Filetype PDF | Posted on 20 Jan 2023 | 2 years ago
Authentication
digital resources
148x Filetype PDF File size 0.63 MB Source: people.duke.edu
Numerical Integration in Structural Dynamics
CEE541. Structural Dynamics
Department of Civil & Environmental Engineering
Duke University
Henri P. Gavin
Fall 2020
Introduction
A damped structural system subjected to dynamic forces and possibly experiencing
nonlinear material behavior is modeled by
Mx¨(t)+Cx˙(t)+Kx(t)+R(x(t),x˙(t)) = fext(t), (1)
where x is a vector of displacements of structural coordinates, M is a positive definite
mass matrix, C is a non-negative definite damping matrix, and K is a non-negative definite
stiffness matrix. The nonlinear restoring forces are given in R(x,x˙) and fext(t) is a vector
of external dynamic loads. At any point in time, t = t =(i+1)h, we may solve for the
i+1
accelerations in terms of the displacements, velocities, and the applied forces.
x¨(t ) = −M−1[Cx˙(t ) +Kx(t ) +R(x(t ), x˙(t )) − fext(t )]: (2)
i+1 i+1 i+1 i+1 i+1 i+1
Given values for accelerations, velocities, displacements, and applied forces at time ti, if we
can extrapolate the velocities and displacements forward in time by a time step h to time
t =t +h, then we may compute the acceleration x¨(t ) using equation 2. The various
i+1 i i+1
numerical integration algorithms described in this document1 2 differ primarily in the manner
in which x(t ) and x˙(t ) are computed from x(t ), x˙(t ), x¨(t ), fext(t ), and fext(t ).
i+1 i+1 i i i i i+1
There are two general classifications of numerical integration methods: explicit and
implicit. In explicit methods, displacements and velocities at ti+1 can be determined in
closed form from displacements, velocities, accelerations, at t , and from external forcing at
i
t and, potentially, t . For structural systems with linear elastic stiffness and linear viscous
i i+1
damping, such discrete-time systems may be written
" x(t ) # " x(t ) #
i+1 =A i +Bfext(t ) (3)
x˙ (t ) x˙ (t ) i
i+1 i
where A is a 2n × 2n discrete time dynamics matrix which depends upon M, C, K, the
time step, h, and some algorithmic parameters. Implicit methods involve the solution of a
set of nonlinear algebraic equations at each time step. For example, the displacements and
velocities at time t , [x(t ), x˙(t )], are determined from the roots of a nonlinear equation
i+1 i+1 i+1
in terms of [x(t ), x˙(t )]. Explicit numerical methods are typically more computationally-
i+1 i+1
efficient than implicit methods.
1Clough, R. and Penzien, J., Dynamics of Structures, 2nd ed., McGraw-Hill, 1993.
2Hanson, R.D., CE 611: Structural Dynamics, class notes, The University of Michigan, 1988
2 CEE 541. Structural Dynamics – Duke University – Fall 2020 – H.P. Gavin
Accuracy and Stability
Numerical methods for integrating equations of motion are assessed and evaluated in
terms of their accuracy and stability. In general, accuracy and stability depend upon the
ratio of the time step, h, to the shortest natural period in the system model. For a system
with many coordinates (n > 103), the shortest natural period can be much shorter than
the fundamental natural period, T /T > 104. Typically, responses of the highest several
n 1
modes of a numerical model are physically meaningless, should be insignificantly small, but
are potentially lightly-damped, and can dominate the errors in numerical integration. The
explicit numerical methods described in these notes can artificially add numerical damping to
suppress instabilities of the higher mode responses. Implicit numerical integration methods
are unconditionally stable.
The Central Difference Method
The central difference approximations for the first and second derivatives are
x˙ (t ) = x˙ ≈ 1 (x −x ), (4)
i i 2h i+1 i−1
x¨(t ) = x¨ ≈ 1 (x −2x +x ): (5)
i i 2 i+1 i i−1
h
Substituting approximations (4) and (5) into equation (1), and re-arranging terms, leads to
1 1 2 1 1 ext
M+ C xi+1= M−K xi+ − M+ C xi−1+f , (6)
2 2 2 i
h 2h h h 2h
which may be written A x =Ax+Ax +fext. AslongasM orC ispositivedefinite,
0 i+1 1 i 2 i−1 i
the matrix A is also positive definite and may be factorized prior to the iterative solution
0
for xi at each time step.
The numerical stability of the central difference method depends on the choice of the
time step interval, h. To obtain a stable solution, h < Ti/π, where Ti is the shortest natural
period of the structural system. If the central difference method is used with a time step
larger than Ti/π the solution will increase exponentially.
3
Astep-by-step procedure for the central difference method may be written as :
1. Input the mass, M, damping, C, stiffness, K, matrices and the time step interval h.
2
2. Initialize x0, x˙0, x¨0 and compute x1 = x0 − hx˙0 + h x¨0:
2
3. Form the matrices A , A and A , and triangularize A using LDLT factorization.
0 1 2 0
4. for i = 1 to N
(a) solve A x =Ax +Ax +fextforx using LDLT back-substitution.
0 i+1 1 i 2 i−1 i i+1
(b) calculate x˙i and x¨i using equations (4) and (5), if required.
(c) write results (t , x , x˙ , x¨ ) to a data file.
i i i i
3K.J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, 1982, pp. 439–449, 499–
506.
cbndH.P.Gavin October 5, 2020
Numerical Integration for Structural Dynamics 3
The Implicit Linear Acceleration Method
Consider the Taylor series expansions for displacement, velocity, and acceleration:
2 3 4
h h ... h ....
x(t ) = x =x +hx˙ + x¨ + x + x +··· (7)
i+1 i+1 i i 2! i 3! i 4! i
2 3
h ... h ....
x˙ (t ) = x˙ =x˙ +hx¨ + x + x +··· (8)
i+1 i+1 i i 2! i 3! i
2
... h ....
x¨(t ) = x¨ =x¨ +hx + x +··· (9)
i+1 i+1 i i 2! i
Rearranging equation (9),
2
... h ....
hxi = x¨i+1 − x¨i − 2! x i − ··· , (10)
and substituting equation (10) into equations (7) and (8) results in
2 2 2 ! 4
h h h .... h ....
xi+1 = xi+hx˙i+ 2 x¨i + 6 x¨i+1 − x¨i − 2! x i − ··· + 4! x i +··· , (11)
2 ! 3
h h .... h ....
x˙ i+1 = x˙i +hx¨i + 2 x¨i+1 − x¨i − 2! x i − ··· + 3! x i +··· (12)
Truncating the fourth time-derivative and higher from these expansions, the resulting finite
difference approximations are
2
x ≈x +hx˙ + h (x¨ +2x¨ ), (13)
i+1 i i 6 i+1 i
and
x˙ ≈x˙ + h(x¨ +x¨ ): (14)
i+1 i 2 i+1 i
These relationships are implicit because x¨i+1 needs to be determined in order to find xi+1 and
x˙ i+1, but x¨i+1 can not be found without knowing xi+1 and x˙i+1. Note that the substitutions
above, have eliminated the third time-derivative of x, and that the method is accurate to
4....
within h x .
This is called the linear acceleration method because the third time derivative of x has
been eliminated. If the rate of change of acceleration within a time-step is truly constant,
then the approximation of truncating the Taylor series at the fourth-order term does not
affect the accuracy of the solution.
cbndH.P.Gavin October 5, 2020
4 CEE 541. Structural Dynamics – Duke University – Fall 2020 – H.P. Gavin
The Implicit Linear Acceleration Method,
Made Explicit for Linear Structural Dynamics
Recall that at time t we can satisfy the equations of motion, by calculating the
i+1
acceleration with equation (2). Substituting equations (13) and (14) for the displacements
and velocities at time t into equation (2),
i+1
2 ext
Mx¨ +C{x˙ +(h/2)(x¨ +x¨ )} +K{x +hx˙ +(h /6)(x¨ +2x¨ )} = f , (15)
i+1 i i i+1 i i i+1 i i+1
where any nonlinearities, R(x,x˙), are assumed to be negligible. Collecting similar derivatives
of x, " # " #
2 2
M+hC+h K x¨ =fext−Kx −[C+hK]x˙ − hC+h K x¨ : (16)
2 6 i+1 i+1 i i 2 3 i
and re-arranging,
" 2 # " 2 #
M+hC+h K x¨ = −hC−h K x¨ −Kx −Cx˙ −hKx˙ +fext : (17)
2 6 i+1 2 3 i i i i i+1
Recall from the equations of motion, that
Mx¨ −fext = −Kx −Cx˙ : (18)
i i i i
Substituting equation (18) into (17), we obtain the closed-form linear acceleration recurrence
relations for structural dynamics simulation.
" 2 # " 2 #
M+hC+h K x¨ = M−hC−h K x¨ −hKx˙ +fext−fext : (19)
2 6 i+1 2 3 i i i+1 i
x˙ =x˙ + h[x¨ +x¨ ]: (20)
i+1 i 2 i+1 i
and 2
x =x +hx˙ + h [x¨ +2x¨ ]: (21)
i+1 i i 6 i+1 i
Theserelationshipsarenowexplicitbecausex¨i+1 canbedeterminedfromthecurrentresponse
values (x and x˙ ), the current dynamic load fext, and the next dynamic load fext. Note
i i i i+1
that within each time step, the dynamic equations of equilibrium are satisfied both at time
t and at time t .
i i+1
cbndH.P.Gavin October 5, 2020
no reviews yet
Please Login to review.
