Introduction
Background and Motivation
- Introduction to Tensegrity Structures
- Dynamics of Tensegrity Structures and Lattices
In the last few years, significant studies of tensegrity structures as potential landers for extraterrestrial applications have been of great interest. Tensegrity structures were interesting in the dynamic regime not only as individual structures, but also as periodic networks.
Goals and Organization
An illustration of this calculation is shown in the x-t diagram of the wave propagation in Figure 4.6. The red line is placed on the upper nodes of the structure in the t = 0 image.
Methods
Numerical Methods
- ABAQUS Finite Element Modeling
- COMSOL Finite Element Modeling
Also, a version of the beam element model is given in Figure 2.3, where (a) shows the wire version and (b) shows the version with the cross-sectional areas shown. Due to the simple nature of the beam element simulation, only 540 elements are required (less than 1% of the number of elements used for the 3D model).
Experimental Methods
- Sample Fabrication
- Static Compression Tests
- Dynamic Impact Testing
- Frequency Transmission Experiments
A schematic overview of the setup and displacement tracking points is shown for the 3D1D grid in Figure 2.11. The forces are thus significantly reduced in the grid when the wave reaches the top of the grids.
Design of Tensegrity-Inspired Structures
Design Methodology
- Target Baseline Tensegrity and Initial Design Iteration
- Conversion Method Between the Fixed and Pin-jointed Struc-
- Examples of the Design With Alternate Materials and Length
The stiffness of the structure with fixed joints is much higher than the stiffness with pins. The effective bending length of the struts is thus significantly shorter than that of the bolt counterpart.
Fundamental Comparison of the Fixed and Pin-Jointed Structures
As a note, the struts in the final shaped spherically jointed structure also satisfy the elastic deflection requirement of Equation 3.1. The relative density of the final designed structure (geometry #2) is 2.5% and the relative density of the corresponding pin stress is 4.9%. The structure is therefore very light, even more so than a bolted tensegrity structure.
We show the force in the strut for the non-prestressed pin-jointed structure to illustrate the onset of buckling load in the strut. For the rest of this thesis, we refer to the designed spherical structure as the 'tensegrity-inspired structure'. The non-mosaic designed structure is called the tensegrity-inspired base structure.
Representative Volume Elements for Tensegrity-Inspired Lattices
For completeness, we found the effective Poisson's ratio for each of the 3 RVEs shown in Figure 3.14. Poisson's ratio in the linear region of the stressβstrain curves is very close to 2 for all three RVEs. The Poisson's ratio increases significantly as the strain increases into the nonlinear region, reaching nearly 5 upon densification.
An effective Poisson's ratio greater than 1 has also been shown for other types of architectural networks [114, 115]. As we will see, this large Poisson's ratio has a significant effect on the dynamics of these tensegrity-inspired structures.
Chapter Summary
In the bending-dominated cases (Kelvin and ours), all members are at a steep angle to the longitudinal direction. Recall that the 3D RVE of the tensegrity-inspired structure consists of 8 basic unit cells. As done in [87], we normalize the dispersion relation frequencies as follows.
In Figure 5.25 we show the time course of the 3D bulk lattice for an impact of 8.5π/π .
Dynamics: Frequency Response
Introduction to Dynamic Frequency Analysis
In Chapter 3, we presented a method to create lightweight tensegrity-inspired structures with unique non-linear buckling properties. Our tensegrity-inspired structure has many advantages, such as low density, high elastic deformation, minimal local stresses, high energy capacity and load limitation. This opens the door to creating new types of tensegrity-inspired materials with unique properties.
The quasi-static features we explored in the previous chapter hold promise for the dynamic regime. In this chapter we begin to investigate how our tensegrity-inspired structure behaves in the dynamic regime by looking at its response to low amplitude frequency excitation.
Unstrained Lattice
- Dispersion Results
- Wave Speed Analysis and Comparison with Other Lattices . 61
Thus, it makes sense that the green coupling modes occur at the intersections of the longitudinal and torsional modes in the dispersion curve. The experimental and numerical results agree, with the numerical dispersion following the maximum of the color map (dark grey/black regions). The low-frequency longitudinal wave velocity of the 1D grid can be easily determined from the experimental and numerical results.
The numerical wave speed is calculated by taking the slope of the dispersion curve in the low-frequency region where the curve is linear. The velocity of the longitudinal waves is twice the velocity of the 1D mesh for both our structure and the tensegrity structure.
Chapter Summary
Here we show the velocity magnitude at the center of two struts, shown by the locations in the inset image of the grid. The first three timestamps show the wave reaching the top of the grid without a clear wavefront. At about 0.0015 π the wave starts to propagate and is localized at the top of the structure.
The stiffness and flexural load of the structure directly affect the energy absorption, i.e. the area under the stress-strain curve. The most striking observation is that the cables carry almost half of the energy in the network through strain energy.
Dynamics: Impact
Introduction of Impact Studies
In the previous chapter, we studied the response of these tensegrity-inspired structures under low amplitude wave excitation. This also confirms recent theoretical and numerical work in the field of tensegrity structures and grids. In the first scenario, a long duration pulse is exerted on the structure of a heavy, falling mass.
In the second scenario, the sample is dropped from such a height that it strikes a rigid surface with an initial impact velocity, π£0. In the third scenario, a short-term high-energy pulse is exerted on top of the structure by a small falling mass and the energy transmission characteristics are studied.
Drop Weight: Long-Duration Impact
- Baseline Unit Cell
Next, it is important to consider the energy loss during impact of the basic unit cell. Images are displayed with increasing time, from 0 to π‘π π π, which is the total impact time of the structure. The total impact time is longer for the experiments due to the smooth discharge characteristic.
The load limitation in the basic unit cell and also in 1D lattices is due to sufficient deformation of the structure so that a region of non-linear buckling is reached. In experiment (d), when the compression reaches about 0.15 strain, a complete disappearance of the compression stages is observed.
Sample Drop Tests
- Experimental Results
- Wave Characteristics
Again, we calculate this based on when the velocity at the end cell (location 6) changes by 1% of the impact velocity. That is, the energy goes much faster to the edges than to the top of the grid. One is at the bottom of the grid near the impact site and the other is 5 cell lengths from the bottom.
Sharp high-amplitude accelerations die out by reaching the top of the grid due to this energy trapping. Faster shear and diagonal wave velocities redirect energy to the lower edges of the grid before the wave reaches the top.
Flatten the Curve: Short Duration Impulse by a Falling Mass
- Wave Propagation
- Force Transmission Reduction
On the right, for each timestamp, we show a graph of the normalized velocity over the horizontal slice area in the transverse directions (X and Z). πΉ/πΉπ is the force normalized by πΉπ, which is the force capacity of the unit cells of the bottom layer. For impact applications, if an object is to be protected, a material such as foam or lattice is desirable to 'spread' the impact pulse through it over time so that by the time the object is reached the forces are reduced and the impulse on the protected object lasts longer.
The impact force and transmitted reaction force for different webs and impact speeds are presented in Figure 5.33. πΉ/πΉπ is the force normalized by πΉπ, which is the force capacity of the unit cells of the bottom layer.
Chapter Summary
The nonlinearity of the compressive response of the structure produces a dramatic evolution of the dynamic characteristics of the web. These tests showed the load limiting characteristic of the structures; i.e., their maximum loads are governed by the buckling load of the base structure. The strain energy can be increased by changing to a configuration where densification occurs much later in compression, such as if the strips run mostly near the outside of the grill.
On the mechanical modeling of the extreme softening/hardening response of axially loaded tension prismsβ. The stress is normalized by the maximum stress of the selected final model (π /ππof 1.44) for comparison.
Conclusions and Future Directions
Summary
Without the need for pin joints or bias, the structure uses geometry to produce tensegrity-like features. Continuous tuning of the bandgap and wave velocities is achieved by increasing the level of global precompression, making these gratings a candidate for energy-focusing acoustic lenses. Since the deformation remains elastic even at large strains, repeatable and active tuning of the lattice response is potentially achievable.
Dissipation due to structural nonlinearity quickly spreads to shock momentum and traveling waves, reducing damaging accelerations and lower velocities. Since the diagonal wave speed is greater than the longitudinal wave speed, the impact energy is redirected towards the edges and prevents most of the energy from propagating towards the bottom of the grating where the protected object might be located.
Future Directions
A disadvantage of the octahedron is that since the struts run through the interior of the lattice, compaction occurs at strains below 50%. The attraction behind this is that the energy absorption of tensegrity structures can be higher for random than for periodically arranged combinations of unit cells. Design of structures for dynamic properties via natural frequency separation: Application to tensegrity structure design.
A physical perspective on length scales in gradient elasticity through the prism of wave dispersion.
MATLAB Codes for Dispersion Experiments
5% wavenumber, ky, second column is frequency in Hz. 6% Email [email protected] if you want data tables.
MATLAB Code for Drop Weight Experiments
Relaxation Tests for Two Materials
Intermediate Geometry Design Iterations
This is important because at a fixed amount of strain (here, strain 0.4), structures with thicker cables result in higher stress concentrations. Plastic deformation from stress concentrations is clearly shown by the reduction in strength for the two lowest ππ /π values. Plastic deformation is not easily seen on the plot
However, the energy input into the grid from the impactor is transferred through the grid via both strain energy and kinetic energy. Here we investigate the stress and kinetic energy distribution in a 10 RVE 3D1D grid under impact by a 10 π impactor moving at 18π/π .