Structural Dynamics
Lecture 3
Outline of Lecture 3
Single-Degree-of-Freedom Systems (cont.)
Forced Vibrations due to Arbitrary Excitation.
D’Alembert’s Principle.
Vibrations due to Movable Support.
Earthquake Excitations.
Vibrations due to Indirectly Acting Dynamic Loads.
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Structural Dynamics
Lecture 3
Single-Degreee-of-Freedom Systems (cont.)
Forced Vibrations due to Arbitrary Excitation
Equations of motion of a linear viscous damped system:
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Box 1 : Equation of momentum
: Impulse (momentum) of external and internal forces, [kg s].
Equation of momentum:
The increase of momentum
of the particle during a time
interval
is equal to the impulse, i.e. the time integral over the same
interval of all external and internal forces.
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The load
is constant for
and zero out-side this
interval. The load is applied at the time , where the displacement is
and the velocity is
. The impulse of
is equal to
for
arbitrary
. Assume that
is small compared to the undamped
eigenperiod of the oscillator. Then, we may let
without any
physical consequences for the response. The shape of the load
is
arbitrary, only its impulse is important.
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Lecture 3
is unaffected by the dynamic load and is continuous for
to the inertia of the mass. This implies that the displacement
time
immediately after the cease of the load is given as:
due
at the
In contrast, the velocity is discontinuous and makes a finite jump. The
velocity
is searched immediately after the load has ceased.
Due to (3) it follows that:
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Structural Dynamics
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Then, for an impulsive dynamic load (short duration, large intensity), it
follows from (2):
An impulsive load for
delta function
:
Especially,
applied at
is denoted a unit impulse, if
can be described by Dirac’s
.
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Lecture 3
Let a unit impulse be applied at
, i.e.
to a SDOF oscillator at rest at
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Structural Dynamics
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Then, from (3) and (5):
The system performs damped eigenvibrations for t > 0 with the initial
conditions (8). The response is given as (see Lecture 2, Eq. (6)):
is denoted the impulse response function.
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Structural Dynamics
Lecture 3
The load history
is divided into differential equivalent impulses of the
magnitude
(hatched area). The response at the time from a
differential impulse applied at the time
to a system at rest becomes:
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Structural Dynamics
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The response from all differential impulses in the time interval [0, t] follows
from the superposition principle:
(12) is a particular integral for (1), known as Duhamel’s integral. (11)
describes the motion, if the impulse is applied at a system at rest at the
time
, i.e.
. Hence, (12) describes the total
response of a system at rest at
, i.e.
.
Then, the total solution of (1) follows from (12) and the complementary
solution (eigenvibration) from the initial values
and
,
cf. Lecture 2, Eq. (6):
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Structural Dynamics
Lecture 3
Example 1 : Undamped SDOF exposed to a ramp load
. Then,
in Eq. (9) reduces to
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The integrals in Eq. (15) may be evaluated analytically.
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Relation between
and
:
The impulse response function
and the frequency response function
describes the same dynamic system exposed to standardized
dynamic loads. There must be a relation between these two system
defining functions.
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Then,
is determined by the inverse Fourier integral transform
Hence,
and
are mutual Fourier transforms.
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Structural Dynamics
Lecture 3
Box 2 : Formal derivation of Fourier’s integral theorem
The complex representation of the Fourier series of
reads, cf. Lecture 1, Eqs. (21), (22):
where
for
is given as
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Lecture 3
for
follows from
.
(18) and (19) may be written as
where
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Structural Dynamics
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As
, so
is described by a continuous frequency
parameter , (20) attains the form
(22) merely represents the result of a formal mathematical operation, i.e.
the convergence of (20) towards (22) is not guaranteed. However, the
convergence can be proven, if
is piece-wise differential and absolute
integrable, i.e. if
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Structural Dynamics
Lecture 3
(22) is known as Fourier's integral theorem, and
is known as the
Fourier transform of
. In a discontinuity point, so
, the 1st relation in (22) converges to:
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Lecture 3
D’Alembert’s Principle
Newton’s 2nd law of motion for the free mass reads:
The equation may be rewritten in the form:
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Structural Dynamics
Lecture 3
Formally, (26) may be interpreted as a statical equilibrium equation, where
a so-called inertial force
is applied in the same direction as
(and
). Its magnitude is
.
d’Alembert’s principle:
Inertial forces
(inertial moments for rotational degrees of
freedom) are applied to all masses along with external and internal
damping and spring forces in the direction of the degrees of freedom, and
the equations of motion are next obtained from static equilibrium
equations of the system.
d’Alembert’s principle is a formal operation easing the formulation of
dynamic equations of motion as an alternative to Newton’s 2nd law of
motion and to analytical dynamics. “Inertial forces” are fictitious quantities
with no physical meaning.
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Structural Dynamics
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Example 2 : Horizontal plane beam
: Bending stiffness, [N m2].
: Mass per unit length, [kg/m].
SDOF system: The degree of freedom is selected as the vertical
displacement
of the mass. The beam is cut free from the damper and
the spring. The internal damping force
and spring force
are
applied as external forces with the signs as shown on Fig. 7b.
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Structural Dynamics
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Use of the equation of moment of momentum:
The equation of moment of momentum is formulated around point
the clock-wise direction, cf. Lecture 1, Eq. (46):
in
Moment of external forces around (no contribution from the unknown
reaction force ) in the clock-wise direction:
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Structural Dynamics
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Use of d’Alembert’s principle:
The inertial force
equilibrium around
The reaction
equation:
is applied as shown on Fig. 7b. Statical moment
provides:
is obtained from the vertical static force equilibrium
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Structural Dynamics
Lecture 3
The undamped angular eigenfrequency and the damping ratio become:
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Vibrations due to Movable Support
Horizontal support motion due to earthquakes or heavy traffic.
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Structural Dynamics
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Plane single-degree-of-freedom shear frame exposed to a horizontal
ground surface motion.
: Mass of infinite rigid storey beam, [kg].
: Total shear stiffness of both massless columns, [N/m].
: Linear viscous damping coefficient, [N s/m].
: Horizontal displacement of storey beam from static equilibrium
state, [m].
: Horizontal ground surface motion, [m].
: Shear force in both columns incl. the damper force, [N].
.
: Reaction force on the ground surface, [N].
.
The ground surface motion
and its time derivatives
and
assumed to be known as a function of time.
may be measured
with an accelerometer or a seismograph.
are
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Structural Dynamics
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The storey mass is cut free from the columns and the damper, and the
shear force
is applied on the free mass with the sign shown on Fig.
8b. Newtons 2nd law of motion provides:
The motion of the storey beam relative to the ground surface becomes:
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Structural Dynamics
Lecture 3
Equation of motion and reaction force on the ground surface formulated in
the relative motion:
(36) is preferred in earthquake engineering, because
available, via measurements.
is directly
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Example 3 : Harmonic motion of vibration isolated mass
: Vertical motion of mass from statical equilibrium state, [m].
: Vertical motion of ground surface from referential position, [m].
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Structural Dynamics
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Equation of motion of free mass:
Let
be harmonically varying with the amplitude , the angular
frequency and the phase . Then, cf. Lecture 2, Eq. (33):
The right-hand side of (38) becomes:
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Structural Dynamics
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Then, the stationary response follows from, see Lecture 2, Eqs. (38), (39):
The dynamic amplification factor becomes:
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Structural Dynamics
Lecture 3
Earthquake Excitations
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(36) is rewritten on the form
represents the ground surface acceleration caused by a given design
earthground. From (47) the maximum relative displacement registered
during the earthquake may be obtained by numerical time integration:
is denoted the spectral displacement. The spectral velocity and the
spectral acceleration are defined based on the spectral displacement as:
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Structural Dynamics
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Fig. 9b shows the variation of the normalized spectral velocity for the NS
component of the El Centro earthquake as a function of the damping ratio
and the undamped angular eigenfrequency
of the structure. Notice
that
is usually different from the maximum relative
acceleration, i.e.
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Structural Dynamics
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In earthquake resistant design
is applied as a horizontal
inertial load on the structure in agreement with d’Alembert’s principle,
and next the structure is designed by usual static analysis. For the SDOF
shear frame in Fig. 8 this provides the maximum relative displacement:
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Structural Dynamics
Lecture 3
Vibrations due to Indirectly Acting Dynamic Loads
Often the dynamic load is acting indirectly on the masses via a transmission
through a linear elastic massless structure.
: Displacement of mass. Degree of freedom of the system.
: Displacement of the structure in the direction of the dynamic
force. Auxiliary degree of freedom.
: Flexibility coefficient. Displacement (rotation) of the degree of
freedom from a unit force (moment) applied at the degree of
freedom .
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Structural Dynamics
Lecture 3
D’Alemberts principle is used. The displacement
is caused by the
quasi-state contribution from the inertial load
at the degree of
freedom
and the external dynamic load
at the degree of freedom
. The linearity of the system (the superposition principle) provides:
(53) is merely an application of the force method of statics.
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Box 3 : Flexibility coefficients
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: Bending stiffness.
: Length of structure.
: Moment distribution from unit force (moment) at the degree
of freedom .
Principle of complementary virtual work:
From the right-hand side of (54) follows:
(55) is known as Maxwell-Betti’s reciprocal theorem.
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Structural Dynamics
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If the degrees of freedom and represents a translation and a rotation,
respectively, the theorem states that the displacement in the degree of
freedom from a unit moment acting in the degree of freedom is equal
to the rotation in the degree of freedom from a unit force acting in the
degree of freedom .
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Structural Dynamics
Lecture 3
Summary of Lecture 3
Forced Vibration due to Arbitrary Excitation
Impulse of loads on a free particle. Equation of momentum.
Impulsive loads (short duration, large intensity). The displacement
is
continuous, whereas the velocity
is discontinuous.
Impulse response function
. Related to the frequency response
by
Fourier’s integral theorem.
Duhamel’s integral. Defines a particular integral (stationary motion) to the
equation of motion.
D’Alembert’s Principle
Inertial force (moment)
is applied to the free particle along with the
internal (
) and external forces (
). Dynamic equations of
motion are formally obtained by static equilibrium equations.
The principle facilitates the formulation of equations of motion. “Inertial
forces” are fictitious quantities. The physical reality is Newton’s 2nd law of
motion.
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Structural Dynamics
Lecture 3
Earthquake Excitations
Spectral displacement,
.
Spectral acceleration,
.
The spectral acceleration is used as an equivalent static load in earthquake
resistant structural design in accordance with d’Alembert’s principle.
Vibrations due to Indirectly Acting Dynamic Loads
Dynamic load are not acting at the mass, but is transferred via a linear elastic
massless structure.
Flexibility coefficients
(Maxwell-Betti’s reciprocal theorem).
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