Dynamic Analysis Of Multi Degrees Of Freedom English Language Essay

Earthquake excitement, in contrast with other kineticss excitements, is applied as support gesture. The analysis of this support gesture can non be given by an exact solution, as it varies, depending to the seismal activity in the geographical country of the construction. This fact increases the trouble in the anticipation of the structural response to temblors. For these grounds, to accomplice a seismal analysis it is needed to take under consideration the previews incidents of the examined country that will supply the necessary informations to order the input seismal gesture. [ 2 ] [ 3 ]

The computation of matching emphasiss and warp, for a individual grade of freedom system theoretical account, could be achieved with an analytical solution. Because the bulk of constructions could be modeled as multiple grades of freedom systems, the solution can be established with the usage of numerical methods, or with Finite component methods.

In this paper a dynamic seismal analysis for model edifices is presented, followed by employment of finite component method. The models will be assumed as rubber bands.

Dynamic Analysis of multi grades of freedom constructions

Eigenvalues Analysis

Eigenvalue analysis helps to change over a multiple grades of freedom system to a amount of individual grade of freedom systems. By utilizing in structural analysis the response spectra of each SDOF and the part of each manner, a really good estimate of floor ‘s supplanting and reactions could be achieved.

Eigenvalue analysis is indispensable to near clip depended jobs of dynamic lading on constructions, such as the non – periodic burden of an temblor excitement. Because of the nature of dynamic jobs, there is more than one reply to their solution. For each system the form of the undamped free quiver could assist depict the behaviour and the response of the construction during a periodic or for a non – periodic burden. The solution of characteristic root of a square matrix job gives the natural frequences ( I» = I‰2 ) and manners of a system.

To find the response of a edifice during an temblor excitement two methods of analysis, based in characteristic root of a square matrixs theory, are by and large used ; response spectrum analysis and response history analysis. [ 4 ]

Mode superposition method

The Mode superposition method is the most effectual and broad used analysis to measure and cipher the response of additive constructions to an temblor excitement. This method simplifies the big set of planetary equilibrium equations to a comparative little figure of uncoupled 2nd order differential equations. This action ends up to an easier and simpler to manage system of equations. The chief advantage of this method is that ends up work outing the free quivers mode forms of uncoupled equations of gesture. The new variables of the uncoupled equations are the average co-ordinates. The solution derives after work outing these average equations independently. A superposition of average co-ordinates gives solution of the original equations.

As an temblor excitement seems to impact merely the lower frequences of a edifice, this method neglects the higher frequences and manner forms, without presenting any important mistakes in the concluding solution [ 5 ]

Response Spectra Analysis

Response spectra analysis uses the rules of Mode Superposition Method, holding as consequence to cut down the dynamic job of temblor excitement to a series of inactive analyses. By utilizing a specific land gesture burden, a complete clip history response of joint supplantings and member forces are calculated. This process can be used merely for linearly elastic analysis. The most of import advantage of RSA is that merely the maximal values of supplantings and members forces of each manner are involved in the solution and this makes the method suitable for the computational analysis of the temblor excitement job. It is needed to be mentioned that response spectra is non accurate for the analysis of no-linear multi grades of freedom constructions. [ 6 ]

By utilizing this attack the clip depended variables are separated from others. So, supplanting, speed and acceleration can be expressed as:

( 1.1 )

( 1.2 )

( 1.3 )

Where I¦ is a ‘n ten N ‘ matrix holding ‘n ‘ spacial vectors which are non variables of clip, and Y ( T ) , are the vectors incorporating variables of clip.

To accomplish a solution for this job the infinite maps need to fulfill the undermentioned mass and stiffness perpendicularity conditions:

( 1.4 )

( 1.5 )

where ‘I ‘ is a diagonal unit matrix and I©2 is besides a diagonal matrix, holding as diagonal footings the. For simpleness it is assumed that vector I¦n is ever normalized, that leads the generalised mass to be equal to integrity ( .

The equation of gesture is given as:

( 1.6 )

By utilizing the above equations, after pre-multiplication by:

a‡’ ( 1.7 )

Where is the average engagement factor for burden map J.

Even though for existent constructions the ‘n x N ‘ matric vitamin D is non diagonal, to decouple the modal equations it is necessary to presume that there is no matching between manners and vitamin D matrix is diagonal, holding as diagonal footings:

( 1.8 )

For a three dimensional seismic job, the typical average equation is:

( 1.9 )

With mode engagement factors:

( 1.10 )

The computation of maximal supplantings and peak forces must follow, for all waies of this 3-D job. Then, it is needed to cipher the response of the system holding the maximal responses of the three constituents present at the same clip.

For one dimensional analysis Eq. ( 1.9 ) takes the signifier:

( 1.11 )

The troubles in the usage of response spectra analysis are refering the big sum of end product information produced which are needed for the complete design checking of the construction, as a map of clip. Besides, reiterating this analysis for several different temblor excitements is needed, to guarantee that all important manners are excited. [ 7 ]

Displacement response spectrum

With the premise that and for a specific land gesture, equation ( 1.11 ) can be solved for different values of I‰ . Using these values, a curve of the maximal peak response can be plotted. This curve represents the supplanting response spectrum.

Figure 1.1: Relative Displacement Spectrum – inches, for Loma Prieta temblor [ 8 ]

Pseudo-velocity and pseudo-acceleration spectrum.

Using the same variables as earlier, plotting and give the pseudo-velocity and the pseudo-acceleration spectrum severally.

Even though the importance of the supplanting response spectrum, the pseudo-velocity and the pseudo-acceleration spectrum is limited, as for their physical values, there is a mathematical relationship that could take us from the pseudo-acceleration spectrum to the entire acceleration spectrum as follows:

( 1.12 )

For an undamped system, utilizing this equation the entire acceleration is equal to. To avoid that, the supplanting response spectrum curve is non plotted as average supplanting versus I‰ , but it is presented in footings of S ( I‰ ) versus a period T in seconds, holding S ( I‰ ) as:

( 1.13 )

( 1.14 )

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Figure 1.2: Pseudo-acceleration spectrum – per centum of gravitation, for Loma Prieta temblor [ 9 ]

These response spectrum curves characterize an temblor at a specific site, so they are non maps of the belongingss of the construction. Choosing the appropriate additive syrupy muffling belongingss of the structural system, a specific response spectrum curve is selected.

Solving ( Eq.13 ) for, the maximal average response for a typical manner N with Tn and matching spectrum response is:

and the maximal average supplanting response is:

Modal combination

For the computation of the peak values of supplanting or forces of a construction, under temblors ‘ excitement burden, the average combination is required. This can be achieved by the undermentioned methods:

Sum of the absolute values of average response.

This is the most conservative method for the computation of the peak values of supplanting or force of a construction under temblor excitement. In this method it is assumed that the maximal modal values, for all manners, occur at the same clip.

Square Root of the Sum of the Squares ( SRSS ) .

In SRSS method it is assumed all of the maximal modal values are statistically independent. For a tree dimensional construction this standard is non valid, as a big figure of frequences are about indistinguishable. [ 10 ]

Complete Quadratic Combination ( CQC ) .

CQC method is based on random quiver theories and it is used in most of the modern computing machine seismal analysis plans. The peak value of a typical force is estimated from the maximal modal values by utilizing the dual summing up equation that follows.

is the average force for manner N.

This average summing up includes all manners. With a similar manner they could be derived the extremum values of supplantings, comparative supplantings, base shear and turn overing minutes.

is the alleged cross-modal coefficient, and for CQC method with changeless damping is given as:

Having as R:

The cross-modal coefficient array is symmetric and all footings are positive.

From these three types of modal combination, CQC is the 1 minimizes the debut of mistakes to this job ‘s solution. [ 11 ] [ 12 ]

Response history analysis

Response history analysis is the analysis of a construction for a specific temblor excitement, by numerical integrating of the equation of gesture. In order to execute response history analysis, it is necessary to hold a digitized land gesture acceleration record. The chief characteristic that makes this process utile is that allows the computation of solutions of the deflected form and force province of the construction at each instant clip during the temblor.

As the consequences of this type of analysis are merely for the reaction of the construction during a specific temblor excitement, to utilize the consequence of forces and supplanting for design it is required the lower limit of at least three temblor records to be analyzed. Having the lower limit of informations required, merely the maximal forces and supplanting from each instance will be used for the design analysis. If the existed records are above seven, so the mean forces and supplanting could be used.

For the design analysis additive response history analysis is used seldom, as response spectrum analysis is more common. [ 13 ]

RHA process

For the response history analysis method the supplanting u derives from the superposition of the average part:

The force distribution can be calculated as the summing up of average inactiveness force distribution SR as:


So, the part of the n-th manner to the excitement vector s = m I? is independent of the standardization of the manner and it is given from the undermentioned equation as:

The equation of gesture is now given as:

( 1.26 )

The solution derives from the above equation, work outing for a individual grade of freedom job, with the belongingss of the n-th manner of the multiple grade of freedom system. By re- composing equation ( 26 ) and replacing u with Dn:

( 1.27 )


= ( 1.28 )

Modal responses

The n-th manners ‘ part to the supplanting is

( 1.29 )

By executing inactive analysis process, the tantamount inactive forces are calculated as:

( 1.30 )


There are two factors that determine the tantamount inactive force, the n-th manner part to the spacial distribution mI? of peff ( T ) and the pseudo-acceleration response of the n-th manner SDF system to the land acceleration.

Generalizing, the n-th manner part to any response measure is calculated by utilizing inactive analysis of the construction, for external burden forces fn ( T ) .

The equation for this part is:

( 1.32 )

The inactive supplanting due to forces tin is:

By utilizing combining weight. 2,

Finally, to happen the entire response of the construction, during an temblor excitement, the summing up of each of the response measures for all manners must be calculated.

The modal supplantings are:

Design Spectrum

The Design Spectrum should be representative of the land gestures that have been recorded during past temblor excitements. If there are no past temblor informations for the site, design spectrum should be calculated based to anchor gestures of similar conditions sites.

Design spectrum is calculated as an norm of many temblors response. A typical design spectrum is shown in the undermentioned figure:

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Figure 1.3: Typical Design Spectrum [ 14 ]

For major constructions normally is developed a site-depended design spectrum, to include the consequence of local dirt conditions and distance to the nearest mistakes, as the Uniform Building Code has defined specific equations for each scope of the spectrum for four different dirt types.

Chapter 2

Elastic model constructing under temblor excitement

The system that will be examined is an n-story model edifice with the same tallness for every narrative ( H ) and bay equal to L. It is assumed that the flexural rigidness of all beams is changeless ( E pound ) and the column rigidness ( E lc ) does non vary with tallness. All floor are assumed to hold the same mass ( m ) . Finally, the muffling ration of the n natural quiver manners is changeless. Besides the premise of lumped mass for each floor has been made, to cut down the grades of freedom of the edifice. The study of the pre-mentioned system follows:

Figure 2.1: Model of a 20 narrative edifice with 2880 DOF and of framework lumped-mass edifice [ 15 ]

There are two cardinal parametric quantities that help in the apprehension of the response of a additive model edifice to an temblor excitement, the beam-to-column stiffness ratio ( I? ) and the cardinal natural quiver period ( T1 ) . By specifying these two parametric quantities the system can be defined wholly. [ 16 ]

Beam-to-column stiffness ratio ( I? )

This parametric quantity is based on the belongingss of the beams and columns in the narrative closest to the mid-height of the edifice. The value of I? derives from the equation that follows:

Pound: length of beams

Lc: length of columns

This summing up involves merely the beams and the columns from the mid-height narrative of the construction.

Beam-to-column stiffness ration indicates how this system behaves during the excitement of an temblor. In the instance that I?=0 the beams does non enforce any restrain on joint rotary motion and the behaviour of the frame is the same as of a flexural beam. If I?=a?z , there is no joint rotary motion, as it is non allowed any from the beams, and the construction behaves as a shear beam with dual curvature bending of the columns in each floor. Finally, for an intermediate value of I? beams and columns manage to undergo flexing distortion with joint rotary motion. Normally, for an temblor immune construction, I? should take a value that insures that the stiffness of the columns is greater than the stiffness of the beams.

The basic parametric quantities of the frame that are related to the beam-to-column ratio are, the cardinal natural period, the comparative intimacy of the natural periods and the form of the natural manners [ 17 ]

Beam-to-column stiffness ratio and cardinal natural period

A diagram that shows the fluctuation of the cardinal period compared to the fluctuation of I? for a five-story model edifice follows.

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Figure 2.2: Cardinal natural quiver period of unvarying five-story frame [ 18 ]

For changeless column stiffness and floor mass, the value of the cardinal natural period is quickly diminishing, as I? additions from 0 to a?z .

Beam-to-column stiffness ratio and natural quiver period ratios

Even though the ratios of the natural periods do non depend on the cardinal natural period, they strongly depend on I? . This consequence is even more noticeable on the higher manner periods. In the undermentioned diagram can be observed that the values of the natural periods of a edifice with little I? are more detached from each other, in comparing with those with larger I? .

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Diagram 2.3: Natrural quiver period ratios of a five-story edifice. [ 19 ]

Beam-to-column stiffness ratio and forms of the natural quiver manners

The values of I? have a important impact on the form of the natural manners. In figure ten are presented the form of the natural quiver manners of a unvarying five-story frame, for I?=0, I?=1/8 and r= .

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Figure 2.4: Natural quiver manners of a five-story edifice [ 20 ]

Cardinal natural quiver period ( T1 )

Cardinal natural quiver period and beam-to-column stiffness ratio and their influence on construction ‘s response.

In the undermentioned diagrams are presented the normalized response measures of a five-story edifice to the cardinal natural period T1, for different values of beam-to-column stiffness ratio. The normalized measures are:

, with the supplanting of the top floor ( five-story edifice ) and the extremum land supplanting

, with the base shear and and is the effectual modal mass for the first manner.

, with the base turn overing minute and the effectual modal tallness of the first manner

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Figure 2.5: Normalized response measures [ 21 ]

From the above diagrams it is noticeable that the consequence of I? on the top floor supplanting is about absent. If the value of the period T1 is high, the top floor supplanting is indistinguishable to the land supplanting, as the floor masses remain stationary, while the temblor occurs.

Normalized shear diagram has the same signifier as the pseudo-acceleration spectrum. In the same manner for little values of T1 the curve goes to 0.5g and for larger values tends to make nothing. There is a fluctuation with I? to the diagram of the normalized overturning minute, but merely for the higher values of T1. Normalized base shear and turn overing minute diagrams are non stand foring for the fluctuation of base shear and turn overing minutes, as and are depending on I? .

Design Spectrum of linearly Elastic Building

The design spectrum analysis belongingss have been introduced to the prevues chapter. A figure of the design spectrum to land gesture of a frame with specified T1 and I? , derived by utilizing the response spectrum analysis follows.


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Figure 2.6: Design spectrum for land gestures [ 22 ]

Modal responses – Modal part factors

To cipher the peak value of the nth-mode part to a response measure r the undermentioned equation is used:

An: ordinate of the pseudo-acceleration response or design spectrum of the nth-mode for natural period Tn and muffling ratio I¶n.

: Modal inactive response



Specifically, the average inactive responses for the base shear, the top-story shear, the base turn overing minute and for the top floor supplanting are:

( 2.6 )

( 2.7 )

( 2.8 )

The average part factors are dimensionless and it does non depend in the method of manners standardization. The summing up the of all factor is equal to one.

Beam-to-column stiffness ratio and its influence on the higher manner response.

For the apprehension of the influence of I? and T1 on the higher manner response, they are plotted in the undermentioned diagram the base shear for all five manners, of a five narrative frame, and the base shear calculated merely from the first manner, for three different values of I? . Besides, in the undermentioned diagrams one can detect that the importance of each of the response measures of the higher manners is limited, in comparing with the significance of the first manners for I?=a?z ( frame behaves like a shear beam ) . This fact changes, as I? is reduced and the response measures of the higher manners have their greatest influence when I?=0 ( frame behaves like a flexural beam ) . The diminishing tendency of I? seems to impact to the opposite manner the values of the higher manners contribution factors for the base shear and the top floor shear particularly for the 2nd manner. Besides, the ratios of T1/Tn additions as I? lessenings. For that ground, the values of Tn are spread out over a wider period scope of the design spectrum. Response measures tend to hold different reaction in the alteration of I? on the higher manner response. For a decreasing I? , the top floor ‘s supplanting higher manners contribution diminishing and the opposite reaction follow for the forces. It needs to be noticed that the little values of the supplantings for the higher manners cut down their importance to this analysis.

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Figure2.7: Higher node response in Vb, V5, Mb and u5 for unvarying five-story frame for three values of I? .

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Figure 2.8: Normalized base shear in unvarying five narrative frame for three values of I? , utilizing RSA, including one or five manners. [ 23 ]

Cardinal natural quiver period and its influence on the higher manner response.

From the above diagram ( figure 2.8 ) derives the decision that the one-mode curves are independent of I? and indistinguishable to the design spectrum. For the higher mode response of a frame and for the acceleration sensitive part of the spectrum, T1 the influence is non important, but it is more effectual, for higher values of T1 in the speed and supplanting sensitive part. This is a consequence of three factors. To get down with, the inactive value of remains the same in all manners. Besides, for a fixed I? the average part factor does non depend on T1. Finally, the lone factor that has a dependance on T1 is the pseudo-acceleration spectrum ordinate, but it is besides depended on period ration Tn/T1, which for a certain value of I? , it is independent from T1. For these grounds, the design spectrum defines the influence of T1 in higher-mode response Figure 2.9.

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Figure 2.9: Natural quiver periods and spectral ordinates [ 24 ]

If the values of T1 and I? remain fixed, there is a difference between the values of the response part of the higher manners, depending to the response measure. To be more specific, the higher-mode response is more important for forces than for supplantings ( for shear frame utop and Mb are indistinguishable, as they are indistinguishable their average part factors ) . Besides, it is more of import for base shear in comparing with base turn overing minute. Finally, the higher-mode response is more important for top-story than for base shear.

To reason, one of the more of import issues in an temblor analysis is the figure of mode one should include. If an exact analysis is to be made, all natural manners should be included. Otherwise, the first few manners are adequate to deduce an accurate consequence for the response of a construction during an temblor excitement. The figure of the manners that should be considered can be decided taking under consideration two of import factors ; The average part factor and the spectral ordinate An.

If merely J manners are included to an temblor analysis, the mistake in inactive solution is given from the equation:

The figure of included manners should be chosen carefully, so that the mistake is little.

By and large, it is needed to include more manners for an accurate analysis for flexural frames, than the manners needed for shear frames.

Chapter 3

Elastic model constructing under temblor excitement and finite component analysis

To deduce the equation of gesture for a frame construction utilizing the finite component method, it is needed to continue utilizing the undermentioned stairss:

Idealize the model edifice:

A frame construction can be idealized as an gathering of beams, columns and walls. The construction will be divided in to E finite component, interconnected merely at the nodal points. The sizes of the elements are arbitrary ; all the elements could hold the same size, or different. For this idealisation it is assumed that the beam and the columns of this construction are one dimensional.

Specify the grades of freedom at the nodes:

The supplantings of the nodes are the grades of freedom. Generally, in a planar two dimensional frame there are three grades of freedom for each node ; The axial distortion in the ten way ( u ) , the warp in the y way ( V ) and the rotary motion in the x-y plane and with regard to the z-axis, I?z. In the amount, each component with two nodes will hold a sum of six grades of freedom. For the instance of a three dimensional frame there are six grades of freedom for each node, three interlingual rendition in the ( x, Y, z constituents ) and three rotary motion ( about the ten, Y and omega axes ) . If it is considered merely a planar supplanting, each node will hold merely two grades of freedom, the transverse supplanting and rotary motion.

The axial distortion of beams is normally neglected in the analysis of edifices. The axial distortion of columns is besides ignored for low-rise edifices.

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Figure 3.1: DOF of a two-dimensional 2-D component: [ 25 ]

Elementss Matrixs and external force constituents

As first measure for this process one has to cipher the stiffness ( ke ) and the mass matrices ( me ) for each finite component and the applied force vector to this elements ( pe ( T ) ) , with mention to the DOF for the component. The force vector is clip depended. The process to measure the equations for the force – supplanting and the inactiveness force – acceleration dealingss follows:

By and large, the external forces on the stiffness constituent of the construction are related to the ensuing displacement uj. If the examined system is additive this relationship derives utilizing the method of superposition and the construct of stiffness influence coefficients.

A unit supplanting is applied along DOF J, while all other supplantings are maintained at nothing. To accomplish that it is needed to use forces along all grades of freedom of the construction.

The stiffness influence coefficient kij is the force applied to DOF I due to unit supplanting at DOF J.

Figure twenty shows the stiffness coefficients for u1=1 and for u4=1 for a two narrative, two bay planar two dimensional frame construction.

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Figure 3.2 [ 26 ]

Using the method of superposition, the force fsi at DOF I and the relation with the supplantings uj ( j=1 aˆ¦ N ) is given from the equation:

( 3.1 )

This equation can be written in a matrix formation:

( 3.2 )


( 3.3 )

For the finite component analysis, the stiffness constituent of the external force for each component is given as:

( 3.4 )

For a beam component the supplanting is clip depended and it is given as:

Where N is the grades of freedom of the component, map is the insertion map that defines the supplanting of the component due to unit supplanting while all other DOFs are zero. For two-dimensional supplanting of a beam component and ( four DOF ‘s ) needs to verify the boundary conditions that follows:

For one = 1:

For one = 2:


For one = 3:

For one = 4:

At Figure 3.3 are presented the grades of freedom for a beam component and at figure 3.4 the form of the insertion maps

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Figure 3.3: DOFs of a beam component, sing two-dimensional supplanting [ 27 ]

The form of could be anything that would fulfill the boundary conditions. It is really hard to deduce the exact deflected forms of a beam component if the flexural rigidness varies through the component ‘s length. To avoid this trouble the shear distortion could be neglected, so the equilibrium equation for a beam that is loaded merely at its terminals is:

From this equation derives a three-dimensional multinomial that describes the supplanting of the saloon.

The changeless variable could be calculated for each of the boundaries conditions and obtain the undermentioned equations for the form maps:

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Figure 3.4: Interpolation maps of a beam component, sing two-dimensional supplanting [ 28 ]

These equations for the insertion are valuable for the preparation of the component matrices for no-uniform elements. This attack is suited for one dimensional beam jobs, but non for two or three dimensional finite elements. For that ground the finite component process is based on false relationships between the supplantings at interior points of the component and the supplanting at the nodes. This attack helps to the simplification of the analysis, but it introduces estimates in the solution.

For a beam component of length L that its flexural rigidness is E I ( ten ) , the stiffness influence coefficient kij is the force in DOF I due to unit supplanting in DOF J. With the usage of the principal of practical supplanting kij can be expressed as:

It is obvious that the symmetric signifier of this equation will take to a diagonal symmetric stiffness matrix. In this equation flexural rigidness could be changed through the length of the beam. This does non go on to the insertion map I? , that it is exact merely for unvarying elements. This fact can present mistakes to the solution. These mistakes could be reduced at any coveted degree by utilizing a finer finite component mesh. For the elastic model constructing a unvarying finite component is assumed ( ) .

Muffling forces

The theoretical account of the energy dissipation of a vibrating construction can be idealized by tantamount syrupy damping. Using this premise it is possible to associate the external forces ( fDj ) moving on the muffling constituent of the construction to the speeds ( ) . A unit speed is assumed along DOF J, holding all the other speeds in the other DOF equal to zero. These speeds will bring forth internal damping forces that resist the speeds. As the equilibrium of the forces should maintained, external forces are needed. The muffling influence coefficient cij is the external force in DOF I due to unit speed in DOF J. Using the method of superposition, the force fDi at DOF I and the relation with the speeds ( j=1 aˆ¦ N ) is given from the equation:

( 3.12 )

This equation can be written in a matrix formation:

( 3.13 )


( 3.14 )

For the finite component analysis, the muffling constituent of the external force for each component is given as:

( 3.15 )

By and large, the coefficients cij are non calculated straight from the dimensions of the construction and the sizes of the structural elements, as it is a really hard procedure. For this ground, the damping is specified by numerical values for the damping ratio, based on experimental informations for similar constructions.

Inertia Forces:

The inactiveness forces fIj, moving on the mass constituent of the construction, are related to the acceleration. Using the same process as old, a unit acceleration is applied to the DOF J, and all accelerations of the other DOF are staying nothing. As a consequence of this action, the fabricated inactiveness forces oppose these accelerations ( D ‘ Alembert ‘s rule ) . To maintain the force equilibrium it is necessary to hold external forces opposites to the inactiveness forces. The mass influence coefficient mij, is the external force that is applied in DOF I due to unit acceleration along DOF J. Using the method of superposition once more, the force fIi at DOF I and the relation with the acceleration ( j=1 aˆ¦ N ) is given from the equation:

( 3.16 )

This equation can be written in a matrix formation:

( 3.17 )


( 3.18 )

m is the mass matrix and it is symmetric ( = )

For the finite component analysis, the mass constituent of the external force for each component is given as:

( 3.19 )

The mass influence coefficient mij for a construction is the force in the I DOF due to unit acceleration in the J DOF. For a beam component utilizing the rule of practical supplanting the undermentioned equation derives:

( 3.20 )

As old with equation for kij, the symmetric signifier of this equation will take to a symmetric mass matrix ( ) . Having the premise of a lumped mass component, where the mass is distributed as point multitudes along the translational DOF at terminals and the two multitudes are calculated utilizing inactive analysis of the beam, the lumped mass matrix takes the signifier:

( 3.21 )

All the off diagonal footings of the lumped mass matrix are zero,

The external force P ( T ) is now expressed to be distributed among the three constituents of the frame and can be calculated for each component as the summing up of the stiffness constituent, the muffling constituent and the mass constituent

Form the transmutation matrix ( ae ) that relates the component supplantings ( ue ) and forces ( pe ) to the supplanting ( u ) and forces ( P ) for the finite element gathering:

and ( 3.22 )

In these equation represents a Boolean matrix dwelling of nothing and 1s. Its necessity is for turn uping the elements of and matrices at the proper locations in the planetary stiffness matrix, mass matrix and applied force vector severally. This transmutation matrix relives as from the duty to transport out the transmutations, and to transform the component stiffness and mass matrices and applied force vector to the nodal supplanting for the gathering.

Start with the calm of the component matrices to find the stiffness and mass matrices and the applied force vector for the gathering of finite elements:

( 3.23 ) ( 3.24 ) ( 3.25 )

The operator A denotes the “ direct assembly ” process for piecing harmonizing to matrix the elements of the stiffness and mass matrices and the elements of the force vector, for each component from vitamin E = 1 to e = N, where N is the entire Numberss of the elements of the construction and are assembled, in to the planetary matrices.

Formulate the equation of gesture for the prescribed finite component gathering:

( 3.26 )

To reason, finite component analysis of a model elastic construction has the same solving process with the supplanting method of analysis for the same construction, holding as a difference the preparation of component stiffness and mass matrix.

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