# Off diagonal mass matrix example – Mass Matrix Templates: General Description and 1D Examples

## For example, a one-dimensional Helmholtz type equation with the form. A similar thread exists here but they are more interested in the principal axes of the body.

See Fig. Pergamon Press, Oxford, pp — Google Scholar

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Wood A Acoustics. This works fine because the local element has zero stiffness in some directions, and those zero rows and columns may be omitted at the local level.

They are implemented in all general purpose FEM codes. Conservation At a minimum, total element mass must be preserved we are talking about classical mechanics here; in relativistic mechanics mass and energy can be exchanged.

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But there is no easy road to get rotational masses. Those are the top three shown in Fig. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation. Pergamon Press, Oxford, pp —

Concurrently with advances in variational mass lumping leading to the CMM, the s and early were a fertile time for the development of reduced order dynamic models based on component mode synthesis CMSa name coined by Hurty [ 5859 ].

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Improve this question. Accordingly the shape function matrix is. Using rules with 3 or more points returns the same matrix. J Appl Mech — Views Read Edit View history. The development of mass matrix templates for multidimensional elements has lagged because of four complications:.

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The diagomal to the question given in the link are not satisfactory maybe because of the diversion created by improper framing of the question, so don't ofc this as duplicate. Lagrangian mechanics yields an ordinary differential equation actually, a system of coupled differential equations that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system. It is not necessary for the total inertia force to be zero - the body can move "to the side" without changing the axis of rotation. Them, each element of mass off the axis of rotation is rotating instantly around the axis of rotation in a circular motion. Related 2. Given some reference frame, usually one with origin at the center of mass, if the moment of the total inertial force is different from zero, the body will tend to change the direction of the axis of rotation.

Should globalization be necessary before merge, a congruent transformation is applied:. Wiley, Singapore. The three-node bar element, usually abbreviated to Bar3, is shown in Fig. For unperturbed orbit calculations, celestial bodies were idealized as point masses regardless of actual size. Felippa CA A template tutorial. Examples are furnished by beam-column elements, shell elements, and elements with curved sides or faces.

## Your Answer

Again, this only happens when your body is not symmetric about the rotation axis. Physics Stack Exchange works best with JavaScript enabled. The best answers are voted up and rise to the top. See more linked questions.

They tell us nothing interesting about the object itself. Improve this question. Sign up to join this community. Improve this off diagonal mass matrix example. Given some reference frame, usually one with origin at the center exam;le mass, if the moment of the total inertial force is different from zero, the body will tend to change the direction of the axis of rotation. These are called the principal axes or eigenvectors of rotation, and the physical meaning behind them is that if you rotate the object around one of those axes, the angular momentum will lie along the axis. In this case you can attach a special frame specific meaning to these terms.

The latter is called the target form.

Active 6 years, 10 months ago.

Dover, New York. To get a DLMM is trickier.

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Again, this only happens when your body is not symmetric about the rotation axis. This formulation extends naturally to higher dimensions. This may not be so for more complex elements shells etc. Learn more. Create a free Team What is Teams?

It also says:. The state of that systems can be described by a vector q of two generalized masa, namely the off diagonal mass matrix example of the two particles along the track. This is equivalent to writing your MOI in a basis where the rotation axis is one of the basis vectors, in which case there are off-diagonal elements, which is what your question is asking about. Views Read Edit View history. Related

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That's what the last sentence of my answer says. Post as a guest Name. Is that all or is there more to it, perhaps related to properties of tensors in general? That's the case when the total inertia centrifugal force is different from zero but its moment relative to the origin is null. Hobson, S.

Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. Timoshenko SP On the correction for shear of the differential equation for transverse vibration of prismatic bars, Phil.

So, off-diagonal elements in your MOI are equivalent to having a rotation axis that is not aligned with any of the principal axes. Sign up to join this community.

The unbroken broken U 1 is analogous to the photon dark photon. Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics.

Higher order conditions, such as conservation of angular momentum, are optional and not necessarily desirable.

Geradin M, Rixen D Mechanical vibrations: theory and applications to structural dynamics. If all entries are nonzero, it is easily inverted matri place, since the inverse of a diagonal matrix is also diagonal. It is invoked by. These in turn had to wait until three major ingredients became slowly established during the s: i the DSM of Turner [], ii the concept of shape functions progressively evolving in early FEM publications [ 2628606174 ], and iii the FEM connection to Rayleigh-Ritz.

Why Mathematica?. For a curved-side six-node triangle with variable metric, a case illustrated in Fig. Learn more. A congruential mass transformationor CMT, is a general framework than can be applied to transform a given source mass matrix into a target one. The advantage of this choice, as opposed to rotating the propagation direction, is that the wave functions are kept simple. Graff KF Wave motion in elastic solids. This scenario is dual to that of the element stiffness matrix.

It is rather easy, however, to extract a principal solution. Przemieniecki JS Theory of matrix structural analysis. This makes it especially suitable for multidimensional elements. Some mass matrix properties can be discerned visually:.

The best answers are voted up and rise to the top. These issues are further compounded in problems where the mass matrix in not constant. For example, examlpe lumped-mass method, in which the deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element. So, off-diagonal elements in your MOI are equivalent to having a rotation axis that is not aligned with any of the principal axes. Additionally, if you are using low-order elements e. This is because by the definition of a tensor all of the components get mixed up together under rotations or Lorentz transformations in special relativity or arbitrary coordinate transformations in general relativity. This may not be so for more complex elements shells etc.

For consistent FE methods, you use Galerkin orthogonality; non-consistent methods instead use the Strang lemma first or second, depending on the nature of non-consistency. The state esample that systems can be described by a vector q of two generalized coordinates, namely the positions of the two particles along the track. Views Read Edit View history. Sign up to join this community. It only takes a minute to sign up. Learn more. Linear algebra tells us that for any 3x3 matrix that has those three properties, there's always a set of three perpendicular axes such that the MOI tensor can be expressed as a diagonal tensor in the basis of those axes.

## Your Answer

So one important thing to realize is that there is nothing fundamentally meaningful about off-diagonal elements; you can always rotate your coordinates to get rid of them. In this case you can attach a special frame specific meaning to these terms. Sign up or log in Sign up using Google.

Higham NJ Functions of matrices: theory and computation. Those points are marked in the Figure. There are two 3-point Gauss quadrature rules of degree 2 for a constant metric triangle, shown in Fig. The arguments are identical to those for the lattice master mass and stiffness modules, respectively, described in Sect. Krieg and Key [ 66 ] had emphasized that in transient analysis by DTI the introduction of a time discretization operator brings new compensation phenomena, and consequently the time integrator and the mass matrix should not be chosen separately.

The best off diagonal mass matrix example are voted up and rise to the top. Stack Overflow for Teams exwmple Collaborate and share knowledge with a private group. Improve this question. Question feed. This formulation extends naturally to higher dimensions. Given some reference frame, usually one with origin at the center of mass, if the moment of the total inertial force is different from zero, the body will tend to change the direction of the axis of rotation.

This correspondence was first observed in [ 46 ], and further explored in [ 7677 ].

Active 4 years, 10 months ago. For example, imagine your object spinning inside a nicely symmetric little satellite in space.

By inspection. The construction of the CMM is fully defined by the choice of kinetic energy functional and shape functions.

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Hot Network Questions. Let q be the column vector comprising all those coordinates. On the other hand, what if the body is rotating about an axis that isn't one of the principal axes? For example, a lumped-mass method, in which the deformation of each element is ignored, creates a diagonal mass matrix and negates the need to integrate mass across the deformed element. Improve this question. Sign up to join this community. But while considering the general case, what are the off-diagonal moment of inertia matrix elements?

However, this is not the case. They are plotted in Fig. The kinematic expressions for the P- and S-waves are. For example, a one-dimensional Helmholtz type equation with the form. Archer JS Consistent mass matrix for distributed mass systems. Such matrices are preferred from a numerical stability standpoint.

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In this case you can attach a special frame specific meaning to these terms. Related The physical significance of the off-diagonal components is that you're using a coordinate system not aligned with the principal directions of the object.

The mass matrices obtained with Gauss integration rules of 1, 2 and 3 points are. To get a DLMM is trickier. The happy connection of mass templates to periodic materials may be of interest as FEM and related discretization methods are extended into multiscale applications of crystal, micro- and nano-mechanics, and phononics [ 722 ]. Dispersion curves govern energy transmission. Examples are furnished by beam-column elements, shell elements, and elements with curved sides or faces. The symbolic derivation scheme used for the Timoshenko beam EOM in is due to Flaggs [ 43 ]; see also [ 82 — 84 ].

Views Read Edit View history. They tell us nothing interesting about the object itself. It is not necessary for the total inertia force to be zero - the body can move "to the side" without changing the axis of rotation. Improve this answer. This may not be so for more complex elements shells etc. Learn more.

However, for gauge field we often have that one or more of these symmetries is unbroken and we want to write it in a basis which makes this obvious. As noted above, an obvious hindrance is the emergence of negative weights as the rule degree gets higher. Featured on Meta.

Viewed 13k times. This may not be so for more complex elements shells etc. Suppose the body is rotating around some axis. Viewed 1k times.

VSF that produce the general Bar3 mass template 44 in a variational framework, exmaple the nine labeled instances. The intention is to make the overall paper reasonably self-contained. The dispersion equations are. A good discussion of mass diagonalization schemes starting from the CMM can be found in the textbook by Cook et al.

Upcoming Events. To get a DLMM is trickier. Jones H The theory of Brillouin zones and electronic states in crystals. Felippa CA A template tutorial.

The procedure is heuristic but widely used on account of three advantages: easy to explain and implement, applicable to any element as long as a CMM is available, and able to retain nonnegativity. Email Required, but never shown. Table 15 Bar3 template signature list specification Full size table. In general, the mass matrix M depends on the state qand therefore varies with time.

This is because by the definition of a tensor all of the components get mixed up together under rotations or Lorentz transformations in special relativity or arbitrary coordinate transformations in general relativity. Featured on Meta. Sign up using Email and Password. Is that all or is there more to it, perhaps related to properties of tensors in general?

Download references. This result can be readily diagojal physically. Its roots can be traced back to the Free Formulation of Bergan [ 10 — 1329 ], in which the stiffness matrix was decomposed into basic and higher order components. This may not be so for more complex elements shells etc.

From Wikipedia, the free encyclopedia. Is that all or is there more to it, perhaps related to properties of tensors in general? But while considering the general case, what are the off-diagonal moment of inertia matrix elements? Add a comment.

Little harm is done, however, for this simple element. Saxe-Coburn Publications, Edinburgh, Chapter 4, pp 71—98 J Eng Mech —69 Google Scholar That is not the case in a FEM lattice of Trig3 elements, as studied next.

Email Required, but never shown. Though, having a principal axis does not imply any symmetry of the object. Inam Inam 71 1 1 silver badge 1 1 bronze badge. For one thing, the angular momentum is not aligned with the angular velocity.

You can see its rotation axis, but if the satellite masw onto the object, it will absorb the angular momentum, and you'll find the satellite spinning on a different axis. Active Oldest Votes. They tell us nothing interesting about the object itself. Question feed. What is the physical significance of the off-diagonal moment of inertia matrix elements?

Clough the consistent mass of that tricubic macroelement was too complicated for hand derivations in

The state of that systems can be described by a vector q of two generalized coordinates, namely the positions of the two particles along the track. For consistent FE methods, you use Galerkin orthogonality; non-consistent methods instead use the Strang lemma first or second, depending on the nature of non-consistency.

Lagrangian mechanics yields an ordinary differential equation actually, a system of coupled differential equations that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system.

This formula can be written in matrix form as.

The null space can be extracted by functions such as NullSpace in Mathematica without the need of computing eigenvalues. The element is endowed with the two freedoms shown in Fig.

They are dictated by the physical matrlx. In this approach, a DLMM is augmented by a scaled version of the stiffness matrix. Two well known discretization methods, off diagonal mass matrix example in FEM textbooks since the late s, lead to diagonally lumped and consistent mass matrices, respectively. Aside from those special cases, it is difficult to draw general conclusions from a glance at Fig. These are the midlevel four pictured in Fig. For the case illustratedin in Fig. Given a Bar3 template instance that is, with all-numeric signature module Bar3DispersionPlotlisted in Fig.

Its roots can be traced back to the Free Formulation of Bergan [ 10 — 1329 ], in which the stiffness matrix was decomposed into basic and higher order components.

So, off-diagonal elements in your MOI are equivalent to having a rotation axis that is not aligned with any of the principal axes.

Renard Y The singular dynamic method for constrained second order hyperbolic equations.

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These are algebraic forms that carry free parameters.

Eample state of that systems can be described by a vector q of two generalized coordinates, namely the positions of the two particles along the track. I've seen an explanation that justifies mass matrix lumping in the context of mechanics where this assumption implies that the mass of the system is concentrated at discrete points, but how does this generalize to more general elliptic PDE problems? Accept all cookies Customize settings. Hand, J. This formula can be written in matrix form as.

Viewed 1k times. Stack Overflow for Teams — Collaborate and share knowledge with a off diagonal mass matrix example katrix. For consistent FE methods, you use Galerkin orthogonality; non-consistent methods instead use the Strang lemma first or second, depending on the nature of non-consistency. It also says:. So one important thing to realize is that there is nothing fundamentally meaningful about off-diagonal elements; you can always rotate your coordinates to get rid of them. On the other hand, what if the body is rotating about an axis that isn't one of the principal axes?

Hot Network Questions. Improve this answer. Sign up to join this community. Ask Question. They tell us nothing interesting about the object itself.

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Sign up using Facebook. The VSF ansatz was obtained after some simplifying initial computations.

In both cases, the general template was considered.

The underlying idea, however, is older.

Accepted : 24 Exampld Kolsky H Stress waves in solids. The main focus is on adjusting the mass matrix. Three reasons may be adduced for the preference: The three-century astronomical heritage of Newtonian Mechanics. Prentice-Hall, Englewood Cliffs. An optional character string to be printed before the plot. Structural and Materials Panel Paper, Aachen.

Question feed. It also says: The physical significance of the off-diagonal components is that you're using off diagonal mass matrix example coordinate system not aligned with the principal diaagonal of the object. Sign up or log in Sign up using Google. That's the case when the initial axis of rotation is not principal axis of rotation and the products of inertia are different from zero they depend on the reference frame selected. Email Required, but never shown. The best answers are voted up and rise to the top.

Bar3 characteristic equation module. Ask Question. Hot Network Questions.

I've seen an explanation that justifies mass matrix lumping in the context of mechanics diagnoal this assumption implies that the mass of the system is concentrated at discrete points, but how does this generalize to more general elliptic PDE problems? Hot Network Questions. Linked Create a free Team What is Teams?

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As a result, diagonal mass matrices are produced. A historical account is provided in a tutorial chapter [ 36 ]. My question is what is the justification of this? A better name for would be stiffness-consistent mass matrix. Springer, London Book Google Scholar

Connect and share knowledge within a single location that is structured and easy to search. Stack Overflow for Teams — Collaborate diet shakes for weight loss whey ultra clean share knowledge with a private group. There are a number of justifications for mass lumping from a practical standpoint - getting the inverse of the mass matrix can be a huge overhead as problems increase in size, becoming completely unfeasible. Them, each element of mass off the axis of rotation is rotating instantly around the axis of rotation in a circular motion. That's what the last sentence of my answer says. Add a comment.

These are the midlevel four pictured in Fig. Template definition argument. The main focus is on adjusting the mass matrix. In fact they dominate all pre work.

For example, a lumped-mass method, in which the deformation of each element is ignored, creates a off diagonal mass matrix example mass matrix and negates the need to integrate mass across the deformed element. Nodes are referred to different coordinate frames in the global system. Hot Network Questions. Templates have the virtue of producing a set of mass matrices that satisfy certain a priori constraint conditions such as symmetry, nonnegativity, invariance and momentum conservation.

The element velocity interpolation is taken to be. AIAA J — If upon rearranging the element DOF so they are grouped node by node:. The function returns tsign as template signature.

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On the other hand, the construction of a DLMM is maes so clear cut, except for simple elements in which the lumping is uniquely defined by conservation and symmetry considerations. One particular form, however, is unique under conditions i - ii. Parlett BN The symmetric eigenvalue problem. Wimp J Sequence transformations and their applications. They are implemented in all general purpose FEM codes. Commun Numer Methods Eng — Stiffness templates for 2D and 3D structural elements have been considered since a modest beginning in the s, as Free Formulation plate elements that incorporated a scaling parameter for the higher order stiffness [ 1329 ].

To accommodate these variations, it is convenient to leave eexample latter parametrized as follows. The happy connection of mass templates to periodic materials may be of interest as FEM and related discretization methods are extended into multiscale applications of crystal, micro- and nano-mechanics, and phononics [ 722 ]. For the CMM, however, repetition is not obvious. What forbids off-diagonal elements in the kinetic terms of the Standard Model? A pair of degree-2, 3-point Gauss quadrature rules for the six-node plane stress triangle with constant metric: a node configuration, b the 3-interior point rule, c the 3-midpoint rule, which is a Lobatto rule for this node configuration. Int J Solids Struct —

The physical significance of the off-diagonal components is that you're using a coordinate system not aligned with the principal directions of the object. Asked 5 years, 1 month ago. Accept all cookies Customize settings. For example, a one-dimensional Helmholtz type equation with the form.

For example, consider a system consisting of two off diagonal mass matrix example masses confined to a straight track. Active 4 years, 10 months ago. I've recently finished an introductory course on the finite element method from a more mathematical perspective following Brenner and Scott and we were introduced to the finite element mass matrix in elliptic problems as the matrix arising from terms without a derivative. If the object has a symmetry axis, then that will be a principal axis.

Felippa CA A study of optimal membrane triangles with drilling freedoms.

They tell us nothing interesting about the object itself.

Felippa, C.

The element has non-translational freedoms; for example node rotations, or displacement derivatives occasionally the rule may work, but that should not be taken for granted.

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Ziman Matris Principles of the theory of solids. The subject is studied in detail in [ 46 ]. If there are rotational freedoms the results can be poor compared to customized templates. To study the performance of an infinite 2D regular lattice such as pictured in Fig. Help Learn to edit Community portal Recent changes Upload file.

This may not matrix example so for more complex elements shells etc. The tensor of moment of inertia contains six off-diagonal matrix exampe, which vanish if we choose a reference frame aligned with the principal axes of the rotating rigid body; the angular momentum vector is then parallel to the angular velocity. An off-diagonal element in the MOI means that if I apply a torque about a certain axis, the object will accelerate its rotation about a different axis. Hand, J. Stefan Bischof 1, 1 1 gold badge 18 18 silver badges 24 24 bronze badges.

On the other hand, what if the body is rotating about an axis that isn't one of the principal axes? Namespaces Article Talk. For example, imagine your object spinning inside a nicely symmetric little satellite in space.

Next consider the 2-node BE plane beam element.

Lagrangian mechanics yields an ordinary differential equation actually, a system of coupled differential equations that describes the evolution of a system in terms of an arbitrary vector of generalized coordinates that completely defines the position of every particle in the system.

Note: this level should be used with caution. Table 14 1D Newton—Cotes integration rules Full size table.

Marcel Dekker, New York. Befitting the overdesigned aircraft structures of the time, the focus is on dynamics and vibrations rather than on statics.

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The happy connection of mass templates to periodic materials may be of interest as FEM and related discretization methods are extended into multiscale applications of crystal, micro- and nano-mechanics, and phononics [ 722 ]. The general form is. However, for gauge field we often have that one or more of these symmetries is unbroken and we want to write it in a basis which makes this obvious. Immediate online access to all issues from Improve this question. Prentice-Hall, Upper Saddle River. The basis of "generations" is presumably precisely the basis in which the mass matrix is diagonal, which makes the question pointless.

The arguments are: numele : Number of elements in lattice. Chandrasekhar S Radiative transfer. Download references. Items in gmpars must be individual symbols, while those in tsign may be symbolic or numeric see examples below.