a novel structural element combining load carrying and energy dissipation capability

1 American Institute of Aeronautics and Astronautics

A Novel Structural Element Combining Load Carrying and Energy

Dissipation Capability

Scott M. Bland1, Robert J. Snyder

2, and Jay Kudva

3

NextGen Aeronautics, Torrance, CA, 90505

Michael E. Pontecorvo4, Silvestro Barbarino

5, and Farhan S. Gandhi

6

Rensselaer Polytechnic Institute, Troy, NY, 12180

and

Edward V. White7

The Boeing Company, Berkeley, Missouri, 63134

This paper reports on the design, fabrication, testing and simulation of a novel structural element that

combines load carrying and energy dissipation capability. The principal components comprise post-buckling

elements (PBEs), and von-Mises trusses (VMTs) coupled to a viscous dashpot, all integrated in a compact

panel-like element. Load carrying capability of the unit comes from the PBEs which provide a high initial

stiffness and very little deformation up to the critical buckling load. Energy dissipation is obtained through

the deformation of VMTs at the top and bottom of a hexagonal cell, connected to the ends of the viscous

dashpot. Under harmonic excitation the VMTs undergo large displacement, stroking the damper in the

process. The paper explains the design procedure of the structural element in detail, and describes a

prototype which is fabricated and tested under harmonic excitation. Under harmonic displacement input the

energy dissipated increased over the frequency range (with loss factor increasing from 0.84 at 0.1 Hz, to 2.69

at 4 Hz). Although simulation predicted both top and bottom VMTs moving simultaneously while the tests

showed one transitioning before the other, the experimental and simulation hysteresis loop areas compared

well. Under harmonic force input, data was obtained only at 0.1 Hz and 0.5 Hz, as the testing machine could

not move at the high-speeds associated with VMT snap-through. The energy dissipated (hysteresis loop areas)

in these tests was slightly lower than that in the displacement controlled tests at the same frequencies, with

loss factors of 0.61 at 0.1 Hz and 0.97 at 0.5 Hz calculated from the measured hysteresis cycles.

The experimental and simulation work done to date establish the basis feasibility of developing novel

structural designs which can carry high static load and also dissipate energy from undesirable dynamic

inputs. Such designs have broad potential applications in aerospace, marine and ground structures.

Nomenclature ALOOP = Hysteresis loop area

C = Linear viscous damping constant

dmax = Allowable axial displacement of the PBB

F = Force

Fsnap = Snap-through load of the von-Mises truss

Kv = von-Mises truss spring stiffness

Lv = von-Mises truss link length

PE = Euler critical buckling load

x = Displacement

η = Loss factor

σp = Proportional stress limit

1 Lead Engineer, AIAA Senior Member

2 Unmanned Systems and Technology Engineer, AIAA Member

3 President, NextGen Aeronautics Inc., AIAA Assoc. Fellow

4 PhD Candidate, Mechanical Aerospace and Nuclear Engineering, AIAA Member

5 Post Graduate Researcher, Mechanical Aerospace and Nuclear Engineering, AIAA Member

6 Professor and Rosalind and John J. Redfern Jr ’33 Endowed Chair in Aerospace Engineering, AIAA Assoc. Fellow

7 Adaptive Structures Technology Focus Team Leader, AIAA Assoc. Fellow


θ0 = Initial von-Mises truss angle

I. Introduction Structural elements that can simultaneously bear load and provide energy dissipation capability under

disturbance are of great interest in many aerospace, mechanical and civil engineering applications. This paper

presents a novel, structural element that supports loads up to a designed limit, beyond which internal von-Mises

truss (VMT) mechanisms snap-through and facilitate energy dissipation. This work builds on a previous study by

Barbarino et al.1

which uses the VMT as an energy dissipation mechanism. Introduced by Mises2 in 1923, the VMT

is composed of two rigid links, pinned together at the vertex, whose endpoints are spring-restrained and constrained

to move along a line perpendicular to the motion of the vertex. VMTs exhibit a nonlinear force-deflection curve that

contains one unstable and two stable equilibrium positions. When a VMT is forced away from one stable

equilibrium position energy is first stored in the system. On passing through the unstable equilibrium position, the

stored energy is released as the VMT moves to the second stable equilibrium position. The system can either display

negative stiffness (under displacement control), or snap-through (under force control), as it transitions from one

stable state to the other.

Several researchers have considered the dynamic behavior of a VMT system. Blair et al.3 investigated trusses

under harmonic forcing, Avramov and Mikhlin4 considered the use of the truss as a vibration absorber, Kounadis et

al.5 studied the truss under impact loading, Padthe et al.

6 presented the hysteretic behavior of the truss, and Murray

and Gandhi7 examined the effect of damping on reducing the transient oscillation associated with snap-through. The

study by Barbarino et al.1 presented computational results showing that the high velocity of the VMT vertex during

snap-through, connected to a damping element, can result in very large system loss factor values (in excess of 0.6).

In Ref. 8, Pontecorvo et al. presented a cell with a three-dimensional equivalent of the VMT at the top and bottom,

connected to a viscous damper for energy dissipation under harmonic loading.

The present study significantly expands on the idea presented in the numerical study by Barbarino et al.1 in two

key areas: (1) the design and fabrication of a panel-like structural element with VMTs and a dashpot integrated

between face-plates, and (2) use of the face-plates as post-buckling elements (PBEs) in parallel with the VMT to

provide high static load-carrying capability. At loads beyond the design static load the element stiffness reduces,

thereby allowing large deformation and energy dissipation under harmonic disturbances, while still preserving its

ability to carry the design static load. The prototype element is tested under harmonic excitation to determine its

energy dissipation capabilities over about a decade of frequency, and the test results are used for validating

simulation results obtained using Simscape.

II. Conceptual Design The quasi-static force-displacement behavior of a VMT is depicted in Fig. 1a. The VMT can carry a maximum

load (denoted on the figure as snap-through force), but under large displacement it loses this load carrying capability

during transition through the negative stiffness region. Introduction of a bi-linear spring with a force-displacement

behavior as shown in Fig. 1b in parallel with the VMT allows the combined system to retain its static load carrying

capability. A bi-linear spring with high initial stiffness, and reduced stiffness beyond a critical load, can be realized

in a number of ways. One approach could be to use a pre-compressed spring as was done by Pontecorvo et al.9.

Another approach, adopted in the current study, is to use a PBE. The PBEs can be tailored to achieve the desired

initial static load-carrying capability, and can further be designed to work in post-buckling regime without failure,

allowing for an overall axial displacement compatible with the stroke of the vertex of the VMT.

The force-displacement curve for the combined system is depicted in Fig. 1c and the VMT curve is seen to be

shifted up on the y-axis by the critical buckling load. Thus the PBE can be designed such that its critical buckling

load matches the static load requirement. The VMT geometry and spring stiffness (Kv), on the other hand, can be

selected to achieve a desired snap-through force (Fsnap) which can be matched to the amplitude of expected

disturbance loads.


(a) (b) (c)

Figure 1. Quasi-static force-vs-displacement behavior of (a) VMT (b) PBEs and (c) the combined assembly.

A physical integration of the VMT and PBEs was envisioned in a compact panel-like element. Figure 2 is a

schematic sketch of the proposed configuration and its operation. Note that many such units can be assembled in-

plane to form a larger panel. The system centers around a pin-jointed hexagonal cell comprised of rigid members

with inclined links of length 𝐿𝑉. Within the cell are two horizontal springs of stiffness 𝐾𝑉 and a linear viscous

damper with a damping constant C. The initial angle of the truss elements is denoted by 𝜃0 and the displacement is

defined by motion of the top vertex of the cell, positive downward. This configuration is based on the hexagonal unit

cell proposed by Pontecorvo et al.10

, which allows for a statically determinate, repeatable structural unit whose

force-displacement behavior is governed by the inclusions within the cell. By replacing the vertical spring within the

cell as proposed in Ref. 10 with the damper as shown in Fig. 2, the hexagonal unit cell under large displacement

now acts as a pair of VMTs connected at their vertices by a damper. This is similar to the single VMT dissipative

element studied by Barbarino et al.1 with the advantage that now the stroke in the damper is doubled, further

increasing the dissipative capability of the system.

The hexagonal cell configuration is placed in parallel with two buckling elements, and the entire unit is

sandwiched between two horizontal rigid blocks. The PBEs act like mechanical switches that reconfigure the load

path in different loading regimes. As the structure is loaded axially, below the critical Euler load 𝑃𝐸 (Fig. 2a), the

load is carried primarily by the stiffer PBEs, as indicated by the open arrows. Beyond the Euler load of the PBEs

(Fig. 2b), the elements buckle, their stiffness reduces dramatically, and the VMTs begin to carry a share of the

incremental load. Although the stiffness of the PBEs has greatly reduced, they continue to carry the load they were

designed to carry before buckling. When the load on the VMTs exceeds the snap-through force each VMT snaps-

through or transitions through the negative stiffness region to its other stable equilibrium position (Fig. 2c), stroking

the dashpot in the process.

(a) (b) (c)

Figure 2. Schematic representation of a VMT, viscous damper and buckling elements.


(a)

(b)

(c)

Figure 3. Static (a), and harmonic response (b and c) when top and bottom VMTs are identical.

(a)

(b)

(c)

(d)

Figure 4. Static (a and b), and harmonic response (c and d) when top and bottom VMTs are not-identical and

transition sequentially.


Another factor governing the system’s ability to dissipate energy is the type of applied input at the vertex of the

cell. Two types of inputs are considered: harmonic displacement control and harmonic force control. In

displacement control, the displacement of the top vertex is prescribed to follow a given path in time, while in force

control the applied force on the top vertex is prescribed. Consider the quasi-static displacement-controlled curve for

the complete system in Fig. 3a (equivalent to Fig. 1c). If a harmonic displacement input was prescribed the damper

in the cell would dissipate energy, which would appear as a widening of the curve (hysteresis) about the quasi-static

path, as seen in Fig. 3b. However, under a harmonic force input, the cell jumps in displacement at constant force and

high velocity as indicated by the dotted lines in Fig. 3a. The result is a hysteresis loop that has a large area (Fig. 3c),

and significantly larger energy dissipation (compare to hysteresis loop area in Fig. 3b).

Figures 3a-3c assume the ideal case where both upper and lower VMTs in the hexagonal cell move in unison. In

reality however, imperfections in the prototype cause one VMT to have a slightly lower critical load than the other,

and therefore that VMT transitions before the other. The corresponding quasi-static force-versus-displacement

behavior under incremental displacement is shown in the solid line on Fig. 4a under compression, and Fig. 4b under

tension. In either loading direction, the first force peak has a smaller magnitude than the second peak. Notice also

that the magnitudes of all the peaks in Figs. 4a and 4b, when the upper and lower VMTs move sequentially, are

lower than the peaks in Fig. 3a when they deform together. The corresponding hysteresis loops under harmonic

displacement and force inputs are shown in Figs. 4c and 4d, respectively.

III. Analysis In the present study, the Matlab-compatible modeling environment Simscape is used to examine the static and

dynamic force/displacement behavior of the structural element. Previous studies by the RPI authors (Refs. 1, 7) used

Hamilton’s principle to derive the governing ODE of the VMT coupled with a damper, and numerically integrated

this ODE. Simscape uses building blocks that model fundamental mechanical functions and these can be assembled

to simulate physical systems. This building-block approach allows the study of larger, more complex systems, and

avoids the need to re-derive governing ODEs when experimenting with configuration changes. Blocks representing

each component (e.g., pin-jointed rigid links, linear springs, dampers, PBEs modeled as nonlinear springs, etc.) are

connected to each other and appropriate boundary conditions and inputs are applied. One such block that was

developed specifically for this study is a bi-linear spring that was used to model the PBEs. The bi-linear spring

model of the PBEs was validated using an analytical Matlab model developed by Barbarino et al.11

based on the

work of Euler12

.

A block diagram of the Simscape model for the prototype element (described in the next section) is shown in

Fig. 5a. At the top of the figure is the input block, where either a harmonic force or displacement input are applied to

the vertex of the top VMT, and the output block, where the displacement or force at the vertex of the top VMT are

recorded. Below these blocks are the top VMT, which is connected to the bottom VMT by two vertical walls.

Between the two walls are the model blocks for the damper and the bi-linear spring (representative of the PBEs). At

the very bottom of the figure, the bottom VMT vertex is grounded. Experimental characterization of the chosen

damper for the prototype revealed that it was highly non-linear. Therefore, the damper block of the model uses the

experimentally measured damping at each frequency in place of an ideal linear viscous dashpot. Figure 5b is a

schematic of the Simscape model shown in its fully extended position on the left, and fully compressed on the right.

All the links in the model are rigid and pinned at their endpoints. The mass of each link is concentrated at its center

of gravity.

From the calculated response to a harmonic force or displacement input by Simscape, force versus displacement

hysteresis loops are generated. Due to nonlinearity of the elements in the system, the hysteresis loops are not the

classical elliptical loops13

obtained for linear systems. Even so, the area of the hysteresis loop, ALOOP , represents the

dissipative capability of the system14

. For a harmonic displacement input, the calculated force response contains

higher harmonics, but retaining only the fundamental frequency of the response while discarding all the higher

harmonics yields classical elliptical hysteresis loops whose area is the same as that of the nonlinear hysteresis loop14

.

From this process and the classical elliptical loops, the storage modulus, the loss modulus, and the loss factor, , a

commonly used metric in damping applications, can be extracted.

A parametric analysis has been performed using the Simscape model to investigate the effect of VMT spring,

dashpot and input frequency variation on the hysteresis loop area. Initial examination determined that increasing the

VMTs initial angle 𝜃0 and link length 𝐿𝑉 (for the same spring stiffness) or increasing the spring stiffness 𝐾𝑉 (for the

same initial angle and link length) are equivalent ways to increase the velocity in the dashpot (and, therefore, the

energy dissipated) at expense of higher required force to snap-through. However, when increasing the initial angle

and link length, the stroke also increases. Thus the initial angle and link length are selected to be compatible with the

PBEs and dashpot capabilities, and parametric change in spring stiffness is used to get the best energy dissipation.


(a)

(b)

Figure 5: (a) Simscape model of the hexagonal cell with VMTs, PBEs and (b) schematic representation of

the Simscape model in the expanded and compressed states.

(a)

(b)

(c)

Figure 6: Tailoring the VMT spring stiffness to the input excitation amplitude: force-displacement plots

(top) and hysteresis loops (bottom) for different VMT spring stiffnesses.


The importance of selecting the optimal VMT parameters for a given loading input is discussed next. Figure 6

shows the energy dissipation performance of a system for three different values of spring stiffnesses, Kv, subjected

to the same harmonic force excitation. The top row of figures represents the static force-displacement curves of the

combined system, equivalent to the curve discussed in Fig. 1c. An optimally designed combined system, Fig. 6b, has

a snap-through force which corresponds to the amplitude of the harmonic force excitation. The bottom row of

figures represents the force-input hysteresis loops of the same three elements (with the initial pre-buckling stiffness

removed to improve clarity). The hysteresis loop area for Fig. 6b is the best possible, maximizing both ALOOP and .

On the other hand, if the VMT spring stiffness is low and the snap-through force is lower than the input excitation

amplitude, Fig. 6a, the hysteresis loop will be sub-optimal. Both the total area of the hysteresis loop as well as the

loss factor are reduced. Conversely, if the VMT spring is too stiff (Fig. 6c), the external excitation is not able to

induce snap-through, and there is no energy dissipation.

IV. Prototype Design and Fabrication A CAD drawing of the prototype that was designed for this study is presented in Fig. 7. The main requirement

for the VMT and PBE combined assembly to work repeatedly without failure is that the displacements of the two

elements be compatible. Each VMT undergoes a maximum displacement ( 𝐿 (𝜃0)) and the sum of the

displacements of the two VMTs should not exceed the allowable axial displacement, , in the PBE. The latter is

limited by the maximum stress level reached in the PBE during post-buckling, which should be below the

proportional (elastic) limit of the material used. The viscous dashpot should be selected such that it has a stroke

of at least . With a larger dashpot stroke leading to more energy dissipation and a larger permissible axial

displacement in the PBE corresponding to a larger device length (Ref. 11), this suggests that the largest possible

device length should be chosen within the available volume constraints.

Figure 7: CAD drawing of the final design for the prototype device.

The design procedure for the proposed assembly can be summarized as follows:

1. The required static load is set, and a critical Euler load is chosen slightly above it

2. The buckling element is sized to satisfy the required initial stiffness and critical Euler buckling load, 𝑃𝐸 ,

and maximize

a. A suitable material is chosen, which sets the proportional limit

b. Boundary conditions are chosen (a pin ended PBE is considered) and analytical equations are

adjusted accordingly


c. Beam length is maximized within the available volume constraints (to maximize )

d. Cross section type and dimensions are chosen to satisfy 𝑃𝐸 and the slenderness ratio (to ensure

is not exceeded)

e. The static behavior of the selected PBE is evaluated

f. The maximum stress level is calculated for increasing axial displacements, and is found

g. Iterations may be necessary if the value of is not satisfactory

3. The VMT is sized to match and the input harmonic loading

a. Truss geometry (𝜃0 and 𝐿𝑉) is adjusted to match the available displacement (note that one

equation is available for 2 unknowns and infinite solutions are possible)

b. The VMT spring stiffness 𝐾𝑉 and initial angle 𝜃0 are chosen to match the input excitation

amplitude (Fig. 6)

c. Iterations may be necessary

4. The viscous dashpot is selected with a maximum stroke equal to and a suitable damping constant C

The material chosen for the PBEs was 301 Stainless Steel due to its high Young’s modulus and maximum

allowable stress. The high Young’s modulus was needed to achieve a high pre-buckling stiffness. A ratio of 1:20

between the cross section of the PBEs and the top area of the device was set as a requirement to achieve a global

stiffness (for the entire device) of at least 10GPa. In turn this set a constraint on the overall packing density of the

device. In this regard, the dimensions of available commercial dampers represented one of the main challenges, and

lead to the choice of an Ace Controls Inc. HB15-50-CC-P damper with a stroke of 50 mm. The combination of these

needs required that the post-buckling components be plate-like. The length of the plates and, therefore, the overall

device height was constrained by the ability to achieve the desired post-buckling displacement without failure.

Figure 8 illustrates the feasible design space for the prototype based on the stated requirements.

Figure 8: Design space for the PBEs.

Having selected the damper and PBEs, the geometry of the two VMTs was defined to match the overall

allowable displacement and fit between the PBBs. The last parameter to be defined was the stiffness of the VMT

spring (𝐾𝑉). These springs were custom designed to minimize the element thickness and integrated to the vertical

walls of the hexagonal cell such that they cannot slide vertically relative to each other. A large number of integrated

spring designs were evaluated using ANSYS Finite Element analysis to ensure structural integrity while allowing for

the necessary lateral displacement during snap-through of the VMTs. Figure 9 shows the FE stress analysis of the

final design adopted for the integrated VMT springs. This integrated spring was laser cut from two sheets of 1.5 mm

thick 301 stainless spring steel sheet and then the two sheets were stacked together in the prototype. Table 1

summarizes the final set of parameters for the fabricated prototype and a picture of the device in its fully compressed

state is presented in Fig. 10.


Figure 9: ANSYS Finite Element calculations of maximum stress in the VMT springs (stress shown in Pa).

Table 1: Design parameters for the experimental prototype.

VMTs

Link length - 𝐿𝑉 31.7 mm

Initial angle - 𝜃0 23 deg

Spring stiffness - 𝐾𝑉 25,000 N/m

Max stroke (both) 50 mm

Damper

Damping constant - C 130 N*s/m

Max stroke (supplier data) 50 mm

Max stroke (actual) 46mm

PBEs

Material Spring steel (E = 205 GPa, = 800 MPa)

Length 254 mm

Thickness 0.635 mm

Width 76.2 mm

Euler critical load - 𝑃𝐸 (both) 102 N

Initial stiffness 78x106 N/m

Secondary stiffness 223 N/m


Figure 10: Fabricated prototype device in fully compressed state.

V. Experimental Testing and Results The fabricated device was tested under harmonic displacement and force input in two separate tests. The first set

of tests was carried out at the Pennsylvania State University (the Rensselaer Polytechnic Institute authors were based

at Penn State until July 2012), and the second set of tests was conducted at the Boeing Company (Philadelphia).

Over 60 tests and several hundred cycles of testing were conducted without failure of any component. Although the

tests were performed using state-of-the-art electro-hydraulic mechanical testing machines, given the large stroke (50

mm) the maximum frequency at which the tests could be conducted was 4 Hz under displacement control, and 0.5

Hz under force control. Tests under force control proved to be challenging due to the sudden change in stiffness

produced by the PBEs in combination with the high velocities and large displacements induced by the snap-through

of the VMTs.

Figure 11 presents experimental data (dashed green) and the corresponding Simscape simulation results (solid

blue) for the four harmonic displacement input tests performed at Penn State. As expected, for the low frequency

cases (Figs. 11a and 11b), the simulation displays the characteristics of the static curve, but with hysteretic effects

(Figs. 3a, 3b). The simulation displays a high initial stiffness below the critical buckling load of the PBEs, after

which the VMT dominated portion of the curve broadens into a hysteresis loop due to energy dissipation in the

viscous damper. Although there are two VMTs in the modeled system no imperfection or dissimilarity between the

top and bottom VMT is seeded into the model, so Simscape predicts simultaneous snap-through of both VMTs. As

the excitation frequency increases (Figs. 11c and 11d), the force generated by the damper begins to dominate the

overall shape of the hysteresis loops.

In contrast to the simulation, the top and bottom VMTs are not exactly identical in the prototype due to

imperfection in fabrication. Therefore, in the experimental loops in Fig. 11 one VMT has a lower critical load and

transitions through the negative stiffness region before the other (as explained in Fig. 4c). The transition of the first

VMT is clearly observed over the negative displacement range in Fig. 11 whereas deformation of the second VMT

occurs over the positive displacement range. It is noted in the experimental loops in Fig. 11 that while the first VMT

transitions completely, the second VMT displaces only partially (compare to Fig. 4c). A displacement of 50 mm

would be required for both VMTs to transition completely, but the dashpot had a stroke of only 0.46 mm (less than

the 50 mm manufacturer specification). As with the simulation, the shape of the experimental hysteresis loops

becomes dominated by the damper as the frequency increases. Despite the differences in shape between the

experimental and simulated hysteresis loops, the hysteresis loop areas (representative of the energy dissipated per

cycle) match well because in both cases the damper is moving the same distance at the same frequency. Also, the

initial portion of the measured loops matches well with the slope and critical load of the simulated loops. The load

spikes in the experimental data are attributed to the dynamic return of the PBEs to their un-buckled state.


(a) 0.1 Hz, Aloop (Exp)=0.90, Aloop (Num)=1.10 (b) 0.5 Hz, Aloop (Exp)=1.13, Aloop (Num)=1.10

(c) 1.0 Hz, Aloop (Exp)=2.07, Aloop (Num)=2.06 (d) 2.0 Hz, Aloop (Exp)=5.15, Aloop (Num)=5.60

Figure 11: Hysteresis loops under harmonic displacement at different input frequencies.

Figure 12 shows the same four experimentally measured hysteresis loops presented in Fig. 11 (dashed green)

along with the additional higher frequency loops measured at the Boeing facility (solid blue). The high stiffness

portion of the curve prior to buckling has been removed in order to focus only on energy dissipation aspects. Along

with the experimentally measured loops, the equivalent ideal elliptical hysteresis loops (obtained by considering

only the first harmonic from the measured force response to the harmonic displacement input) are presented in light

blue, and these are used to estimate the loss factor. Both hysteresis loop area and loss factor increase with increasing

frequency. Fig. 12d compares data from experiments under a 2 Hz harmonic displacement input from the Penn State

and Boeing facilities. The agreement between the hysteresis loops from both tests highlights the repeatability of the

device.

The results of experiments under harmonic force input were conducted at The Boeing Company and are

presented in Fig. 13 (displacement and force time histories) and Fig. 14 (hysteresis loops). These tests are

challenging to conduct as snap-through involves large displacement at high velocity, and the testing machines

struggle to keep up. The non-linear behavior produced by the change in stiffness of the buckling elements further

adds to the complexity of the experiments. In view of the above, data under harmonic force input could only be

obtained up to 0.5 Hz. Even at these low frequencies problems were inherent. From the force time histories in Fig.

13 it is evident that the machine could not deliver a clean harmonic input force even though that is what was

prescribed. The occurrence of snap-through can be identified on the displacement time histories, and the force is

seen to peak at the end of snap-through as the damper bottoms out. The hysteresis loops in Fig. 14 show qualitative

similarity to the expected loop in Fig. 4d. However, instead of flat plateaus in the hysteresis loops corresponding to

snap-through, a slight negative stiffness is observed as the testing machine does not allow the VMT to move as

rapidly as it otherwise would. This is one factor that contributes to the lower than expected measured hysteresis loop

areas in Fig. 14 compared to Figs. 12a and 12b, for the same frequencies under a harmonic displacement input

(recall, from Figs. 4c and 4d the expected hysteresis loop area would be higher under harmonic force excitation). A


(a) 0.1 Hz, Aloop=0.90, η=0.84 (b) 0.5 Hz, Aloop=1.13, η=1.09

(c) 1.0 Hz, Aloop=2.07, η=1.50 (d) 2.0 Hz: Boeing test, Aloop=5.33, η=2.13

PSU test, Aloop=5.15, η=1.9

(e) 3.0 Hz, Aloop=11.07, η=2.09 (f) 4.0 Hz, Aloop=14.33, η=2.69

Figure 12: Experimental and idealized hysteresis loops under harmonic displacement at frequencies

between 0.1-4.0 Hz.


second possible factor contributing to the lower hysteresis areas in Fig. 14 is that the harmonic force amplitude was

larger than optimal (as explained in Fig. 6). This larger harmonic force, along with the impulse at the end of snap-

through both increase the effective stiffness (or storage modulus of the hysteresis loop) thereby resulting in a lower

loss factor.

A summary of measured loss factors for all the device tests are presented in Fig. 15. Loss factors are seen to

increase from about 0.6 at 0.1 Hz, to about 2.7 at 4 Hz. The displacement and force control tests show similar loss

factors at low frequencies (0.1 and 0.5 Hz), and the trends from the Penn State displacement control tests (0.1-2.0

Hz) and the Boeing displacement control tests (from 2-4 Hz) appear to be consistent. Overall, the prototype shows

good energy dissipation capability over the frequency range considered.

(a) 0.1 Hz (b) 0.5 Hz

Figure 13: Displacement and force time histories under harmonic force input.

(a) 0.1 Hz (b) 0.5 Hz

Figure 14: Hysteresis loops under harmonic force input.

VI. Conclusions This paper reports on the design, fabrication, testing and simulation of a novel structural element that combines

load carrying and energy dissipation capability. The principal components comprise PBEs, and VMTs coupled to a

viscous dashpot, all integrated in a panel-like element that can be assembled into larger structures, if desired. Load

carrying capability of the unit comes from the PBEs which provide a high initial stiffness and very little deformation

up to the critical buckling load. Even after the elements buckle under a harmonic disturbance superposed on the

static load, they continue to carry the design load. With the PBEs softened in the post-buckling regime, load

transfers to the VMT-dashpot load-path. Energy dissipation is obtained through the deformation of two VMTs at the

top and bottom of a hexagonal cell, connected to the ends of the viscous dashpot. Under harmonic displacement


input each VMT transitions through a negative stiffness region, whereas under harmonic force input each VMT

undergoes snap-through. In either case, the large stroking produced in the viscous damper leads to significant energy

dissipation. The paper explains the design procedure of the structural element in detail – the selection of the PBE

parameters to carry a design static load, its axial displacement in the post-buckling regime constrained by the

maximum allowable stresses in the elements, the axial displacement in the PBEs matched to the displacements in the

VMTs, the spring stiffness of the VMTs selected for an expected harmonic disturbance amplitude, and the damper

stroke matched to the displacements in the VMTs and the PBEs. A prototype device is fabricated and tested under

harmonic excitation.

Under harmonic displacement input, data was obtained at frequencies ranging from 0.1 Hz to 4 Hz. The

hysteresis loop area (and energy dissipation) increased with frequency, and loss factor increased from 0.84 at 0.1 Hz,

to 2.69 at 4.0 Hz. Although simulation predicted both top and bottom VMTs moving simultaneously while the tests

showed one transitioning before the other, the experimental and simulation hysteresis loop areas compared well.

Under harmonic force input, data was obtained only at 0.1 Hz and 0.5 Hz, as the testing machine struggled to move

at high-speeds associated with VMT snap-through. Even at these low test frequencies, a perfect harmonic load could

not be applied. But snap-through was observed in the time histories. The energy dissipated (hysteresis loop areas) in

these tests was slightly lower than that in the displacement controlled tests as the same frequencies, with loss factors

of 0.61 at 0.1 Hz and 0.97 at 0.5 Hz calculated from the measured hysteresis cycles.

Figure 15: Summary of experimental loss factor measurements on the device.

Acknowledgments

The authors gratefully acknowledge the sponsorship of these research activities within the DARPA sponsored

program at NextGen Aeronautics entitled “Structural Logic,” Contract # HR0011-10-C-0189. The views expressed

are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S.

Government.

References 1Barbarino, S., Pontecorvo, M.E., and Gandhi, F.S., “Energy Dissipation of a Bi-stable von-Mises Truss under Harmonic

Excitation,” Proceedings of the 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference,

Honolulu, HI, April 23-26 2012, AIAA-2012-1712. 2Mises R., “Uber die Stabilitatsprobleme der Elastizitatstheorie,” Zeitschrift Angewandte Mathematik und Mechanik, Vol. 3,

406, 1923. 3Blair, K. B., Krousgrill, C. M., and Farris, T. N., “Nonlinear Dynamic Response of Shallow Arches,” Proceedings of the

33rd AIAA/ASME/ASCE/ASC Structures, Structural Dynamic and Material Conference, 13-15 April 1992, pp. 2376-2384. 4Avramov, K. V., and Mikhlin, Y. V., “Snap-through Truss as an Absorber of Forced Oscillations,” J. of Sound and

Vibration, Vol. 290, No. 3-5, 2006, pp. 705-722. 5Kounadis, A. N., Raftoyiannis, J., and Mallis, J., “Dynamic Buckling of an Arch Model under Impact Loading,” J. of Sound

and Vibration, Vol. 134, No. 2, 1989, pp. 193–202.


6Padthe, K., Chaturvedi, N. A., Bernstein, D. S., Bhat, S. P., and Waas, A. M., “Feedback Stabilization of Snap-through

Buckling in a Preloaded Two-bar Linkage with Hysteresis,” Int. J. Non-Linear Mech., Vol. 43, 2008, pp. 277–291. 7Murray, G. J., and Gandhi, F. S., “The Use of Damping to Mitigate Violent Snap-through of Bistable Systems,” Proceedings

of the ASME 2011 Conference on Smart Materials, Adaptive Structures & Intelligent Systems (SMASIS), Phoenix, Arizona, USA,

18 September 2011, paper SMASIS2011-4997. 8Pontecorvo, M. E., Barbarino, S., and Gandhi, F., Bland, S., Snyder, R., Kudva, J., and White, E., “A Three-Dimensional

Multi-Stable Unit Cell for Energy Dissipation,” Proceedings of 22nd AIAA/ASME/AHS Adaptive Structures Conference,

National Harbor, MD, Jan 13-17 2014. 9Pontecorvo, M., Barbarino, S., Gandhi, F., Bland, S., Snyder, R., and Kudva, J., “A Load-Bearing Structural Element with

Energy Dissipation Capabality under Harmonic Excitation,” Proceedings of the ASME 2013 Conference on Smart Materials,

Adaptive Structures and Intelligent Systems, SMASIS2013, Sept. 16-18, 2013, Snowbird, Utah, SMASIS2013-3060. 10Pontecorvo, M.E., Barbarino, S., Gandhi, F.S., “Cellular Honeycomb-Like Structures with Internal Inclusions in the Unit-

Cell", Proceedings of the ASME 2012 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Stone

Mountain, Georgia, USA, September 19-21, 2012. 11Barbarino, S., Pontecorvo, M.E., Gandhi, F., “Cellular Honeycomb-Like Structures with Internal Buckling and Viscous

Elements for Simultaneous Load-Bearing and Dissipative Capability,” Proceedings of the ASME 2012 Conference on Smart

Materials, Adaptive Structures and Intelligent Systems (SMASIS 2012), September 19-21 2012, Stone Mountain, Georgia. 12Euler, L., “Methodus Inveniendi Lineas Curves Maximi Minimive Propriete Gaudentes, Additamentum I. De Curvis

Elasticis,” Lausanne and Geneva, 1774. 13Lazan, B.J., Damping of Materials and Members in Structural Mechanics, 1st ed., Oxford: Pergamon, 1969. 14Gandhi, F. and Wolons, D., “Characterization of the Pseudoelastic Damping Behavior of Shape Memory Alloy Wires using

Complex Modulus”, Smart Materials and Structures, Vol. 8, 1999, pp. 49–56.

a novel structural element combining load carrying and energy dissipation capability

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