Proceedings of the IMAC-XXVIII
February 1–4, 2010, Jacksonville, Florida USA
©2010 Society for Experimental Mechanics Inc.
Substructuring using Impulse Response Functions for Impact Analysis
Daniel J. Rixen
Delft University of Technology, Faculty of Mechanics, Maritime and Material Engineering
Department of Precision and Microsystems Engineering, Engineering Dynamics
Mekelweg 2, 2628 CD Delft, The Netherlands
[email protected]
ABSTRACT
In the present paper we outline the basic theory of assembling substructures for which the dynamics is described
as impulse response functions. The assembly procedure computes the time response of a system by evaluating per
substructure the convolution product between the impulse response functions and the applied forces, including the
interface forces that are computed to satisfy the interface compatibility. We call this approach the Impulse Based
Substructuring method since it transposes to the time domain the Frequency Based Substructuring approach. In the
Impulse Based Substructuring technique the impulse response functions of the substructures can be gathered either
from experimental test using a hammer impact or from time-integration of numerical submodels. In this paper the
implementation of the method is outlined for the case when the impulse responses of the substructures are computed
numerically. A simple bar example is shown in order to illustrate the concept. Future work will concentrate on
including in the assembly measured substructure impulse responses. The Impulse Based Substructuring allows fast
evaluation of impact response of a structure when the impulse response of its components are known. It can thus
be used to efficiently optimize designs of consumer products by including impact behavior at the early stage of the
design.
Keywords: Experimental Substructuring, assembly, impulse response functions, time integration, impact, impulse
based substructuring
NOMENCLATURE
dof
FRF
IRF
FBS
IBS
u
f
H(t)
⋆(s)
Ns
B
λ
M , K, C
dt
⋆n
[⋆]i
β, γ
degrees of freedom
Frequency Response Functions
Impulse Response Functions
Frequency Based Substructuring
Impulse Based Substructuring
array of degrees of freedom
array of external forces
matrix of Impulse Response function
pertaining to substructure s
number of substructures in the system
signed Boolean matrix defining compatibility constraints
Lagrange multipliers on interface
mass, stiffness and damping matrix of a linear(ized) system
time-step size
pertaining to time-step n
component i of an array
parameters of the Newmark time-integration scheme
1
INTRODUCTION
Substructuring techniques allow combining the dynamics of different components. The dynamics of the components is
described either using experimental data (Frequency Response Functions in the frequency domain) or with a numerical
model (in terms of system matrices, computed Frequency Response Functions or component modes). See for instance 5
for a review of substructuring concepts.
In the Frequency Based Substructuring approach one can assemble the Frequency Response Functions (FRFs) of
substructures that were measured or that were obtained by numerical simulation. This technique can provide accurate
results on simple systems 11 and gives interesting qualitative information when applied to complex engineering systems
such as cars 2,8 . When the substructures are described by FRFs obtained experimentally tremendous care must be
taken to ensure a high degree of accuracy, for instance it must satisfy reciprocity, passivity and artifacts like additional
mass effects or location/orientation errors in the sensors must be very small. In practice such errors often introduce in
the assembled FRFs spurious peaks 10 and non-physical properties 1 that renders the obtained assembled model useless.
To clean-up the measured FRFs of the substructures before assembly one can apply modal identification techniques in
order to fit a pole-residue model to the data. When combining identified modes of the substructures with a numerical
model of the measured components high quality assembled models can be obtained 7 .
Obtaining high quality measured FRFs is delicate since, unless slow and costly sine sweeps are used, the dynamic
properties in the frequency domain are obtained through several processing steps (anti-aliasing filters, windowing,
Fourier transforms) which will unavoidably alter the information contained in the measurements. Furthermore using
modal identification and FRF synthesis to obtain clean FRFs is a very labour-intensive and error-prone process, and
it assumes that a pole-residue model can be fitted. When high damping is present, proper identification of the poles
and residues becomes difficult and if non-viscous damping (visco-elastic damping) is present the pole-residue model
cannot correctly represent the frequency domain response of the components. Finally we note that if substructuring
is used to simulate impact responses, working in the frequency domain requires considering a large frequency band
which makes all the Frequency Based Substructuring strategies expensive and badly suited.
Applying substructuring techniques to simulate impact responses of structures is a very attractive idea since it would
allow to efficiently predict impact behavior at the early design stage in many fields. For instance when designing the
structure of a mobile phone or a notebook , many COTS (components of the shelf) are used. If the dynamics of
the COTS are characterized properly either numerically or experimentally, one could use susbtructuring techniques to
rapidly optimize the housing, frame and connections (screws, rubber pads ...) and thereby guarantee an improved
life-time and reliability of valuable mass-produced appliances.
Since Frequency Based Substructuring is not well suited for setting up a model for impact simulation (see discussion
above), we propose in this paper an alternative substructuring technique. We use the same concepts as in other
substructuring approaches (i.e. admittance representation of the components and dual assembly) but consider for the
substructures directly the impulse response functions in the time domain measured for the input and interface degrees
of freedom, instead of the modal properties or the FRFs. The method will be called Impulse Based Substructuring
or IBS. This paper outlines the basic principle of the method and shows that the theory can be easily applied when
the impulse response functions are computed through direct time integration of a numerical model. Future work will
investigate the combination of numerical and experimental sub-models in the Impulse Based Substructuring strategy.
2
THEORY OF IMPULSE RESPONSE SUPERPOSITION FOR PARTITIONED PROBLEMS
Let us call H(t) the matrix of the responses to a unit impulse at t = 0 for a linear system that is initially at rest. In
other words a coefficient [H(t)]ij of the impulse response matrix represents the response of degree of freedom (dof)
i to a unit impulse on dof j. The response of the linear system to an applied force f (t) can then be evaluated by the
convolution product (Duhamel’s integral) between the impulse response function matrix and the applied forces:
Z t
H(t − τ )f (τ )dτ
(1)
u(t) =
0
This is a classical result of time analysis of linear systems, usually obtained using Laplace transforms. This convolution
product can be interpreted as follows: the response at time t is an infinite sum of the responses to the infinitesimal
impulses f (τ )dτ before time t (see figure 1). Each impulse at time τ gives a contribution through the impulse response
from τ to t, that is H(t − τ ).
φ(τ )
∆τ
τ
Figure 1: Forcing function as a series of impulses
The impulse responses can be obtained either experimentally or numerically. If obtain by measurements, an impact
hammer can be used:
R t+
• One evaluates the impulse I = 0 δ(t)dt, where δ(t) is the measured force and t+ the duration of the force
assumed to be very short for a hammer impact. In fact the hammer impact is only an approximation of a Dirac
function, but if the duration t+ is much smaller then the characteristic time of the expected response u(t) to
the applied force f (t) in (1) the obtained impulse response can be used in the convolution product. In other
words the duration of the hammer impulse must be much shorter then the period of the frequency of the modes
having a significant contribution to the response.
• Compute the impulse response by scaling the measured time response to the hammer impact by the impulse I.
The impulse response can also be obtained from a closed-form solution of a mathematical model or by time-integration
of a numerical model. In the later case, the time-step must be small enough so that the initial impulse can be considered
as an impulse for the dynamic response one wants to compute with the Duhamel integral. See the next section for
further discussion on how to compute the impulse response of a numerical model.
Let us now assume that the problem has been decomposed into N s sub-structures. The response of each substructure
can be obtained using the convolution product (1), but for the solutions to be the responses of the substructures as
part of a full system the coupling forces on the interface between the substructures must be included in the forcing
function. The interface forces are unknown beforehand, but we know that those interface forces coupling the interface
dofs must be such that the interface is compatible in the assembled problem.
Calling B (s) the signed Boolean matrices localizing the interface dofs (see for instance 3,9 ), the compatibility condition
on the interface, namely the condition stating that the dofs on each side of the interface are equal, can be written as
s
N
X
B (s) u(s) = 0
(2)
s=1
Hence the extension of Duhamel’s integral (1) to a partitioned problem writes

Z t

(s)
(s)
(s)
(s)T

u
(t)
=
H
(t
−
τ
)
f
(τ
)
+
B
λ(τ
)
dτ


0
Ns
X



B (s) u(s) (t) = 0

(3)
s=1
T
where B (s) λ represent the interface forces, namely the reactions associated to interface compatibility constraint, λ
being the Lagrange multipliers.
The Impulse Base Substructuring (IBS) method proposed in this paper relies on (3): if the impulse response functions
are known per substructures it allows computing the impulse response (or the response to any external force) for
the assembled problem. It is thus clearly a transposition to the time domain of the Frequency Based Substructuring
(FBS). In practice the formulation (3) must be applied by discretizing the time integral in order to evaluate the
convolution product. Indeed the impulse response H(t) is generally known only for discrete time instances, either
from measurements or from numerical modelling. A way to evaluate the integral is explained in the next section.
3
IMPULSE BASED SUBSTRUCTURING WITH NUMERICAL MODELS
In this section we will show how the IBS method can be applied, in particular how Duhamel’s integral can be discretized
in time. Here we will assume that the dynamics of the substructures are described by a numerical model. First we
will explain how an impulse response is computed, then we outline how impulse response superposition can be applied
for a single (non-decomposed system). We will extend the method to assemble several numerical models described
by their IRFs and finally illustrate the strategy with a simple application example.
3.1
Impulse Response computation
For a numerical model the impulse response is obtained by numerical time-integration. The linear dynamic equilibrium
at time tn can generally be represented by the matrix equation
M ün + C u̇n + Kun = fn
(4)
where M , C, K are the linear(ized) mass, damping and stiffness matrices, u is the set of degrees of freedom, and f
are the applied forces. The subscript n indicates the time-step at which the accelerations, velocities and displacement
are considered. Let us call dt the time-step size (assumed for simplicity to be constant during the time-integration)
such that t = n dt = tn .
Given the initial conditions u0 , u̇0 , the initial acceleration can be computed by
ü0 = M −1 (f0 − Ku0 − C u̇0 )
(5)
To solve (4) one needs to approximate time derivatives by well chosen finite differences. In structural dynamics one
classically uses the Newmark time-integration scheme 4,6 stating that
un
u̇n
=
=
un−1 + dtu̇n−1 + (0.5 − β)dt2 ün−1 + βdt2 ün
u̇n−1 + (1 − γ)dtün−1 + γdtün
(6)
(7)
where β and γ are parameters used to build integration schemes with different properties. For instance when γ =
1/2, β = 1/4 one obtains an implicit and unconditionally stable scheme (equivalent to the trapezoidal integration
rule). If γ = 1/2, β = 0 the scheme is explicit but conditionally stable (equivalent to the central difference).
Replacing the discretized time derivatives (6,7) in the dynamic equation (4)
M + γdtC + βdt2 K ün = fn − K ũn − C ũ˙ n
(8)
where ũn and ũ˙ are the predictors
ũn
ũ˙ n
=
un−1 + dtu̇n−1 + (0.5 − β)dt2 ün−1
=
u̇n−1 + (1 − γ)dtün−1
For an unit impulse at time t = 0 the dynamic response can be computed in three different ways.
Initial velocity step To compute the impulse response we can first compute the velocity jump at time t = 0 due
to a unit impulse. Integrating the dynamic equation in an infinitesimal interval [0− , 0+ ] results in the momentum
equation
M (u̇0+ − u̇0− ) =
Z
0+
f (t)dt
0−
and since just before t = 0 the system is at rest, the initial velocity resulting from a unit impulse on dof j is
M u̇0 = j
(9)
where j is a vector with a unit coefficient for dof j. Hence the impulse response can be computed by setting the
applied force f (t) to zero and starting the time integration with the initial conditions
u0
u̇0
ü0
= 0
= M −1 j
= M
−1
(10)
(−C u̇0 )
then continuing the integration with (8).
Initial applied force Another manner to compute the impulse response is to use the initial conditions and applied
force
u0
u̇0
f0
fn>0
=
=
=
=
0
0
j
0
(11)
hence ü0 = M
−1
j
Note that, in the time integration scheme, this is equivalent to an impulse generated by a force being suddenly unity
at time t = 0 and decreasing linearly to 0 at time t = dt: the IRF so obtained is in fact for an impulse equal to
dt/2. The numerical impulse response is thus the time response obtained for the settings (11) divided by dt/2. In the
limit where the time-step goes to zero this method is obviously converging to the impulse response as computed with
the initial velocity step (see above). If the Newmark time integration scheme γ = 1/2, β = 1/4 is used the impulse
response obtained by this method is actually identical to the response obtained with an initial velocity jump even for
a finite time-step size.
Applied force at the second time-step The two methods outlined above to compute the impulse response require
factorizing the mass matrix. This is not a problem when the mass matrix is diagonal (as found for an explicit time
integration), but for a consistent (non-diagonal) mass matrix the factorization cost could be significant. It that case
one can avoid factorizing the mass matrix by applying a unit force on the second time-step, namely
u0
u̇0
ü0
f1
=
=
=
=
0
0
0
j
(12)
(13)
fn6=1 = 0
This computation represents in fact a force increasing linearly to j between t0 and t1 , then decreasing to zero between
t1 and t2 . The related impulse value is thus dt and the unit impulse response is obtained by dividing the obtained time
response by dt. Again In the limit where the time-step goes to zero this method is equivalent to the two strategies
explained above. For a finite time-step the obtained impulse response is slightly different.
3.2
Impulse superposition for a single structure
The impulse responses computed at time tn by one of the methods described in the previous section are stored in
the Impulse Response matrix Hn . The coefficient [Hn ]ij is the response for dof i at time t = n dt to an unit
impulse at time t = 0 on a dof j. So H, containing the full time history of the impulse responses, can be seen as a
three-dimensional matrix of dimension N × p × nmax , calling N the number of degrees of freedom of the system (or
the number of output considered), p the number of excitation (input) locations, and nmax the number of time-steps
for which the impulse response has been computed. Note that this is similar to the Frequency Response Functions
(FRFs) of a system, except that for the FRFs the third dimension is the frequency. Obviously, theoretically speaking,
the FRFs of a system are the Fourier transform of the IRFs.
The time response for a general applied force f (t) can then be computed by approximating the convolution integral
(1) by the finite sum
n−1
X
Hn−i fi dt
(14)
un =
i=0
Note that H0 is not present in this series since the displacement response to an impulse is null at the instant when the
impulse is applied, meaning that H0 = 0. The graphical interpretation of the discretized convolution (14) is given in
figure 2.
f
u
τ
H(t-τ)
t-τ
t
Figure 2: Discretization of Duhamel’s integral
In figure 2 it is seen that the applied force, in the time-integration scheme, is approximated by piece-wise linear forces
between the time steps. Hence during a time step from tn to tn+1 the applied forces are coming for one half from
fn and for the other half from fn+1 . Let us then consider in figure 2 the response at time t1 . According to (14) the
response at t1 is solely due to the impulse created by the force at t0 , while in fact the force at time t1 also produced
an impulse between t0 and t1 . Hence one can say the the response at t1 is due to an impulse equal to
It0 ,t1 = f0 dt/2 + f1 dt/2 =
f0 + f1
dt
2
indicating that to evaluate the impulse the force should in fact be taken at the middle of the time step. Therefore it
is more accurate to consider the following discretization of Duhamel’s integral:
un =
n−1
X
Hn−i (fi + fi+1 ) dt/2
(15)
i=0
In practice however, since the time step dt is small, the difference between (15) and (14) is negligible.
3.3
Impulse superposition and assembly of substructures
The matrices of the impulse responses for each substructure can be computed as indicated in the section 3.1. In order
to assemble the substructures the IRFs between all interface dofs are required, in addition to the IRFs for the dofs
where the external forces f (s) are applied. The convolution product and the compatibility condition of (3) can then
be discretized as

X (s) (s)
 (s) n−1

(s)T

dt
+
B
λ
f
H
u
=

i
n
i
n−i

i=0
s
(16)
N
X




B (s) u(s) = 0

s=1
where λi are the Lagrange multipliers related to the compatibility condition. They represent the impulse between the
interface dofs needed to ensure the interface compatibility.1
In the dual assembly formula (16) the solution at time tn is determined by the external forces and the interface
impulses for t = 0 up to t = n − 1. Hence the compatibility condition at time tn determines the interface impulse at
time tn−1 . Let us rewrite (16) as
 (s)
(s) (s)T
(s)

λn−1
 uns = ũn + H1 B
N
X
(17)

B (s) u(s) = 0

s=1
(s)
where ũn is the predicted displacement when λn−1 = 0, namely
ũ(s)
n =
n−2
X
i=0
T
(s)
(s)
(s) (s)
Hn−i fi dt + B (s) λi + H1 fn−1 dt
From (17) the Lagrange multiplier is computed by solving the dual interface problem
!
Ns
Ns
X
X
T
(s)
(s)
(s)
λn−1 = −
B (s) ũ(s)
B H1 B
n
s=1
(18)
(19)
s=1
which is very similar to the dual interface problem of the Frequency Based Substructure (see e.g. 5 ).
Equations (19,17) constitute the stepping algorithm for the Impulse Based Substructuring strategy (IBS) proposed in
this paper.
3.4
Numerical example
To illustrate the Impulse Based Substructuring technique let us consider the bar structure described in figure 3 excited
by a load at its end. The structure is divided in 2 substructures of equal length, each substructure being modeled
by 25 bar finite elements (the consistent mass matrices are used here). The bar is made of steel (E = 2.1 1011 Pa,
ρ = 7500 kg/m3 ), has a uniform cross-section of A = 3.14 10−4 m2 and each substructure has a length of L = 0.5 m.
In the model damping has been introduced by constructing C = 2 10−6 K.
1 Equation (16) can also be written for the forces at half time-step as in (15). This does not modify the basics of the algorithm and
provides slightly more accurate results.
λ
0
f
0
L
L
Figure 3: Example of a beam with two substructures
Assuming the dofs of the substructures are numbered from left to right, the Boolean constraining matrices are
B (1)
=
[ 0
···
0 −1 ]
(2)
=
[ 1
···
0 0 ]
B
First we compute the Impulse Response Functions as indicated in section 3.1. Here a unit force at time t = 0 is used.
The implicit, unconditionally stable Newmark γ = 1/2, β = 1/4 scheme is used. The time-step is chosen equal to
3hcrit where hcri t is the critical time step, namely the stability limit if the integration scheme would be explicit. This
critical time-step is given by the CFL condition and is equal to 4
hcrit = 2/ω
for ω the highset eigenfrequency in the model.
The obtained IRFs are plotted in figure 4 for inputs on the interface and on the end of the bar. On the right of that
figure the IRFs are zoomed.
−4
(1)
HLL
1
−4
x 10
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
0.005
0.01
0.015
0.02
−1
x 10
0
0.5
1
1.5
2
2.5
−3
x 10
−3
(2) 0.015
2
H00
x 10
1.5
0.01
1
0.005
0.5
0
0
0.005
0.01
0.015
0.02
0
0
0.5
1
1.5
2
2.5
−3
x 10
−3
(2) 0.015
2
HL0
x 10
1.5
0.01
1
0.005
0.5
0
0
0.005
0.01
t (s)
0.015
0.02
0
0
0.5
1
1.5
t (s)
Figure 4: IRFs for the bar substructures (zoomed on the right)
2
2.5
−3
x 10
Let us assume that one is interested in computing the dynamic response at the end of the bar when a force is applied
on it. The IBS expression (17) and the dual interface problem (19) are
(2)
ũn[ L]
=
n−2
X
(2)
(2)
(2)
(2)
(2)
Hn−i[LL] fi[L] dt + Hn−i[L0] λi + H1[LL] fn−1[L] dt
i=0
(1)
ũn[L]
(2)
− ũn[L]
λn−1
=
u(2)
n[L]
= ũ(2)
n[L] + H1[LL] λn−1
(20)
(2)
(1)
H1[LL] + H1[00]
(2)
First we will apply the IBS technique to compute the response of the full bar to an impulse at its end: applying a unit
force at the end of the second substructure, and dividing the obtained response by dt/2 we obtain the IRF shown in
figure 5. If this impulse response is computed with a non-decomposed model, exactly the same IRF is found.
−4
f ull
H2L,2L
1
−4
x 10
1
0.5
0.5
0
0
−0.5
−0.5
−1
0
0.005
0.01
0.015
−1
0.02
x 10
0
0.5
1
t (s)
1.5
2
2.5
−3
x 10
t (s)
Figure 5: IRF for the full bar computed by IBS (zoomed on the right)
(2)
Finally let us apply a step load at the end of the bar (fL (t) = 1 for t ≥ 0). Using again the IBS approach to
compute the response based on the impulse responses of the substructures one obtains the response plotted in figure
6. Again the same response would have been found if a model of the complete bar would have been used in a direct
time-integration. It is observed that the solution converges in time to the correct static solution.
−8
(2)
uL
(1)
3
−8
x 10
3
(2) 2
uL = u0
(1)
udx
2
1
1
0
0
−1
x 10
0
0.01
0.02
0.03
t (s)
0.04
−1
0
0.5
1
1.5
t (s)
2
2.5
−3
x 10
Figure 6: Dynamic response to a step load at the end of the bar, computed by IBS (zoomed on the right).
The solutions are shown for the end of the bar, the middle point (on the interface between the
substructures) and on the first node next to the fixed end.
4
CONCLUSIONS AND FUTURE WORK
In this paper we propose a method that transposes the Frequency Based Substructuring technique to the time domain.
It allows computing the response of a system by computing the responses of its substructures with a discretization of
the Duhamel integral and enforcing the interface compatibility at every time step. Hence if the substructure impulse
responses are known for all interface and input degrees of freedom, the method proposed in this paper allows predicting
the dynamics of the full system. The method is named Impulse Based Substructuring or IBS.
The IBS (in the time domain) is the equivalent of the FBS (in the frequency domain), but we believe that when
applied to the computation of impact response and shocks, the IBS can be significantly more effective since it allows
using directly the impact responses of the susbstructures.
The IBS approach could be used for instance when predicting the impact response of new designs. It can enable
engineers to optimize their designs during the very initial design phase, if the impulse response of the components are
known either from models or tests. In many cases the designer uses standard components which would then need to be
characterized once, the work of the designer being then to optimize the housing and the link between the components.
In this paper we outline the basic principle of the method and show that the theory can be easily applied when
the impulse response functions are computed through direct time integration of numerical models. Future work
will investigate the applicability of the approach when the impulse responses of the substructures are obtained from
experimental tests. We will also investigate the combination of numerical and experimental sub-models in the IBS
method. Finally we will perform research to combine substructures described by impulse response functions with
non-linear components that need to be time-integrated simultaneously with the time-stepping in the IBS for the linear
parts.
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