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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𝑣 𝑥, 𝑡 =𝑥

𝑙𝑧(𝑡) → ሶ𝑣 𝑥, 𝑡 =

𝑥

𝑙ሶ𝑧(𝑡) → ሷ𝑣 𝑡 =

𝑥

𝑙ሷ𝑧(𝑡)

𝐹𝐼 = ഥ𝑚𝑥

𝑙ሷ𝑧(𝑡)𝐹𝑠 = 𝑘

𝑥

𝑙𝑧(𝑡) 𝐹𝐷 = 𝑐

𝑥

𝑙ሶ𝑧(𝑡)

𝑀𝑜 = 0 → 𝑀𝐼 +𝑀𝐷 +𝑀𝑆 = 𝑀𝑃 → න0

𝑙

𝐹𝐼 𝑥, 𝑡 𝑥𝑑𝑥 + 𝐹𝐷𝑙

4+ 𝐹𝐾

3𝑙

4= 𝑃 𝑡 𝑙 → 𝑀∗ ሷ𝑍 𝑡 + 𝐶∗ ሶ𝑍 𝑡 + 𝐾∗𝑍 𝑡 = 𝑃(𝑡)

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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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𝑣 𝑥, 𝑡 =𝑥

𝑙𝑧(𝑡) → ሶ𝑣 𝑥, 𝑡 =

𝑥

𝑙ሶ𝑧(𝑡) → ሷ𝑣 𝑡 =

𝑥

𝑙ሷ𝑧(𝑡)

𝐹𝐼 = ഥ𝑚𝑥

𝑙ሷ𝑧(𝑡)𝐹𝑠 = 𝑘

𝑥

𝑙𝑧(𝑡) 𝐹𝐷 = 𝑐

𝑥

𝑙ሶ𝑧(𝑡)

𝑀𝑜 = 0 → 𝑀𝐼 +𝑀𝐷 +𝑀𝑆 −𝑀𝑁 = 𝑀𝑃

→ න0

𝑙

𝐹𝐼 𝑥, 𝑡 𝑥𝑑𝑥 + 𝐹𝐷𝑙

4+ 𝐹𝐾

3𝑙

4− 𝑁𝑍(𝑡) = 𝑃 𝑡 𝑙

→ 𝑀∗ ሷ𝑍 𝑡 + 𝐶∗ ሶ𝑍 𝑡 + 𝐾∗𝑍 𝑡 = 𝑃(𝑡)

𝐾∗ =9

16𝐾1 −

𝑁

𝐿

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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, 𝜓(𝑥) and the amplitude of the motion relative to the moving base will be

represented by the generalized coordinate 𝑍(𝑡); 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 𝛿𝑊𝑖 is performed by the real internal

moments 𝑀(𝑥, 𝑡) acting through their corresponding virtual changes in curvature 𝛿[𝜕2𝑣 𝑥

𝜕𝑥2]

𝑣" =𝜕2𝑣 𝑥

𝜕𝑥2

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

ഥ𝑚 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 𝜓 0 = 𝜓′ = 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

𝜓(𝑥) 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 𝑖 (𝑖 = 1; 2; … . ) as

denoted by 𝑚𝑖

External damping is provided by distributed

dashpots varying in accordance with 𝑐(𝑥) and by

discrete dashpots as denoted by the 𝐶𝑖 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 𝑘(𝑥), from discrete springs

as denoted by the 𝑘𝑖 values, and from

distributed flexural stiffness given by 𝐸𝐼(𝑥).

External loadings are applied to the system in both

discrete and distributed forms as indicated by the

time independent axial forces 𝑞(𝑥) and the time

dependent lateral forces 𝑝(𝑥; 𝑡) and 𝑝𝑖 (𝑡). These

loadings produce internal moment distributions

𝑀(𝑥; 𝑡), 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 𝜓(𝑥) 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 𝑝(𝑥) (see Fig. 8-7) is merely the weight of the beam, that is,

𝑝(𝑥) = 𝑚(𝑥) 𝑔, where 𝑚(𝑥) is the mass distribution and 𝑔 is the acceleration of gravity. The frequency then is

evaluated on the basis of the deflected shape 𝑣𝑑 (𝑥) 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 𝜓(𝑥) = 𝑣𝑑 (𝑥)/𝑍0 is the shape function computed from the dead load.