17.4. Harmonic Response Analyses

Theory Reference

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17.4. Harmonic Response Analyses

The following harmonic response analysis topics are available:

Assumptions and RestrictionsDescription of AnalysisComplex Displacement OutputNodal and Reaction Load ComputationSolutionVariational Technology MethodAutomatic Frequency SpacingRotating Forces on Rotating Structures

The harmonic response analysis (ANTYPE,HARMIC) solves the time-dependent equations of motion ( ) for linear structures undergoing steady-state vibration.

17.4.1. Assumptions and Restrictions

1. Valid for structural, fluid, magnetic, and electrical degrees of freedom (DOFs). Thermal DOFs may be present in a coupled field harmonic response analysis using structural DOFs.

2. The entire structure has constant or frequency-dependent stiffness, damping, and mass effects.

3. All loads and displacements vary sinusoidally at the same known frequency (although not necessarily in phase).

4. Element loads are assumed to be real (in-phase) only, except for:

a. current densityb. pressures in SURF153 and SURF154

17.4.2. Description of Analysis

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Consider the general equation of motion for a structural system ( ).

where:[M] = structural mass matrix

[C] = structural damping matrix

[K] = structural stiffness matrix

{ } = nodal acceleration vector

{ } = nodal velocity vector

{u} = nodal displacement vector

{Fa} = 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:

where:umax = maximum displacement

i = square root of -1

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

t = time

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= displacement phase shift (radians)

Note that umax and may be different at each DOF. The use of complex notation allows a

compact and efficient description and solution of the problem. can be rewritten as:

or as:

where:

{u1} = {umax cos } = real displacement vector (input as VALUE on D command, when

specified)

{u2} = {umax sin } = imaginary displacement vector (input as VALUE2 on D command,

when specified)

The force vector can be specified analogously to the displacement:

where:Fmax = force amplitude

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= force phase shift (radians)

{F1} = {Fmax cos } = real force vector (input as VALUE on F command, when specified)

{F1} = {Fmax sin } = imaginary force vector (input as on VALUE2 on F command, when

specified)

Substituting and into gives:

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

The solution of this equation is discussed later.

17.4.3. Complex Displacement Output

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

1. The same form as u1 and u2 as defined in (selected with the

command HROUT,ON).2. The form umax and (amplitude and phase angle (in degrees)), as defined in

(selected with the command HROUT,OFF). These two terms are computed at each DOF as:

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

17.4.4. Nodal and Reaction Load Computation

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

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

where:

[Me] = element mass matrix

{u1}e = element real displacement vector

{u2}e = element imaginary displacement vector

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

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where:

[Ce] = element damping matrix

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 {u1}e. The imaginary static

load is computed also the same way, using the imaginary part {u2}e. Note that the imaginary

part of the element loads (e.g., {Fpr}) 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.

17.4.5. Solution

Four methods of solution to are available: full, reduced, mode superposition, and Variational Technology and each are described subsequently.

17.4.5.1. Full Solution Method

The full solution method (HROPT,FULL) solves directly. may be expressed as:

where c denotes a complex matrix or vector. is solved using the same sparse solver used for a static analysis in Equation Solvers, except that it is done using complex arithmetic.

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17.4.5.2. Reduced Solution Method

The reduced solution method (HROPT,REDUC) uses reduced structure matrices to solve the equation of motion ( ). This solution method imposes the following additional assumptions and restrictions:

1. No element load vectors (e.g., pressures or thermal strains). Only nodal forces applied directly at master DOF or acceleration effects acting on the reduced mass matrix are permitted.

2. Nonzero displacements may be applied only at master DOF.

This method usually runs faster than the full harmonic analysis by several orders of

Substructuring Analysis is used so that the matrix representing the system will be reduced to the essential DOFs required to characterize the response of the system. These essential

discussed in Automatic Master Degrees of Freedom Selection and guidelines for their manual selection are given in Modal Analysis of the Structural Analysis Guide. The reduction of

for the reduced method results in:

where the ^ denotes reduced matrices and vectors. These equations, which have been

reduced to the master DOFs, are then solved in the same way as the full method. may contain prestressed effects (PSTRES,ON) corresponding to a non-varying stress state, described in Stress Stiffening.

17.4.5.2.1. Expansion Pass

The reduced harmonic response method produces a solution of complex displacements at

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(EXPASS,ON). As in the full method, both a real and imaginary solution corresponding to { 1)

and { 2) can be expanded (see ) and element stresses obtained (HREXP,

ALL).

Alternatively, a solution at a certain phase angle may be obtained (HREXP,ANGLE). The solution is computed at this phase angle for each master DOF by:

where:

max = amplitude given by

= computed phase angle given by

' = input as ANGLE (in degrees), HREXP Command

This solution is then expanded and stresses obtained for these displacements. In this case, only the real part of the nodal loads is computed.

17.4.5.3. Mode Superposition Method

The mode superposition method (HROPT,MSUP) uses the natural frequencies and mode shapes to compute the response to a sinusoidally varying forcing function. This solution method imposes the following additional assumptions and restrictions:

1. Nonzero imposed harmonic displacements are not allowed.2. There are no element damping matrices. However, various types of system damping

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The development of the general mode superposition procedure is given in Mode Superposition Method. The equation of motion ( ) is converted to modal form, as described in Mode Superposition Method. is:

where:yj = modal coordinate

j = natural circular frequency of mode j

i = fraction of critical damping for mode j

fj = force in modal coordinates

The load vector which is converted to modal coordinates ( ) is given by

where:

{Fnd} = nodal force vector

s = load vector scale factor, (input as FACT, LVSCALE command)

{Fs} = load vector from the modal analysis (see Mode Superposition Method).

For a steady sinusoidal vibration, fj has the form

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where:fjc = complex force amplitude

For to be true at all times, yj must have a similar form as fj, or

where:yjc = complex amplitude of the modal coordinate for mode j.

Differentiating , and substituting and into ,

Collecting coefficients of yjc and dividing by (eit)

solving for yjc,

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The contribution from each mode is:

where:

{Cj} = contribution of mode j (output if Mcont = ON, on the HROUT command)

{ j} = mode shape for mode j

Finally, the complex displacements are obtained from as

where:{uc} = vector of complex displacements

If the modal analysis was performed using the reduced method (MODOPT,REDUC), then the

vectors {} and {u c} in the above equations would be in terms of the master DOFs (i.e.

and { c}).

In the case of the QR damped mode extraction method, one substitutes for , so becomes:

Solving the above equation and multiplying by the eigenvectors, one can calculate the

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17.4.5.3.1. Expansion Pass

The expansion pass of the mode superposition method involves computing the complex displacements at slave DOFs (see ) if the reduced modal analysis was used ( MODOPT,REDUC)) and computing element stresses. The expansion pass is the same as the reduced method discussed in the previous section.

17.4.6. Variational Technology Method

A common way to compute the harmonic response of a structure is to compute the normal modes of the undamped structure, and to use a modal superposition method to evaluate the response, after determining the modal damping. Determining the modal damping can be based on modal testing, or by using empirical rules. However, when the structure is non-metallic, the elastic properties can be highly dependent on the frequency and the damping can be high enough that the undamped modes and the damped modes are significantly different, and an approach based on a real, undamped modes is not appropriate.

One alternative to straight modal analysis is to build multiple modal bases, for different property values, and combine them together over the frequency range of the analysis. This technique is complex, error prone, and does not address the problem of determining the modal damping factors. Another alternative is a direct frequency response, updating the elastic properties for every frequency step. This technique give a much better prediction of the frequency response, but is CPU intensive.

The variational technology method (HROPT,VT) is available as the harmonic sweep capability of the VT Accelerator add-on. You can define the material elastic properties as being frequency-dependent (using TB,ELASTIC and TB,SDAMP) and efficiently compute the frequency response over an entire frequency range. For the Variational Technology theory, see Guillaume([333.]) and Beley, Broudiscou, et al.([360.]).

17.4.6.1. Viscous or Hysteretic Damping

When using the Variational Technology method, the user has a choice between viscous and

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Viscous Damping

Consider a spring-damper-mass system subjected to a harmonic excitation. The response of the system is given by:

Due to the damping, the system is not conservative and the energy is dissipated. Using viscous damping, the energy dissipated by the cycle is proportional to the frequency, . In a single DOF spring-mass-damper system, with a viscous damper C:

where:U = change of energy

C = viscous damper

Hysteretic Damping

Experience shows that energy dissipated by internal friction in a real system does not depend

on frequency, and approximately is a function of :

where:

= frequency-dependent damping (input using TB,SDAMP command)

damping is known as structural or hysteretic damping. It can be included in the elastic

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where:E = Young's modulus (input using TB,ELASTIC command)

Using this kind of representation, the equations of motion of the system become:

where:[M] = structural mass matrix

[K] = structural stiffness matrix

[H] = structural damping matrix

{ } = nodal acceleration vector

{u} = nodal displacement vector

{Fa} = applied load vector

17.4.7. Automatic Frequency Spacing

In harmonic response analysis, the imposed frequencies that involve the most kinetic energy are those near the natural frequencies of the structure. The automatic frequency spacing or

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 17.2: Frequency Spacing shows two typical resonance peaks and the imposed frequencies chosen by this method, which are computed from:

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gives frequencies slightly less than the natural circular frequency j.

gives slightly higher frequencies. The spacing parameter aij is defined as:

where:i = modal damping as defined by . (If i is computed as 0.0, it is redefined to

be 0.005 for this equation only).

N = 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.

j = 1, 2, 3, ... N

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

Figure 17.2: Frequency Spacing

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17.4.8. Rotating Forces on Rotating Structures

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

General rotating asynchronous forces 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 ( ) except for the coefficients of the damping matrix [C] which will be a function of the rotational velocity of the structure (see the CORIOLIS command). [C] will be updated for each excitation frequency step using the following rotational velocity:

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where: = rotational velocity of the structure (rd/s)

= frequency of excitation (rd/s)

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

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

17.4.8.1. 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:

where:F = amplitude of force

Using complex notations, the equations become:

where:

i = square root of -1 Fa = Fcos = real force in Y-direction; also, negative of imaginary force in Z-direction

(input as VALUE on F command, label FY; input as VALUE2 on F command, label FZ)

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direction (input as VALUE on F command, label FZ; input as VALUE2 on F command, label FY)

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

17.4.8.2. 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 17.3: Mass Unbalance at Node I

Figure 17.3: 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:

where:m = mass unbalance

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e = 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

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:

where:F = me

Using complex notations, it can be written as:

Note: The multiplication of the forces by 2 is done internally at each frequency step.

Release 12.0 - © 2009 SAS IP, Inc. All rights reserved.

17.4. Harmonic Response Analyses

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