic sub-sequences of {θ_i}^N₁ are partitioned by vertical dash lines).

Extending these monotonic sub-sequences of {θ_i}^N₁ such that the boundary terms of any two successive sub-sequences overlay,we can still obtain p monotonic sub-sequences since the monotonicity of the original sub-sequences alternates. Now,in each of these sub-sequences, {θ_i}^N₁ reverses its sign at most once.

However,if there is a non-negative local minimum term θ_j,which is defined as a non- negative ending term of a decreasing sequence and a starting term of an increasing sequence, or a non-positive local maximum term,which can be defined similarly,then the two successive monotonic sub-sequences θ_jdo not reverse their signs,as shown in Fig. 5(a). Figure 5(b) shows a degenerated condition in which θ_jincreases according to its previous definition. Though the series decreases in the illustrated part,θ_jis a positive local minimum term. If there are totally c₁non-negative local minima and non-positive local maxima under the k forces,then the sequence {θ_i}^N₁ reverses its sign no more than (p − 2c₁) ( k − 2c₁) times.

Fig. 5. Non-negative local minimum term of {θi }N1 (hollow points) and corresponding moment under normal and degenerated conditions.

Figure options

Now,we claim that if

Click Browse formula

then either θ_i or θ_i+1 is a non-negative local minimum or a non-positive local maximum. In fact, the given conditions imply that θ_i+1 − {θ_i,i+1} θ_i− {θ_i,i+1} > 0. On the other hand,we have M_iM_i+1<0 (see Fig. 6). Otherwise,{θ_i,i+1} should be somewhere between θ_iand θ_i+1 according to Assumption (i) in Theorem 1. Taking the strain energy inequality (15) into account,we have

Click Browse formula

Fig. 6. Deflection of beam segment under action of two opposite-sign nodal moments.

Figure options

Without loss of generality,suppose {θ_i,i+1}>0, θ_i≤0, and θ_i+1≤0. Then,(16) leads to M_i> 0 and M_i+1<0,i.e.,the series {θ_i}^N₁reverses its monotonicity from increasing to decreasing either at θ_ior θ_i+1. Since θ_i≤0 and θ_i+1≤0,according to our definition,one of them is a non-positive local maximum (see Fig. 7).

Fig. 7. Sketch of nodal slopes and moments when θi < 0,θi+1 < 0,and {θi,i+1} > 0 (not degenerated condition).

Figure options

Consider a sign-definite sub-sequence {θ_i}ⁿ_j . If

Click Browse formula

holds for i = j,j + 1,• • • ,n − 1,then the average rotation sub-sequence {θ_i,i+1}ⁿ⁻¹_j is also sign-definite. But if {θ_i,i+1} ≠ 0 and {θ_i,i+1}(θ_i+ θ_i+1) ≤0 at some node {x_i},the sequence {θ_i,i+1}ⁿ⁻¹_j reverses its sign at most twice for each such i. Since there are (k − 2c₁) sign-definite sub-sequences of {θ_i}^N₁ ,the sequence {θ_i,i+1}^N−1₁ reverses its sign no more than (k − 2c₁+ 2c₂) times,in which c₂denotes the number of i such that {θ_i,i+1} ≠ 0 and {θ_i,i+1}(θ_i+ θ_i+1)≤0 with the condition θ_iθ_i+1≤0. As mentioned above,this happens only if θ_ior θ_i+1 is a non-positive local maximum or a non-negative local minimum. Therefore,c₂≤c₁.

Up to now,we prove that the sequence {θ_i,i+1}^N−1₁=reverses its sign at most k ( k − 2c₁+ 2c₂) times. Because of Lemma 3,the u-line as the integral of the average slope reverses its sign no more than (k − 1) times.

Now,it is explained that if k = 1,then w_i≠ 0 (i = 2,3,• • • ,N − 1). In this condition, without loss of generality,assume that M₁= M_N= 0 and M_i> 0 (i = 2,3,• • • ,N − 1). Assumption (i) in Theorem 1 indicates that the equality in (13) never holds simultaneously. Therefore,

Click Browse formula

If w₂≥0,then (17) leads to w₃> 0,and further w₄> 0,• • • ,w_N> 0,which is against w_N= 0. Thus,w₂< 0. In the same way,w_N-1 < 0. On the other hand,the series {θ_i,i+1}reverses its sign at most one time. Thus,the series {w_i} first decreases and then increases. This together with w₂< 0 and w_N-1 < 0 leads to wi< 0 (i = 2,3,• • • ,N − 1).

According to the definition,the discrete beam possesses the SOP.

The SOP of the discrete beams under the first four boundary conditions in Table 1 can be established similarly,while that of the discrete beams with the last two boundary conditions can be discussed by employing a conjugated transformation first. 4 SOP/OP of discrete beam models

In this section,Theorem 1 is employed to establish the SOP/OP of different beam models by verifying that they satisfy the assumptions within Theorem 1. 4.1 SOP/OP of analytic beams only with concentrated masses

One of the discrete beam models is the analytic beam only with concentrated masses. Taking the points with concentrated masses as nodes,the assumptions of Theorem 1 are satisfied for this model. Therefore,the SOP/OP holds. 4.2 SOP/OP of FD beams

It has been proved that a 5-point FD beam is equivalent to a spring-mass-bar system (see Fig. 8)^[14].

Fig. 8. Equivalent model to FD beam.

Figure options

In the spring-mass-bar system,the nodal slope {θ_i } can be defined as (14). Substituting (14) into Hooks’ law,one obtains

Click Browse formula

It implies that Assumption (i) of Theorem 1 holds. Substituting (14) and Hooks’ law into (15),it follows that

Click Browse formula

Assumption (ii) of Theorem 1 holds. At last,the system in Fig. 8 is a physical system,and Assumption (iii) of balance holds,too.

Corollary 1 5-point FD beams (with lumped mass matrices) possess the SOP (and hence the OP) for arbitrary sectional parameters and meshes. 4.3 SOP/OP of HR-FE beams

Because of the characteristics of the FEM,Assumptions (ii) and (iii) of Theorem 1 are always satisfied. For the FE interpolations,the rotational angles are usually employed as nodal variables. Thus,there is no need to define additional nodal slopes {θ_i}^N₁.

Let I and J be the two ends of a single element. For most practical FE beam elements, the strain energy is determined by the nodal moments M_I and M_J at the two ends if there is no load inside the element. Then,the relative rotational angles,i.e.,Δθ_I= θ_I− θ_I,J and Δθ_J= θ_J− θ_I,J,which are work-conjugate with MIand MJ,are determined by MIand MJ, too. In other words,Δθ_Iand Δθ_Jdo not depend on w_I and w_J. In that condition,Assumption (i) of Theorem 1 can be verified by examining a simply supported beam element and checking whether the corresponding flexibility matrix,i.e.,

Click Browse formula

is positive definite and satisfies

Click Browse formula

or equivalently,the stiffness matrix of the beam element is positive definite and satisfies

Click Browse formula

For a simply supported FE beam segment,the equilibrium equation derived by the Heilinger- Reissener (HR) principle can be expressed as

Click Browse formula

where EI (x) is the bending stiffness and is positive over the segment, is the length of the segment,and N^F = (1 − ξ,−ξ) is the shape function of M (x). The flexibility matrix in (23) can be further expressed as

Click Browse formula

Obviously,every component of the matrix is sign-definite,and (21) is always satisfied for any positive bending stiffness EI (x).

Corollary 2 2-node HR-FE beams (with lumped mass matrices) possess the SOP (and hence the OP) for arbitrary sectional parameters and meshes. 4.4 SOP/OP of PE-FE beams

The equilibrium equation of a simply supported FE beam segment derived by the potential energy principle can be expressed as

Click Browse formula

Here,

Click Browse formula

The stiffness matrix in (25) can be further expressed as

Click Browse formula

Obviously,for the positive bending stiffness EI(ξ),(22) holds if and only if

Click Browse formula

Then,Corollary 2 leads to Corollary 3.

Corollary 3 2-node PE-FE beam models (with lumped mass matrices) possess the SOP (and hence the OP) if (27) holds for all of its elements.

Note that Corollary 3 is a sufficient condition,not a necessary one. But it includes most practical conditions since EI(ξ) can be linear,quadratic,or monotonic in each element. In fact, by illustrating the weight function of (27),i.e.,,in Fig. 9,one can easily find that the condition (27) fails only if EI (ξ) is averagely much larger in (1/3,2/3) than in [0,1/3] ∪ [2/3,1]. Obviously,this seldom happens in practical conditions.

Fig. 9. Weighting function in (27).

Figure options

For analytic beams,only those based on the Euler beam theory show the SOP/OP. How- ever,in our proof to the SOP/OP of discrete beam models,the Euler-Bernoulli formula is not employed. Thus,the SOP/OP may exist for discrete beam models based on other theories. In this subsection,the SOP/OP of a few typical 2-node FE Timoshenko beam models is discussed by the proposed approach.

The stiffness matrix of a simply supported Timoshenko beam element can be derived via the HR principle or potential energy (PE) principle. For the former,the flexibility matrix is

Click Browse formula

in which GA(ξ) and k stand for the shear stiffness and the shear coefficient,respectively. Obviously,the inequality (21) is satisfied if and only if

Click Browse formula

Thus,we obtain Corollary 4.

Corollary 4 2-node HR-FE Timoshenko beams (with lumped mass matrices) possess the SOP (and hence the OP) if (29) holds for all of its elements.

For the Timoshenko element derived via the PE principle,we assume that EI and GA are element-wise constant for simplicity. If the element displacement w(ξ) is interpolated by cubic functions,while the cross sectional rotation θ(ξ) is interpolated by quadratic functions,the stiffness matrix of a simply supported element after static condensation is^[15]

Click Browse formula

It can be found that (22) holds if and only if

Click Browse formula

Thus,for such a PE-FE Timoshenko beam,if (31) is satisfied for all of its elements,the SOP/OP holds.

When EI and GA are element-wise constant,(29) is equivalent to (31). Physically,the left-hand side of (31), ,is the slenderness ratio of the element. In practice,it is supposed to be small such that the beam theory is applicable. Thus,in practice,(31) is usually satisfied. 5 SOP/OP of discrete multibearing beams

The proposed approach can also be employed to discuss relevant qualitative properties of discrete multibearing beams. In fact,the qualitative property of the multibearing FD beams proposed in literature^[9] also holds for any discrete multibearing beam satisfying the assumptions of Theorem 1. In this section,we shall take a beam with one overhang for example (see Fig. 10(a)) to demonstrate that.

Fig. 10. ue-line of overhang beam.

Figure options

Let (u_i,u_o) be the displacement vector of the overhang beam,in which u_i denotes the displacements of the nodes between the two supports,while u_o is the displacement of the overhang nodes. If we employ generalized coordinates in the overhang ,which are the opposites of the nodal displacements (see the hollow points in Figs. 10(b) and 10(c)),and employ the original displacement coordinates between the two supports,then the vibration equation can be expressed as

Click Browse formula

where is the flexibility matrix with respect to ,and can be expressed as

Click Browse formula

Denote -line as the polyline connecting successive deflected free nodes (xi,_i) (see Figs. 10(b) and 10(c)). It is found that for the FD models of such an overhang beam, is oscillatory. Thus,if the mode shapes in P₁-P₅ are replaced with -lines,P₁-P₅ also hold^[9].

More generally,for a multibearing FD beam consisting of s spans, can be defined as

Click Browse formula

in which uⁱ denotes the displacement vector in the ith span. Then,the -lines possess the same properties. This conclusion is known as the OP of multibearing beams due to its relation with P₁-P₅. By employing the -line,the concept of the SOP of multibearing beams is also obtained.

Now,the SOP/OP of the discrete overhang beam is established again under the assumptions of Theorem 1 by the proposed approach (see Fig. 10). It is sufficient to show that when the beam is under the action of k concentrated forces,the -line reverses its sign no more than (k − 1) times.

First,under the action of the k forces,the bending moments {M_i }^N₁ can be divided into no more than (k + 2) monotonic sub-sequences because the monotonicity of {M_i }^N₁ varies only at the (k + 1) points subjected to the forces (including the reacting force of the inner support).

Then,through exact the same progress as in the proof of Theorem 1,we have that {θ_i,i+1}^N₁ reverses its sign no more than (k + 1) times. Thus,the u-line can be divided into no more than (k + 1) monotonic sub intervals.

Finally,the sign reversion of the -line can be obtained by studying the sign reversion of the u-line. Under the k forces,the deflection of the overhang beam falls into one of the following categories:

(I) {θ_i,i+1} ^N₁ does not reverse its sign across the inner support (see Fig. 10(b)). As the u-line can be divided into no more than (k + 1) monotonic sub intervals and vanishes at the outer support,it reverses its sign no more than k times. Because the u-line reverses its sign over the inner support in this condition,the -line does not reverse its sign over the support. The rest roots of the u-line are also roots of the -line,and vice versa. Thus,the -line reverses its sign one time less than the u-line does,i.e.,no more than (k − 1) times.

(II) {θ_i,i+1}^N₁reverses its sign over the inner support (see Fig. 10(c)). Then,the u-line does not reverse its sign in its two successive monotonic intervals clamping the inner support. Thus, the u-line reverses its sign no more than (k − 2) times. In this condition,the -line reverses its sign over the inner support. Thus,it reverses one time more than the u-line does,i.e.,at most (k − 1) times.

In a word,the -line reverses its sign no more than (k− 1) times subjected to k concentrated forces with the assumptions of Theorem 1. Therefore,we have the following corollary.

Corollary 5 A simply supported beam model with one overhang (and lumped mass matrix) possesses the SOP (and hence the OP) if the assumptions of Theorem 1 are satisfied.

In the same way,the SOP/OP of other multibearing beams can also be obtained under the assumptions of Theorem 1. 6 Conclusions

In this paper,a new approach is proposed to discuss the OP of discrete beams. It focuses on a static property closely related to the OP instead of studying the OP directly.

A unified proof of the SOP/OP of discrete beam models is given under three local assump- tions,which are obtained from the classical Euler-Bernoulli beam theory. Compared with the known approaches,the proof focuses on the static properties of a local segment instead of the whole models,and thus no stiffness matrix assembling or factorization is required. Benefiting from that,the verification of the SOP/OP is almost straight forward.

Through the proposed approach,2-node HR-FE beams with concentrated mass matrices and 5-point FD beams are verified to hold the SOP/OP unconditionally,i.e.,beams with arbitrary section parameters and grids always possess the SOP/OP. That agrees with the literatures^{[8, 10]}. For 2-node PE-FE beams,we greatly widen the cases possessing the SOP/OP by providing a sufficient condition (27),which is satisfied by most practical PE-FE elements.

Besides,the proposed approach also shows that FE beams based on the Timoshenko beam theory possess this property,too,if the slenderness ratio is small in each element. This con- clusion corroborates that the results derived via the Euler beam theory and the Timoshenko beam theory are similar under the condition of small slenderness.

Moreover,the proposed approach is generalized to discuss the SOP/OP of multibearing beams. The property of beams with one overhang is also established uniformly with the same assumptions in Theorem 1.