Vibrational energy flow analysis of corrected flexural waves in Timoshenko beam – Part II : Application to coupled Timoshenko beams

This paper presents the methodology for the energy flow analysis of coupled Timoshenko beam structures and various numerical applications to verify the developed methodology. To extend the application of the energy flow model for corrected flexural waves in the Timoshenko beam, which is developed in the other companion paper, to coupled structures, the wave transmission analyses of general coupled Timoshenko beam systems are performed. First, power transmission and reflection coefficients for all kinds of propagating waves in the general, coupled Timoshenko beam structures are derived by the wave transmission approach. In numerical applications, the energy flow solutions using the derived coefficients agree well with the classical solutions for various exciting frequencies, damping loss factors, and coupled Timoshenko beam structures. Additionally, the numerical results for the Timoshenko beam are compared with those for the Euler-Bernoulli beam.


Introduction
The Energy Flow Analysis (EFA) has been developed as a promising tool for predicting acoustic and vibrational responses of built-up structures in the medium-to-high frequency ranges over recent decades, and has the advantage in these frequency ranges over the traditional vibro-acoustic analytic tools such as the traditional finite element method (FEM), boundary element method (BEM), and the statistical energy analysis (SEA) [1,2].
Until now, most researches on EFA have been restricted to the analysis of simple structures such as rod, Euler-Bernoulli beams, membrane and Kirchhoff plates in structural elements [1,2,[4][5][6][7].Especially, the research on the coupling relationships for EFA of the coupled Timoshenko beam or Mindlin plate structure has never been properly performed.Though EFA is a more suitable method in the high than low frequency range, the traditional energy flow models were not able to consider the effects of rotatory inertia and shear distortion, which are important at high frequencies [3].Therefore, to improve the EFA vibrational predictions of coupled beam structures at high frequencies, the development of the energy flow model for the Timoshenko beam including these effects and the researches on coupling relationships for the energy flow model are needed [9].
In this paper, first, to extend the application of the energy flow model, which is developed in the other companion paper, into the coupled Timoshenko beam system, the wave transmission analyses for the general three-dimensional joint of coupled Timoshenko beam structures are performed.Finally, to verify the accuracy and validity of the developed energy flow model and coupling relationships, the energy flow solutions and classical solutions for the coupled Timoshenko beam structures are compared for several different conditions, and the results of the Timoshenko beam model are also compared with those of the Euler-Bernoulli beam model.

Wave transmission analysis of N Timoshenko beams coupled in three-dimensional joint
Figure 1 shows N semi-infinite undamped Timoshenko beams coupled in a three-dimensional joint.In this case, two types of flexural waves, one longitudinal wave, and one torsional wave exist in each beam.Two types of flexural waves oscillate respectively in the y-and z-direction, which are orthogonal in the cross-section of beam.Each type of flexural wave in the Timoshenko beam consists of two kinds of wave components, which are bending dominant flexural wave (BDFW) and shear dominant flexural wave (SDFW), having different wavenumbers [9].Therefore, above the critical frequency, a total of six kinds of propagating waves can exist maximally in one Timoshenko beam.
The general solutions for the homogeneous undamped problem of corrected flexural motion in a Timoshenko beam can be expressed by [9] where v is the transverse displacement, α is the angle of rotation due to bending, A is the cross sectional area of the beam, I is the second moment of area of the beam's cross section, ρ is the density of the beam, C ij is the constant coefficient, κ is the shear factor, ω is the angular excitation frequency, G and E are the shear and Young's modulus respectively, (2) By Eq. ( 1), the y-directional transverse displacement v I and the angle of rotation about z-axis due to bending moment α zI in the incident beam can be represented as where "I" in the subscript denotes the incident beam, and k 1yI and k 2yI are wavenumbers of y-directional transverse displacement in the incident Timoshenko beam.Additionally, the z-directional transverse displacement w I and the angle of rotation about y-axis due to bending moment α yI in incident beam are represented as where k 1zI and k 2zI are wavenumbers of the z-directional transverse displacement in the incident Timoshenko beam.
The longitudinal displacement u I in the incident beam can be represented as where k lI is the longitudinal wavenumber of the incident beam (k lI = ω ρ I /E I ).
Unlike the case of two-dimensional joint, this case has a torsional displacement θ I , which can be represented as where k tI is the torsional wavenumber in the incident beam and is expressed as k tI = ω ρ I /G I for the circular section.
The two types of flexural waves, one longitudinal wave, and one torsional wave in beam i coupled with the incident beam at three-dimensional joint also are represented, respectively, as where "i" in the subscript denotes the beam i.When the excitation frequency is higher than the critical frequency of the incident beam, six kinds of propagating waves can be maximally incident upon the joint.If the BDFW with wavenumber k 1yI in the incident beam is incident upon the joint, the displacements in the incident beam are composed of only the wave components outgoing from joint and can be expressed, respectively, as Because the incident waves do not exist in beam i, the two types of flexural waves, one longitudinal wave, and one torsional wave, which are traveling in the +x i -direction, can be rewritten, respectively, as When one kind of propagating wave is incident upon the three-dimensional joint in the incident beam, six unknown coefficients exist in the incident beam.The same unknown coefficients are in each of the others beams.Therefore, when the number of N beams including the incident beam are coupled at a three-dimensional joint, a total of 6N unknown coefficients exist, and the same equilibrium conditions are required to solve these unknown coefficients.In this case, the 6N equilibrium conditions consist of three moment equilibrium and three force equilibrium conditions, the 3 (N − 1) continuities of displacement, and the 3 (N − 1) continuities of slope.
The three moment equilibrium conditions about the three orthogonal directions in the incident beam can be represented, respectively, as where M tj , M yyj , and M zzj are the torsional moment (M tj = T j (dθ j /dx j )), moment about y-axis (M yyj = E j I zj (dα zj /dx j )), and moment about z-axis (M zzj = E j I yj (dα yj /dx j )) in the local coordinates of beam j, respectively.The sign convention of the moments in each beam is shown in Fig. 2. β pqmn denotes the angle between the p-axis of beam m and the q-axis of beam n, shown in Fig. 1.
The three force equilibrium conditions about three orthogonal directions in the incident beam are also represented, respectively, as where F xj , V yj , and V zj are the axial force in the x-direction (F xj = E j A j (∂u j /∂x j )), shear force in the y-direction (V yj = κ j G j A j (∂v i /∂x j − d zj )), and shear force in the z-direction (V zj = κ j G j A j (∂w j /∂x j − d yj )) in the local coordinates of beam j, respectively.The sign convention of the forces in each beam is shown in Fig. 2. The 3 (N − 1) continuities of displacement must be enforced in the x-, y-, and z-direction in the local coordinates of the incident beam, respectively; where u j , v j , and w j are the local x-, y-, and z-directional displacements in beam j, respectively.The 3 (N − 1) continuities of slope must be also enforced about the x-, y-, and z-axes in the local coordinates of incident beam, respectively; where θ tj , α zj , and α yj are the torsional displacement, and the angles of rotation about the y-and z-axis due to bending moment, respectively.When all 6N conditions at the joint are applied, the 6N unknown coefficients can be solved numerically.Using the coefficients solved, the time-averaged far-field flexural powers in the incident beam can be expressed, respectively, as [9] P in fy = 1 2 where P ij ± fr denote the time-averaged far-field power of the r-direction flexural wave propagating with wavenumber k ir in the ±x j direction in beam j.
The time-averaged far-field longitudinal and torsional powers in the incident beam can be represented, respectively, as [7] where T is the torsional stiffness, k t is the wavenumber of the torsional wave, and P i ± l and P i ± t are the timeaveraged far-field powers of the longitudinal and torsional waves propagating in the ±x i directions in beam i, respectively.
The time-averaged far-field flexural, longitudinal, and torsional powers in beam i, as in the incident beam, can be represented, respectively, as From the powers calculated above, all the power transmission and reflection coefficients can be expressed, respectively, as where τ and γ are the power transmission and reflection coefficients, respectively, and the subscript in τ fy1I,f z2i means the power transmission coefficient of the transmitted z-directional SDFW with wavenumber k 2z in beam i due to the incident BDFW with wavenumber k 1y in incident beam.All power coefficients in Eq. ( 52) can be summed according to the principle of conservation of energy; If other waves are incident upon at the joint in addition to the BDFW with wavenumber k 1y in the incident beam, the power transmission and reflection coefficients can be obtained by the same procedure.

Wave transmission analysis of two Timoshenko beams joined at an arbitrary angle
For a numerical example for the wave transmission analysis of the coupled semi-infinite Timoshenko beam structure, as shown in Fig. 3, a simple structure of two semi-infinite Timoshenko beams joined at an arbitrary angle is applied.For the energy flow analysis of this coupled Timoshenko beam structure, the power transmission and reflection coefficients among all propagating waves in each semi-infinite Timoshenko beam are required and can be calculated by the procedure of Section 2.
In the first case, two coupled semi-infinite Timoshenko beams of the same material and geometric properties are joined at various coupling angles.The material properties of both beams are those of steel and the dimensions of the cross-sections of both rectangular beams are B × H = 0.01 m × 0.01 m.The critical frequencies of both beams are about 158.73 kHz. Figure 4 shows the power transmission and reflection coefficients for the case in which the BDFW with wavenumber k 1 , the SDFW with wavenumber k 2 , and the longitudinal wave are incident upon the joint of 45 • .In all sub-figures of Fig. 4, the sum of all power coefficients is one, regardless of the excitation frequency by the principle of conservation of energy.The power transmission and reflection coefficients change severely near the critical frequency.Because SDFW components with wavenumber k 2 are near-field terms below the critical frequency, the power transmission and reflection coefficients, such as τ f 11,f 22 ,τ f 21,f 12 , and so forth, related to the SDFW with wavenumber k 2 in each beam, are zero in the Fig. 4. In Figs 4(a) and 4(b), when a kind of flexural wave (BDFW or SDFW) is incident upon the joint in beam 1, the power transmission coefficient, for instance τ f 11,f 12 , between flexural waves having the same kind of wavenumber is much larger than the power transmission coefficient, for instance τ f 11,f 22 , between flexural waves having a different kind of wavenumber.In Fig. 4(c), when the longitudinal wave is incident upon the joint, the incident longitudinal wave is principally transmitted to the BDFW with wavenumber k 1 out of two kinds of flexural waves, and transmitted to the wave of the same type, which is a longitudinal wave, in beam 2. Additionally, though the BDFW with wavenumber k 1 and the longitudinal wave are well transformed mutually through the joint as shown in Figs 4(a) and 4(c), the incident SDFW with wavenumber k 2 is principally transmitted to only the wave of the same kind, which is the SDFW with wavenumber k 2 , in Fig. 4(b).Figure 5 shows the power transmission and coefficients in the structure of two coupled semi-infinite Euler-Bernoulli beams having the same material and geometrical properties as those of the first case shown in Fig. 4. In Fig. 5, because the critical frequency does not exist, the transition region of the power coefficients shown in Fig. 4 does not appear.In Figs 5(a Figure 6 shows the power transmission and reflection coefficients for all propagating waves if the BDFW with wavenumber k 1 , the SDFW with wavenumber k 2 , and the longitudinal wave are incident upon the joint of 90 • .The sum of all power coefficient values is one, regardless of the excitation frequency, by the principle of conservation of energy as shown in Fig. 6.As expected, because two beams are joined at 90 • , the incident BDFW with wavenumber k 1 and longitudinal wave in beam 1 are principally transmitted to the longitudinal wave and BDFW with wavenumber k 1 in Figs 6(a) and 6(c), respectively.However, if the SDFW with wavenumber k 2 is incident, though the joint angle is normal, the power transmission coefficient τ f 21,f 22 of the transmitted SDFW having the same kind of wavenumber as the incident flexural wave is larger than τ f 21,l2 of the transmitted longitudinal wave in Fig. 6(b).Therefore, as shown in Figs 4 and 6, the bending and longitudinal motions are well transformed each other as the geometric characteristics of the structure but the shear motion is principally generated by the same shear motion.
In the second case, two coupled semi-infinite Timoshenko beams of different material and geometry properties are examined at 90 • coupling angle.The material properties of beam 1 and beam 2 are those of steel and aluminum, by the principle of conservation of energy in Fig. 7.In Fig. 7(a), because the critical frequencies of the two beams are different, the two transition regions of the power coefficients exist.Additionally, because the SDFW with wavenumber k 2 in each beam is a near-field term below the critical frequency, the power transmission and coefficients related to this wave are zero in this frequency region of each beam in Fig. 7.In Fig. 7, the power reflection coefficients due to each incident wave roughly become larger than those of the first case because the two coupled semi-infinite Timoshenko beams are heterogeneous and have widely different impedances at the joint.
When the flexural and longitudinal waves in beam 2 are incident upon the joint, all the power transmission and reflection coefficients can be obtained by the same procedure.

Wave transmission analysis of a three-dimensional joint
For the second numerical example of the wave transmission analysis, three semi-infinite Timoshenko beams coupled at a three-dimensional joint as shown in Fig. 8 are applied.As mentioned in the previous section, two types of flexural waves, one longitudinal and one torsional wave exist in each semi-infinite Timoshenko beam coupled at the three-dimensional joint.For the energy flow analysis of this Timoshenko beam structure coupled at threedimensional joint, the power transmission and reflection coefficients for all propagating waves in each semi-infinite Timoshenko beam are required and can be calculated by the procedure of Section 2.
In the coupled Timoshenko beam structure shown in Fig. 8, three semi-infinite Timoshenko beams are joined at right angles each other.The material properties of all beams are those of steel and the dimensions of cross-sections of all rectangular beams are B × H = 0.01 m × 0.01 m.The critical frequencies of all beams are about 158.73 kHz.Above the critical frequencies of all beams, a total of eighteen propagating waves exist.Therefore, whenever one propagating wave is incident upon the joint, eighteen power transmission and reflection coefficients can be obtained.Figure 9 shows the power transmission and reflection coefficients when each propagating wave is incident upon the joint.Because the cross-sections of all beams are square, the y-and z-direction wavenumbers in each beam are identical.In Figs 9(a) and 9(c), the incident y-and z-directional BDFWs with wavenumber k 1 in beam 1 are well transmitted to the longitudinal wave of beams 2 and 3, respectively, like the first example.Additionally, the incident longitudinal wave in beam 1 is principally transmitted to the y-directional BDFWs with wavenumber k 1 in beams 2 and 3, and the power transmission coefficients, τ l1,f y12 and τ l1,f y13 , are the same as shown in Fig. 9(e).However, the incident y-and z-directional SDFWs with wavenumber k 2 are principally transmitted to not the SDFWs with the same kind of wavenumber k 2 , but the torsional waves, as shown in Figs 9(b) and 9(d).When the torsional wave is incident upon the joint in beam 1, the incident torsional wave is mostly reflected to the torsional wave in beam 1 below the critical frequency at which the propagating SDFWs with wavenumber k 2 do not exist, but is principally transmitted to the SDFWs with wavenumber k 2 in beams 2 and 3 above the critical frequency as shown Fig. 9(f).Generally, the BDFW with wavenumber k 1 and the longitudinal wave are well transformed mutually through the joint, and the SDFW with wavenumber k 2 is well transformed into the torsional wave.

Energy flow analysis of two finite Timoshenko beams joined at an arbitrary angle
To verify the power transmission and reflection coefficients derived for the energy flow analysis of a coupled Timoshenko beam structure, numerical analyses are performed for the two finite Timoshenko beams which are joined two kinds of energetics as mentioned [9]; where c gf 1 and c gf 2 are the group velocities of BDFW and SDFW with wavenumbers k 1 and k 2 , respectively, and Π f 1,in and Π f 2,in are the input powers of two kinds of flexural components.The energy flow model for the longitudinal wave in the Timoshenko beam can be represented as [7] ( where c gl is the group velocity of the longitudinal wave in the beam and represented as c gl = E/ρ , and Π l,in is the input power of the longitudinal component. To obtain the accurate energy flow solution of the coupled Timoshenko beam structure, the time-averaged input power by the external force shown in Fig. 10 must be calculated as well as the power transmission and reflection coefficients of all propagating waves.In Eqs (54) and (55), though a point force is applied to the structure, the time-averaged total input power is separated into powers by two kinds of flexural waves unlike the energy flow model of the Euler-Bernoulli beam.When ω > ω c , the propagating transverse displacement by Eq. ( 1) can be expressed as, where A, B, C, and D are complex coefficients, k 1c and k 2c are the complex wavenumbers of BDFW and SDFW including the hysteretic damping η respectively.By separating the flexural waves into the BDFW and SDFW, Eq. ( 57) can be rewritten as, where v 1 (x, t) = A e −jk1cx + B e jk1c x e jωt and v 2 (x, t) = C e −jk1cx + D e jk1cx e jωt .Therefore, the time-averaged input powers in Eqs ( 54) and (55), Π f 1,in and Π f 2,in , can be calculated, respectively, as follows: The time-averaged total input power can be express as the summation of the input powers by the two kinds of flexural waves: The time-and locally space-averaged far-field energy density and power for the flexural wave in the homogeneous region i of each Timoshenko beam can be expressed, respectively, as where A i , B i , C i , and D i are the coefficients determined by the boundary conditions of the homogeneous region i, ēi f 1 and ēi f 2 are the time-and locally space-averaged far-field energy densities of flexural waves with wavenumbers k 1 and k 2 , respectively, and ω ci is the critical frequency of each beam ( The time-and locally space-averaged far-field energy density and power when ω > ω ci can be rewritten, respectively, as (64,65) The φ f 1i and φ f 2i in Eq. ( 62) are defined, respectively, as where η i is hysteretic damping loss factor, c gf 1i and c gf 2i are the group velocities of the BDFW and SDFW respectively, in region i of each Timoshenko beam.
The energy flow solutions for the longitudinal waves in the homogeneous region i of each Timoshenko beam similar to those of each Euler-Bernoulli beam, can be expressed as [2] ēi l = M i e −φ li x + N i e φ li x (i = 1, 2, 3), and where φ li is defined as φ li = η i ω/c gli .
To determine the unknowns in Eqs (62) and (68), the energy and power boundary conditions are required.Since the energy outflows are zero at the simply supported edges, the following equations are obtained; From the conservation of power and the continuity of energy density between regions 1 and 2, the following relations are obtained;   At the joint of the two beams, the various incident waves are converted to other types of waves.From the wave transmission analysis discussed in previous sections, the relations among the powers of waves existing in the beams are expressed in terms of the power transmission and reflection coefficients, +τ l2,f 13 q2 where τ f 22,f 13 means the power transmission coefficient of the transmitted BDFW in beam 3 due to the incident SDFW in beam 2, the superscripts (+) and (−) represent wave propagation in the +x and −x directions, respectively, and the power can be expressed as q ± = c g ē ± in terms of the energy density.Using the upper boundary conditions, the unknowns in Eqs (62) and (68) are calculated and the time-and locally space-averaged far-field energy densities and powers of each wave are obtained by Eqs (62), ( 63), (68), and (69).The classical solutions for this model can be obtained by the similar procedure which is described in the other companion paper.
In the first example, when the joint angle is θ = 45 • , and the hysteretic damping is η = 0.01, the energy flow solutions and classical solutions of the Timoshenko beam model are compared with those of the Euler-Bernoulli beam model at various excitation frequencies.Figure 11 shows the results for the case of f = 1 kHz.In all sub-figures of Fig. 11, the energy flow solutions of the Timoshenko beam and the Euler-Bernoulli beam models represent well the global variation of the classical solutions of respective models.Because the critical frequencies of both beams in this example are about f c = 159 kHz, the excitation frequency is much lower than the critical frequency of each beam.As expected, the time-and locally space-averaged far-field energy densities and powers between the Timoshenko beam model and the Euler-Bernoulli beam model are not greatly different, as shown in Fig. 11.However, as the excitation frequency increases, because the effects of shear distortion and rotatory inertia become dominant, the energy density and power of the Timoshenko beam model become significantly different from those of the Euler-Bernoulli beam model.In Fig. 12, the excitation frequency is set to 250 kHz, which is higher than the critical frequency.The energy flow solutions of the two beam models show better approximation of the classical solutions of respective beam models than the corresponding results obtained for f = 1 kHz.The flexural energy density and power of the Timoshenko beam model show different distributions from those of the Euler-Bernoulli beam model in Figs 12(a) and 12(c).Additionally, because the energetics of longitudinal waves used in the two beam models are the same, the longitudinal energy densities and powers of the two beam models show the same distributions due to the same group velocities in Figs 12(b) and 12(d).However, the levels of the longitudinal energy densities and powers of the two beam models are not same because the flexural energy, and the power transmission and reflection coefficients of the two beam models have different values.
In the second example, the hysteretic damping loss factor is changed into η = 0.1.Figures 13 and 14 show the results for f = 1 kHz and f = 50 kHz, respectively.In Figs 13 and 14, the energy flow solutions and classical solutions of each model show very good agreement with the corresponding results obtained for η = 0.01, regardless of frequency.The energy flow solutions of the Timoshenko beam model agree well with the classical solutions as the damping loss factor increases, like the Euler-Bernoulli beam model [7].In Figs 13(a) and 13(c), because the excitation frequency is much lower than the critical frequency, the flexural energy density and power of the Timoshenko beam model have the almost same distribution and level of those of the Euler-Bernoulli beam model.However, the levels of the longitudinal energy density and power of each model are only slightly different, as shown in Figs 12(b) and 12(d).In Fig. 14, though the excitation frequency is lower than the critical frequency, the results of the Timoshenko beam model are considerably different from those of the Euler-Bernoulli beam model because the structural damping term of the Timoshenko beam model, which can be expressed as ηω/c gf , is larger than that of the Euler-Bernoulli beam model.Based on these results, generally, as the excitation frequency increases in the high frequency ranges, the peak flexural energy density level of the Timoshenko beam model becomes higher than that of the Euler-Bernoulli beam model, but the flexural energy density and power decrease rapidly due to large structural damping loss as the distance from the excitation point grows longer.
In the final example, the joint angle and hysteretic damping are set to θ = 90 • and η = 0.1 respectively.As expected, in all sub-figures of Fig. 15, the energy densities and powers at the joint change more than those in the Fig. 14 of θ = 45 • .In Fig. 15(a), the flexural energy density is nearly zero decibel at the end of the coupled Timoshenko beam because of the large damping loss of beam and attenuation in the joint.

Conclusion
For the improved vibrational energy flow analysis of coupled Timoshenko beam structures, the wave transmission analysis on the general three-dimensional joints in the Timoshenko beam was performed.The derived power transmission and reflection coefficients of all propagating waves in the Timoshenko beam model were compared with those of all waves in the Euler-Bernoulli beam model and were verified for the various conditions.
To verify the developed energy flow model and coupling relationships, numerical analyses for various examples were performed.In the coupled Timoshenko beam structure, energy flow analyses were successfully performed for the various coupled Timoshenko beam structures by using the derived power transfer coefficients and energy governing equations of each kind of propagating wave.The energy flow solutions and classical solutions for coupled Timoshenko beam structures show a good agreement at various frequencies and damping loss factors.Generally, as the excitation frequency and damping loss factor increase, the results from the energy flow models between the Timoshenko beam and Euler-Bernoulli beam were remarkably different.
The developed energy flow model for the finite coupled Timoshenko beam can be used as the advanced tool for the prediction of vibrational energy density and power distributions of built-up structures composed of beams in the high frequency ranges.
Fig. 1.N semi-infinite undamped Timoshenko beams coupled in three-dimensional joint and the indication of angles among coordinates axes.

Fig. 2 .
Fig. 2. The sign convention of the shear forces and moments of two Timoshenko beams coupled in three-dimensional joint.

Fig. 4 .
Fig. 4. The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with the same material and geometrical properties joined at 45 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.

Fig. 4 ,
Fig. 4, continued.The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with the same material and geometrical properties joined at 45 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.

Fig. 5 .
Fig. 5.The power transmission and reflection coefficients of two semi-infinite Euler-Bernoulli beams with the same material and geometrical properties joined at 45 • : (a) Case of the incident flexural wave, (b) Case of the incident longitudinal wave.

Fig. 6 .
Fig. 6.The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with the same material and geometrical properties joined at 90 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.

Fig. 6 ,
Fig. 6, continued.The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with the same material and geometrical properties joined at 90 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.

Fig. 7 .
Fig. 7.The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with different material and geometrical properties joined at 90 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.

Fig. 7 ,
Fig. 7, continued.The power transmission and reflection coefficients of two semi-infinite Timoshenko beams with different material and geometrical properties joined at 90 • : (a) Case of the incident BDFW with wavenumber k 1 , (b) Case of the incident SDFW with wavenumber k 2 , (c) Case of the incident longitudinal wave.
) and 5(b), the incident flexural and longitudinal waves are well converted to transmitted longitudinal and flexural waves, respectively, due to the 45 • joint, as shown in Figs 4(a) and 4(c).Below the critical frequency of Fig. 4, because only the BDFW with wavenumber k 1 is a far-field term, the power transmission and reflection coefficients of Timoshenko beams in Figs 4(a) and 4(c) have the similar values of distribution and level with those of Euler-Bernoulli beams in Figs 5(a) and 5(b), respectively.However, above the critical frequency, because two kinds of flexural waves are far-field terms, the power coefficients of Timoshenko beams shown in Fig. 4 are very different from those of Euler-Bernoulli beams shown in Fig. 5.

Fig. 9 .
Fig. 9.The power transmission and reflection coefficients of three semi-infinite Timoshenko beams with the same material and geometrical properties joined at 90 • : (a) Case of the incident y-directional flexural wave with wavenumber k 1y , (b) Case of the incident y-directional flexural wave with wavenumber k 2y , (c) Case of the incident z-directional flexural wave with wavenumber k 1z , (d) Case of the incident z-directional flexural wave with wavenumber k 2z , (e) Case of the incident longitudinal wave, (f) Case of the incident torsional wave.

Fig. 9 ,Fig. 9 ,
Fig. 9, continued.The power transmission and reflection coefficients of three semi-infinite Timoshenko beams with the same material and geometrical properties joined at 90 • : (a) Case of the incident y-directional flexural wave with wavenumber k 1y , (b) Case of the incident y-directional flexural wave with wavenumber k 2y , (c) Case of the incident z-directional flexural wave with wavenumber k 1z , (d) Case of the incident z-directional flexural wave with wavenumber k 2z , (e) Case of the incident longitudinal wave, (f) Case of the incident torsional wave.

Fig. 10 .
Fig. 10.The coupled finite Timoshenko beam structure joined at an arbitrary angle for EFA.