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# On the Effective Dynamic Mass of Mechanical Lattices with Microstructure

We present a general formalism for the analysis of mechanical lattices with microstructure using the concept of effective dynamic mass. We first revisit a classical case of microstructure being modeled by a spring-interconnected mass-in-mass cell. The frequency-dependent effective dynamic mass of the cell is the sum of a static mass and of an added mass, in analogy to that of a swimmer in a fluid. The effective dynamic mass is derived using three different methods: momentum equivalence, dynamic condensation, and action equivalence. These methods are generalized to mechanical systems with arbitrary microstructure. As an application, we calculate the effective dynamic mass of a 1D composite lattice with microstructure modeled by a chiral spring-interconnected mass-in-mass cell. A reduced (condensed) model of the full lattice is then obtained by lumping the microstructure into a single effective dynamic mass. A dynamic Bloch analysis is then performed using both the full and reduced lattice models, which give the same spectral results. In particular, the frequency bands follow from the full lattice model by solving a linear eigenvalue problem, or from the reduced lattice model by solving a smaller nonlinear eigenvalue problem. The range of frequencies of negative effective dynamic mass falls within the bandgaps of the lattice. Localized modes due to defects in the microstructure have frequencies within the bandgaps, inside the negative-mass range. Defects of the outer, or macro stiffness yield localized modes within each bandgap, but outside the negative-mass range. The proposed formalism can be applied to study the odd properties of coupled micro-macro systems, e.g., active matter.

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## Comments

## Effective mass and effective mass matrix

Dear Yavari,

You have cited in your paper the effective mass with an analogy of a swimmer in a fluid.

In structural dynamics, the effective mass matrix is the generalyzed mass matrix obtained from the normalyzed eigenvectors pre-multiplication and post-multiplication of the structural mass matrix. The result is an identity matrix called the effective mass matrix.

What is different between these two definitions and what defines the lattice dynamics in this example ?

Mohammed Lamine

## Dear Mohammed: Please have a

Dear Mohammed: Please have a look at the paper by Milton and Willis (On modifications of Newton’s second law and linear continuum elastodynamics. Proceedings of the Royal Society A, 463(2079):855–880, 2007.) for some background on this problem. You have a system with microstructure. Under harmonic loads the effective mass of the outer system becomes frequency dependent. Regards,Arash

## Newton's law in rotation is efficient and more convenient

Dear Arash thank you for the details.

In his paper, Milton has defined P(t) a time momentum of the whole bar presented in figure 1 where the rotation is not shown. But if he considers the angular momentum vector of a rotating bar the momentum is

P=I*omegawhereIis the inertia moment. The reader can find in figure 2 a well explained rotating model of the microstructure mechanism.In dynamics, there are two forms of the Newton’s law: in translation

∑(F)=m*aand in rotation∑(Mext)=deltawhere delta is the dynamic momentum at the same point of the external moments calculation and its simplest form isdelta=time derivative(kinetic momentum). One can include the substructures sum.Instead of modifying the Newton’s law it is efficient to apply this second form of Newton’s law which is more convenient to include rotational effects.

Best regards.

Mohammed Lamine

## Dear Mohammed: In continuum

Dear Mohammed: In continuum mechanics one has balance of linear and angular momenta. Perhaps by “second form of Newton’s law” you mean the balance of angular momentum. I am not suggesting that Newton’s law of motion should be modified. In this paper, we study dynamic effective mass for lattices with general microstructures. The constitutive equations are assumed to be linear (linear springs) and loads are assumed to be harmonic. This is what Milton and Willis (and many other researchers) have also considered. As a side note, a balance law should be independent of constitutive equations. It is certainly interesting to study exotic properties of linear systems. However, any modification of any balance law should be applicable to nonlinear systems as well. Regards,Arash

## Moments

Dear Arash,

To get the strain energy, one can use the stresses which are an expression of the internal forces and this energy develops the mechanism stiffness. These stresses can also be used in the equilibrium equations to consider the internal effects of deformable mechanisms.

The Newton’s law second form calculates the moments of the external forces. Here, one uses only the external forces. Its general form includes the internal forces when their sum is not zero. This Newton’s law second form is not the balance of angular momentum.

There are several methods to get the motion equation of a mechanism in rotation. One can use the kinetic energy theorem, the Lagrange’s equations or the Newton’s fundamental law in rotation which uses the inertia tensor of the mechanism. This last method is a law and it is fundamental because it is trivial.

Regards.

Mohammed Lamine