mech dynamics 14.5 l05 harmonic analysis

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14.5 Release

Lecture 5 Harmonic Analysis

ANSYS Mechanical Linear and Nonlinear Dynamics

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Harmonic Analysis

Topics Covered

A. What is Harmonic Analysis

B. Theory and Terminology

C. Contact in Harmonic Analysis

D. Full Harmonic Analysis

E. Damping in Full Harmonic Analysis

F. Loads and Boundary Conditions

G. Analysis settings – Full Harmonic

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Harmonic Analysis

… Topics Covered

H. Mode-superposition Harmonic Analysis

I. Damping in Mode-superposition Harmonic Analysis

J. Analysis settings – Mode Superposition anslysis

K. Workshop 5

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A. What is Harmonic Analysis

• Input:

– Harmonic loads (forces, pressures, and imposed displacements) of known magnitude and frequency.

– May be multiple loads all at the same frequency.

– Forces and displacements can be in-phase or out-of phase.

– Body loads can only be specified with a phase angle of zero.

• Output:

– Harmonic displacements at each DOF, usually out of phase with the applied loads.

– Other derived quantities, such as stresses and strains.

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... What is Harmonic Analysis

• Assumptions and Restrictions:

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

– No nonlinearities are permitted.

– Transient effects are not calculated.

– Acceleration, bearing, and moment loads are assumed to be real (in-phase) only.

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... What is Harmonic Analysis

• Assumptions and Restrictions:

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

– All loads and displacements, both input and output, are assumed to occur at the same frequency.

– Calculated displacements are complex if: – damping is specified, or

– applied load is complex.

angle phase

freqency

amplitude where

sin

F

iii tFF

---- F1

---- F2

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• Governing equation for a mass-spring-damper system, subject to a sinusoidal force is

tfkuucum sin

21 nd

22221 nn

kfu

2

1

1

2tan

n

n

B. Theory and Terminology

/

uk/f

/

u

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... Theory and Terminology

/

uk/f

/

u • When the imposed frequency approaches a natural frequency in the direction of excitation, resonance occurs.

• an increase in damping decreases the amplitude of the response for all imposed frequencies,

• a small change in damping has a large effect on the response near resonance, and

• the phase angle always passes through ±90° at resonance for any amount of damping.

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... Theory and Terminology

• The governing equation for a linear structure is:

• Assume {F} and {u} are harmonic with frequency :

Note: The symbols an differentiate the input from the output: = input (imposed) circular frequency

= output (natural) circular frequency

FuKuCuM

tii

tii

eeuu

eeFF

max

max

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... Theory and Terminology

• Take two time derivatives:

• Substitute and simplify:

• This can then be solved using one of two methods.

ti

ti

ti

euiuu

euiuiu

euiuu

21

2

21

21

2121

2 FiFuiuKCiM

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... Theory and Terminology

• Solution Techniques:

– Full Harmonic Response Analysis

– solves a system of simultaneous equations directly using a static solver designed for complex arithmetic:

– Mode Superposition Response Analysis

– expresses the displacements as a linear combination of mode shapes.

ccc

FuK

FuK

FiFuiuKCiM

ccc

2121

2

jcjcjjj fyi

FiFuiuKCiM

22

2121

2

2

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• Contact regions are available in harmonic analysis; however, since this is a purely linear analysis, contact behavior will differ for the nonlinear contact types, as shown below:

• Contact behavior will reduce to its linear counterparts.

Contact Type Static Analysis

Linear Dynamic Analysis

Initially Touching Inside Pinball

Region Outside Pinball Region

Bonded Bonded Bonded Bonded Free

No Separation No Separation No Separation No Separation Free

Rough Rough Bonded Free Free

Frictionless Frictionless No Separation Free Free

Frictional Frictional = 0, No Separation

> 0, Bonded Free Free

C. Contact in Harmonic Analysis

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D. Full Harmonic Analysis

• Exact solution.

• Generally slower than MSUP.

• Supports all types of loads and boundary conditions.

• Solution points must be equally distributed across the frequency domain

• Solves the full system of simultaneous equations using the Sparse matrix solver for complex arithmetic.

ccc

FuK

FuK

FiFuiuKCiM

ccc

2121

2

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E. Damping in Full Harmonic Analysis

1. Rayleigh Damping:

– Alpha damping and Beta damping are used to define Rayleigh damping constants α and β. The damping matrix [C] is calculated by using these constants to multiply the mass matrix [M] and stiffness matrix [K]:

Equivalent damping KMC 22

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... Damping in Full Harmonic Analysis

1. Material Damping:

• Material damping is inherently present in a material (energy is dissipated by internal friction), so it is typically considered in a dynamic analysis.

• Energy dissipated by internal friction in a real system does not depend on the cyclic frequency.

• The simplest device to represent it is to assume the damping force is proportional to velocity and inversely proportional to frequency

g = constant structural damping ratio

KgC

2

gEquivalent damping

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• The complete expression for the structural damping matrix, [C], is

• g is constant damping.

dampingicViscoelast

1

dampingGyroscopic

1

dampingElement

1

dampingStructural

1

dampingMass

1

1

22

vge

mb

ma

N

l

m

N

l

l

N

k

k

N

j

jj

m

j

N

i

i

m

i

CGC

KgKg

MMC

... Structural Damping Matrix [C]

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The value of g, and can be input using the following:

[1] Material-dependent damping value

(Mass-Matrix Damping Multiplier, and k-Matrix Damping

Multiplier)

…. Structural Damping Matrix [C]

mbma N

j

jj

m

j

N

i

i

m

i KgMC11

2 gi

i

i 22

Equivalent damping

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[2] Directly as global damping value

(Details section of Analysis Settings)

…. Structural Damping Matrix [C]

KgMC

2 gi

i

i 22

Equivalent damping

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F. Loads and Boundary Conditions

• Structural loads and supports may also be used in harmonic analyses with the following exceptions:

• Loads Not Supported:

– Gravity Loads

– Thermal Loads

– Rotational Velocity

– Pretension Bolt Load

– Compression Only Support (if present, it behaves similar to a Frictionless Support)

• Remember that all structural loads will vary sinusoidally at the same excitation frequency

• Loads can be out of phase with each other.

• Transient effects are not calculated.

• Remote Force, Moment, and Acceleration loads may be defined, although these loads are assumed to act at a phase angle of zero.

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... Loads and Boundary Conditions

• A list of supported loads are shown below:

• Not all available loads support phase input. Accelerations, Bearing Load, and Moment Load will have a phase angle of 0°.

– If other loads are present, shift the phase angle of other loads, such that the Acceleration, Bearing, and Moment Loads will remain at a phase angle of 0°.

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... Loads and Boundary Conditions

• Specifying harmonic loads requires:

1. Amplitude Fimax

2. phase angle , and

3. Frequency

angle phase

freqency

amplitudemax

where

sinmax

iF

iii tFF

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... Loads and Boundary Conditions

• Amplitude and phase angle • The load value (magnitude) represents the amplitude (F1max and F2max).

• Phase angle is the phase shift between two or more harmonic loads.

• is not required if only one load is present.

---- F1

---- F2

Amplitude

Phase Angle

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F. Analysis Settings – Full Harmonic

• Analysis Settings > Options

– Frequency Range: Specified in cycles per second (Hertz)

• Range Minimum >> Minimum Frequency

• Range Maximum >> Maximum Frequency

– Solution Intervals

– Solution Method

A range of 0-500 Hz with 10 solution intervals gives solutions at frequencies of 50, 100, 150, …, 450, and 500 Hz. Same range with 1 substep gives one solution at 500 Hz.

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F. Analysis Settings – Full Harmonic

• Analysis Settings > Options

– Solution Intervals

Evenly-spaced

frequency points

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... Analysis Settings – Full Harmonic

• Analysis Settings > Output Controls

• Analysis Settings > Damping Controls

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... Full Harmonic Analysis

Analysis Setting > Solution Method > Full

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... Results- Frequency Response

• Frequency Response:

• display how the response varies with frequency

Frequency (Hz)

Frequency (Hz)

Am

plit

ud

e (

m)

Ph

ase

An

gle

(o)

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... Results- Phase Response

• Phase Response:

• show how much a response lags behind the applied loads.

Angle (o)

----- Force ----- Output

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... Results – Contour Plots

• Contour plots include: – stress,

– elastic strain, and

– deformation.

• For these results, you must specify a frequency and phase angle.

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... Results – Contour Plots

• A contour result can be created from a Frequency Response.

• The Phase Angle of the contour result has the same magnitude as the frequency result type but an opposite sign.

• The sign of the phase angle is reversed so that the response amplitude of the frequency response plot for that frequency and phase angle matches with the contour results.

RMB

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Note: The sign of the phase angle

in the contour result is reversed so

that the response amplitude of the

frequency response plot for that

frequency and phase angle

matches with the contour results.

RMB

1. By Frequency

... Results – Contour Plots

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RMB

2. By: Maximum Over Frequency

... Results – Contour Plots

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RMB

3. By: Frequency of Maximum

... Results – Contour Plots

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RMB

4. By: Maximum over Phase

... Results – Contour Plots

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RMB

5. By: Phase of Maximum

Note: The sign of the

phase angle is reversed.

... Results – Contour Plots

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14.5 Release

Mode Sup Harmonic Analysis

ANSYS Mechanical Linear and Nonlinear Dynamics

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G. Mode-superposition Harmonic Analysis

• Approximate solution; accuracy depends on whether an adequate number of modes have been extracted.

• Generally faster than FULL.

• Does not support nonzero imposed harmonic displacements.

• Solution points may be either equally distributed across the frequency domain or clustered about the natural frequencies of the structure.

• Solves an uncoupled system of equations by performing a linear combination of orthogonal vectors (mode shapes).

jcjcjjj fyi

FiFuiuKCiM

22

2121

2

2

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… Mode Superposition Method • Example:

– Here, the sum of mode shape 1 and mode shape 2 approximates the final response. Since mode shapes are relative, the coefficients y1 and y2 are required.

– Mode shapes (eigenvectors) are also known as generalized coordinates, and in this case, coefficients y1 and y2 are the DOF.

y1 y2 + =

1 2

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22

i

i

md

ii

Stiff. Coef.

Mass Coef. Constant

ratio

H. Damping in Mode-Sup Harmonic Analysis

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I. Analysis Settings – Mode-Sup Harmonic

• Analysis Settings > Options

– Frequency Range

• Range Minimum >> Minimum Frequency

• Range Maximum >> Maximum Frequency

– Solution Intervals

– Solution Method > Mode Superposition

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• Analysis Settings > Options

– Cluster Results > Yes

Without Cluster Option

… Analysis Settings – Mode-Sup Harmonic

With Cluster Option

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• Analysis Settings > Options

– Include Residual Vector

• In MSUP analysis, the dynamic response will be approximate when the applied loading excites the higher frequency modes of a structure.

• The residual vector method:

– employs additional modal transformation vectors in addition to the eigenvectors in the modal transformation .

– accounts for high frequency dynamic responses with fewer eigen-modes.

... Analysis Settings – Mode-Sup Harmonic

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... Mode-superposition Harmonic Analysis

• Setup a mode-sup transient analysis in the schematic by:

1. linking a modal system to a transient structural system at the solution level.

• Notice in the transient branch, the modal analysis result becomes an initial condition.

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... Mode-superposition Harmonic Analysis

2. Or, Analysis Setting > Solution Method > Mode Superposition

(Standalone Analysis)

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14.5 Release

ANSYS Mechanical Linear and Nonlinear Dynamics

Workshop 5 Harmonic Response (Fixed-Fixed Beam)