## Abstract

The frequency-dependent mass and stiffness matrices of a Timoshenko–Ehrenfest beam are developed through extensive application of symbolic computation. Explicit algebraic expressions for the frequency dependent shape functions and each of the independent elements of the frequency-dependent mass and stiffness matrices are presented concisely. The ensuing frequency-dependent mass and stiffness matrices of the Timoshenko–Ehrenfest beam are applied with particular reference to the Wittrick–Williams algorithm to investigate the free vibration characteristics of an individual Timoshenko–Ehrenfest beam and a framework. The results are discussed with significant conclusions drawn. The proposed method retains the exactness of results like the dynamic stiffness method, but importantly, it opens the possibility of including damping in the analysis.

## 1 Introduction

The original idea of the frequency dependency of mass and stiffness properties of structural elements for free vibration analysis was developed by Przemieniecki [1,2] who formulated the frequency dependent mass and stiffness matrices of a Timoshenko–Ehrenfest beam and provided series expansions of the matrices by retaining two frequency dependent terms. Przemieniecki’s work was further developed by subsequent researchers [36] who also relied on power series expansion of the mass and stiffness matrices and truncated the series at some point. By contrast, explicit algebraic expressions for the elements of the frequency-dependent mass and stiffness matrices of a Bernoulli–Euler beam using symbolic computation were published recently [7] which circumvented the limitation of earlier research by including all terms of the infinite series implicitly. This technical brief extends the work of Ref. [7] to a Timoshenko–Ehrenfest beam through skillful application of symbolic computation [810]. The resulting frequency-dependent mass and stiffness matrices of the Timoshenko–Ehrenfest beam are related to its dynamic stiffness matrix which is finally utilized to compute the natural frequencies of two illustrative examples with particular reference to the Wittrick–Williams algorithm [11].

## 2 Frequency-Dependent Exact Shape Functions

Figure 1 shows the coordinate system and notations for a Timoshenko–Ehrenfest beam of length L with its node 1 located at the origin O and node 2 at the other end at a distance L from the origin.

Fig. 1
Fig. 1
Close modal
In the usual notation, the governing differential equations of motion in free vibration in terms of the flexural displacement w(x, t) and bending rotation θ(x, t) are [12]
$−ρA∂2w∂t2+kAG∂∂x(∂w∂x−θ)=0$
(1)
$−ρI∂2θ∂t2+EI∂2θ∂x2+kAG(∂w∂x−θ)$
(2)

In Eqs. (1) and (2), A is the area, and I is the second moment of area of the beam cross-section, E, G, and ρ are Young’s modulus, modulus of rigidity (shear modulus), and density of the beam material, respectively, and k is the shear correction factor (also known as shape factor) for the beam cross-section which accounts for the nonuniform shear stress distribution through the beam cross-section, approximately.

Introducing the nondimensional length ξ = x/L and assuming harmonic oscillation with circular or angular frequency ω, Eqs. (1) and (2) can be solved to give the amplitudes of flexural displacement W and bending rotation Θ as [12]
$W(ξ)=A1coshαξ+A2sinhαξ+A3cosβξ+A4sinβξ$
(3)
$Θ(ξ)=B1sinhαξ+B2coshαξ+B3sinβξ+B4cosβξ$
(4)
where α and β are given by
$α2=−b2(r2+s2)2+b22(r2+s2)2+4b2(1−b2r2s2)$
(5)
$β2=b2(r2+s2)2+b22(r2+s2)2+4b2(1−b2r2s2)$
(6)
with
$b2=ρAω2L4EI;r2=IAL2;s2=EIkAGL2$
(7)
and A1A4 and B1B4 are two different sets of constants, related as follows [12]:
$B1=kαLA1;B2=kαLA2;B3=−kβLA3;B4=kβLA4$
(8)
where
$kα=(α2+b2s2α);kβ=(β2−b2s2β)$
(9)
By eliminating the constants A1A4 and B1B4 from Eqs. (3) and (4) with the help of nodal end conditions at ξ = 0 and ξ = 1, respectively, the shape function N relating the displacements within the element (δ) to the nodal displacements (δN) at node 1 (W1 and Θ1) and at node 2 (W2 and Θ2) is given by
$δ=NδNor[δ]=[NwNθ][δ1δ2],i.e.,[WΘ]=[Nw1Nθ1Nw2Nθ2Nw3Nθ3Nw4Nθ4][W1Θ1W2Θ2]$
(10)
Through extensive application of symbolic computation using REDUCE [10], explicit expressions for the shape functions (Nw1, Nw2, Nw3, Nw4) and (Nθ1, Nθ2, Nθ3, Nθ4) are derived and given below
$Nw1=(μ1kβcoshαξ+μ2kβsinhαξ+μ3kαcosβξ−μ2kαsinβξ)/ΔNw2=(−μ4coshαξ+μ3sinhαξ+μ4cosβξ+μ1sinβξ)L/ΔNw3=(μ6kαkβcoshαξ−μ5kβsinhαξ−μ6kαkβcosβξ+μ5kαsinβξ)/ΔNw4=(μ7coshαξ+μ6kβsinhαξ−μ7cosβξ−μ6kαsinβξ)L/ΔNθ1=(μ2coshαξ+μ1sinhαξ−μ2cosβξ−μ3sinβξ)kαkβ/(ΔL)Nθ2=(μ3kαcoshαξ−μ4kαsinhαξ+μ1kβcosβξ−μ4kβsinβξ)/ΔNθ3=(−μ5coshαξ+μ6kαsinhαξ+μ5cosβξ+μ6kβsinβξ)kαkβ/(ΔL)Nθ4=(μ6kαkβcoshαξ+μ7kαsinhαξ−μ6kαkβcosβξ+μ7kβsinβξ)/Δ}$
(11)
with
$μ1=kα(1−ChαCβ)−kβShαSβμ2=kαShαCβ+kβChαSβμ3=kαShαSβ+kβ(1−ChαCβ)μ4=kαChαSβ−kβShαCβμ5=kαShα+kβSβμ6=Chα−Cβμ7=kαSβ−kβShαΔ=(kα2−kβ2)ShαSβ+2kαkβ(1−ChαCβ)}$
(12)
where
$Chα=coshα;Shα=sinhα;Cβ=cosβ;Sβ=sinβ$
(13)

## 3 Frequency-Dependent Mass Matrix

The shape functions derived in Sec. 2 can now be used to derive the frequency-dependent mass matrix m of the Timoshenko–Ehrenfest beam as follows [2,13]:
$m=∫VρNTNdv$
(14)
where T denotes a transpose.
Using Eq. (10), the frequency dependent mass matrix of Eq. (14) can be expressed as
$m(ω)=L∫01[Nw1Nw2Nw3Nw4Nθ1Nθ2Nθ3Nθ4][ρA00ρI][Nw1Nθ1Nw2Nθ2Nw3Nθ3Nw4Nθ4]dξ=[m11m12m13m14m12m22m23m24m13m23m33m34m14m24m34m44]$
(15)
Substituting Eqs. (11) and (12) into Eq. (15) and carrying out the matrix multiplications and integration algebraically with the help of REDUCE [10] yielded explicit expressions for the elements of the frequency-dependent mass matrix m(ω) as follows:
$m11(ω)=m33(ω)=ρALΦ12αα1βΔ2$
(16)
$m12(ω)=−m34(ω)=ρALΦ2L2αα1βΔ2$
(17)
$m13(ω)=ρALΦ32αα1βΔ2$
(18)
$m14(ω)=−m23(ω)=ρALΦ4L2αα1βΔ2$
(19)
$m22(ω)=m44(ω)=ρALΦ5L22αα1βΔ2$
(20)
$m24(ω)=ρALΦ6L22αα1βΔ2$
(21)
where
$Φ1=2αα1β2μ2μ3kα2Cβ2+4αβμ2kαkβ(ζ1−ζ2r2kαkβ)CβChα−αα1β2δ1kα2CβSβ+4αβkαkβ(λ1−κ4r2kαkβ)CβShα+2α1ββ1μ1μ2kβ2Chα2−4αβkαkβ(λ1r2kαkβ+κ4)ChαSβ+α1ββ1γ1kβ2ChαShα−4αβμ2kαkβ(ζ2+ζ1r2kαkβ)SβShα+Γ1$
(22)
$Φ2=−αα1β2ε3kαCβ2+2αβ{kβ(η3r2kα2+κ3)+kα(η1r2kβ2+κ2)}CβChα+αα1β2ψ4kαCβSβ−2αβν1(ζ3r2kαkβ+ζ4)CβShα+α1ββ1ε3kβChα2+2αβν1(ζ4r2kαkβ−ζ3)ChαSβ−α1ββ1ε2kβChαShα−2αβ{r2kαkβ(κ2kα+κ3kβ)−(η3kα+η1kβ)}SβShα+Γ2$
(23)
$Φ3=−αα1β2ν3kα2Cβ2−2αβ{μ2μ6kαkβ(α1r2kαkβ−α4)−kαkβ(β3μ1μ5−β4μ3μ5)}CβChα−αα1β2ν4kα2CβSβ+2αβkαkβ(β4ξ1+2β3μ2μ5)CβShα+α1ββ1ξ2kβ2Chα2−2αβkαkβ(β3ξ1−2β4μ2μ5)ChαSβ+α1ββ1ϕ1kβ2ChαShα−2αβkαkβ{μ2μ6(α4r2kαkβ+α1)−μ5(β3μ3+β4μ1)}SβShα+Γ3$
(24)
$Φ4=αα1β2ρ4kαCβ2−2αβ{μ6kαkβ(β3μ1−β4μ3)+μ2μ7(α1r2kαkβ−α4)}CβChα−αα1β2ξ3kαCβSβ+2αβ(β4ξ4−2β3μ2μ6kαkβ)CβShα+α1ββ1ϕ2kβChα2−2αβ(2β4μ2μ6kαkβ+β3ξ4)ChαSβ+α1ββ1ν2kβChαShα−2αβ{μ6kαkβ(β3μ3+β4μ1)+μ2μ7(α4r2kαkβ+α1)}SβShα+Γ4$
(25)
$Φ5=−2αα1β2μ1μ4Cβ2−αα1β2δ2CβSβ−4αβμ4(ζ2r2kαkβ−ζ1)CβChα+4αβ(η4r2kαkβ−κ1)CβShα−2α1ββ1μ3μ4Chα2+4αβ(κ1r2kαkβ+η4)ChαSβ+α1ββ1(μ32+μ42)ChαShα−4αβ(λ3r2kαkβ+ψ1)SβShα+Γ5$
(26)
$Φ6=αα1β2(μ4μ6kα+μ1μ7)Cβ2+2αβ{μ4μ6(α1r2kαkβ−α4)+μ7(β3μ1−β4μ3)}CβChα+αα1β2ϕ3CβSβ−2αβ(β3ξ1−2β4μ4μ7)CβShα−α1ββ1λ2Chα2−2αβ(β4ξ1+2β3μ4μ7)ChαSβ+α1ββ1ϕ4ChαShα+2αβ{μ4μ6(α4r2kαkβ+α1)+μ7(β3μ3+β4μ1)}SβShα+Γ6$
(27)
with
$α1=α2+β2;α2=α2−β2;α3=−αkα+βkβ;α4=βkα−αkβ$
(28)
$β1=1+kα2r2;β2=1−kβ2r2;β3=αkαkβr2−β;β4=βkαkβr2+α$
(29)
$θ1=1−kα2r2;θ2=1+kβ2r2;θ3=kα2−kβ2;θ4=μ32+μ42$
(30)
$γ1=μ12+μ22;γ2=μ22+μ32;γ3=μ12−μ22;γ4=μ12−μ32$
(31)
$δ1=μ22−μ32;δ2=μ12−μ42;δ3=μ32−μ42;δ4=μ12+μ42$
(32)
$ε1=μ1μ4+μ2μ3;ε2=μ1μ4−μ2μ3;ε3=μ1μ3−μ2μ4;ε4=μ1μ2−μ3μ4$
(33)
$ζ1=αμ3+βμ1;ζ2=αμ1−βμ3;ζ3=αμ2+βμ4;ζ4=αμ4−βμ2$
(34)
$η1=αμ12+βμ2μ4;η2=αμ22−βμ2μ3;η3=βμ32+αμ2μ4;η4=−βμ42+αμ1μ3$
(35)
$κ1=αμ42+βμ1μ3;κ2=αμ32−βμ2μ4;κ3=−βμ12+αμ2μ4;κ4=αμ22−βμ1μ3$
(36)
$λ1=αμ1μ3+βμ22;λ2=kβμ4μ6−μ3μ7;λ3=αμ3μ4+βμ1μ4;λ4=αμ1μ7−βμ3μ7$
(37)
$ψ1=αμ1μ4−βμ3μ4;ψ2=αμ3μ5+βμ1μ5;ψ3=αμ3μ6+βμ1μ6;ψ4=μ1μ2+μ3μ4$
(38)
$ν1=kαμ3−kβμ1;ν2=kβμ2μ6+μ1μ7;ν3=kβμ2μ6+μ3μ5;ν4=kβμ3μ6−μ2μ5$
(39)
$ξ1=kαμ3μ6−kβμ1μ6;ξ2=kαμ2μ6−μ1μ5;ξ3=kαμ2μ6+μ3μ7;ξ4=kαμ3μ7−kβμ1μ7$
(40)
$ρ1=αkαμ32−βkβμ12;ρ2=βkαμ32+αkβμ12;ρ3=βμ22r2kβ2−μ2μ3;ρ4=kαμ3μ6−μ2μ7$
(41)
$ϕ1=kαμ1μ6−μ2μ5;ϕ2=kβμ1μ6+μ2μ7;ϕ3=kαμ1μ6−μ4μ7;ϕ4=kβμ3μ6−μ4μ7$
(42)
and
$Γ1=−αα1βγ4r2kα2kβ2+2αα1ρ3kα2+2α2μ2ζ1r2kα2kβ2+αα1βγ2kα2−4αβμ2ζ1kαkβ+αα1βγ3kβ2−2α1βμ1μ2kβ2$
(43)
$Γ2=−αα1β(ε1θ1kβ+ε4θ2kα)+α1μ1μ3(αβ2kα−ββ1kβ)−α2μ2μ4(αβ2kα+ββ1kβ)−2αβ(ρ2r2kαkβ+ρ1)$
(44)
$Γ3=αα1βμ6kαkβ(μ1θ1kβ−μ3θ2kα)+α2μ2μ6kαkβ(ββ1kβ+αβ2kα)−αμ3μ5kα2(α2r2kβ2−α1)−βμ1μ5kβ2(α2r2kα2−α1)−αα1βμ2μ5θ3−2αβkαkβ(α1μ2μ5r2kαkβ−ψ2)$
(45)
$Γ4=αμ3μ6kα2(α2r2kβ2−α1)+βμ1μ6kβ2(α2r2kα2−α1)+αα1β(μ1μ7θ1kβ+μ2μ6θ3)+α2μ2μ7(ββ1kβ+αβ2kα)−αα1βμ3μ7θ2kα+2αβkαkβ(α1μ2μ6r2kαkβ−ψ3)$
(46)
$Γ5=αα1βδ3r2kα2+2βμ3μ4(α1r2kα2−α2)+αα1β(δ4r2kβ2+γ4)−2αμ1μ4(α1r2kβ2−α2)+2αβμ4(2ζ2r2kαkβ+α1μ4)$
(47)
$Γ6=−αα1βμ6(μ3θ1kβ+μ1θ2kα)−α2μ4μ6(ββ1kβ+αβ2kα)+αα1βμ4μ7θ3r2−βμ3μ7(α1r2kα2−α2)+αμ1μ7(α1r2kβ2−α2)−2αβ(λ4r2kαkβ+α1μ4μ7)$
(48)

## 4 Frequency-Dependent Stiffness Matrix

The shape functions derived in Sec. 2 can likewise be used to derive the frequency-dependent stiffness matrix of the Timoshenko–Ehrenfest beam by using the following relationship [2,13]:
$k(ω)=L∫01[(Nθ′)T(Nw′−Nθ)T][EI00kAG][Nθ′Nw′−Nθ]dξ$
(49)
Substituting Nw and Nθ from Eq. (10) into Eq. (49) gives
$k(ω)=L∫01[Nθ1′Nw1′−Nθ1Nθ2′Nw2′−Nθ2Nθ3′Nw3′−Nθ3Nθ4′Nw4′−Nθ4][EI00kAG][Nθ1′Nw1′−Nθ1Nθ2′Nw2′−Nθ2Nθ3′Nw3′−Nθ3Nθ4′Nw4′−Nθ4]dξ=[k11k12k13k14k12k22k23k24k13k23k33k34k14k24k34k44]$
(50)
Performing all matrix operations and subsequent integration using REDUCE [10], explicit algebraic expressions of the elements of the frequency-dependent stiffness matrix k(ω) are generated as follows:
$k11(ω)=k33(ω)=(EIL3)Ψ12αα1βs2Δ2$
(51)
$k12(ω)=−k34(ω)=(EIL3)Ψ2L2αα1βs2Δ2$
(52)
$k13(ω)=(EIL3)Ψ32αα1βs2Δ2$
(53)
$k14(ω)=−k23(ω)=(EIL3)Ψ4L2αα1βs2Δ2$
(54)
$k22(ω)=k44(ω)=(EIL3)Ψ5L22αα1βs2Δ2$
(55)
$k24(ω)=(EIL3)Ψ6L22αα1βs2Δ2$
(56)
where
$Ψ1=2αα1α¯4μ2μ3kα2Cβ2−4αβμ2kαkβ(αβζ1s2kαkβ+ζ2gαgβ)CβChα−αα1α¯4δ1kα2CβSβ−4αβkαkβ(αβλ1s2kαkβ+κ4gαgβ)CβShα+2α1α¯1βμ1μ2kβ2Chα2+4αβkαkβ(αβκ4s2kαkβ−λ1gαgβ)ChαSβ+α1α¯1βγ1kβ2ChαShα+4αβμ2kαkβ(αβζ2s2kαkβ−ζ1gαgβ)SβShα+Γ¯1$
(57)
$Ψ2=−αα1α¯4ε3kαCβ2+2αβ(−αβκ2s2kα2kβ−αβκ3s2kαkβ2+η3gαgβkα+η1gαgβkβ)CβChα+αα1α¯4ψ4kαCβSβ+2αβ{αβζ4s2kαkβ(μ3kα−μ1kβ)−ζ3gαgβ(μ3kα−μ1kβ)}CβShα+α1α¯1βε3kβChα2+2αβν1(αβζ3s2kαkβ+ζ4gαgβ)ChαSβ−α1α¯1βε2kβChαShα−2αβ{αβs2kαkβ(η3kα+η1kβ)+gαgβ(κ2kα+κ3kβ)}SβShα+Γ¯2$
(58)
$Ψ3=−αα1α¯4ν3kα2Cβ2−2αβkαkβ{αβs2kαkβ(α4μ2μ6−μ5ζ1)+gαgβ(α1μ2μ6−ζ2μ5)}CβChα−αα1α¯4ν4kα2CβSβ−2αβkαkβ{βμ6ν1(α2s2kαkβ−gαgβ)−2αβ¯3μ2μ5}CβShα+α1α¯1βξ2kβ2Cha2−2αβkαkβ{αμ6ν1(β2s2kαkβ+gαgβ)+2ββ¯2μ2μ5}ChαSβ+α1α¯1βϕ1kβ2ChαShα+2αβkαkβ{αβs2kαkβ(α1μ2μ6−μ5ζ2)−gαgβ(α4μ2μ6−μ5ζ1)}SβShα+Γ¯3$
(59)
$Ψ4=αα1α¯4ρ4kαCβ2−2αβ{μ6kαkβ(ββ¯2μ3+αβ¯3μ1)+αα4βμ2μ7s2kαkβ+α1μ2μ7gαgβ}CβChα−αα1α¯4ξ3kαCβSβ−2αβ{2αβ¯3μ2μ6kαkβ+ββ¯2μ7(μ3kα−μ1kβ)}CβShα+α1α¯1βϕ2kβChα2+2αβ{2ββ¯2μ2μ6kαkβ−αβ¯3μ7(μ3kα−μ1kβ)}ChαSβ+α1α¯1βν2kβChαShα+2αβ{μ6kαkβ(ββ¯2μ1−αβ¯3μ3)+μ2μ7(αα1βs2kαkβ−α4gαgβ)}SβShα+Γ¯4$
(60)
$Ψ5=−2αα1α¯4μ1μ4Cβ2−4αβμ4(ββ¯2μ3+αβ¯3μ1)CβChα−αα1α¯4δ2CβSβ+4αβ(ββ¯2μ42+αβ¯3μ1μ3)CβShα−2α1α¯1βμ3μ4Chα2+4αβ(αβ¯3μ42−ββ¯2μ1μ3)ChαSβ+α1α¯1βθ4ChαShα+4αβμ4(ββ¯2μ1−αβ¯3μ3)SβShα+Γ¯5$
(61)
$Ψ6=αα1α¯4(μ4μ6kα+μ1μ7)Cβ2+2αβ{μ4μ6(αβ¯3kα−ββ¯2kβ)+μ7(αβζ1s2kαkβ+ζ2gαgβ)}CβChα+αα1α¯4ϕ3CβSβ−2αβ{αβ¯3μ6(μ3kα−μ1kβ)+2ββ¯2μ4μ7}CβShα−α1α¯1βλ2Chα2+2αβ{ββ¯2μ6(μ3kα−μ1kβ)−2αβ¯3μ4μ7}ChαSβ+α1α¯1βϕ4ChαShα−2αβ{μ4μ6(ββ¯2kα+αβ¯3kβ)+μ7(ββ¯2μ1−αβ¯3μ3)}SβShα+Γ¯6$
(62)
with
$α¯1=α2s2kα2+gα2;α¯2=α2s2kα2−gα2;α¯3=β2s2kβ2+gβ2;α¯4=β2s2kβ2−gβ2;gα=α−kα;gβ=β−kβ$
(63)
$β¯1=αkαgβ2+βkβgα2;β¯2=α2s2kαkβ−gαgβ;β¯3=β2s2kαkβ+gαgβ;β¯4=kα2gβ2+kβ2gα2$
(64)
and
$Γ¯1=αα1α¯2βμ12kβ2−α3βs2kα2kβ2(α2μ22−β2μ32)−2αα2βs2kα2kβ2(αμ1μ2−βμ2μ3)+αα¯3β3γ2kα2+α3βγ2gβ2kα2+2αα1μ2μ3gβ2kα2+4αβgαgβkαkβ(αμ1μ2−βμ2μ3)+αα1βμ22gα2kβ2−2α1βμ1μ2gα2kβ2$
(65)
$Γ¯2=−αα1α¯2βε1kβ+αα1α3βμ1μ3s2kαkβ+αα1α2βμ2μ4s2kαkβ+2α2β2ρ1s2kαkβ−αα1α¯3βε4kα−α1β¯1ε3−2αβη3gαgβkα−2αβη1gαgβkβ$
(66)
$Γ¯3=αα1α¯2βμ1μ6kαkβ2−αα1α2βμ2μ6s2kα2kβ2−αα1β3μ3μ6s2kα2kβ3+αα1α2βμ2μ5s2kα2kβ2+α2α2βμ1μ5s2kα2kβ2−αα2β2μ3μ5s2kα2kβ2−αα1μ6gβ2kα2kβ(μ2+βμ3)+2αβgαgβkαkβ(α1μ2μ6−ζ2μ5)−αα1ββ¯4μ2μ5−αα1μ3μ5gβ2kα2−α1βξ2gα2kβ2$
(67)
$Γ¯4=−αα1α2βμ2μ6s2kα2kβ2−αα2βμ6ζ2s2kα2kβ2+αα1α¯2βμ1μ7kβ−αα1α2βμ2μ7s2kαkβ+αα1ββ¯4μ2μ6+αα1μ3μ6gβ2kα2−αα1β3μ3μ7s2kαkβ2+2αβgαgβ(μ6ζ2kαkβ+α1μ2μ7)−αα1μ7gβ2kα(μ2+βμ3)−α1βϕ2gα2kβ$
(68)
$Γ¯5=−α3α1βδ3s2kα2+2α1α¯1βμ3μ4+4αβμ4(αβζ1s2kαkβ+ζ2gαgβ)+αα1β3δ4s2kβ2+2αα1α¯4μ1μ4+αα1βδ4gβ2+αα1βδ3gα2$
(69)
$Γ¯6=−αα1α¯2βμ3μ6kβ+αα1α2βμ4μ6s2kαkβ−αα1α¯2βμ4μ7−α1α¯1βμ3μ7−αα1α¯3βμ1μ6kα−2αβμ7(αβζ1s2kαkβ+ζ2gαgβ)+α1β¯1μ4μ6−2αα1βμ4μ6gαgβ−αα1α¯4μ1μ7−αα1α¯3βμ4μ7$
(70)
and all the rest of the terms have already been defined before.

## 5 Application of the Frequency-Dependent Mass and Stiffness Matrices

The frequency-dependent mass and stiffness matrices in axial motion already exist in the literature [2,10] which can be combined with the above theory for free vibration analysis of frame structures consisting of Timoshenko–Ehrenfest beam elements. In the expressions for the mass and stiffness elements given above, the frequency ω must not be set to exactly zero, but a small value, e.g., ω = 10–4 or 10–5 rad/s can be used for most of the practical problems so that any numerical overflow or ill-conditioning can be avoided and yet the degenerate case giving the frequency-independent mass and stiffness matrices that are generally used in the finite element method (FEM) can be obtained.

The frequency-dependent mass and stiffness matrices for the Timoshenko–Ehrenfest beam m(ω) and k(ω) derived in Secs. 3 and 4 can be related to its dynamic stiffness matrix kD(ω) by the following relationship [7]:
$kD(ω)=k(ω)−ω2m(ω)$
(71)

Equation (71) can now be applied either to an individual Timoshenko–Ehrenfest beam or to a plane or space frame when investigating the free vibration characteristics. For frameworks, the frequency-dependent mass and stiffness matrices in axial motion [2,7] must be incorporated into the corresponding flexural mass and stiffness matrices given by Eqs. (16)(21) and Eqs. (51)(56), and then for all individual elements in the frame, the k(ω) and m(ω) matrices should be assembled using conventional transformation based on the orientation of the elements, as commonly employed in FEM. Once the overall frequency-dependent mass and stiffness matrices K(ω) and M(ω) and hence the overall dynamic stiffness matrix KD(ω) of the final structure are constructed the natural frequencies and mode shapes follow from the application of the Wittrick–Williams algorithm [11].

## 6 Numerical Results and Discussions

The theory developed above was applied to a wide range of problems and it was ascertained that the frequency-dependent mass and stiffness matrices give the same results that can be obtained by the conventional dynamic stiffness method which uses a single matrix containing both the mass and stiffness properties, rather than using separate mass and stiffness matrices. Two illustrative examples are presented here. The first example focuses on the free vibration analysis of a Timoshenko–Ehrenfest beam with simple support–simple support (S–S), clamped–free (C–F), and clamped–simple support (C–S) boundary conditions for which Chen et al. [13] quoted numerical results up to eight significant figures for the natural frequencies although Huang [12] gave exact expressions for the frequency equation of each of the above boundary conditions from which results to any desired accuracy can be computed. The data used were taken from Chen et al. [13] which are Young’s modulus E = 210 GPa, density ρ = 7850 kg/m3, length L = 0.4 m, width b = 0.02 m, depth or height h = 0.08 m, shear correction factor k = 2/3 and Poisson’s ratio ν = 1/3. The shear modulus G was calculated by relating it to E through Poisson’s ratio ν to give G = 3E/8 as reported in Ref. [13]. Using these data, the beam parameters were worked out as bending rigidity EI = 1.792 × 105 N m2, shear rigidity kAG = 8.4 × 107 N, and mass per unit length ρA = 12.56 kg/m. The first nine natural frequencies of the beam with S–S, C–F, and C–S boundary conditions computed using the present theory are shown in Table 1. (To be consistent with Ref. [13], the axial natural frequencies are not included in the analysis.)

Table 1

Natural frequencies of a Timoshenko–Ehrenfest beam for S–S, C–F, and C–S boundary conditions

Boundary conditions
S–SC–FC–S
16838.83362529.49279741.9469
223,190.82713,279.90526,150.251
343,443.49331,044.79145,545.510
464,939.18650,825.83466211.994
586,710.89971,565.04787,376.643
6108,431.3491,994.824108,601.14
7111,981.29110,975.98114,295.44
8120,647.24119,244.57128,739.40
9130,003.61131,606.52131,610.63
Boundary conditions
S–SC–FC–S
16838.83362529.49279741.9469
223,190.82713,279.90526,150.251
343,443.49331,044.79145,545.510
464,939.18650,825.83466211.994
586,710.89971,565.04787,376.643
6108,431.3491,994.824108,601.14
7111,981.29110,975.98114,295.44
8120,647.24119,244.57128,739.40
9130,003.61131,606.52131,610.63

The natural frequencies shown in Table 1 agreed with the exact natural frequencies quoted in Ref. [13]. These results were further checked using the exact frequency equations reported by Huang [12] and again complete agreement was found. Since symbolic algebra has been used, such a high degree of accuracy in results is certainly important so that interested readers can check their own theory or computer programs in future research.

The second example is a plane frame shown in Fig. 2 which is that of Ref. [14]. Each element of the frame has the same uniform geometrical, cross-sectional and material properties and the data used in the analysis for each element are EI = 4.0 × 106 N m2, EA = 8.0 × 108 N, kAG = 2.0 × 108 N, ρA = 30 kg/m, ν = 1/3, k = 2/3. Using the present theory, nine natural frequencies were computed within the low, medium, and high frequency ranges which are 1st, 3rd, 4th, 50th, 70th, 90th, 150th, 200th, and 250th natural frequencies. The results are shown in Table 2. For comparison purposes, the corresponding natural frequencies using the Bernoulli–Euler theory-based dynamic stiffness theory [14] were also computed and shown in Table 2. As expected, the Bernoulli–Euler theory gives poor results for higher-order natural frequencies.

Fig. 2
Fig. 2
Close modal
Table 2

Natural frequencies of a plane frame

Natural frequency number (i)Natural frequency ωi (rad/s)
Timoshenko–Ehrenfest theoryBernoulli–Euler theory
1222.29030224.75854
3263.74414267.38209
4318.72268322.91808
503553.54943776.0281
705209.27375870.9164
906967.86128175.5854
15013,518.98417,167.288
20019,153.40025,695.288
25024,771.93035,091.591
Natural frequency number (i)Natural frequency ωi (rad/s)
Timoshenko–Ehrenfest theoryBernoulli–Euler theory
1222.29030224.75854
3263.74414267.38209
4318.72268322.91808
503553.54943776.0281
705209.27375870.9164
906967.86128175.5854
15013,518.98417,167.288
20019,153.40025,695.288
25024,771.93035,091.591

## 7 Conclusions

Explicit algebraic expressions for the elements of the frequency-dependent mass and stiffness matrices of a Timoshenko–Ehrenfest beam have been derived by extensive application of symbolic computation. The investigation allows exact free vibration analysis of Timoshenko–Ehrenfest beams and frameworks, but importantly it paves the way for the inclusion of damping in exact free vibration analysis of such structures. Numerical examples by applying the Wittrick–Williams algorithm as solution technique are given to demonstrate the capability of the theory.

## Acknowledgment

The author is grateful to EPSRC, UK, for an earlier grant (GR/R21875/01) and to Leverhulme Trust, UK, for a recent grant (EM-2019-061) which made this work possible.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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