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Dynamics of structures
Fall 2019
University of Qom
By:
A. Shahiditabar
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CHAPTER 6
GENERALIZED
SINGLE DEGREE OF FREEDOM
SYSTEMS
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The analysis of most real systems requires the use of more complicated idealizations, even when they can be included in
the generalized single degree of freedom category. In this chapter we will discuss these generalized SDOF systems,
GENERALIZED PROPERTIES: ASSEMBLAGES OF RIGID BODIES
In formulating the equations of motion of a rigid body assemblage, the elastic forces developed during the SDOF
displacements can be expressed easily in terms of the displacement amplitude because each elastic element is a
discrete spring subjected to a specified deformation. Similarly the damping forces can be expressed in terms of the
specified velocities of the attachment points of the discrete dampers On the other hand, the mass of the rigid bodies
need not be localized, and distributed inertial forces generally will result from the assumed accelerations. However,
for the purposes of dynamic analysis, it usually is most effective to treat the rigid body inertial forces as though the
mass and the mass moment of inertia were concentrated at the center of mass.
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π£ π₯, π‘ =π₯
ππ§(π‘) β αΆπ£ π₯, π‘ =
π₯
παΆπ§(π‘) β α·π£ π‘ =
π₯
πα·π§(π‘)
πΉπΌ = ΰ΄₯ππ₯
πα·π§(π‘)πΉπ = π
π₯
ππ§(π‘) πΉπ· = π
π₯
παΆπ§(π‘)
ππ = 0 β ππΌ +ππ· +ππ = ππ β ΰΆ±0
π
πΉπΌ π₯, π‘ π₯ππ₯ + πΉπ·π
4+ πΉπΎ
3π
4= π π‘ π β πβ α·π π‘ + πΆβ αΆπ π‘ + πΎβπ π‘ = π(π‘)
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For the purposes of dynamic analysis, it usually is most effective to treat the rigid body inertial forces as
though the mass and the mass moment of inertia were concentrated at the center of mass.
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Example E82. As a second example of the formulation of the equations of motion for a rigid body assemblage, the
system shown in Fig. E84 will be considered.
The small amplitude motion of this system can
be characterized by the downward displacement
of the load point Z(t),
and all the system forces resisting this motion
can be expressed in terms of it:
The equation of motion can be written directly by expressing
the equilibrium of moments about the plate hinge:
Dividing by the length a and substituting the above
expressions for the forces, this equation becomes
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π£ π₯, π‘ =π₯
ππ§(π‘) β αΆπ£ π₯, π‘ =
π₯
παΆπ§(π‘) β α·π£ π‘ =
π₯
πα·π§(π‘)
πΉπΌ = ΰ΄₯ππ₯
πα·π§(π‘)πΉπ = π
π₯
ππ§(π‘) πΉπ· = π
π₯
παΆπ§(π‘)
ππ = 0 β ππΌ +ππ· +ππ βππ = ππ
β ΰΆ±0
π
πΉπΌ π₯, π‘ π₯ππ₯ + πΉπ·π
4+ πΉπΎ
3π
4β ππ(π‘) = π π‘ π
β πβ α·π π‘ + πΆβ αΆπ π‘ + πΎβπ π‘ = π(π‘)
πΎβ =9
16πΎ1 β
π
πΏ
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8-3 GENERALIZED PROPERTIES: DISTRIBUTED FLEXIBILITY
If the bars could deform in flexure, the system would have an infinite number of degrees of freedom. A simple SDOF
analysis could still be made, however, if it were assumed that only a single flexural deflection pattern could be developed
The essential properties of the tower (excluding damping) are its flexural stiffness
EI(x) and its mass per unit of length m(x). It is assumed to be subjected to horizontal
earthquake ground motion excitation vg(t), and it supports a constant vertical load N
applied at the top.
To approximate the motion of this system with a single degree of freedom, it is
necessary to assume that it will deform only in a single shape. The shape function will
be designated, π(π₯) and the amplitude of the motion relative to the moving base will be
represented by the generalized coordinate π(π‘); thus
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The equation of motion of this generalized SDOF system can be formulated conveniently only by work or energy
principles, and the principle of virtual work will be used in this case.
Since the structure in this example is flexible in flexure, internal virtual work πΏππ is performed by the real internal
moments π(π₯, π‘) acting through their corresponding virtual changes in curvature πΏ[π2π£ π₯
ππ₯2]
π£" =π2π£ π₯
ππ₯2
If it is assumed that damping stresses are developed in proportion to the strain velocity, a uniaxial stress strain relation of the form
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Using the full set of external forces, the external virtual work is given by
Equating Eqs. (812) in accordance with Eq. (84) yields the generalized equation of motion
(generalized mass)
(generalized damping)
damping
(generalized flexural stiffness )
(generalized effective load)
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Example E83. Assumed that the tower of Fig. 8-2 has constant flexural stiffness EI and constant mass distribution
ΰ΄₯π along its length and damping in accordance with Eq. (8-8).
Also, its deflected shape in free vibrations will be assumed as
which satisfies the geometric boundary conditions π 0 = πβ² = 0,
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8-4 EXPRESSIONS FOR GENERALIZED SYSTEM PROPERTIES
Consider an arbitrary one dimensional system
assumed to displace only in a single shape
π(π₯) with displacements expressed
Part of the total mass of the system is distributed
in accordance with m(x) and the remainder is
lumped at discrete locations π (π = 1; 2; β¦ . ) as
denoted by ππ
External damping is provided by distributed
dashpots varying in accordance with π(π₯) and by
discrete dashpots as denoted by the πΆπ values, and
internal damping is assumed to be present in
flexure as controlled by the uniaxial stress strain
relation of Eq. (8-8)
Applying the procedure of virtual work
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The elastic properties of the system result
from distributed external springs varying in
accordance with π(π₯), from discrete springs
as denoted by the ππ values, and from
distributed flexural stiffness given by πΈπΌ(π₯).
External loadings are applied to the system in both
discrete and distributed forms as indicated by the
time independent axial forces π(π₯) and the time
dependent lateral forces π(π₯; π‘) and ππ (π‘). These
loadings produce internal moment distributions
π(π₯; π‘), respectively.
Applying the procedure of virtual work to this general SDOF system in the same manner as it was applied to the
previous example solutions, one obtains the following useful expressions for the contributions to the generalized
properties:
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8-5 VIBRATION ANALYSIS BY RAYLEIGH'S METHOD
Now Rayleigh's assumed shape concept will be extended further to develop an approximate method of evaluating the
vibration frequency of the member.
The basic concept in the Rayleigh method is the principle of conservation of energy; the energy in a freely vibrating
system must remain constant if no damping forces act to absorb it. Consider the free vibration motion of the undamped
spring mass system shown in Fig. 8-5a. With an appropriate choice of time origin, the displacement can be expressed
(Fig. 85b) by
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As explained above, this assumption may be expressed by Eq. (82), or noting
the harmonic variation of the generalized coordinate in free vibrations
The assumption of the shape function π(π₯) effectively reduces the beam to a
SDOF system.
The strain energy of this
flexural system is given by
The kinetic energy of the
nonuniformly distributed mass is
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8-6 SELECTION OF THE RAYLEIGH VIBRATION SHAPE
from which
This second frequency is significantly less than the first (actually
almost 20 percent less); thus it is a much better approximation.
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One common assumption is that the inertial loading π(π₯) (see Fig. 8-7) is merely the weight of the beam, that is,
π(π₯) = π(π₯) π, where π(π₯) is the mass distribution and π is the acceleration of gravity. The frequency then is
evaluated on the basis of the deflected shape π£π (π₯) resulting from this dead weight load. The maximum strain
energy can be found very simply in this case from the fact that the stored energy must be equal to the work done
on the system by the applied loading:
The kinetic energy is given still by Eq. (8-29),
in which π(π₯) = π£π (π₯)/π0 is the shape function computed from the dead load.