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GENERALIZED SDF SYSTEMS Ch 8 Chopra
Two types of systems that can be classified as generalized SDF
systems:
1. Rigid body assemblages
2. Systems with distributed mass and elasticity
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Setting the sum of the moments of all forces about O to zero
gives:
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Example E8-1 (From Clough & Penzien)
A representative example of a rigid body assemblage, shown in
Fig. E8-1, consists of two rigid bars connected by a hinge at E and
supported by a pivot at A and a roller at H. Dynamic excitation is
provided by a transverse load p(x,t) varying linearly along the
length of bar AB. In addition, a constant axial force N acts
through the system, and the motion is constrained by discrete
springs and dampers located as shown along the lengths of the bars.
The mass is distributed uniformly through bar AB, and the
weightless bar BC supports a lumped mass m2 having a centroidal
mass moment of inertia j2.
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Solution
It is more convenient to use a work or energy formulation. A
virtual work analysis will
be used.
1. Assume a deformed shape (see Fig E8-2). Show all the forces
acting on the
structure.
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2. Express all required displacements in terms of the chosen
generalized coordinate.
The hinge motion Z(t) may be taken as the basic quantity
(generalized coordinate)
and all other displacements expressed in terms of it
BB(t) = Z(t)/4 DD(t) = 3Z(t)/4, FF(t) = 2Z(t)/3 etc.
3. Determine the force components acting on the system
(exclusive of the axial
applied force N, which will be considered later). Each resisting
force component
can be expressed in terms of Z(t) or its time derivatives, as
follows:
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4. Principle of Virtual Work: Work done by all forces during an
arbitrary virtual
displacement Z is zero.
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5. Consider now the externally applied axial force N.
The virtual work done by this force during the virtual
displacement Z is Ne. The
displacement e is made up of two parts, e1 and e2, associated
with the
rotations of the two bars.
e1 = (Z(t)/4a)Z e2 = (Z(t)3a)Z
thus the total displacement is
and the virtual work done by
the axial force N is
(d)
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Other possible shape functions are:
Fictitious inertia forces:
Using Principle of Virtual Work: WE = W
I
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Equivalent Static Forces, fS(x), = external forces that would
cause displacements, u(x)
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