Structural Dynamics

Lecture 2

Outline of Lecture 2

� Single-Degree-of-Freedom Systems (cont.)

� Linear Viscous Damped Eigenvibrations.

� Logarithmic decrement.

� Response to Harmonic and Periodic Loads.

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� Single-Degreee-of-Freedom Systems (cont.).

� Linear Viscous Damped System

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� Linear Viscous Damped Eigenvibrations

. Division with :

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� : Damping ratio. Non-dimensional viscous damping coefficient. Four qualitatively different cases to be considered.

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1) : Undamped system.

2) : Undercritically damped system.

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2) : Undercritically damped system.

3) : Critically damped system.

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4) : Overcritically damped system.

Characteristic values of damping ratio:

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Characteristic values of damping ratio:

� Typical value: (mechanical systems are lightly damped).

� Offshore jacket structure :

� Wind turbine rotor (aerodynamic damping) :

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Undercritically damped systems:

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(6) can be written as

� : Damped angular eigenfrequency, [s-1].

� : Damped eigenvibration period, [s].

� : Phase angle.

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� : Phase angle.

Proof:

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� Logarithmic decrement:

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� Logarithmic decrement:

: Logarithmic decrement.

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� Example 1 : Identification of and from eigenvibration test

Fig. 2 shows the decay of an eigenvibration of a SDOF system.

1) is measured on the curve as the time interval between two succeeding upcrossings of the time axis.

2) Displacements and with the time interval are measured on the

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2) Displacements and with the time interval are measured on the

curve. Then, .

3) Next, follows from (12):

4)

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� Example 2 : Eigenvibrations of a rigid drive train of a wind turbine with a synchronous generator

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A synchronous generator acts as a linear elastic rotational spring with the spring constant , [Nm/rad], for small rotations relative to a referential rotation with the (“nominal”) angular frequency of the generator rotor.

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� : Angular velocity (“rotational speed”) of rotor, [s-1].

� : Angular velocity of generator rotor, [s-1].

� : Mass moment of inertia of rotor, [kg m2].

� : Mass moment of inertia of generator rotor, [kg m2].

� : Mass moment of inertia and radius of gear wheel 1, [kg m2], [m].

� : Mass moment of inertia and radius of gear wheel 2, [kg m2], [m].

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� : Gear ratio.

Single-degree-of-freedom system:

� : Rotational angle of rotor.

� : Auxiliary degrees of freedom of gear wheels and generatorrotor.

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Kinematic constraints:

Lagrange’s equation of motion:

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Lagrange’s equation of motion:

(Lagrange’s function)

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Undamped angular eigenfrequency:

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Undamped angular eigenfrequency:

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� Example 3 : Undamped eigenvibrations of a geared system

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Determine the equation of motion of the system shown in Fig. 4, formulated in the displacement of the mass , and determine the undamped angular eigenfrequency.

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Kinematic constraints of auxiliary degrees of freedom and :

Torsional stiffness of shaft:

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Lagrange equation of motion:

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Undamped angular eigenfrequency:

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� Response to Harmonic and Periodic Loads

Equation of motion:

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Solution:

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Complementary solution for, undercritically damped system (arbitrary eigenvibration):

� : Undamped angular eigenfrequency, [s-1].

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: Undamped angular eigenfrequency, [s ].

� : Damping ratio.

� : Damped angular eigenfrequency, [s-1].

Let the external dynamic force be harmonically varying with the amplitude , the angular frequency , and the phase angle :

� : Complex force amplitude.

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Physical observation:

The stationary motion (the particular solution) after dissipation of eigenvibrations from the initial conditions becomes harmonically varying with the same angular frequency as the excitation and with different amplitude and phase angle :

ω

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Determination of and by insertion of Eqs. (33), (35) and (36) in Eq. (27):

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� : Frequency response function.

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Denominator of :

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Amplitude and phase angle of response, Eqs. (34), (36), (39):

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� : Dynamic amplification factor.

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Dynamic amplification factor :

� Max. value for (exists for ).� Resonance, i.e. : .

� Quasi-static response : .

� High-frequency response : .

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Phase angle :

represents the phase delay of relative to .

� Resonance : .

� Quasi-static response : . ( and in phase).

� High-frequency response : . ( and in counter-phase).

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Stationary response to periodic varying load:

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Physical observation: The stationary motion (the particular solution)due to a periodically varying dynamic load

becomes periodic with the same period , i.e.

Fourier expansions of and :

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Fourier expansions of and :

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Determination of , , :

The mean response represents the static response from themean load :

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The harmonic response component is caused by the harmonic load component with the angular frequency

. From Eqs. (37), (41), (42):

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� Example 4 : Machine with an unbalanced rotating mass

� : Total mass of machine.� : Rotating unbalanced mass.

Eccentricity: .

� Vertical displacement of balancedmass : .

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mass : .� Vertical displacement of unbalanced

mass : .

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Lagrange’s equation of motion:

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From Eq. (33):

From Eqs. (35), (40), (41) and (42):

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� Example 5 : Resonance of undamped SDOF system

Determine the motion of the undamped SDOF system:

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At first, the stationary motion for is determined, and next the limit passing is performed.

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Summary of Lecture 2

� Linear Viscous Damped Eigenvibrations.

� Depends on the damping ratio .

� Undercritically damped structures, .

Structures are lightly damped, .

� Logarithmic decrement .

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� Logarithmic decrement .

� Response to Harmonic and Periodic Loads.

� Determination of a particular integral (stationary motion).

� Response, . Large amplitudes. Dynamic amplification factor

. Rapid change of phase in resonance region from to

.