Frequency Response of a Viscoelastic Tensegrity Model: Structural Rearrangement Contribution to Cell Dynamics

Spatial view of the viscoelastic tensegrity structure (VTS) studied (6 bars and 24 viscoelastic cables). At the reference state (no external force applied to the structure), the four nodes {1, 2, 4, 8} are anchored and fixed in their spatial positions (●). The rectangular base {x,y,z} is the referential system. Oscillating forces (Fz and −Fz) are applied at nodal points {6, 11} along the z axis. The overall deformation of the VTS model is defined by the displacement along the z axis of the two nodes {6, 11} normalized by the bar length Lb which characterizes the size of the structure.

Oscillatory deformation of the overall VTS as a function of the normalized forced frequency f* in a double logarithmic scale. Three distinct zones could be distinguished: a low frequency zone (f*<0.1) of roughly constant oscillatory deformation amplitude where elastic effects are predominant; a transitional zone (0.1<f*<10) of a rapid change in the amplitude of the deformation (with a negative logarithmic slope (−0.83); and a satisfactory correlation coefficient R2=0.99) where elastic and viscous effects are balanced and a high frequency zone (f*>10) of low oscillatory deformation where viscous effects are predominant. The contribution of the spatial redistribution of the tensegrity structure appears to decrease when the frequency increases.

Normalized time constant τ* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant (i) elastic (f*<1) and (ii) viscous (f*>10) effects, as well as the transitional zone (0.1<f*<10). The normalized time constant of the overall tensegrity model decreases nonlinearly with the oscillatory frequency in the upper part of the transitional zone, following the power law τ*∼f*−0.42 with a correlation coefficient R2(=0.97). The decreasing contribution of spatial redistribution of VTS elements contribute to the frequency-dependent solidifyinglike process for the VTS model.

Normalized viscosity modulus η* and elasticity modulus E* of the overall VTS model as a function of the normalized forced frequency f* in a double logarithmic scale with the two zones of predominant elastic (f*<1) and viscous (f*>10) effects, and the transitional zone (0.1<f*<10). The normalized elasticity modulus E*(•) increases in the transitional zone with a logarithmic slope (+0.18) and the normalized viscosity modulus η*(▴) decreases with a logarithmic slope (−0.24), the two with a correlation coefficient R2(=0.99).

Normalized time constant τ* of the overall VTS as a function of the normalized internal tension (corresponding to the initial strain of the elastic cable) T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. τ* decreases following a mean power law given by (τ*∼T*−0.42; R2=0.98) whatever L*. The values of τ* decrease within a small range of [0.1-1] when T* increases by two orders of magnitude, whatever the value of L*.

Normalized elasticity modulus E* of the overall viscoelastic tensegrity structure as a function of the normalized internal tension T* for three different normalized element length L* (=100, 1000, 10,000) in the transitional zone. E* increases with T* following a power law E*∼T*0.52(R2=0.97) whatever the value of L*. Note that the values of E* decreases proportionally to the inverse of (L*2) in the overall range of T*.

Normalized viscosity modulus η* of the overall tensegrity model as a function of the initial internal tension T* for three different values of L* (=100, 1000, 10,000) in the transitional zone. Viscosity modulus η* is shown to increase nonsignificantly (logarithmic slope ≤+0.1;R2=0.98) with increasing T* for the three values of L*.

Normalized elasticity modulus E* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.

Normalized viscosity modulus η* as a function of the normalized length L* for five different values of the internal tension T*(=◆0.005,∎0.05,▴0.25,◆0.5,•0.75) in the transitional zone. Note that the curves show a strictly negative logarithmic slope (−2) whatever the value of T*.