Recursive formulae and performance comparisons for first mode dynamics of periodic structures

The first structural arrangement to be considered is referred to as PZ. Also known in the literature as simply 'zigzag', PZ structures are made up of a series of beams connected in an alternating arrangement. The first beam is fixed at its base while the base of the second beam is attached to the tip of the first, and so on. Each new beam is oriented in the same plane either +90° or −90° relative to the previous beam thus forming the structure illustrated in figure 2.

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Up to four loads are considered for each beam member, (1) distributed load $W$ caused by beam mass; (2) tip shear load $S$ from all masses attached to the tip including point masses (${M}_{t}$ and ${M}_{c}$) and previous distributed masses; (3) tip moment $M;$ and (4) tip torque $T.$ The first three loading conditions induce bending deflection $z$ where positive values of $W,$ $S,$ and $M$ result in positive deflection. The fourth load (torque) is in the axial direction of the beam and results in twisting deflection $\beta .$ Location and direction of these four forces are shown in figure 2(b).

The effective length of each beam member is approximated by representing the structure as a series of lines rather than beams. This 1D representation is shown in figure 2 as the dashed line. Each line segment travels down the center of each beam and intersects with the centerline of the adjacent beam. Effective beam lengths ${{\ell }}_{1}$ and ${{\ell }}_{2}$ in figure 2 are the lengths of each centerline segment. This effective beam length approximation accounts for 100% of the total mass and for the fact that the connections between beam segments are not rigid.

For this part of the analysis, beam 1 is considered to be the free beam. All loads acting on beam 1 include only shear $S$ due to a point tip mass and distributed load $W$ due to the beam mass. Given a unit acceleration, the distributed and concentrated loads due to these masses are defined as,

$$\begin{eqnarray}&&{S}_{1}={M}_{t},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{W}_{1}={m}_{1},\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&{T}_{1}={M}_{1}=0.\end{eqnarray} \tag{ 6 }$$

Notice that the distributed load $W$ for each segment is equal to the linear mass density of that segment. Now considering all forces acting on beam 2, we have shear from beam 1 including the weight of beam 1, distributed mass load, and now tip torque ${T}_{2}$ from the loads acting on beam 1. These loads are defined as,

$$\begin{eqnarray}&&{S}_{2}={S}_{1}+{M}_{c}+{m}_{1}{{\ell }}_{1},\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{T}_{2}={M}_{t}{{\ell }}_{1}+\displaystyle \frac{1}{2}{m}_{1}{{\ell }}_{1}^{2},\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&{M}_{2}=0,\end{eqnarray} \tag{ 9 }$$

where ${M}_{c}$ is incremental mass located at each joint and acts as a point tip mass to each beam segment (see figure 2). This incremental mass is used to account for mass that may be added or lost if beams of different thicknesses are joined together.

Continuing with all forces acting on beam segment 3 we have,

$$\begin{eqnarray}&&{S}_{3}={S}_{2}+{M}_{c}+{m}_{2}{{\ell }}_{2},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{T}_{3}={S}_{2}{{\ell }}_{2}+\displaystyle \frac{1}{2}{m}_{2}{{\ell }}_{2}^{2}+{T}_{1},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{M}_{3}=-{T}_{2},\end{eqnarray} \tag{ 12 }$$

where it is now possible to use fundamentals of static beam analysis to define the current shear, moment, and torque as functions of previous forces, geometry, and beam properties. The resulting recursive equations for shear and torque are,

$$\begin{eqnarray}&&{S}_{i}={S}_{i-1}+{M}_{c}+{m}_{i-1}{{\ell }}_{i-1},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{T}_{i}={S}_{i-1}{{\ell }}_{i-1}+\displaystyle \frac{1}{2}{m}_{i-1}{{\ell }}_{i-1}^{2}-{(-1)}^{i}{T}_{i-2}\end{eqnarray} \tag{ 14 }$$

for $i\geqslant 3$ where it is no longer necessary to define tip moment $M$ because it will always be equal to or opposite the torque in the previous beam segment depending on the beam orientation.

With all local forces defined, superposition can be used to determine the bending and torsion deflection of each cantilever beam segment. Beginning with bending given as,

$$\begin{eqnarray}{z}_{i}(r) & = & \mathop{\underbrace{\displaystyle \frac{{S}_{i}{r}_{i}^{2}}{6E{I}_{i}}(3{{\ell }}_{i}-{r}_{i})}}\limits_{{\rm{T}}{\rm{i}}{\rm{p}}\,\,\,\,\,\,{\rm{S}}{\rm{h}}{\rm{e}}{\rm{a}}{\rm{r}}}+\mathop{\underbrace{\displaystyle \frac{{m}_{i}{r}_{i}^{2}}{24E{I}_{i}}({r}_{i}^{2}-4{{\ell }}_{i}{r}_{i}+6{{\ell }}_{i}^{2})}}\limits_{{\rm{D}}{\rm{i}}{\rm{s}}{\rm{t}}{\rm{r}}{\rm{i}}{\rm{b}}{\rm{u}}{\rm{t}}{\rm{e}}{\rm{d}}\,\,{\rm{L}}{\rm{o}}{\rm{a}}{\rm{d}}}\\ & & +\mathop{\underbrace{\displaystyle \frac{{(-1)}^{i}{T}_{i-1}{r}_{i}^{2}}{2E{I}_{i}}}}\limits_{{\rm{T}}{\rm{i}}{\rm{p}}\,\,\,{\rm{M}}{\rm{o}}{\rm{m}}{\rm{e}}{\rm{n}}{\rm{t}}}\end{eqnarray} \tag{ 15 }$$

the total local deflection $z$ is found by simply adding the contributions due to shear load, distributed load, and moment respectively, where ${r}_{i}$ is the length coordinate of the $i{\rm{th}}$ beam with values from zero to the length of the $i{\rm{th}}$ beam $({{\ell }}_{i}).$ Recall that beam number 1 is the free beam. The torsional deflection of each beam is defined as,

$$\begin{eqnarray}&&{\beta }_{i}(r)=\displaystyle \frac{{T}_{i}{r}_{i}}{G{J}_{i}},\end{eqnarray} \tag{ 16 }$$

where $GJ$ is the torsional stiffness of the $i{\rm{th}}$ beam segment which is a function of the shear modulus $G$ and the torsional constant $J.$ Considering the rectangular cross-section of the beams $J$ can be defined as,

$$\begin{eqnarray}&&J={k}_{t}b{h}^{3},\end{eqnarray} \tag{ 17 }$$

where ${k}_{t}$ is a function of the ratio of beam thickness $h$ to width $b$ and is given as,

$$\begin{eqnarray}&&k{}_{t}=\displaystyle \frac{1}{3}-0.21\displaystyle \frac{h}{b}\left[1-\displaystyle \frac{1}{12}{\left(\displaystyle \frac{h}{b}\right)}^{4}\right].\end{eqnarray} \tag{ 18 }$$

Now that all local deflections are known, it is possible to define global deflections for each beam. Except for the fixed beam, continuity conditions require that the global deflection of all beams depends on the global deflection of the beam it is attached to. Enforcing these continuity conditions while defining global deflections requires beginning with the fixed beam deflection which is simply equal to the local deflection given as,

$$\begin{eqnarray}&&{\psi }_{1}={z}_{1}\end{eqnarray} \tag{ 19 }$$

which and has zero slope and deflection at the fixed end. Global deflection of the next beam (beam 2) is defined as,

$$\begin{eqnarray}&&{\psi }_{2}={z}_{2}+{\psi }_{1}({\ell })+{\beta }_{1}({\ell }){r}_{2}\end{eqnarray} \tag{ 20 }$$

which is equal to its local deflection; however, the base of this beam has a nonzero displacement and angle which are equal to the tip deflection and twist angle of the first beam. Similar to beam 2, global deflection of beam 3 is given as,

$$\begin{eqnarray}&&{\psi }_{3}={z}_{3}+{\psi }_{2}({\ell })+\left[{\beta }_{2}({\ell })-{\displaystyle \frac{{\rm{d}}{\psi }_{1}}{{\rm{d}}{r}_{1}}\right|}_{r={\ell }}\right]{r}_{3}\end{eqnarray} \tag{ 21 }$$

which is equal to its local deflection added to the global tip deflection of beam 2. The base angle of beam 3 is now a function of both the twist angle of beam 2 and the global tip angle of beam 1. Finally, the recursive expression for global displacements can be given as,

$$\begin{eqnarray}&&{\psi }_{j}={z}_{j}+{\psi }_{j-1}({\ell })+\left[{\beta }_{j-1}({\ell })+{(-1)}^{j}{\displaystyle \frac{{\rm{d}}{\psi }_{j-2}}{{\rm{d}}{r}_{j-2}}\right|}_{r={\ell }}\right]{r}_{j}\end{eqnarray} \tag{ 22 }$$

for $j\geqslant 3$ where the subscript $j$ denotes the beam segment number beginning with the fixed beam which is opposite of subscript $i$ used for the local deflections in equations (13) through (16). The final theoretical mode shape determined with equations (19) through (22) for the 5-beam PZ structure shown in figure 2 is given in figure 3 where the shape with dashed lines indicates the neutral position.

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Given the estimated fundamental mode shape defined in equations (19) through (22), Rayleigh's Quotient can be used to determine the equivalent mass, stiffness, and natural frequency of the structure. Rayleigh's Quotient for the PZ structure is defined as,

$$\begin{eqnarray}&&{\omega }_{0}^{2}=\displaystyle \frac{{\psi }_{N{\ell }}^{2}{k}_{{\rm{P}}{\rm{Z}}}}{{\psi }_{N{\ell }}^{2}{m}_{{\rm{P}}{\rm{Z}}}}\\ &&\,=\displaystyle \frac{\displaystyle {\sum }_{j=1}^{N}\displaystyle {\int }_{0}^{{{\ell }}_{j}}\left[E{I}_{j}{\left(\displaystyle \frac{{{\rm{d}}}^{2}{\psi }_{j}}{{\rm{d}}{r}_{j}^{2}}\right)}^{2}+G{J}_{j}{\left(\displaystyle \frac{{\rm{d}}{\beta }_{j}}{{\rm{d}}{r}_{j}}\right)}^{2}\right]\,{\rm{d}}{r}_{j}}{\displaystyle {\sum }_{j=1}^{N}\left\{{m}_{j}\displaystyle {\int }_{0}^{{{\ell }}_{j}}{\psi }_{j}^{2}{\rm{d}}{r}_{j}+[{M}_{c}+{\delta }_{jN}({M}_{t}-{M}_{c})]{\psi }_{j}^{2}({\ell })\right\}},\end{eqnarray} \tag{ 23 }$$

where ${\omega }_{0}$ is the natural frequency (rad s⁻¹), ${k}_{{\rm{P}}{\rm{Z}}}$ is the equivalent stiffness, ${m}_{{\rm{P}}{\rm{Z}}}$ is the equivalent mass, ${\psi }_{N{\ell }}$ is the tip displacement of the free beam, and $N$ is the total number of beam segments. The numerator of equation (23) contains two familiar strain energy terms associated with beam bending and torsion which must be integrated over the length of each beam segment. The denominator of equation (23) has kinetic energy terms that include the mass of each beam segment along with incremental point masses $({M}_{c})$ at the end of each beam and a tip mass $({M}_{t})$ at the end of the free bream. This point mass localization is achieved using the Kronecker delta function ${\delta }_{jN}$ which is equal to one if j = n and is zero otherwise.

The second type of periodic structure considered in this analysis is called an OS. Similar to the PZ structures discussed in the previous section, OS structures consist of a series of coplanar beams each attached to the tip of the previous beam as illustrated in figure 4. The difference between PZ and OS designs is that each new beam of the OS structure is oriented 90 degrees relative to the previous but in the same direction i.e., all are at +90° or all are at −90° to form a rectangular spiral.

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Following a procedure similar to that for the PZ in section 2.1, the individual forces acting on the free beam (beam 1) and beam 2 are identical to those defined in equations (4) through (9). Forces on beam 3 of the OS are defined as,

$$\begin{eqnarray}&&{S}_{3}={S}_{2}+{M}_{c}+{m}_{2}{{\ell }}_{2},\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{T}_{3}={S}_{2}{{\ell }}_{2}+\displaystyle \frac{1}{2}{m}_{2}{{\ell }}_{2}^{2}-{T}_{1},\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&{M}_{3}=-{T}_{2},\end{eqnarray} \tag{ 26 }$$

where the only difference between these and equations (10) through (12) is the sign of previous torque ${T}_{1}$ in the definition of ${T}_{3}.$ The recursive force equations for $i\geqslant 3$ can therefore be defined as,

$$\begin{eqnarray}&&{S}_{i}={S}_{i-1}+{M}_{c}+{m}_{i-1}{{\ell }}_{i-1},\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&{T}_{i}={S}_{i-1}{{\ell }}_{i-1}+\displaystyle \frac{1}{2}{m}_{i-1}{{\ell }}_{i-1}^{2}-{T}_{i-2},\end{eqnarray} \tag{ 28 }$$

where the sign of the previous torque ${T}_{i-2}$ is constant because each beam is attached with the same angle relative to the previous beam unlike the PZ structure. Local bending deflection can be defined as,

$$\begin{eqnarray}{z}_{i}(r) & = & \displaystyle \frac{{S}_{i}{r}_{i}^{2}}{6E{I}_{i}}(3{{\ell }}_{i}-{r}_{i})+\displaystyle \frac{{m}_{i}{r}_{i}^{2}}{24E{I}_{i}}\\ & & \times ({r}_{i}^{2}-4{{\ell }}_{i}{r}_{i}+6{{\ell }}_{i}^{2})-\displaystyle \frac{{T}_{i-1}{r}_{i}^{2}}{2E{I}_{i}}\end{eqnarray} \tag{ 29 }$$

and the local torsion deflection as,

$$\begin{eqnarray}&&{\beta }_{i}(r)=\displaystyle \frac{{T}_{i}{r}_{i}}{G{J}_{i}},\end{eqnarray} \tag{ 30 }$$

where ${r}_{i}$ is the length coordinate of the $i{\rm{th}}$ beam with values from zero to the length of the $i{\rm{th}}$ beam $({{\ell }}_{i})$ and torsional stiffness $GJ$ can be defined using equations (17) and (18).

Global deflections for the first three beams of the OS structure (beginning with the fixed beam as beam 1) are equal to equations (19) through (21). The resulting recursive expression for global deflections of the OS can be defined as,

$$\begin{eqnarray}&&{\psi }_{j}={z}_{j}+{\psi }_{j-1}({\ell })+\left[{\beta }_{j-1}({\ell })-{\displaystyle \frac{{\rm{d}}{\psi }_{j-2}}{{\rm{d}}{r}_{j-2}}\right|}_{r={\ell }}\right]{r}_{j}\end{eqnarray} \tag{ 31 }$$

for $j\geqslant 3$ where the global index $j$ denotes beam number where the fixed beam is beam 1 and is opposite the numbering convention of local index $i$ which starts with the free beam as beam 1. It is important to note that force and deflection equations for the OS structure are reversible depending on the location of the fixed and free beams as long as the index convention is consistent. In other words, the same local and global equations for the OS can be used for either center beam fixed or outer beam fixed as illustrated in figures 4(a), (b). Finally, given the global deflections, Rayleigh's quotient for the OS structure can be defined as,

$$\begin{eqnarray}&&{\omega }_{0}^{2}=\displaystyle \frac{{\psi }_{N{\ell }}^{2}{k}_{{\rm{O}}{\rm{S}}}}{{\psi }_{N{\ell }}^{2}{m}_{{\rm{O}}{\rm{S}}}}\\ &&\,=\displaystyle \frac{\displaystyle {\sum }_{j=1}^{N}\displaystyle {\int }_{0}^{{{\ell }}_{j}}\left[E{I}_{j}{\left(\displaystyle \frac{{{\rm{d}}}^{2}{\psi }_{j}}{{\rm{d}}{{r}}_{j}^{2}}\right)}^{2}+G{J}_{j}{\left(\displaystyle \frac{{\rm{d}}{\beta }_{j}}{{\rm{d}}{{r}}_{j}}\right)}^{2}\right]\,{\rm{d}}{{r}}_{j}}{\displaystyle {\sum }_{j=1}^{N}\left\{{m}_{j}\displaystyle {\int }_{0}^{{{\ell }}_{j}}{\psi }_{j}^{2}{\rm{d}}{{r}}_{j}+[{M}_{c}+{\delta }_{jN}({M}_{t}-{M}_{c})]{\psi }_{j}^{2}({\ell })\right\}},\end{eqnarray} \tag{ 32 }$$

where the lumped parameter equivalent mass and stiffness are ${k}_{{\rm{O}}{\rm{S}}}$ and ${m}_{{\rm{O}}{\rm{S}}}$ respectively. Figure 5 shows the fundamental theoretical mode shapes found using equation (31) for the 7-beam OS structures in figure 4 where the dashed lines indicate the neutral position. Figure 5(a) is for the case of the outer beam fixed while figure 5(b) is for the case of the center beam fixed.

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The third and final type of periodic structure discussed in this research is called a SZ which is also referred to in the literature as a fan folded structure. Similar to the previously discussed PZ and OS structures, the SZ is a series of attached beams. The first beam is fixed at its base and its tip is attached to the base of the next beam with a rigid joint block. Unlike the coplanar design of the PZ and OS, adjacent beams of the SZ are positioned in orthogonal planes as illustrated in figure 6.

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Beginning with the free beam as shown in figure 6, because the uniform loading is acting in the axial direction along the beam, there are no out of plane forces to cause local deflection. This would not be the case however, with an even number of beam segments, so it is necessary to define two sets of recursive equations. Also, notice that unlike the PZ and OS structures, all deflections in the SZ structure are bending only i.e., there is no torsional deflection.

Given an odd number of beam segments, the forces acting on the free beam (beam 1) are,

$$\begin{eqnarray}&&{S}_{1}={M}_{1}={W}_{1}=0,\end{eqnarray} \tag{ 33 }$$

where tip shear $S,$ tip moment $M,$ and distributed force $W$ are all zero. Forces on beam 2 are defined as,

$$\begin{eqnarray}&&{S}_{2}={M}_{c}+{M}_{t}+{m}_{1}{{\ell }}_{1},\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{M}_{2}=({M}_{c}+{M}_{t}+{m}_{1}{{\ell }}_{1}){{\ell }}_{e},\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&{W}_{2}={m}_{2},\end{eqnarray} \tag{ 36 }$$

where all mass of beam 1 along with the tip mass ${M}_{t}$ and joint block mass ${M}_{c}$ contribute to the tip shear and moment. The tip moment arm ${{\ell }}_{e}$ is the distance from the tip of the second beam to the center of gravity of the block mass which is also the center of gravity of the first beam and tip mass. See figure 6 for reference. Continuing with the analysis, all forces on beam 3 can be expressed as,

$$\begin{eqnarray}&&{S}_{3}={W}_{3}=0,\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&{M}_{3}={M}_{2}+{S}_{2}({{\ell }}_{2}+{{\ell }}_{e})+{m}_{2}{{\ell }}_{2}\left(\displaystyle \frac{{{\ell }}_{2}}{2}+{{\ell }}_{e}\right),\end{eqnarray} \tag{ 38 }$$

where again the tip shear and distributed loads are zero as with beam 1; however, the tip moment is now a function of the previous tip moment ${M}_{2},$ the previous shear ${S}_{2},$ and the mass of the previous beam. The forces acting on beam 4 are,

$$\begin{eqnarray}&&{S}_{4}={S}_{2}+2{M}_{c}+{m}_{2}{{\ell }}_{2}+{m}_{3}{{\ell }}_{3},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{M}_{4}={M}_{3}+{S}_{4}{{\ell }}_{e},\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}&&{W}_{4}={m}_{4}\end{eqnarray} \tag{ 41 }$$

which are similar to the forces on beam 2 defined in equations (34) through (36) except that the previous moment ${M}_{3}$ and shear ${S}_{2}$ are now nonzero and must be accounted for.

Now that all forces can be defined with known constants and previous values, the recursive force equations can be determined. Upon first considering the odd numbered beams for $i=\mathrm{3,5,7..}.$ the forces are,

$$\begin{eqnarray}&&{S}_{i}={W}_{i}=0,\end{eqnarray} \tag{ 42 }$$

$$\begin{eqnarray}&&{M}_{i}={M}_{i-1}+{S}_{i-1}({{\ell }}_{i-1}+{{\ell }}_{e})+{m}_{i-1}{{\ell }}_{i-1}\left(\displaystyle \frac{{{\ell }}_{i-1}}{2}+{{\ell }}_{e}\right)\end{eqnarray} \tag{ 43 }$$

and the even numbered beam forces $(i=4,\,6,\,8...)$ are,

$$\begin{eqnarray}&&{S}_{i}={S}_{i-2}+2{M}_{c}+{m}_{i-2}{{\ell }}_{i-2}+{m}_{i-1}{{\ell }}_{i-1},\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{M}_{i}={M}_{i-1}+{S}_{i}{{\ell }}_{e},\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}&&{W}_{i}={m}_{i}.\end{eqnarray} \tag{ 46 }$$

Recall that equations (33) through (46) are for an SZ structure with an odd number of beam segments and oriented as shown in figure 6. Given this same orientation and following a similar procedure for the case of an even number of beam segments one can arrive at an identical set of recursive force equations where the odd and even indices are swapped. In other words, equations (42) and (43) would apply to even numbered beams $(i=4,\,6,\,8\,\cdots )$ and equations (44) through (46) would apply to odd numbered beams $(i=3,\,5,\,7\cdots ).$ Forces on the free beam become,

$$\begin{eqnarray}&&{S}_{1}={M}_{t},\end{eqnarray} \tag{ 47 }$$

$$\begin{eqnarray}&&{M}_{1}={M}_{t}{{\ell }}_{e},\end{eqnarray} \tag{ 48 }$$

$$\begin{eqnarray}&&{W}_{1}={m}_{1}\end{eqnarray} \tag{ 49 }$$

which are the same as those for a simple cantilever with a large tip mass, and the forces on beam 2 are,

$$\begin{eqnarray}&&{S}_{2}={W}_{2}=0,\end{eqnarray} \tag{ 50 }$$

$$\begin{eqnarray}&&{M}_{2}={M}_{t}({{\ell }}_{1}+{{\ell }}_{e})+{m}_{1}{{\ell }}_{1}\left(\displaystyle \frac{{{\ell }}_{1}}{2}+{{\ell }}_{e}\right).\end{eqnarray} \tag{ 51 }$$

The equations for forces on beam 3 are identical to those given in equations (39) through (41) after reducing all indices by 1. Similarly, the equations for forces on beam 4 are identical to those given in equations (37) and (38) after subtracting 1 from every index.

The local deflections for SZ structures can be defined as,

$$\begin{eqnarray}{z}_{i}(r) & = & \displaystyle \frac{{S}_{i}{r}_{i}^{2}}{6E{I}_{i}}(3{{\ell }}_{i}-{r}_{i})+\displaystyle \frac{{W}_{i}{r}_{i}^{2}}{24E{I}_{i}}\\ & & \times ({r}_{i}^{2}-4{{\ell }}_{i}{r}_{i}+6{{\ell }}_{i}^{2})+\displaystyle \frac{{M}_{i}{r}_{i}^{2}}{2E{I}_{i}}\end{eqnarray} \tag{ 52 }$$

which is similar to equations (15) and (29) with the major difference being that the distributed load $W$ alternates between zero and the linear mass density of the current beam segment. Given all local deflections and geometry, it is possible to determine the global deflections. Beginning with the clamped beam as beam 1, its global deflection is,

$$\begin{eqnarray}&&{\psi }_{1}={z}_{1}\end{eqnarray} \tag{ 53 }$$

which is simply equal to its local deflection. Global deflection of beam 2 can be defined as,

$$\begin{eqnarray}&&{\psi }_{2}={z}_{2}+\left({\displaystyle \frac{{\rm{d}}{\psi }_{1}}{{\rm{d}}{r}_{1}}\right|}_{r={\ell }}\right)({r}_{2}+{{\ell }}_{e})\end{eqnarray} \tag{ 54 }$$

which is equal to the local deflection including the influence of the rotated join block which adds an angle equal to the tip angle of beam 1 and an offset caused by the distance ${{\ell }}_{e}.$ Global deflection of beam 3 can be expressed as,

$$\begin{eqnarray}&&{\psi }_{3}={z}_{3}+\left({\displaystyle \frac{{\rm{d}}{\psi }_{2}}{{\rm{d}}{r}_{2}}\right|}_{r={\ell }}\right)({r}_{3}+{{\ell }}_{e})-\left[{\psi }_{1}({\ell })+{{\ell }}_{e}{\displaystyle \frac{{\rm{d}}{\psi }_{1}}{{\rm{d}}{r}_{1}}\right|}_{r={\ell }}\right]\end{eqnarray} \tag{ 55 }$$

which accounts for the joint block angle as with equation (54) but also must include the deflection of the first beam which is in the opposite global direction of beam 3. Continuing with these definitions leads to the recursive set of global deflection equations,

$$\begin{eqnarray}{\psi }_{j} & = & {z}_{j}+\left({\displaystyle \frac{{\rm{d}}{\psi }_{j-1}}{{\rm{d}}{r}_{j-1}}\right|}_{r={\ell }}\right)({r}_{j}+{{\ell }}_{e})+{(-1)}^{j}\\ & & \times \left[{\psi }_{j-2}({\ell })+{{\ell }}_{e}{\displaystyle \frac{{\rm{d}}{\psi }_{j-2}}{{\rm{d}}{r}_{j-2}}\right|}_{r={\ell }}\right]\end{eqnarray} \tag{ 56 }$$

for $j\geqslant 3$ where the global index $j$ begins at 1 for the clamped beam and goes to the number of beam segments $N$ for the free beam which is in the reverse order of local index $i.$

Figure 7 shows the theoretical fundamental mode shape of the 7-beam SZ structure in figure 6 found using equations (53) through (56) where the dashed lines indicate the neutral position. The gaps between corners of each line segment in figure 7 are where the beams are joined by rigid joint blocks. For simplicity and clarity, these blocks were not included in the mode shape plots.

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Again, Rayleigh's quotient can be used to define the natural frequency as,

$$\begin{eqnarray}&&{\omega }_{0}^{2}=\displaystyle \frac{{\psi }_{N{\ell }}^{2}{k}_{{\rm{S}}{\rm{Z}}}}{{\psi }_{N{\ell }}^{2}{m}_{{\rm{S}}{\rm{Z}}}}\\ &&=\displaystyle \frac{\displaystyle {\sum }_{j=1}^{N}\left[\displaystyle {\int }_{0}^{{{\ell }}_{j}}E{I}_{j}{\left(\displaystyle \frac{{{\rm{d}}}^{2}{\psi }_{j}}{{\rm{d}}{r}_{j}^{2}}\right)}^{2}\,{\rm{d}}{r}_{j}\right]}{\displaystyle {\sum }_{j=1}^{N}\left[{m}_{j}\displaystyle {\int }_{0}^{{{\ell }}_{j}}{\psi }_{j}^{2}{\rm{d}}{r}_{j}+{{\rm{\Gamma }}}_{j}{\eta }_{j}+(1-{\delta }_{j1})({m}_{j}{{\ell }}_{j}+{{\rm{\Gamma }}}_{j}){\eta }_{j-1}\right]},\end{eqnarray} \tag{ 57 }$$

where the equivalent lumped parameter mass and stiffness are ${m}_{{\rm{S}}{\rm{Z}}}$ and ${k}_{{\rm{S}}{\rm{Z}}}$ respectively and the two special functions ${\rm{\Gamma }}$ and $\eta$ have been included for convenience. The first function is a mass term defined as,

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{j}=[{M}_{c}+{\delta }_{jN}({M}_{t}-{M}_{c})],\end{eqnarray} \tag{ 58 }$$

where the Kronecker delta function ${\delta }_{jN}$ is used to replace the join block mass ${M}_{c}$ with the tip mass ${M}_{t}$ on the free beam (for j = N). The second special function is given as,

$$\begin{eqnarray}&&{\eta }_{j}={\left[{\psi }_{j}({\ell })+{{\ell }}_{e}\left({\displaystyle \frac{{\rm{d}}{\psi }_{j}}{{\rm{d}}{r}_{j}}\right|}_{r={\ell }}\right)\right]}^{2}\end{eqnarray} \tag{ 59 }$$

which is the square of the deflection at the center of mass (defined by ${\rm{\Gamma }}$) attached to the end of the $j{\rm{th}}$ beam segment. For simplicity, it is assumed that the center of mass for all join block masses and the tip mass is located at the same distance ${{\ell }}_{e}$ from the end of the beam.

The denominator of Rayleigh's quotient in equation (57) includes three terms beginning with the familiar distributed mass term where the square of the beam deflection must be integrated over the length of each beam segment. The second term is a concentrated mass term associated with the tip deflection of each beam segment. The third and final term in the denominator accounts for in-plane rigid-body-motion of each beam and mass caused by deflection of the previous beam. Of course, the fixed beam does not experience this rigid-body-motion, so the Kronecker delta is used to eliminate the third term for $j=1.$

The following analysis considers four structures of similar overall size, material properties, and beam segment dimensions. The structure types are those of the previously discussed PZ, OS, and SZ. For convenience, OS structures will be classified as two types: outer beam fixed (OSo) and inner beam fixed (OSi) as illustrated in figure 4. All structures are made with aluminum 6061-T6 and have a beam segment width and thickness of 5.08 mm and 1.52 mm respectively. The distance between adjacent PZ and OS beam segments is 1.02 mm. There are no tip masses and the SZ joint block masses are 2.55 g.

One performance metric considered for both metastructure and energy harvesting applications examines how strain is distributed throughout the entire structure. It is a more efficient use of volume and mass to have strain evenly distributed through the entire structure rather than locally concentrated. Given the mode shape $\psi ,$ the single-sided absolute surface strain of each beam segment can be defined as,

$$\begin{eqnarray}&&{\varepsilon }_{j}^{{\rm{a}}{\rm{b}}{\rm{s}}}=\displaystyle \frac{{h}_{j}}{2}\displaystyle {\int }_{0}^{{{\ell }}_{j}}\left|\displaystyle \frac{{{\rm{d}}}^{2}{\psi }_{j}}{{\rm{d}}{r}_{j}^{2}}\right|{\rm{d}}{r}_{j},\end{eqnarray} \tag{ 60 }$$

where ${h}_{j},$ ${r}_{j},$ and ${{\ell }}_{j}$ are the thickness, length coordinate, and length of the $j{\rm{th}}$ beam segment respectively. Upon normalizing ${\varepsilon }^{{\rm{a}}{\rm{b}}{\rm{s}}}$ by the maximum absolute strain and dividing by the number of beam segments, the average strain distribution can be defined as,

$$\begin{eqnarray}&&\bar{\varepsilon }=\displaystyle \frac{1}{N{\varepsilon }_{\max }^{{\rm{a}}{\rm{b}}{\rm{s}}}}\displaystyle \sum _{j=1}^{N}{\varepsilon }_{j}^{{\rm{a}}{\rm{b}}{\rm{s}}}\end{eqnarray} \tag{ 61 }$$

which is equal to one if the absolute surface strain in all segments is equal. This strain distribution metric is low if the highest strains are experienced in a smaller percentage of the structure. For comparison, the first mode of a uniform cantilever beam has a strain distribution value of 0.393.

Figure 12 shows a series of strain distribution values calculated using equation (61) while increasing the number of beam segments over a range from 3 to 30. The best (largest value) strain distribution was initially achieved with the OSi; however, the PZ performs better for $7\leqslant N\leqslant 13.$ Compared to all other designs, the OSi has the best strain distribution for $N\geqslant 15$ which appears to stabilize at 0.54 while that of the PZ, OSo, and SZ structures decays steadily as $N$ increases. Strain distribution for the OSo and SZ structures is already less than that of a EB beam at only 5 and 3 beam segments respectively.

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This vast difference between the two OS-type structures can be explained by considering which beam segments must support the most weight under a static loading condition. The OSi has the best overall strain distribution because the shorter stiffer beams (nearest the fixed beam) support the entire structure. Unfortunately for the OSo, a majority of the strain is in the longest softest beam segment while segments closer to the center of the structure experience negligible strains.

Although the SZ and OSo structures have poor strain distributions, they should not be ignored nor considered poor designs overall. There may be instances where strain distribution is not the most important design factor. For example, the SZ structures considered in this section have a natural frequency approximately an order of magnitude lower than the OSo and OSi structures and nearly half that of the PZ structure. While it has such low natural frequencies, the SZ structure has the largest overall volume which is nearly an order of magnitude greater than that of the OSo and OSi depending on the number of beam segments.

Given these various advantages and disadvantages of each structure type, another design metric is proposed in an attempt to provide a fair performance comparison. This design metric is a measure of total surface strain normalized by the maximum strain, natural frequency, and volume. This total strain density metric can be expressed as,

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{n}}{\rm{o}}{\rm{r}}{\rm{m}}}^{V}=\displaystyle \frac{1}{V{\omega }_{0}{\varepsilon }_{\max }}\displaystyle \sum _{j=1}^{N}{\varepsilon }_{j}^{{\rm{a}}{\rm{b}}{\rm{s}}},\end{eqnarray} \tag{ 62 }$$

where $V$ is the overall volume required to encase the structure, ${\omega }_{0}$ is the natural frequency, ${\varepsilon }_{\max }$ is the maximum surface strain in the structure, and ${\varepsilon }^{{\rm{a}}{\rm{b}}{\rm{s}}}$ is the single-sided absolute surface strain defined in equation (60). The strain density metric can be related to the energy density in energy harvesting terms. Also, both energy harvester and metastructure designs seek to minimize resonator volume and natural frequency which are well-known competing design goals. Therefore, this strain density metric increases with increasing total normalized strain, decreasing natural frequency, and decreasing volume.

Figure 13 summarizes the total strain density calculations made with equation (62) for the PZ, OS, and SZ structure types. Comparing these strain density results reveals that the PZ structure is the best choice for $N\leqslant 9$ where it is then surpassed by the OSi structure for $N\geqslant 10,$ then the OSo for $N\geqslant 12,$ then lastly by the SZ for $N\geqslant 19.$

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In order to demonstrate the efficiency of the proposed models, a simple comparison was performed on a 12-beam OS beam using ANSYS finite element analysis software and MATLAB. Calculations were performed on a computer having 16 GB of RAM and a quad core processor running at 3.6 GHz. ANSYS was setup to solve for the natural frequency and mode shape of the first mode first for a fine mesh having 4608 elements then a coarse mesh having 735 elements. Results show that ANSYS required 4.2 and 0.5 s for the fine and coarse mesh solutions respectively (not including time required to generate the mesh). Reducing from fine to coarse meshes caused the frequency estimate to increase by 0.65%. The proposed model was implemented in MATLAB where the beam segments were discretized to have the same number of elements as the two previous ANSYS models. Results of these calculations show that the complete solution (including discretization) only took 6.6 ms and 5.3 ms respectively, and caused the frequency estimate to increase by only 0.29%. In other words, the proposed model could produce 94 complete design iterations during the time it took the FEA model to produce one solution.