Consider the general equation of motion for a structural system (Equation 15–5).

(15–58)

where:

= structural mass matrix

= structural damping matrix

= structural stiffness matrix

= nodal acceleration vector

= nodal velocity vector

= nodal displacement vector

= applied load vector

As stated above, all points in the structure are moving at the same known frequency, however, not necessarily in phase. Also, it is known that the presence of damping causes phase shifts. Therefore, the displacements may be defined as:

(15–59)

where:

= maximum displacement

= imposed circular frequency (radians/time) =

= imposed frequency (cycles/time) (input as FREQB and FREQE on the HARFRQ command)

= time

= displacement phase shift (radians)

Note that and may be different at each DOF. The use of complex notation allows a compact and efficient description and solution of the problem. Equation 15–59 can be rewritten as:

(15–60)

or as:

(15–61)

where:

= real displacement vector (input as VALUE on D command, when specified)

= imaginary displacement vector (input as VALUE2 on D command, when specified)

The force vector can be specified analogously to the displacement:

(15–62)

(15–63)

(15–64)

where:

= force amplitude

= force phase shift (radians)

= real force vector (input as VALUE on F command, when specified)

= imaginary force vector (input as on VALUE2 on F command, when specified)

Substituting Equation 15–61 and Equation 15–64 into Equation 15–58 gives:

(15–65)

The dependence on time () is the same on both sides of the equation and may therefore be removed:

(15–66)

The solution of this equation is discussed later.

The complex displacement output at each DOF may be given in one of two forms:

(15–67)

(15–68)

Note that the response lags the excitation by a phase angle of .

Inertia, damping and static loads on the nodes of each element are computed.

The inertia load is expressed as:

(15–69)

where:

= vector of element inertia forces

= element mass matrix

= element displacement vector

The real and imaginary inertia load parts of the element output are then computed by:

(15–70)

(15–71)

where:

= vector of element inertia forces (real part)

= element real displacement vector

= vector element inertia forces (imaginary part)

= element imaginary displacement vector

The damping load is expressed as:

(15–72)

where:

= vector of element damping forces

= element damping matrix

The real and imaginary damping loads part of the element output are then computed by:

(15–73)

(15–74)

where:

= vector of element damping forces (real part)

= vector of element damping forces (imaginary part)

The real static load is computed the same way as in a static analysis (Solving for Unknowns and Reactions) using the real part of the displacement solution . The imaginary static load is computed also the same way, using the imaginary part . Note that the imaginary part of the element loads (e.g., ) are normally zero, except for current density loads.

The nodal reaction loads are computed as the sum of all three types of loads (inertia, damping, and static) over all elements connected to a given fixed displacement node.

In a harmonic analysis, the imposed frequencies that involve the most kinetic energy are those near the natural frequencies of the structure. The automatic frequency spacing or "cluster" option (Clust = ON, on the HROUT command) provides an approximate method of choosing suitable imposed frequencies. The nearness of the imposed frequencies to the natural frequencies depends on damping, because the resonance peaks narrow when the damping is reduced. Figure 15.3: Frequency Spacing shows two typical resonance peaks and the imposed frequencies chosen by this method, which are computed from:

(15–91)

(15–92)

Equation 15–91 gives frequencies slightly less than the natural circular frequency . Equation 15–92 gives slightly higher frequencies. The spacing parameter is defined as:

(15–93)

where:

= modal damping. (If is computed as 0.0, it is redefined to be 0.005 for this equation only).

= integer constant (input as NSBSTP, NSUBST command) which may be between 2 and 20. Anything above this range defaults to 10 and anything below this range defaults to 4.

= 1, 2, 3, ... N

Each natural frequency, as well as frequencies midway between, are also chosen as imposed frequencies.

Figure 15.3: Frequency Spacing

The logarithm frequency spacing can be defined over an n^th-octave band or a general-frequency band in a harmonic analysis (the HARFRQ command with LogOpt).

The octave band is defined by the lower-end frequency:

(15–94)

and upper-end frequency:

(15–95)

where the central frequency of the octave band is specified at 16, 31.6, 63, 125, 250, 500, 1000, 2000, 4000, 8000, 16000 Hz for 11 octave bands, and n = 1 (octave band), 2 (1/2 octave band), 3 (1/3 octave band), 6 (1/6 octave band), 12 (1/12 octave band), and 24 (1/24 octave band).

The recursive relationships between the central frequencies of the specified octave bands are:

(15–96)

(15–97)

The logarithm frequency increment is given by:

(15–98)

Where NSBSTP is input by the NSUBST command.

The imposed working frequency is calculated by:

(15–99)

The imposed working frequency is , if NSBSTP = 1. For a general logarithm frequency span, the beginning frequency and ending frequency are defined by the FREQB and FREQE parameters of the HARFRQ command respectively.

If a structure is rotating, forces rotating synchronously or asynchronously with the structure are of interest.

General rotating asynchronous forces in a stationary reference frame are described in General Asynchronous Rotating Force. A specific synchronous force: mass unbalance is shown in Specific Synchronous Forces: Mass Unbalance.

In both cases, the equation solved for harmonic analysis is the same as (Equation 15–66) except for the coefficients of the damping matrix which will be a function of the rotational velocity of the structure (see the CORIOLIS command). will be updated for each excitation frequency step using the following rotational velocity:

(15–100)

where:

= rotational velocity of the structure (rd/s)

= frequency of excitation (rd/s)

= ratio between and (s = 1 for synchronous excitations) (input as RATIO on the SYNCHRO command).

The right-hand term of the equation is given below depending on the force considered.

All developments below apply to equations expressed in a stationary reference frame. See Rotating Structures for more information about reference frames. In a rotating reference frame, a general rotating asynchronous force is the same as in a stationary reference frame except it is calculated replacing s by s’ with:

where:

= rotating reference frame ratio ( = 0 for synchronous excitations) (input as RATIO on the SYNCHRO command).

As a consequence, in this case, a mass unbalance force in a rotating reference frame is a static force.

 

15.4.8.1. General Asynchronous Rotating Force

If the structure is rotating about X axis, then an asynchronous force having its direction in the plane perpendicular to the spin axis is expressed as:

(15–101)

(15–102)

where:

= amplitude of force

Using complex notations, the equations become:

(15–103)

(15–104)

where:

= real force in Y-direction; also, negative of imaginary force in Z-direction. Input as VALUE on F, label FY; input as VALUE2 on F, label FZ

= negative of real force in Z-direction; also, negative of imaginary force in Y-direction. Input as VALUE on F, label FZ; input as VALUE2 on F, label FY

The expression of the forces for structures rotating about another direction than X are developed analogously.

 

15.4.8.2. Specific Synchronous Forces: Mass Unbalance

Consider a structure rotating about the X axis. The mass unbalance m situated at node I with the eccentricity e may be represented as shown in Figure 15.4: Mass Unbalance at Node I

Figure 15.4: Mass Unbalance at Node I

If we only consider the motion in the plane perpendicular to the spin axis (YZ plane), the kinetic energy of the unbalanced mass is written as:

(15–105)

where:

= mass unbalance

= distance from the mass unbalance to the spin axis

= amplitude of the rotational velocity vector of the structure (input as OMEGA or CMOMEGA command). It is equal to the frequency of excitation .

= phase of the unbalance

= instantaneous velocity along Y

= instantaneous velocity along Z

Because the mass unbalance is much smaller than the weight of the structure, the first two terms are neglected. The third term being constant, will have no effect on the final equations.

Applying Lagrange's equations, the force vector is:

(15–106)

(15–107)

where:

Using complex notations, it can be written as:

(15–108)

(15–109)

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Note:  The multiplication of the forces by is done internally at each frequency step.

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Standard harmonic analysis with ocean loading includes the added mass of the water outside of line elements (such as LINK180, BEAM188, BEAM189, PIPE288, and PIPE289). The load vector is calculated based on the loads at a given time, but the standard analysis method usually misses some important effects, as all peak wave loads rarely occur at the same time.

All relevant ocean wave loading effects are accounted for via a specialized variation of the harmonic analysis, the harmonic ocean wave procedure (HOWP).

The HOWP applies to regular waves only (Airy and Wheeler one-component waves, as well as Stokes and Deans Stream Function waves), and works only with the full-solution harmonic analysis method (HROPT,FULL).

The HOWP is accessed via the HROCEAN command. The frequency is obtained automatically, directly from the ocean information (via OCDATA and OCTABLE commands). As with a standard harmonic analysis, a damping matrix must be added separately if one is needed. Ocean loads are calculated with the assumption that the structure is stationary.

For every component of every element, HOWP-specific calculations obtain the real and imaginary loads caused by ocean waves. All other loads (such as those related to pressure, temperature, gravity and buoyancy) are removed. Stated differently, all loads are sinusoidal at the given frequency, but the magnitude and phase angle of every component of every element must be determined. To determine the magnitude and phase angle, a series of static analyses is performed along the phase angle ranging from 0 to 360 degrees. (The number of these analyses is controlled via the HROCEAN command.) The result is a roughly sinusoidal force pattern, as shown in this figure:

Figure 15.5: Force History

From this force history, the maximum and minimum forces are calculated by fitting the highest and lowest points. Then:

(15–110)

(15–111)

(15–112)

so that is the coefficient on the real load vector and is the coefficient on the imaginary load vector.