adams vibration theory

15Adams/Vibration Theory Manual

Adams/Vibration Theory Manual

Adams/Vibration

16

Introduction to Adams/Vibration

Using Adams/Vibration, you can compute system response in the frequency domain. You can perform

two types of analyses:

1. Normal-modes analysis

2. Forced response analysis

A normal modes analysis computes eigen-values and eigenvectors of your model at an operating point

you specify. This analysis is effective in understanding natural modes of vibration for the model and to

determine the basic dynamic characteristics of your model. Although the result of an eigenvalue analysis

is independent of specific excitation, they are useful in predicting the effects of applying dynamic loads

on your model.

Normal modes analysis is relevant in many scenarios. In one scenario you may need to assess dynamic

interaction between parts in an MD Adams model. For example, if you are designing a washing machine,

it is necessary to determine if the operating rotational frequency of the tub is close to one or more natural

frequencies of the supporting structure and electronic components. If they are then ordinary operation of

the washing machine may lead to damage of the supporting structure and/or premature failure of

electrical and electronic components in the machine.

If you are setting up a physical test, a normal-modes analysis is useful in determining the best location

on your systems to attach strain gauges and/or accelerometers. After the test, test results can be correlated

with the results of the normal-modes analysis.

Frequency-response analysis is an efficient method for finding the steady-state model response to

sinusoidal excitation. In this analysis the loading is in the form of a sine wave for which you specify the

frequency, amplitude, and phase. Adams/Vibration performs frequency response analysis using

linearized MD Adams models. Several different types of inputs can be applied to the model and force

and kinematic output measured.

This document describes the theory and modeling constructs used in Adams/Vibration. It is to be used as

a companion to the on-line product documentation. It is assumed that you are familiar with using the

product in its interactive or batch environments. If not, consult the on-line product documentation for

Adams/Vibration before reading this document.

The next section presents linearization theory for MD Adams model. This is followed by definition of

vibration modeling entities available in Adams/Vibration. Solutions types available in Adams/Vibration

are described in Section 4. The last section, Section 5, gives a list of references you may want to consult

for more details on the respective topics.


Linearization in Adams/VibrationNonlinear MD Adams models are represented by implicit equations of the form:

(0-1)

where:

• z = vector of states of the model.

• G = system of first-order differential and algebraic equations.

States of the model include:

• Displacement and velocity states from mass-bearing elements PART, POINT_MASS, and

FLEX_BODY

• First-order states introduced by the DIFF, LSE, GSE, and TFSISO modeling elements

• Forces variables due to force elements

Lagrange multipliers due to kinematic constraints (JOINT, JPRIM, MOTION, PTCV and CVCV) also

contribute to the vector of states.

This system of equations is linearized about an operating point z0. An operating point is defined by an

initial-conditions analysis or static or dynamic analysis [Sohoni, 1986; �� ]. Using a process of eliminating algebraic equations from the linearized representation of , ordinary differential

equations along with algebraic output equations of the following state-space form are obtained:

(0-2)

where:

• x = Vector of states of the linear model

• y = Outputs from the linear model

• u = Inputs to the linear model

• [A,B,C,D] = State matrices for the linear model

Specific details of the linearization process implemented in Adams/Solver (FORTRAN) are given by

Sohoni [1986]. �� have given details of linearization process implemented in Adams/Solver (C++).

States of the linear model, x in (0-2)are a subset of states of the nonlinear model, represented by z in (0-1).

When performing normal-modes analysis, specification of inputs and outputs is not required. Therefore,

the problem is reduced to finding a solution to the homogeneous equation:

(0-3)

Representing the solution to (0-3) as:

G z· z t, ,( ) 0=

x· Ax Bu+=

y Cx Du+=

x· Ax=

Adams/Vibration

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where:

• = ith eigenvector

• = ith eigenvalue

Substituting in (0-3) gives:

(0-4)

Equation (0-4) is the classic eigenvalue problem. This problem is solved in Adams/Vibration using the

well-known QR method [Press et. al., 1994] for computing eigen-values and eigen-vectors.

Note that Adams/Vibration always computes λ and ζ as complex quantities. The complete eigen-solution

to (0-3) can be expressed in the form [Ewins, 1995],

(0-5)

where:

• System mode shape matrix .

• the ith diagonal element of diagonal system eigenvalue matrix is

• , the ith natural frequency.

•

• * indicates the complex conjugate

Differences between proportional and general viscous damping – Adams/Vibration implements

general viscous damping. In these cases, damping matrix is no longer guaranty to be proportional to mass

x t( ) ζieλit

=

Aζi

λiζi

=

*

*

* *

0

0

ϒ=

=

Λ

Λ

Φ ΦQ

ΦΛ Φ Λ

λi

ωi

ξi

– i 1 ξi

( )2

–+ ≡ λ r

iiλ ii

+=

ωi

λ ri

( )2

λ ii

( )2

+=

ξi

λ riωi

⁄–=


and/or stiffness matrices. Following ways are generally referred in modal space to identify a presence

of non-proportional damping in the Adams/Vibration model:

• The real-valued modal matrix of the un-damped model no-more diagonals the damping

matrix.

• For fixed k, the phase is no-more constant for all i=l,...,n.

• The real and imaginary parts of the kth eigenvector are now linearly independent.

• The modal matrix cannot be normalized to become real-valued.

Forced-Response Analysis

Inputs and outputs to the linear model are defined by means of input and output channels in

Adams/Vibration. Input channels contribute to the B matrix. Output channels contribute to the C matrix.

The D matrix represents direct interaction between input and output channels. For more details, see

Modeling of Vibration Entities.

Using Laplace transform, (0-2) can be expressed in the form [De Silva, 2000]:

(0-6)

where:

• s = Laplace variable

From (0-6), transfer function in the state-space form is given as:

(0-7)

Transformation to Modal Space

Using the model eigenvectors as a basis for the solution,(0-6) is transformed to modal space using the

modal transformation:

(0-8)

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where:

• = Matrix of eigenvectors

• = Array of modal coordinates

The ith column of Ζ corresponds to eigenvector of the model.

Substituting from (0-8) in (0-6) gives:

(0-9)

Simplifying (0-9) gives:

(0-10)

where:

•

•

•

Therefore, the transfer function for the model in modal space is given as:

(0-11)

When you ask for the transfer function of a model to be computed, you specify the range of values for s

and the number of steps. At each step in the range of s, transfer function H(s) is computed from

expression in the right-hand side of (0-11).

Transfer function can be computed in two forms, that is, using (0-7) or (0-11). Solution of the transfer

function in the form of (0-7) is called the direct solution, while solution in the form (0-11) is called the

modal solution.

Adams/Vibration uses the following steps to compute the transfer function for the model:

Ζ

( )sq

iζ

Am Z1–AZ=

Bm Z1–B=

Cm CZ=


1. From the given input/output channel specification,

2. Compute state matrices in the state space forms as in (0-2).

3. Compute eigen-solution for the model using (0-4).

4. Transform to modal space using (0-10).

5. Compute transfer function from (0-11).

Due to degeneracy in the eigen-solution, for some models, Z may not be invertible. In such cases, the

transfer function is computed using the direct solution of (0-7). The modal solution for transfer function

computation is much faster than the direct solution. If the transfer function is computed using the direct

method, it will not be possible to plot model coordinates and participations or perform vibration

animation for the model. However, normal mode shapes can still be animated.

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Modeling of Vibration Entities

This section presents the formulation for modeling the following vibration entities:

a. Vibration Input Channels

b. Vibration Actuators

c. Vibration Output Channels

d. Frequency-dependent (FD) Modeling Elements

Vibration Input Channels

A vibration input channel defines the location, orientation, and type of forcing function to be applied.

There are three types of input channels you can specify in Adams/Vibration:

1. Force-type input channel applies a force at the specified marker. The expression for the force is

as specified by the vibration actuator.

2. User-specified state variable applies the vibration actuator to a state variable that you may have

created in your model. This input channel is useful in applying vibratory input to models that are

represented by general dynamical elements, such as GSE/LSE/TFSISO.

3. Kinematic input channel applies displacement, velocity, or acceleration input. This form of the

input channel imposes a kinematic constraint in the frequency domain at the specified marker.

This constraint will result in removal of one degree of freedom at the maker at which it is applied.

Vibration Actuators

A vibration actuator defines the magnitude and phase of the applied forcing function. Vibration actuators

are required for modeling forcing function in forced response analysis. Phase angle, as defined in a

vibration actuator, is with respect to the positive direction of the marker in the vibration input channel on

which this actuator is defined.

Vibration actuators are applied at the input channel after the model is linearized. Therefore, vibration

actuators are only in effect for frequency domain analysis and have no effect on the operation point

analysis for the model.

Swept Sine Actuator

Force applied in a swept-sine actuator is defined as:

Note: Due to this constraint, the frequency response peaks may be shifted from the frequencies

of the normal modes of the model.


where:

• f= force magnitude

• θ=phase angle

The phase angle θ is measured with respect to positive direction of the marker axis for the input channel

on which this actuator is acting.

Rotating Mass Actuator

Force Type

The force rotating mass actuator is represented by two actuators. The Leading and Lagging actuators are

represented as:

Figure 1 Rotating Mass Actuator

where:

• = Force applied in leading actuator

• = Force applied in lagging actuator

• m = Unbalanced mass

• r = Radial offset

• ω = Frequency of excitation

)(sleΡ

)(slaΡ

Adams/Vibration

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When you want to incorporate vibratory effects due to an unbalanced mass in your vibration analysis,

you must create two vibration actuators acting on the same input channel. The first actuator must be

defined as a Leading actuator and the second as the Lagging actuator.

Moment Type

The moment type of rotating mass actuator is applicable for simulating moments due to radial, as well as

axially offset unbalanced masses. The leading and lagging actuators are represented as:

Figure 2 Rotating Mass Actuator - Moment Type

where:

• Mle = Moment applied in the leading actuator

• Mla = Moment applied in the lagging actuator

• d = Axial offset between unbalanced masses

As with the force type rotating mass actuator, to incorporate vibratory moment effects due to an

unbalanced mass in your vibration analysis, you must create two vibration actuators acting on the same

input channel. The first actuator must be defined as a Leading actuator and the second as the Lagging

actuator.

PSD Actuator

The PSD actuator is for defining a spectrum that you want to apply to the model. For more information

on power spectral density, see Reference 4. The PSD actuator is specified using a spline. You can create

a spline using the Function Builder in Adams/View. In Adams/Vibration, you can specify the

autocorrelation PSD of an input channel to itself, as well as the cross correlation PSD to other input

channels. Section 4d provides details of how auto and cross correlations are used for computing PSD

response.


PSD type actuators cannot be mixed with swept sine, rotating mass, or user-type actuators in the same

vibration analysis. Response solution for PSD actuator is different from the force vibration response

analysis.

Vibration Output Channels

Vibration output channels are defined using run-time expressions. These expressions can consist of

kinematic or force expression or some combination of the state variables you define.

Frequency-dependent (FD) Modeling Elements

FD modeling elements are useful for incorporating compliant components with frequency-domain force

characteristics in your model. These force characteristics may have been measured using a bench test in

a laboratory. Several flavors of frequency-dependent modeling elements are available within

Adams/Vibration. From given force versus frequency characteristics, you can use system identification

tools, such as described in article 1-KB12433 (see MSC.Software Knowledge Base) for identifying

parameters needed for defining FD element.

The FD modeling element is represented by a combination of spring and dampers in series and parallel.

Each FD element adds two internal auxiliary states for each direction. These differential and output

equations are contained within customized GSE subroutines in Adams/Vibration.

A GFORCE element is used to apply forces on the I- and J-bodies between which you are applying the

FD element. When you add a 3D FD element to your model, Adams/Vibration automatically creates a

GSE and GFORCE element in your model and instantiates the associated customized user subroutines

for evaluation of the force in the FD element.

FD modeling elements are usable in frequency domain, as well as time domain analysis.

http://datanet.internal.mscsoftware.com/tools/siebel/frm_solutions_form.cfm?fuseaction=solutions&action=display&siebel_ID=1-KB12433&s_flag=on

Adams/Vibration

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Figure 3 Scematic of One-dimentional FD Element


Vibration Analyses

Adams/Vibration performs two types of vibration analyses:

• Normal Mode Analysis to compute eigen-values and eigenvectors for a model

• Forced Vibration Analysis to compute the forced response of the model to vibratory inputs

applied using vibration input channels and actuators on the model. Eigen-values and

eigenvectors are also available computed during the forced-response analysis.

Normal Mode Analysis

The general equation of motion for a MDOF system with general viscous damping and harmonic

excitation is:

(1)

This can be arranged in the form of equation (0-2) as a state-space representation. For normal mode

analysis, matrix A in equation (0-3) takes the form [Ewins, 1995],

(2)

Representing solution to (0-3) as ,

(3)

The solution to this equation constitutes a complex eigen-problem. In this case there are 2N eigen-values

and now those occur in complex conjugate pairs. This is an inevitable (unavoidable) result of the fact

that all coefficients in the matrices are real and thus any characteristic values, or roots, must either be

real or occur in complex conjugate pairs). As a result, eigen-vector corresponding to these eigen-values

also occurs as complex conjugates. Hence the eigen-solution needs to be described as:

(4)

Express each eigen-value in the form:

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(5)

Where, is ‘natural frequency’ and is the critical damping ratio for that mode. Sometimes this

natural frequency is referred to as ‘un-damped natural frequency’ but this is not strictly correct except in

the case of proportional damping (or, of a single degree of freedom system). In Adams/Vibration, the

general viscous damping is implemented which may not necessarily proportional, and therefore is

referred as ‘natural frequency’ instead ‘un-damped natural frequency’.

The pair of orthogonal equations is shown to be:

(6)

For complex conjugate pairs, we will get a very important result,

(7)

Where denotes the Hermitian (complex conjugate) transpose, from which we can obtain:

(8)

Here, (all real) may be described as modal mass, stiffness and damping parameters

respectively although the meaning is slightly different from those used in other systems (un-damped,

hysteretic damping).

Rewriting second equation of (6) in energy format,

ωr ξr

ωr

{ }H

mr krandcr,


(9)

This shows that the Kinetic modal energy equals Potential modal energy. It should be noted that with

little algebra it can be shown that the modal energies thus calculated are time averaged over one cycle.

Adams/Vibration computes eigen-values and eigenvectors as complex numbers. When you perform

normal-modes analysis in Adams/View, you can display the results in the form as shown in Table 1

EIGEN VALUES (Time = 0.000000)

FREQUENCY UNITS: Hz

Table 1 Eigenvalue Table Reported in Adams/View

Quantities presented in this table are computed in this table are computed as:

(10)

where:

• = Natural frequency, column 2 of table

• = Damping ratio, column 3 of table

• = Real part of the eigenvalue, column 4 of table

Mode Number

Undamped Natural Frequency

Damping Ratio Real

Imaginary

1 0.00617426 1 -0.00617426 0

2 16.6257 1 -16.6257 0

3 0.270866 0.23569 -0.0638404 +/-0.263235

4 0.712246 0.580427 -0.413406 +/-0.57999

ωr ξ λ r λ i

ωr

ξ

λ r

Adams/Vibration

30

• = Imaginary part of the eigenvalue, column 5 of table

Correspondence between the number of modes in the table and the number of states in the model is as

follows. The number of modes conforms to the following identity:

2*Nc + Nr = 2* Nm + Nd (11)

where:

• Nc = Number of complex eigen-values, that is, eigen-values with nonzero real and imaginary

parts

• Nr = Number of real eigen-values, that is, eigen-values with zero imaginary parts

• Nm = Number of mechanical degrees of freedom

• Nd = Number of states due to supplementary differential states contributed by modeling

elements, such as DIFF/LSE/GSE/TFSISO

For the three-DOF model (Nm =3, Nd =0) from which Table 1 is obtained, there are two complex (Nc

=2) and two real (Nr =2) eigen-values. Using these values, it can be verified that the identity in (11) holds.

Eigen-values can be plotted as a complex scatter on a real-imaginary axis, as shown in Figure 4. The real

part of eigen-values corresponds to damping in a model. The imaginary part corresponds to stiffness in

the model. A mode with an eigenvalue of negative real part is considered stable. Conversely, a mode with

eigen-values of positive real part is unstable. As shown in Figure 4, eigen-values lying in the left half

plane are stable and those in the right half plane are unstable.

λ i


Figure 4 Complex Scatter Plot

A model with all stable eigen-values, if perturbed, will return to its original unperturbed configuration

on removal of the perturbation. A perturbation applied to a model with unstable eigenvalue(s) may cause

unstable modes to be excited. In this case, the model will have an unbounded response to this

perturbation. For more information, see Reference 4.

Forced Vibration Analysis

To perform a forced-response analysis, Adams/Vibration requires you to construct a vibration analysis.

A vibration analysis consists of: a collection of input and output channels, the operating point

specification, the frequency range and number of steps in this range.

Adams/Vibration

32

Figure 5 Input and Output Channels

Each input channel results in an action-only SFORCE being created in the model along with a

VARIABLE. The VARIABLE is used to define the SFORCE. The expression for the variable is zero.

Each output channel results in one VARIABLE being created. Function expression for this entity depends

on the type of output channel you define. On defining a vibration analysis, the variable corresponding to

the input channels are collected into a PINPUT and variables corresponding to the output channels are

collected into a POUTPUT. The model is linearized using this PINPUT and POUTPUT pair. As shown

in Figure 5, each input channels contributes one column to the B matrix. Each output channel defines one

rows of the C matrix.

For output channels representing acceleration outputs, there will be nonzero entries in the corresponding

rows in the D matrix.

Forced vibration response is the response of a model to vibration input channels and is computed as:

(1)

where:

• = Vector of vibration actuators evaluated at si

• H(si)= Transfer function as defined in(0-11), evaluated at si

• n = Number of steps in the frequency range

Transfer Function

The transfer function plot is computed from (1) with unit inputs applied at all input channels. For this plot

the vibration actuator definition is ignored.

PSD Response Computation

For actuators of the PSD type, the PSD output is computed using the following:

( )isΡ


(2)

where:

• = Output PSD

• = Matrix input PSD

• = Transfer function

• = Hermitiam transpose of

For multiple PSD input channels:

(3)

where:

• Pk(s) = PSD function corresponding to PSD actuator k

• Pkj(s) = Cross-correlation of PSD actuator k due to PSD actuator j

When plotting PSD response, input channels with actuators of type PSD cannot be mixed with actuators

of force type in one vibration analysis.

Modal Coordinates

The plots of modal coordinates identify the response of coordinates corresponding to modes of the

model. They are computed as:

(4)

where:

• = Vector of modal coordinates for the model

( )out isP

( )in isP

( )isH

( )H

isH ( )isH

( )

1 12 1

21 2 2

1 2

...

...

... ... ... ...

...

n

n

in

n n n

P P P

P P Ps

P P P

=

P

( )c sM

Adams/Vibration

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At a given frequency, the modal coordinates identify how much each of the respective system modes is

excited. As shown in the Figure 6, the frequency response has three peaks.

Figure 6 Plot of Modal Coordinates

Modal coordinates for Modes 1, and 2 show that the first peak corresponds to the first peak in the modal

coordinate response for Mode 1. While at the second peak in the frequency response, Mode 2 is the one

with the most excitation.

Modal Participation

While the modal coordinates define how active a certain mode is at a given frequency, the modal

participation identifies how much given modes contribute to the response at a specified frequency.

Figure 7 Plot of Modal Participations


Modal participation is computed as:

(1)

where:

• = vector of modal participations

Modal Energy Computation

i. Kinetic energy distribution – Mass bearing modeling such as PART, POINT_MASS, and

FLEX_BODY contribute to kinetic energy distribution computation. The modal kinetic energy

distribution in a system is defined with respect to degree of freedom of various modeling elements

mentioned above.

From equation , the modal kinetic energy of ith mode can be calculated as

(2)

( )p sM

Adams/Vibration

36

where:

• ith column mode shape vector of system mode shape matrix

• = System mass matrix

To find modal energy contributions from various parts and degrees of freedom, it is convenient to

substructure this system level energy equation to part level equations. Finally, these sub-structured part

level energies will be summed to calculate the system level modal kinetic energy.

Kinetic energy contribution of PART j to individual mode i can therefore be written as,

(3)

where:

• ith column vector of matrix , which is condensed mode shape matrix from system

mode shape matrix , corresponding to six degrees of freedom of PART j

• = Mass matrix of PART j

Adams/Vibration breaks this modal kinetic energy contribution of PART j in nine components to present

it in modal energy tables:

(4)

(5)

iε = Φ

Μ

j

i =ζ jΨ

Φ

jΜ

*

1

*

2

*

3

1(1) (1)

2

1(2) (2)

2

1(3) (3)

2

j j j

i i

j j j

i i

j j j

i i

K m

K m

K m

=

=

=

ζ ζ

ζ ζ

ζ ζ

4

6

*

*

5

*

1(4) (4)

2

1(5) (5)

2

1(6) (6)

2

i i

i i

i i

j j j

xx

j j j

yy

j j j

zz

K I

K I

K I

=

=

=

ζ ζ

ζ ζ

ζ ζ


(6)

Equation (3) now equivalently expressed as,

(7)

Total kinetic energy for mode i can then be computed as,

(8)

where:

• = Total kinetic energy of mode i

• n = Number of parts in the model

• Kinetic energy contribution of PART j to mode i

Percentage distribution of PART j mode i kinetic energy in direction e is defined as:

(9)

Above equation shows that the percentage kinetic energy contribution of any degree of freedom to a

particular mode is independent on natural frequency of that mode and depends on the product of the mass

or inertia and square of the mode shape amplitude (mass and inertia terms) or product of mode shape

amplitudes (product of inertia terms) of the mode in that directions. As the sum of the kinetic energy

contributions of all degree of freedoms in that mode shape (not part degrees of freedom) is made equal

to 100, through the definition in equation , it is easy to identify the relative importance of various degrees

of freedoms. This will help to identify which particular degree of freedom is contributing more or less

to the overall modal kinetic energy.

Percentage kinetic energy contributed by PART j to mode i is defined as:

7

8

9

*

*

*

(4) (5)

(4) (6)

(5) (6)

i i

i i

i i

j j j

xy

j j j

xz

j j j

yz

K I

K I

K I

= −

= −

= −

ζ ζ

ζ ζ

ζ ζ

1

nj

i i

j

τ=

Γ =∑

iΓ

j

iτ =

Adams/Vibration

38

(10)

where:

• = Percentage kinetic energy contributed by PART j to mode i

Equation (10) shows that the percentage kinetic energy contribution of any part to a particular mode is

independent on natural frequency of that mode and depends on the sum of product of mass (or inertia)

and square of the mode shape amplitude (mass and inertia terms) or product of mode shape amplitudes

(product of inertia terms) of that mode in all nine directions. As the sum of the kinetic energy

contributions of all parts is made equal to 100, it is easy to identify the relative importance of various

parts over each other. This will help to identify which particular part of model is contributing more or

less to the overall modal kinetic energy.

It should be noted that the large kinetic energy contribution can be due to large mass (inertia) or motion

of that degree of freedom. Therefore, the modal kinetic energy density needs to be calculated to remove

this ambiguity and make it dependent only on mode shape. The modal kinetic energy density for a given

mode is defined as the ratio of modal kinetic energy contribution of a degree of freedom to the rigid body

mass of the part.

Sometimes it is advantageous to normalize the modal kinetic energy contributions with respect to

generalized mass of that mode. This will make these modal kinetic energy distributions independent of

mode shape normalization.

Adams/Vibration computes the kinetic, strain, and dissipative energy distribution within modes of your

model. Table 1 shows an example of these distribution tables. These distributions are important

indicators of which components of the model have the greatest contribution to a given mode.

As in this example, PART_2 has the greatest contribution to the kinetic energy in this mode. The Strain

energy tables indicate that SPRING_2 stores the most amount of strain energy in this mode. Similarly,

from the dissipative energy table, SPRING_2 dissipates the most amount of energy in this mode.

Table 1 Modal Energy Distribution

j

iT


ii. Strain energy distribution - Compliant components, such as SPRINGDAMPER, SFORCE,

BUSHING, and so on, contribute to the computation of strain energy in a mode. In addition,

FLEX_BODYs in the model contribute to strain energy computation.

Strain energy due to a compliant element p is:

(11)

where:

• = Strain energy in complaint element p for mode i

• = Matrix of size equal to model stiffness matrix, with all zero entries except stiffness matrix

entries contributed by compliant element p.

Percentage distribution of strain energy for mode i is given as:

ipσ

pΚ

Adams/Vibration

40

; p=1, …,m (12)

where:

• = Percentage strain energy contribution by compliant element p to mode i

• m = Number of compliant elements in the model

iii. Dissipative energy distribution is computed in a similar manner to the strain energy distribution. In

(11) and (12), the damping matrix is substituted for the stiffness matrix.

Stress Recovery

With Adams/Vibration you can recover stresses and strains on flexible bodies. Recovering stresses on

flexible bodies is called Modal Stress Recovery (MSR). Adams/Vibration allows you to export modal-

coordinates corresponding to flexible body modes that can later be used as input to FEM software like

NASTRAN to recover the stresses in the flexbody.

The linear deformation (y) of the flexbody nodes can be approximated as a linear combination of M

orthogonalized mode shape vectors ( ) of that flexbody.

(13)

The basic advantage of the modal superposition is that the deformation behavior of a component with a

very large number of nodal DOF can be captured with a much smaller number of modal DOF.

Adams/Vibration exports the modal coordinates ( ) in a Modal Deformation File (MDF) format

(binary). FEM software like NASTRAN then does the summation to get linear deformation of flexbody

nodes and then recovers stresses and strains as follows,

(14)

where:

• is the strain vector

• is the stress vector

• B is a function matrix of the FE geometry relating strains to displacements

∑=

×=

m

p

ip

ip

ipS

1

100

σ

σ

ipS

ΨΨΨΨ

q

γγγγ

σ


• E is the stress-strain relationship matrix.

Note that it is necessary to create a super-element of flexbody without any constraints (free-free

boundary condition). The restart run is necessary in NASTRAN to recover the stresses and strains.

Adams/Vibration

42

Adams/Vibration References1. De Silva, C.W., Vibration: Fundamentals and Practice, CRC Press, Boca Raton, (2000).

2. D. J. Ewins., Modal Testing: Theory and Practice, John Willey and Sons Inc., New York, (1995).

3. Lyon, R.H., and Dejong, R.G. Theory and applications of Statistical Energy Analysis (second

edition), Butterworth-Heinemann, London, (1995).

4. Munjal, M. L., Fundamentals of vibration (Chapter 2), In Workshop on Noise, Vibration and

Harshness (NVH) for Automotive Engineering, SAE Western section, ARAI, Pune-India, (2002).

5. Negrut, D and Ortiz, J.L., On An Approach For The Linearization of The Differential Algebraic

Equations of MultiBody Dynamics, ASME/IEEE International Conference on Mechatronic and

Embedded Systems and Applications, (2005).

6. Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. "QR Decomposition."

§2.10 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge,

England: Cambridge University Press, pp. 91-95, (1992).

7. Sohoni, V.N and Whitesell, J., Automatic Linearization of Constrained Dynamical Systems,

ASME J, of Mechanisms, Transmissions and Automation in Design, Vol.108, No. 3, pp. 300-304,

(1986).

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