iMechanica - Comments for "On the Effective Dynamic Mass of Mechanical Lattices with Microstructure"
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Comments for "On the Effective Dynamic Mass of Mechanical Lattices with Microstructure"enMoments
https://www.imechanica.org/comment/30895#comment-30895
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<p><em>In reply to <a href="https://www.imechanica.org/node/26779">On the Effective Dynamic Mass of Mechanical Lattices with Microstructure</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Arash,</p>
<p class="MsoNormal">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.</p>
<p class="MsoNormal">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.</p>
<p class="MsoNormal">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.</p>
<p>Regards.</p>
<p>Mohammed Lamine</p>
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</ul>Sat, 22 Jul 2023 15:07:17 +0000mohammedlaminecomment 30895 at https://www.imechanica.orgDear Mohammed: In continuum
https://www.imechanica.org/comment/30894#comment-30894
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<p><em>In reply to <a href="https://www.imechanica.org/comment/30893#comment-30893">Newton's law in rotation is efficient and more convenient</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>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</p>
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</ul>Fri, 21 Jul 2023 22:12:22 +0000arash_yavaricomment 30894 at https://www.imechanica.orgNewton's law in rotation is efficient and more convenient
https://www.imechanica.org/comment/30893#comment-30893
<a id="comment-30893"></a>
<p><em>In reply to <a href="https://www.imechanica.org/node/26779">On the Effective Dynamic Mass of Mechanical Lattices with Microstructure</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p class="MsoNormal">Dear Arash thank you for the details.</p>
<p class="MsoNormal">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 <strong>P=I*omega</strong> where <strong>I</strong> is the inertia moment. The reader can find in figure 2 a well explained rotating model of the microstructure mechanism.</p>
<p class="MsoNormal">In dynamics, there are two forms of the Newton’s law: in translation <strong>∑(F)=m*a</strong> and in rotation <strong>∑(Mext)=delta</strong> where delta is the dynamic momentum at the same point of the external moments calculation and its simplest form is <strong>delta=time derivative(kinetic momentum)</strong>. One can include the substructures sum.</p>
<p class="MsoNormal">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.</p>
<p><span>Best regards.</span></p>
<p><span>Mohammed Lamine</span></p>
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</ul>Fri, 21 Jul 2023 19:21:17 +0000mohammedlaminecomment 30893 at https://www.imechanica.orgDear Mohammed: Please have a
https://www.imechanica.org/comment/30892#comment-30892
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<p><em>In reply to <a href="https://www.imechanica.org/comment/30891#comment-30891">Effective mass and effective mass matrix</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>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</p>
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</ul>Thu, 20 Jul 2023 22:37:16 +0000arash_yavaricomment 30892 at https://www.imechanica.orgEffective mass and effective mass matrix
https://www.imechanica.org/comment/30891#comment-30891
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<p><em>In reply to <a href="https://www.imechanica.org/node/26779">On the Effective Dynamic Mass of Mechanical Lattices with Microstructure</a></em></p>
<div class="field field-name-comment-body field-type-text-long field-label-hidden"><div class="field-items"><div class="field-item even"><p>Dear Yavari,</p>
<p>You have cited in your paper the effective mass with an analogy of a swimmer in a fluid.</p>
<p>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.</p>
<p>What is different between these two definitions and what defines the lattice dynamics in this example ?</p>
<p> </p>
<p>Mohammed Lamine</p>
<p> </p>
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</ul>Thu, 20 Jul 2023 21:37:36 +0000mohammedlaminecomment 30891 at https://www.imechanica.org