Abstract
A bilinear formulation of elasto-dynamics is offered which includes, as a special case, “Hamilton's law of varying action”. However, the more general bilinear formulation has several advantages over Hamilton's law. First, it admits a larger class of initial-value and boundary-value problems. Second, in its variational form, it offers physical insight into the so-called “trailing terms” of Hamilton's law. Third, numerical applications (i.e., finite elements in time) can be proven to be convergent under correct application of the bilinear formulation, whereas they can be demonstrated to diverge for specific problems under Hamilton's law. Fourth, the bilinear formulation offers automatic convergence of the “natural” velocity end conditions; while these must be constrained in present applications of Hamilton's law. Fifth, the bilinear formulation can be implemented in terms of a Larange multiplier that gives an order of magnitude improvement in the convergence of velocity. This implies that, in this form, the method is a hybrid finite-element approach.
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Abbreviations
- b :
-
arbitrary constant
- A i, A′ i :
-
vector of integrals, i = 0, j
- A(ν):
-
linear operator on ν
- Ā(ν):
-
Hamilton's form of A
- B (u, ν):
-
bilinear operator u, ν
- B (u, ν):
-
Hamilton's form of B
- B i,j , B ij , B′ ij :
-
matrix of integrals
- C :
-
constant, N/m
- c :
-
number of floating-point operations per coef. evaluation
- f, f(x):
-
force per unit length, N/m
- F, F 0, F L :
-
forces, N
- J :
-
number of functions in series for û
- k :
-
spring rate per unit length, N/m2
- K :
-
spring rate, N/m
- K max :
-
maximum value of K
- L a :
-
Lagrangian, non-dimensional
- L :
-
length of beam, m
- m :
-
mass per unit length, kg/m
- M :
-
mass, kg
- M max :
-
maximum value of M
- n :
-
number of functions in series for \(\hat v\)
- N :
-
number of elements in domain
- p :
-
momentum density, kg/sec
- P, P 0, P T :
-
momentum, kg-m/sec
- q i :
-
generalized coordinates
- r j :
-
coefficients of ψ j
- t :
-
time, sec
- t 0, t 1 :
-
limits of action integral, Hamilton's law
- T :
-
end of time period, sec
- u :
-
solution for displacement, m
- û :
-
approximation to u, m
- u 0 :
-
initial value for u, m
- ν:
-
test function, m
- \(\hat v\) :
-
limited class of ν, m
- x :
-
spatial coordinate, m
- β:
-
flapping angle, rad
- γ:
-
Lock number
- Δ:
-
time increment, sec
- λ:
-
Lagrange multiplier
- μ:
-
longitudinal stiffness EA, N (Eqs. 1–18)
- μ:
-
advance ratio of rotor (Eqs. 33–34 and figures)
- ϕ i , ψ r :
-
polynomial functions
- ψ:
-
non-dimensional time, azimuth angle
- δ():
-
variation of ( )
- δW :
-
virtual work
- ( )′:
-
d ( )/dx
- ( .):
-
d ( )/dt
- (*):
-
d/dψ
- [ ]:
-
matrix
- { }:
-
column vector
- 〈〉:
-
row vector
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Communicated by S. N. Aduri 18 February 1987
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Peters, D.A., Izadpanah, A.P. hp-version finite elements for the space-time domain. Computational Mechanics 3, 73–88 (1988). https://doi.org/10.1007/BF00317056
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DOI: https://doi.org/10.1007/BF00317056