ANSYS Mechanical APDL Fluids Analysis Guide
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Table of Contents
1. Introduction ............................................................................................................................................ 1
2. Elements for Modeling Thin Films .......................................................................................................... 3
3. Squeeze Film Analysis ............................................................................................................................. 5
3.1. Static Analysis Overview ................................................................................................................... 6
3.2. Harmonic Analysis Overview ............................................................................................................. 6
3.3. Transient Response Analysis Overview .............................................................................................. 7
3.4. Flow Regime Considerations ............................................................................................................. 7
3.5. Modeling and Meshing Considerations ............................................................................................. 8
3.6. Analysis Settings and Options ........................................................................................................... 8
3.7. Loads and Solution ........................................................................................................................... 9
3.8. Review Results .................................................................................................................................. 9
3.9. Sample Harmonic Analysis .............................................................................................................. 10
3.10. Simplified Transient Analysis ......................................................................................................... 13
3.11. Complex Transient Analysis ........................................................................................................... 15
4. Modal Projection Method for Squeeze Film Analysis ........................................................................... 21
4.1. Modal Projection Method Overview ................................................................................................ 21
4.2. Steps in Computing the Damping Parameter Using the Modal Projection Technique ........................ 22
4.2.1. Modeling and Meshing .......................................................................................................... 23
4.2.2. Perform Modal Analysis .......................................................................................................... 23
4.2.3. Extracting Eigenvectors .......................................................................................................... 23
4.2.4. Performing a Harmonic Analysis and Extracting Damping Parameters ..................................... 23
4.3. Example Problem Using the Modal Projection Method .................................................................... 24
4.4. Damping Extraction for Large Signal Cases ...................................................................................... 28
5. Slide Film Damping ............................................................................................................................... 29
5.1. Slide Film Damping Example ........................................................................................................... 31
Index .......................................................................................................................................................... 35
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List of Figures
3.1. Moving Plate above a Fixed Wall .............................................................................................................. 5
3.2. Perforated Plate Structure ....................................................................................................................... 5
3.3. Pressure Distribution on a Plate at a Low Driving Frequency .................................................................... 6
3.4. Damping and Squeeze Stiffness Coefficients vs. Frequency ...................................................................... 7
3.5. Coupling of Nodes at the Hole Periphery with the Center Node of a FLUID138 Element ............................ 8
3.6. Pressure Distribution (Real Component) ................................................................................................ 11
3.7. Pressure Distribution (Imaginary Component) ....................................................................................... 11
3.8. Applied Voltage .................................................................................................................................... 16
3.9. Displacement Time History ................................................................................................................... 16
3.10. Pressure Time History .......................................................................................................................... 17
4.1. Modal Projection Technique for Damping Characterization .................................................................... 21
4.2. Damping and Squeeze Stiffness Parameters for a Rectangular Plate ....................................................... 22
4.3. Fluidic Cross-Talk between Transverse and Rotational Motion ................................................................ 24
4.4. Time-Transient Response from Voltage Pulse ......................................................................................... 28
5.1. Slide Film Damping at Low and High Frequencies .................................................................................. 29
5.2. Viscous Slide Film Element FLUID139 ..................................................................................................... 31
5.3. Comb Drive Resonator .......................................................................................................................... 31
5.4. Displacement of Central Mass ............................................................................................................... 32
5.5. Real and Imaginary Current ................................................................................................................... 32
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List of Tables
2.1. Thin Film Fluid Elements ......................................................................................................................... 3
3.1. Load Options for Thin Film Fluid Elements ............................................................................................... 9
3.2. Beam Model Results Considering Perforated Holes ................................................................................ 10
4.1. Modal Damping Parameters for First Two Eigenfrequencies ................................................................... 24
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Chapter 1: Introduction
A thin film is a small gap of fluid between moving surfaces. This thin layer of fluid can alter the structural response of the structure by adding stiffness and/or damping to the system. Movement normal
to the gap produces a squeeze film effect. Movement tangential to the gap produces a slide film effect.
Thin film effects are important in macrostructures, in hydrodynamic bearings, for example, as well as
in microstructures where the damping and stiffening effects of thin layers of air can significantly affect
the behavior of devices used in micro-electromechanical systems (MEMS). Squeeze film effects are important in devices such as accelerometers and micromirrors. Slide film effects are important in devices
such as comb drives.
Hydrodynamic bearing simulation methods are described in the Rotordynamic Analysis Guide. The information presented in this guide only concerns microstructures.
One method for assessing the effect of thin films for microstructures is to use thin film fluid elements
based on the Reynolds number (which is known from lubrication technology and rarified gas physics)
to calculate the stiffening and damping effects. These effects can then be added to the overall system
model. Separate element types are used to assess squeeze and slide film effects.
If the following conditions are not satisfied, the effects of the thin film cannot be assessed using thin
film elements:
• The governing Reynolds equation limits the application of thin film analyses to structures with lateral dimensions much greater than the gap separation.
• The pressure change across the gap must be much smaller than the ambient (surrounding) pressure.
• Any viscous heating effects must be ignored.
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Chapter 2: Elements for Modeling Thin Films
The ANSYS program has three elements for modeling thin film effects. FLUID136 and FLUID138 are
used to model squeeze film effects. FLUID139 is used to model slide film effects.
Table 2.1: Thin Film Fluid Elements
Element
Dimens.
Shape or Characteristic
DOF
FLUID136
3-D
Quadrilateral, 4 or 8-node
PRES
FLUID138
3-D
Line, 2-node
PRES
FLUID139
1-D
Line, 2-node
UX or UY or
UZ
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Chapter 3: Squeeze Film Analysis
Squeeze film analysis simulates the effects of fluid in small gaps between fixed surfaces and structures
moving perpendicular to the surfaces. Depending on the operating frequencies, the fluid can add stiffening and/or damping to the system. At low frequencies, the fluid can escape before it compresses.
Therefore, the fluid only adds damping to the system. At high frequencies, the fluid compresses before
it can escape. Therefore, the fluid adds both stiffening and damping to the system.
A static analysis is used to determine the damping effects at low frequencies. A harmonic analysis is
used to determine the stiffening and damping effects at high frequencies.
The FLUID136 element is used to model the fluid domain between a fixed surface and a structure
moving normally to that surface. The fluid domain is modeled by overlaying FLUID136 elements on the
moving surface of the structure (i.e., lower surface of the structure adjacent to the fixed wall).
Figure 3.1: Moving Plate above a Fixed Wall
The FLUID138 element is used to model the pressure drop through holes in a moving structure. Microstructures are commonly perforated to reduce the damping effects of the fluid layer. FLUID138 elements
are used in conjunction with FLUID136 elements to obtain a consistent pressure distribution, including
the effects of holes.
Figure 3.2: Perforated Plate Structure
A squeeze film analysis can be performed on structures with known velocities or unknown velocities.
If the velocity profile is known, you can apply it directly to the fluid elements. If the velocity profile is
not known (or is too complicated), it can be determined from the mode-frequency response of the
structure (model projection method). This section deals with directly applying the velocities. Modal
Projection Method for Squeeze Film Analysis (p. 21) covers the modal projection method.
The following squeeze film analysis topics are available:
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Squeeze Film Analysis
3.1. Static Analysis Overview
3.2. Harmonic Analysis Overview
3.3.Transient Response Analysis Overview
3.4. Flow Regime Considerations
3.5. Modeling and Meshing Considerations
3.6. Analysis Settings and Options
3.7. Loads and Solution
3.8. Review Results
3.9. Sample Harmonic Analysis
3.10. Simplified Transient Analysis
3.11. Complex Transient Analysis
3.1. Static Analysis Overview
Use a static analysis to calculate the pressure distribution for low operating frequencies, where the
compression of the fluid (and thus the stiffening effect) is negligible. The resulting pressure distribution
can then be used to extract a damping coefficient that reflects the damping effect of the fluid.
F is the total pressure force at the fluid structure interface and vz is the normal velocity component of
the moving structure. A typical pressure distribution on a rectangular plate is shown in Figure 3.3: Pressure
Distribution on a Plate at a Low Driving Frequency (p. 6).
Figure 3.3: Pressure Distribution on a Plate at a Low Driving Frequency
3.2. Harmonic Analysis Overview
Use a harmonic analysis to calculate the complex pressure distribution for high operating frequencies.
The complex pressure distribution can be used to extract a damping coefficient and a stiffness coefficient.
The phase shift between the pressure and the velocity is caused by compression of the fluid. The real
pressure component contributes to the damping of the system. The imaginary pressure component
contributes to the stiffening of the system.
FRe is the real component of the pressure force. FIm is the imaginary component of the pressure force.
Omega (ω) is the frequency (rad/sec). The damping and stiffness coefficients are frequency-dependent.
Figure 3.4: Damping and Squeeze Stiffness Coefficients vs. Frequency (p. 7) illustrates the frequency
dependency for a typical microsystem application.
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Flow Regime Considerations
Figure 3.4: Damping and Squeeze Stiffness Coefficients vs. Frequency
The DMPEXT command is used to extract frequency-dependent damping parameters (p. 22) for use
with the MDAMP, DMPRAT, ALPHAD, and BETAD command inputs.
3.3. Transient Response Analysis Overview
A transient analysis is often necessary (for example, in problems where a moving structure closes the
gap). When there is contact, it is prudent to use a nonlinear analysis to model the contact more accurately.
A linear transient analysis can be performed when the gap between a moving structure and a fixed
surface remains relatively large. For information on the linear and nonlinear Reynolds equations, refer
to Flow Between Flat Surfaces in the Mechanical APDL Theory Reference.
For transient analysis example problems, see Simplified Transient Analysis (p. 13) and Complex Transient
Analysis (p. 15).
3.4. Flow Regime Considerations
The Reynolds squeeze film approach assumes a continuous fluid flow regime. To ensure a continuous
flow regime, the characteristic length (i.e., gap thickness) must be more than one hundred times larger
than the mean free path of the fluid particles.
Lm(po) is the mean free path of the fluid at the operating pressure P0, and L0 is the mean path at reference
pressure P0. If the fluid is air at atmospheric pressure (1.01325*105), then by definition L0 is approximately
64*10-9 meters (64 nm). So, for continuum theory to be directly applicable (without modification) to air
at atmospheric pressure, the gap should be greater than 6.4 μm (64*10-9 meters * 100).
The applicability of the continuum theory is generally assessed using the Knudsen number, which is
equal to the mean free fluid path divided by the gap. For continuum theory to be valid, the Knudsen
number should be less than 0.01.
For high Knudsen numbers, the continuum theory is not valid. However, the dynamic viscosity can be
adjusted to simulate the high Knudsen number flow regime. The default flow regime for FLUID136 and
FLUID138 is continuum theory (KEYOPT(1) = 0). Set KEYOPT (1) = 1 to specify the high Knudsen number
flow regime. For FLUID136, set KEYOPT(1) = 2 to specify the high Knudsen number flow regime with
surface accommodation factors.
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Squeeze Film Analysis
The type of reflection of the gas molecules at the wall interface is specified using accommodation
factors. Squeeze film models assume diffuse reflection of the gas molecules at the wall interface (accommodation factor = 1). This assumption is valid for most metals, but is less accurate for micromachined
surfaces, particularly those fabricated from silicon. Materials such as silicon cause specular reflection.
Typical accommodation factors for silicon are between 0.80 and 0.90. For more information on accommodation factors, see Flow Between Flat Surfaces in the Mechanical APDL Theory Reference.
3.5. Modeling and Meshing Considerations
The methods used to model and mesh the fluid domain depend on the squeeze film analysis method.
If the velocities are directly applied to the fluid elements, the fluid elements can be independent of the
structure. If the velocities are determined from the mode-frequency response of the structure (modal
projection method), the fluid elements cannot be independent of the structure.
In the modal projection method, the FLUID136 elements must lie on the surface of the structural mesh.
Use the ESURF command to overlay FLUID136 elements on the surface of an existing structural mesh.
FLUID136 supports 4-node and 8-node options, along with degenerative triangles.
The FLUID138 element is used to model the pressure drop in the channels (holes) in a structure as
shown in Figure 3.2: Perforated Plate Structure (p. 5). The element supports a circular hole configuration
and a rectangular hole configuration (KEYOPT(3)). The element works in conjunction with FLUID136 to
correctly compute the pressure distribution on the structure's surface and through the channels. To
correctly model the FLUID138 element:
1.
Use a single FLUID138 element for each hole.
2.
Align the FLUID138 element at the center of the hole. The element should extend through the depth of
the hole and have a length equal to the depth of the hole.
3.
Assume the pressure at the center of the hole (flush with the plate surface in the thin film region) is at
the same pressure as the nodes modeled at the periphery of the hole. Couple the nodes of the FLUID136
element at the whole periphery with the node of the FLUID138 element as shown in Figure 3.5: Coupling
of Nodes at the Hole Periphery with the Center Node of a FLUID138 Element (p. 8).
Figure 3.5: Coupling of Nodes at the Hole Periphery with the Center Node of a FLUID138 Element
3.6. Analysis Settings and Options
A thin film analysis using FLUID136 and FLUID138 can be a linear or nonlinear solution for both a static
and a transient analysis. No special analysis options or load step options are required. See the Basic
Analysis Guide for general guidelines on defining the analysis type and analysis options.
Harmonic analysis is a linear analysis. For a harmonic analysis, you must use the full method.
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Review Results
3.7. Loads and Solution
The fluid elements support pressure loads at the nodal PRES degrees of freedom and velocity body
loads (FLUE) at the node or element as shown in Table 3.1: Load Options for Thin Film Fluid Elements (p. 9). Pressure loads are applied using the D family of commands. The velocity load is applied
using the BF family of commands. For the velocity load, a positive velocity load will cause a positive
pressure. For a harmonic analysis, a positive velocity load will cause a positive in-phase pressure load.
See Applying Body Loads for details on applying body loads.
Note
Velocity body loads (FLUE) are the velocities of the moving wall in a direction normal to the
wall. They are not in-plane velocities of the fluid.
Table 3.1: Load Options for Thin Film Fluid Elements
Load
Type
ANSYS
Label
Pressure
Nodal
DOF
PRES
Normal
Velocity
Body
Load
FLUE
The recommended equation solvers for the fluid elements are the Sparse Direct Solver (SPARSE), the
Jacobi Conjugate Gradient Solver (JCG) and the Incomplete Cholesky Conjugate Gradient Solver (ICCG).
3.8. Review Results
Results from a thin film fluid analysis are stored on the Jobname.RST file. The primary result from the
fluid analysis is the nodal pressures. For a harmonic analysis, the pressures are stored as complex values
(Real and Imaginary components). The General Postprocessor (POST1) discusses reading results into the
database and viewing the nodal degree of freedom (pressure) results.
Each fluid element computes and stores results for the mid-plane fluid velocity (PG) and effective viscosity. Use the element table to retrieve and view these results.
To compute the damping and squeeze stiffness coefficients from the resulting pressure distribution on
the surface of a structure, use the element table tools to compute the force on a per-element basis,
then sum over all elements to compute the coefficients. The following command sequence demonstrates
how to do this task. This example assumes parameters “omega” and “velo” exist (frequency and normal
velocity (FLUE loading), respectively).
set,1,1
etable,presR,temp
etable,earea,volu
smult,forR,presR,earea
ssum
*get,Fre,ssum,,item,forR
set,1,1,,1
etable,presI,temp
smult,forI,presI,earea
ssum
*get,Fim,ssum,,item,forI
!
!
!
!
!
!
!
!
!
!
!
store Real solution
extract "Real" pressure
extract element area
compute "Real" force
sum over all elements
get the total "real" force
store Imaginary solution
extract "Imaginary" pressure
compute "Imaginary" force
sum over all elements
get the total "imaginary" force
K=abs(Fim*omega/velo)
C=abs(Fre/velo)
! Compute squeeze stiffness coefficient
! Compute damping coefficient
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9
Squeeze Film Analysis
/com, ** Equivalent squeeze stiffness coefficient **
*stat,K
/com, ** Equivalent damping coefficient **
*stat,C
3.9. Sample Harmonic Analysis
Problem Description
A rectangular beam with perforated holes under transverse motion is modeled to compute the effective
damping and squeeze stiffness coefficients. The thin film surface of the structure is modeled with FLUID136 elements. FLUID138 elements are used to model the hole regions. By altering the boundary
condition of the free FLUID138 node, we can simulate different pressure boundary conditions of the
hole region. The purpose of the analysis is to compute the equivalent squeeze stiffness and damping
coefficient for an assumed uniform plate velocity. A harmonic analysis is performed at 150 kHz. A previous modal analysis indicated that the pertinent eigenfrequency was 150 kHz. Since the analysis is
linear, the magnitude of the velocity can be arbitrary for computing the coefficients.
Three cases were considered:
1. Holes modeled with no resistance (pressure = 0 at hole location, no FLUID138 elements ).
2. Holes modeled with finite resistance (FLUID138 elements modeled, pressure set to zero on fluid node at
top of plate).
3. Holes modeled with infinite resistance (no pressure specification at hole location, no FLUID138 elements).
Case 1 is typical for large diameter holes compared to the hole depth. Case 3 is typical for very high
flow resistance, which happens in the case of narrow and long holes. Case 2 is the most accurate case
where the fluid pressure drop is taken into account by the FLUID138 elements.
Results
Table 3.2: Beam Model Results Considering Perforated Holes (p. 10) lists the damping and squeeze
coefficient results. Figure 3.6: Pressure Distribution (Real Component) (p. 11) and Figure 3.7: Pressure
Distribution (Imaginary Component) (p. 11) illustrate the real and imaginary pressure distribution. The
input file for case 2 is listed.
Table 3.2: Beam Model Results Considering Perforated Holes
Frequency
(kHz.)
Hole option
Damping
Coefficient
Squeeze stiffness coefficient
150
Infinite
resistance
2.016e-5
1.201
150
Finite resistance
1.325e-5
.5465
150
No resistance
1.165e-5
.4607
10
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Sample Harmonic Analysis
Figure 3.6: Pressure Distribution (Real Component)
Figure 3.7: Pressure Distribution (Imaginary Component)
Command Listing
The input file for this example is shown below for the finite-resistance Case 2
/batch,list
/PREP7
/title, Damping and Squeeze film stiffness calculations for a rigid
/com,
plate with holes
/com
uMKS units
ET, 1,136,1
ET, 2,138,1
! 4-node option, High Knudsen Number
! Circular hole option, Hugh Knudsen Number
s_l=100
s_l1=60
s_w=20
s_t=1
c_r=3
d_el=2
pamb=.1
visc=18.3e-12
velo=2000
freq=150000
pi=3.14159
omega=2*pi*freq
pref=.1
mfp=64e-3
Knud=mfp/d_el
!
!
!
!
!
!
!
!
!
!
Half Plate length (um)
Plate hole location
Plate width
Plate thickness
Hole radius
Gap
ambient pressure (MPa)
viscosity kg/(um)(s)
arbitrary velocity (um/s)
Frequency (Hz.)
!
!
!
!
Frequency (rad/sec)
Reference pressure (MPa)
mean free path (um)
Knudsen number
mp,visc,1,visc
mp,visc,2,visc
! Dynamic viscosity gap
! Dynamic viscosity holes
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11
Squeeze Film Analysis
r,1,d_el,,,pamb
rmore,pref,mfp
! Real constants - gap
r,2,c_r,,,pamb
rmore,pref,mfp
! Real constants - hole
! Build the model
rectng,-s_l,s_l,-s_w,s_w
pcirc,c_r
agen,3,2,,,-s_l1/3
agen,3,2,,,s_l1/3
ASBA, 1, all
TYPE, 1
MAT, 1
smrtsize,4
AMESH, all
! Mesh plate domain
! Begin Hole generation
*do,i,1,5
nsel,all
*GET, numb, node, , num, max
N, numb+1,-s_l1+i*s_l1/3,,
N, numb+2,-s_l1+i*s_l1/3,, s_t
TYPE,2
MAT, 2
REAL,2
NSEL, all
E, numb+1, numb+2
ESEL, s, type,,1
NSLE,s,1
local,11,1,-s_l1+i*s_l1/3
csys,11
NSEL,r, loc, x, c_r
NSEL,a, node, ,numb+1
*GET, next, node, , num, min
CP, i, pres, numb+1, next
nsel,u,node, ,numb+1
nsel,u,node, ,next
CP, i, pres,all
csys,0
*enddo
! End hole generation
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
nsel,a,loc,y,-s_w
nsel,a,loc,y,s_w
nsel,r,loc,z,-1e-9,1e-9
d,all,pres
nsel,all
esel,s,type,,2
nsle,s,1
nsel,r,loc,z,s_t
d,all,pres,0
dlist,all
! Plate domain
! Hole domain
! Create nodes for link elements
! Define 2-D link element
! Select all nodes on the hole circumference
!Coupled DOF set for constant pressure
! Fix pressure at outer plate boundary
! P=0 at top of plate
allsel
bfe,all,flue,,velo
fini
finish
/solu
antyp,harm
harfrq,freq
solve
finish
/post1
12
! Apply arbitrary velocity
! Full Harmonic analysis
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Simplified Transient Analysis
esel,s,type,,1
set,1,1
etable,presR,pres
etable,earea,volu
smult,forR,presR,earea
ssum
*get,Fre,ssum,,item,forR
set,1,1,,1
etable,presI,pres
smult,forI,presI,earea
ssum
*get,Fim,ssum,,item,forI
! extract "Real" pressure
! compute "Real" force
! extract "Imaginary" pressure
! compute "Imaginary" pressure
K=abs(Fim*omega/velo)
C=abs(Fre/velo)
! Compute equivalent stiffness
! Compute equivalent damping
/com, ******* Equivalent stiffness ************************
*stat,K
/com, ******* Equivalent damping **************************
*stat,C
finish
3.10. Simplified Transient Analysis
Problem Description
A simplified transient analysis is performed for a rectangular plate fixed at two ends with a hole in the
center. A stepped pressure load is applied to the top surface of the plate. SOLID185 elements model
the structure. FLUID136 elements model the thin film surface and a FLUID138 element models the hole
region. Displacement and pressure time history results are determined at the bottom of the plate on
the circumference of the hole.
Command Listing
The input file for this example is shown below.
/PREP7
/title, Simplified Transient Analysis
/com
uMKS units
ET, 1,136,1
keyop,1,3,2
et,2,138,1
ET,3,185,,3
!
!
!
!
4-node option, High Knudsen Number
PRES + UZ + UY + UZ DOFs, implicit treatment
Hole
Structural element
s_l=20
s_w=20
s_t=10
c_r=6
d_el=2
pamb=.1
visc=18.3e-12
pref=.1
mfp=64e-3
Knud=mfp/d_el
!
!
!
!
!
!
!
!
!
!
Half Plate length (um)
Plate width
Plate thickness
Hole radius
Gap
Ambient pressure (MPa)
viscosity kg/(um)(s)
Reference pressure (MPa)
Mean free path (um)
Knudsen number
mp,visc,1,visc
mp,visc,2,visc
mp,ex,3,79e3
mp,dens,3,19300e-18
mp,nuxy,3,.1
! Dynamic viscosity gap
! Dynamic viscosity hole
! Gold
r,1,d_el,,,pamb
rmore,pref,mfp
! Real constants - FLUID136
r,2,c_r,,,pamb
! Real constants - FLUID138
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13
Squeeze Film Analysis
rmore,pref,mfp
block,-s_l,s_l,-s_w,s_w,0,s_t
cyl4,0,0,c_r,,c_r,,s_t
vsbv,1,2
numcmp,all
type,3
mat,3
real,3
mshape,1,3d
esize,,3
vmesh,1
! SOLID185 mesh
nsel,s,loc,z,0
esln,s
TYPE, 1
MAT, 1
real,1
esurf
allsel
! FLUID136 mesh
!make the nodes/elem for 138
*get,numb,node,,num,max
n,numb+1,0,0,0
n,numb+2,0,0,s_t
type,2
mat,2
real,2
e,numb+1,numb+2
! FLUID138 mesh
!couple hole pres eq
esel,s,type,,1
nsle,s,1
local,11,1,0,0,0
csys,11
nsel,r,loc,x,c_r
nsel,a,node,,numb+1
cp,1,pres,all
csys,0
allsel
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
nsel,a,loc,y,-s_w
nsel,a,loc,y,s_w
nsel,r,loc,z,-1e-9,1e-9
nsel,a,node,,numb+2
d,all,pres
nsel,all
dlist
! Fix pressure at outer plate boundary
!esel,s,type,,1
!nsle,s,1
!nsel,u,cp,,1
!cm,FLUN,node
!allsel
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
d,all,ux
d,all,uy
d,all,uz
allsel
! Fix two ends of SOLID185 elements
nsel,s,loc,z,s_t
sf,all,pres,1.e+1
allsel
! Apply pressure loading on SOLID185 top surface
fini
/solu
antype,trans
14
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Complex Transient Analysis
kbc,1
nropt,full
time,180e-6
!time,6e-6
deltim,2e-6
outres,all,none
outres,nsol,all
outres,esol,all
cnvtol,pres,,1.e-6
cnvtol,u
,,1.e-6
save
solve
fini
! Specify a stepped loading
! Specify time step size
! Specify convergence values
/post26
nl=node(c_r,0,0)
nsol,2,nl,uz
nsol,3,nl,pres
prvar,2,3
fini
/post1
set,last
esel,s,type,,1
etable,_gap,nmisc,2
pretab,_gap
fini
! Print time history results
! Retrieve results at last substep
! FLUID136
! Print GAP value
3.11. Complex Transient Analysis
Problem Description
A complex transient analysis is performed for a rectangular top plate fixed at two ends with a hole in
the center. A voltage is applied to the bottom surface of the top plate for a specified time and the
voltage is then set to a release value. The top plate does not make contact with the bottom plate.
SOLID185 elements model the top plate. TRANS126 transducer elements are created between the plates.
FLUID136 elements model the thin film surface and a FLUID138 element models the hole region. The
applied voltage is shown in Figure 3.8: Applied Voltage (p. 16). Displacement and pressure results at
the bottom of the top plate on the circumference of the hole are shown in Figure 3.9: Displacement
Time History (p. 16) and Figure 3.10: Pressure Time History (p. 17).
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15
Squeeze Film Analysis
Results
Figure 3.8: Applied Voltage
Figure 3.9: Displacement Time History
16
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Complex Transient Analysis
Figure 3.10: Pressure Time History
Command Listing
The input file for this example is shown below.
/batch,list
/title, Complex Transient Analysis
!/nerr,,,-1
/PREP7
/title, Complex Transient Analysis
/com
uMKS units
ET, 1,136,1
keyop,1,3,2
keyop,1,4,0
keyop,1,5,2
keyop,1,6,1
et,2,185
ET,3,185
! 4-node option, High Knudsen Number
!implicit
!compressible
!fluid gap option
!mech gap option
s_l=100
s_w=20
s_t=1
d_el=2
pamb=.1
visc=18.3e-12
pref=.1
mfp=64e-3
Knud=mfp/d_el
mingapf=1e-2
mech_mingapf=1e-2
app_volt=23.281
app_pres=0
!
!
!
!
!
!
!
!
!
!
!
!
!
mp,visc,1,visc
mp,ex,3,79e+3
mp,dens,3,19300e-18
mp,nuxy,3,.1
! Dynamic viscosity gap
! Gold
! Structural element
Half Plate length (um)
Plate width
Plate thickness
Gap
ambient pressure (MPa)
viscosity kg/(um)(s)
Reference pressure (MPa)
mean free path (um)
Knudsen number
Minimum gap for real constanst on element 136
Mechanical minumum gap for real constant on element 136
Applied voltage
Applied pressure at boundaries
r,1,d_el,,,pamb
! Real constants - gap
rmore,pref,mfp,0,0,1,mech_mingapf
rmore,mingapf,1.e+18,0.e-2,-4.e+0,-0.e-2,1.e-0
!
mingapf,penpar,deadpres,deadspr,deaddam,deadeps
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17
Squeeze Film Analysis
!top plate
block,-s_l,s_l,-s_w,s_w,0,s_t
lsel,s,line,,4,5,1
lsel,a,line,,2,7,5
lesize,all,,,19,-.1
lsel,s,line,,1,8,7
lsel,a,line,,3,6,3
lesize,all,,,11,-.1
lsel,s,line,,9,12,1
lesize,all,,,2
type,3
mat,3
real,3
!mshape,1,3d
mshape,1,3d
cyl4,0,0,s_w/50,,,,s_t+1
vsbv,1,2
numcmp,volu
vmesh,1
!hole
n,10000,0,0,0
n,10001,0,0,s_t
et,6,138,1,,0
r,10,s_w/200,,,pamb,
rmore,pref,mfp
type,6
real,10
mat,1
e,10000,10001
lsel,s,line,,13,16
nsll,s,1
nsel,a,node,,10000
cp,next,pres,all
d,10001,pres,0
allsel
!INSTALL TOP PLATE
!Set element depth
!Mesh top plate with Gold material
!Create a hole in top plate
!Specify nodes for hole element (138)
!Define hole shape
!Define hole diameter and pressures
!Tie top plate hole edge to hole element.
!else emtgen computes INCORRECT areas
nsel,s,loc,z,0
cm,trun,node
allsel
emtgen,'TRUN',,,'uz',-d_el,mingapf*d_el,1.e-18
!Create transducer elements 126 using emtgen
!apply gap information
!fluid
nsel,s,loc,z,0
!Define Fluid region
esln,s
TYPE, 1
MAT, 1
real,1
esurf
!Surface the solid top plate with 136 squeeze film elements
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
nsel,a,loc,y,-s_w
nsel,a,loc,y,s_w
nsel,r,loc,z,-1e-9,1e-9
d,all,pres
nsel,all
!Fix pressure at outer plate boundary
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
d,all,ux
d,all,uy
d,all,uz
allsel
nsel,s,loc,z,s_t
sf,all,pres,app_pres
allsel
18
!Apply pressure at boundary
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Complex Transient Analysis
!fix bottom node voltage/disp
nsel,s,loc,z,-d_el
d,all,ux
d,all,uy
d,all,uz
d,all,volt
allsel
!fix top node voltage
cmsel,s,trun
d,all,volt,app_volt
allsel
!Set voltage/displacement at bottom plate
!V(bottom)=0V
!displacement(bottom)=(0,0,0)
!Set voltage on top side to applied voltage value
!for pulldown
fini
/solu
antype,trans
kbc,1
time,11.e-5
!autots,on
deltim,1.e-5,
nsubst,10,1000,2
outres,all,none
outres,nsol,2
outres,rsol,2
outres,esol,2
cnvtol,pres,,5.e-6
cnvtol,u
,,5.e-6
nropt,full
neqit,500
save
solve
time,20e-5
nropt,full
neqit,100
outres,all,none
outres,nsol,2
outres,rsol,2
outres,esol,2
cnvtol,pres,,5.e-6
cnvtol,u
,,5.e-6
solve
cmsel,s,trun
d,all,volt,1.5
allsel
time,20.4e-5
outres,all,none
outres,nsol,20
outres,rsol,20
outres,esol,20
cnvtol,pres,,5.e-6
cnvtol,u
,,5.e-6
cnvtol,pres,,1.e-4
cnvtol,u
,,1.e-4
autots,on
deltim,.005e-5,0.001e-5,1e-5
solve
time,30e-5
autots,on
deltim,.5e-5,0.02e-5,1e-5
solve
fini
!Pulldown stage
!enforce regular timesteps
!Reduce number of equilibrium iterations to save time.
!Setup release time
!Set release voltage across
!only limited time here
!Long grinding period here, reduce result steps so we don't go
!over 1000 results.
!Let Autotimestep run here
!to better capture the overshoot movement on release
!After initial snapback,
!increase timestep as overshoot decays.
/POST1
allsel
GG=node(0,-0.4,0)
SET,LIST,999
SET,,, ,,, ,30
!*
!*
PLESOL, NMISC,13, 0,1.0
presol,nmisc,13
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19
Squeeze Film Analysis
/out,
fini
!*list,closes,lis1
!*list,closes,lis2
/post26
NUMVAR,200
SOLU,191,NCMIT
STORE,MERGE
FILLDATA,191,,,,1,1
REALVAR,191,191
NSOL,2,GG,U,Z,UZ_2
NSOL,3,GG,PRES,,PRES_3
NSOL,4,GG,VOLT,,VOLT_4
plvar,2
plvar,3
plvar,4
fini
20
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Chapter 4: Modal Projection Method for Squeeze Film Analysis
Modal projection techniques provide an efficient method for computing damping parameters for flexible
bodies. The Modal Projection Technique is the process of calculating the squeeze stiffness and damping
coefficients of the fluid using the eigenvectors of the structure. In the modal projection method, the
velocity profiles are determined from the mode-frequency response of the structure.
The following modal projection method topics are available:
4.1. Modal Projection Method Overview
4.2. Steps in Computing the Damping Parameter Using the Modal Projection Technique
4.3. Example Problem Using the Modal Projection Method
4.4. Damping Extraction for Large Signal Cases
4.1. Modal Projection Method Overview
For a structure undergoing flexible body dynamics, the displacement and velocity vary along the
structure. For a thin-film fluid adjacent to the structure, there is a true dependency between the structural velocity and the fluid pressure. A modal analysis of a structure provides information on the fundamental eigenfrequencies and mode shapes (eigenvectors). For each mode, i, we can impress the distributed eigenvector values as velocity loads and compute the resulting pressure distribution and hence
the fluid force acting on mode j. Therefore, we can compute a matrix of modal squeeze stiffness coefficients, Kij, and modal damping coefficients, Cij. The diagonal terms of these matrices are the coefficients
of each mode, the off-diagonal terms represent the "cross-talk" coefficients between modes. Figure 4.1: Modal Projection Technique for Damping Characterization (p. 21) illustrates the modal projection
technique.
Figure 4.1: Modal Projection Technique for Damping Characterization
From the modal squeeze and stiffness coefficients for the main diagonal terms (source mode = target
mode), other useful damping parameters can be calculated. For example, the modal damping ratio, ξ
, and the modal squeeze to stiffness ratio, kri, can be computed by:
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21
Modal Projection Method for Squeeze Film Analysis
Where mi is the modal mass and ωi the eigenfrequency. The damping ratio can be used to compute
ALPHAD and BETAD parameters for Rayleigh damping (see ABEXTRACT command), or to specify constant
and modal damping by means of the DMPRAT or MDAMP commands.
The squeeze to stiffness ratio represents the relative stiffening of the mechanical system due to fluid
compression and resonance shift. If the squeeze-to-stiffness ratio is small (e.g., <.02), then the squeeze
stiffness affects the structural stiffness by less than 2%, which may be negligible. At higher frequencies,
the ratio can be much larger, indicating a significant stiffening of the structure.
The stiffness and damping coefficients may be strongly frequency-dependent. Figure 4.2: Damping and
Squeeze Stiffness Parameters for a Rectangular Plate (p. 22) shows a typical frequency dependency of
the squeeze and damping forces on a flat plate, and the resulting stiffness and damping constant. The
frequency where the forces intersect is known as the cut-off frequency. Below this frequency, structures
may be accurately characterized by a constant damping coefficient, and stiffness effects can be ignored.
Structures which operate far above the cut-off can be described by a constant stiffness coefficient, and
damping effects can be ignored.
Figure 4.2: Damping and Squeeze Stiffness Parameters for a Rectangular Plate
4.2. Steps in Computing the Damping Parameter Using the Modal Projection Technique
The basic steps in performing the analysis are as follows:
1.
Build a structural and thin-film fluid model and mesh.
2.
Perform a modal analysis on the structure.
3.
Extract the desired mode eigenvectors.
4.
Select the desired modes for damping parameter calculations.
5.
Perform a harmonic analysis on the thin-film elements.
6.
Compute the modal squeeze stiffness and damping parameters.
7.
Compute modal damping ratio and squeeze stiffness coefficient.
8.
Display the results.
Steps 4-7 have been automated using the DMPEXT command macro. Step 8 is available through the
MDPLOT command macro.
22
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Steps in Computing the Damping Parameter Using the Modal Projection Technique
4.2.1. Modeling and Meshing
Model preparation for the modal projection method for damping parameter extraction follows very
closely to the method outlined in Modeling and Meshing Considerations (p. 8). Both the structural
model and the thin-film fluid model need to be defined simultaneously. Group the fluid nodes on the
structure surface into a node component named "FLUN". This component should only include nodes
attached to FLUID136 elements. This component defines the nodes for which the eigenvector values
will be extracted from the mode-frequency analysis.
4.2.2. Perform Modal Analysis
A modal analysis of the structure is required to obtain the eigenvectors of the desired modes. The
modal analysis should be set up with the following options:
1. Use the Block Lanczos, Supernode, Subspace, or PCG Lanczos (if applicable) method (MODOPT command).
2. Extract the eigenmodes of interest (MODOPT command).
3. Expand the eigenmodes of interest (MXPAND command).
The expanded eigenmodes will compute the necessary eigenvectors required for the modal projection
technique. Note: The thin-film fluid elements may be left active in the model when performing the
modal analysis. These elements will not contribute to the modal solution.
4.2.3. Extracting Eigenvectors
The eigenvectors from the modal analysis must be retrieved and stored in a format appropriate for use
by the thin-film fluid elements for a subsequent harmonic fluid analysis. Use the RMFLVEC command
in POST1 to retrieve the eigenvectors. The data is stored in the file Jobname.EFL.
4.2.4. Performing a Harmonic Analysis and Extracting Damping Parameters
The command macro DMPEXT performs a harmonic analysis with the thin-film fluid elements using
the eigenvector information for the desired mode(s). The macro also extracts damping parameter information for use in subsequent dynamic structural analysis. Damping parameters extracted include
the modal damping coefficient, modal squeeze stiffness coefficient, damping ratio, and the squeeze-tostiffness ratio.
DMPEXT requires a "source" mode number and a "target" mode number. The source mode is the mode
for which the eigenvectors are used to impart a velocity field on the fluid elements. The target mode
is the mode which is acted on by the pressure solution (fluid forces). For transverse oscillations and a
uniform gap, there is little mode interaction and hence only diagonal terms are required (source mode
(i) = target mode (j)). For transverse motion with nonuniform gaps, or asymmetric plate motion, mode
interaction effects (cross-talk) can be significant and should be computed (source mode .ne. target
mode). Typical cross-talk scenarios are shown in Figure 4.3: Fluidic Cross-Talk between Transverse and
Rotational Motion (p. 24).
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23
Modal Projection Method for Squeeze Film Analysis
Figure 4.3: Fluidic Cross-Talk between Transverse and Rotational Motion
Damping parameter extraction can be selected over a desired frequency range or at the eigenfrequencies
of the structure. Damping results are computed and stored in an array parameter.
Computed damping ratios may be used in subsequent dynamic structural analyses. A constant damping
ratio can be applied using DMPRAT for a subsequent harmonic or mode-superposition transient analysis. Mode damping ratios may be applied using the MDAMP command for use in mode-superposition
harmonic and transient analyses. Note: Subsequent structural analysis using these damping parameters
do not require the thin-film fluid elements in the model.
The command macro ABEXTRACT (like DMPEXT) extracts alpha and beta damping parameters for use
in the ALPHAD and BETAD commands. These commands define Rayleigh damping and can be used
in time-transient structural analysis to model damping effects. The damping parameters are computed
from two modes defined by the user.
The command macro MDPLOT may be used to display the frequency-dependent damping parameters.
4.3. Example Problem Using the Modal Projection Method
A rectangular beam with perforated holes is modeled to compute the effective damping coefficient,
stiffness coefficient, and damping ratio. The surface of the structure is modeled with FLUID136 elements
to represent the thin film. The structure is modeled with SOLID185 elements. This example computes
the Rayleigh damping coefficients (α and β) and modal damping ratios ξi for use in a transient dynamic
analysis.
The problem is defined as an extension to the earlier squeeze film example (p. 10). After modeling the
fluid domain, the structure of the beam is modeled and meshed with SOLID185 elements. A modal
analysis is performed on the structure and the first two eigenvalues and eigenvectors are computed.
RMFLVEC is used to extract the eigenvectors for use in the modal projection method to compute the
thin-film damping parameters. The command macro ABEXTRACT is used to compute the Rayleigh
parameters from the lowest two eigenmodes. Modal damping ratios are also provided by the ABEXTRACT
macro for the two eigenmodes.
Table 4.1: Modal Damping Parameters for First Two Eigenfrequencies (p. 24) lists the results from the
damping parameter extraction. Computed Rayleigh parameters are: ALPHAD=65212, BETAD=1.829e-8.
The input file for this example is shown below.
Table 4.1: Modal Damping Parameters for First Two Eigenfrequencies
Frequency
(Hz.)
Modal Damping
Coefficient
Modal Squeeze Stiffness Damping
Coefficient
Ratio
Stiffness
Ratio
55294
67419
.2758e9
.2285e-2
24
.9703e-1
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Example Problem Using the Modal Projection Method
Frequency
(Hz.)
Modal Damping
Coefficient
Modal Squeeze Stiffness Damping
Coefficient
Ratio
Stiffness
Ratio
157497
83123
.3568e10
.3644e-2
.4200e-1
The input file for this example is shown below.
/batch,list
/PREP7
/title, Damping Ratio and Rayleigh Damping Calculations for a Perforated Plate
/com
uMKS units
ET, 1,136,1
ET, 2,138,1
ET,3,185,,3
! 4-node option, High Knudsen Number
! Circular hole option, Hugh Knudsen Number
! Structural element
s_l=100
s_l1=60
s_w=20
s_t=1
c_r=3
d_el=2
pamb=.1
visc=18.3e-12
pref=.1
mfp=64e-3
Knud=mfp/d_el
!
!
!
!
!
!
Half Plate length (um)
Plate hole location
Plate width
Plate thickness
Hole radius
Gap
! ambient pressure (MPa)
! viscosity kg/(um)(s)
! Reference pressure (MPa)
! mean free path (um)
! Knudsen number
mp,visc,1,visc
! Dynamic viscosity gap
mp,visc,2,visc
! Dynamic viscosity holes
mp,ex,3,79e3
! Gold
mp,dens,3,19300e-18
mp,nuxy,3,.1
r,1,d_el,,,pamb
rmore,pref,mfp
r,2,c_r,,,pamb
rmore,pref,mfp
! Real constants - gap
! Real constants - hole
! Build the model
rectng,-s_l,s_l,-s_w,s_w
pcirc,c_r
agen,3,2,,,-s_l1/3
agen,3,2,,,s_l1/3
ASBA, 1, all
TYPE, 1
MAT, 1
smrtsize,4
AMESH, all
! Plate domain
! Hole domain
! Mesh plate domain
! Begin Hole generation
*do,i,1,5
nsel,all
*GET, numb, node, , num, max
! Create nodes for link elements
N, numb+1,-s_l1+i*s_l1/3,,
N, numb+2,-s_l1+i*s_l1/3,, s_t
TYPE,2
MAT, 2
REAL,2
NSEL, all
E, numb+1, numb+2
! Define 2-D link element
ESEL, s, type,,1
NSLE,s,1
local,11,1,-s_l1+i*s_l1/3
csys,11
NSEL,r, loc, x, c_r
! Select all nodes on the hole circumference
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25
Modal Projection Method for Squeeze Film Analysis
NSEL,a, node, ,numb+1
*GET, next, node, , num, min
CP, i, pres, numb+1, next
nsel,u,node, ,numb+1
nsel,u,node, ,next
CP, i, pres,all !Coupled DOF set for constant pressure
csys,0
*enddo
! End hole generation
esize,,2
type,3
mat,3
real,3
vext,all,,,,,s_t
! Extrude structural domain
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
nsel,a,loc,y,-s_w
nsel,a,loc,y,s_w
nsel,r,loc,z,-1e-9,1e-9
d,all,pres
! Fix pressure at outer plate boundary
nsel,all
esel,s,type,,2
nsle,s,1
nsel,r,loc,z,s_t
d,all,pres,0
dlist,all
! P=0 at top of plate
esel,s,type,,1
nsle,s,1
nsel,u,cp,,1,5
cm,FLUN,node
allsel
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
d,all,ux
d,all,uy
d,all,uz
allsel
fini
/solu
antype,modal
modopt,lanb,2
eqslv,sparse
mxpand,2
solve
fini
! Modal analysis
! Extract lowest two eigenmodes
! Expand lowest two eigenmodes
/post1
RMFLVEC
fini
! Extract eigenvectors
/solu
abextract,1,2
finish
! Extract damping ratios and Rayleigh constants
A transient dynamic analysis on the switch can be performed using the extracted damping parameters.
The following input illustrates a coupled electrostatic-structural time-transient solution. The electrostatic
field is assumed to be normal to the plate in the gap, and fringing effects are ignored. TRANS126 elements
are used to model the coupled electrostatic-structural interaction with the switch. The EMTGEN command
is used to generate the transducer elements. A step voltage pulse is applied over a short duration (65
micro-seconds) then released. Rayleigh damping parameters, when used, are applied to the global mass
26
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Example Problem Using the Modal Projection Method
(ALPHAD parameter) and stiffness (BETAD parameter) matrices. We want to apply these only to the
mass and stiffness matrix of the SOLID185 elements, and not the TRANS126 elements, because doing
so may overdamp the system. The TRANS126 elements produce a damping matrix and a stiffness matrix
only. Hence, we can use the ALPHAD parameter for the global mass matrix since there is no contribution
from the TRANS126 elements. To isolate the SOLID185 elements for beta damping, we can apply the
beta damping via a material damping option (MP,BETD). Figure 4.4: Time-Transient Response from
Voltage Pulse (p. 28) illustrates the displacement of a node near the center of the plate over time. The
maximum amplitude of displacement (0.11 microns) is small enough to ignore large deflection damping
effects.
/batch,list
/PREP7
/title, Damped Transient Dynamic Response of an RF MEMS switch
/com
uMKS units
/com,
Small deflection assumption
et,1,200,6
ET,3,185,,3
! Structural element
s_l=100
s_l1=60
s_w=20
s_t=1
c_r=3
d_el=2
pamb=.1
visc=18.3e-12
pref=.1
mfp=64e-3
Knud=mfp/d_el
!
!
!
!
!
!
Plate length (um)
Plate hole location
Plate width
Plate thickness
Hole radius
Gap
! ambient pressure (MPa)
! viscosity kg/(um)(s)
! Reference pressure (MPa)
! mean free path (um)
! Knudsen number
mp,ex,3,79e3
mp,dens,3,19300e-18
mp,nuxy,3,.1
mp,betd,1,1.829e-8
! Build the model
! Gold
! Material damping (from squeeze film results)
rectng,-s_l,s_l,-s_w,s_w
pcirc,c_r
agen,3,2,,,-s_l1/3
agen,3,2,,,s_l1/3
ASBA, 1, all
! Plate domain
! Hole domain
TYPE, 1
MAT, 1
smrtsize,4
AMESH, all
! Mesh plate domain
esize,,2
type,3
mat,3
real,3
vext,all,,,,,s_t
! Extrude structural domain
nsel,s,loc,x,-s_l
nsel,a,loc,x,s_l
d,all,ux
d,all,uy
d,all,uz
allsel
aclear,all
save
nsel,s,loc,z
cm,base,nodes
emtgen,'base',,,'uz',-d_el
allsel,all
nsel,s,loc,z,-d_el
! generate Transducer elements
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27
Modal Projection Method for Squeeze Film Analysis
d,all,uz,0
d,all,volt,0
cmsel,s,base
d,all,volt,10
allsel,all
fini
/solu
antyp,trans
alphad,65211
! alpha damping computed from Squeeze-film theory
kbc,1
time,.000065
deltime,2.0e-6
outres,all,none
outres,nsol,all
solve
time,.0002
cmsel,s,base
d,all,volt,0
allsel,all
solve
fini
n1=node(0,2,0)
/post26
nsol,2,n1,uz
prvar,2
plvar,2
finish
Figure 4.4: Time-Transient Response from Voltage Pulse
4.4. Damping Extraction for Large Signal Cases
If a structure does not undergo significant deformation, the gap between the structure and the fixed
wall may be assumed to be constant (small signal case). If the structural deformation in the gap is significant, the change in the gap may effect the damping of the structure (large signal case). The gap is
defined as a real constant in the thin-film fluid elements. Use SETFGAP to automatically define and/or
update the gap real constants for the thin-film FLUID136 elements. The command can be used to:
• Create real constant table entries for every selected FLUID136 element (large signal case).
• Update an existing real constant table according to structural displacements for every FLUID136 element
(large signal case).
This option allows you to check the dependence of the damping parameters over the deflection range
of the structure.
28
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Chapter 5: Slide Film Damping
Slide film damping occurs if two surfaces separated by a thin fluid film move tangentially with respect
to each other, because energy is dissipated due to viscous flow. Typical applications are damping
between the fingers of comb drives and horizontally-moving seismic masses.
FLUID139 is used to model slide film damping. FLUID139 can model Couette and Stokes flow (Figure 5.1: Slide Film Damping at Low and High Frequencies (p. 29)). Couette flow assumes a constant
velocity gradient across the fluid gap. Tangentially moving surfaces at low frequencies produce a nearly
constant velocity gradient in the fluid, so Couette flow is applicable to low frequencies.
Stokes flow assumes that the velocity gradient is not constant across the fluid gap. Tangentially moving
surfaces at high frequencies do not produce a constant velocity gradient, so the Stokes flow is applicable
to high frequencies. The transition from Couette to Stokes flow occurs near the cut-off frequency.
where ρ is the fluid density, η the dynamic viscosity and d the gap separation.
Couette flow does not occur at high frequencies, because inertial effects are important and viscous
friction is only able to accelerate a thin fluid layer near the moving surface. The penetration depth is
given by
Figure 5.1: Slide Film Damping at Low and High Frequencies
FLUID139 models the effect of viscous flow in the thin film with a series of mass-damper elements, as
shown in Figure 5.2: Viscous Slide Film Element FLUID139 (p. 31). Couette flow is modeled using a two-
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of ANSYS, Inc. and its subsidiaries and affiliates.
29
Slide Film Damping
node option. Stokes flow is modeled using a 32-node option. In the 32-node option, the end nodes
(node 1 and 32) are connected to the structural portion of the model with structural degrees of freedom,
and the series of mass-damper elements are contained within "auxiliary" nodes (nodes 2-31).
The gap separation distance (GAP) and the overlap area of the surfaces (AREA) are specified as real
constants for FLUID139. If the overlap area changes during the analysis, the area change rate (DADU)
must also be specified as a real constant. A changing overlap area is typical in lateral comb drives.
AREA is the initial overlap area, DADU is the change in the overlap area with respect to the surface
displacement, and un and ui are the location of the interface nodes (1 and 32). Note: For a constant
surface area, DADU is the width of the overlap surface.
FLUID139 may be used for a prestress static analysis in conjunction with a prestress harmonic analysis,
as well as a full transient analysis. Prestress analysis is required if the plate area changes due to the
structural displacement. A typical command sequence for a prestress harmonic analysis would include
the following:
/solu
antyp,static
pstress,on
.
.
solve
fini
/solu
antyp,harmonic
pstress,on
.
.
solve
finish
KEYOPT(3) specifies the continuous flow options. For a no-slip assumption at the wall, use the continuum
theory (KEYOPT(3) = 0). For slip flow where the Knudsen number is much less or much greater than
1.0, use the first order slip flow condition (KEYOPT(3) = 1). For slip flow with a Knudsen number of approximately 1.0, use the extended slip flow option (KEYOPT(3) = 2).
30
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Slide Film Damping Example
Figure 5.2: Viscous Slide Film Element FLUID139
5.1. Slide Film Damping Example
Comb drive resonators are a common application for slide film damping. Figure 5.3: Comb Drive Resonator (p. 31) illustrates a typical comb drive assembly consisting of fixed and moving comb drives, springs,
and a central mass.
Figure 5.3: Comb Drive Resonator
Details on the comb drive geometry are given in the input file listed below. The electrostatic behavior
of the moving comb drives are represented by two TRANS126 transducer elements with prescribed
Stroke vs. Capacitance data obtained from finite element runs. The structural behavior of the crab legs
is modeled by a spring element. The inertial effects of the central mass are modeled with a mass element.
Damping occurs between the comb teeth (lateral dimension), and between the comb teeth and the
substrate (vertical dimension). Four FLUID139 elements are used to model the lateral and vertical
damping (two per comb drive). The cut-off frequency is well below the frequency range for the analysis;
therefore, Couette flow is assumed. Since the comb drive effective surface area changes with displacement, the DADU real constant is used (equal to the width of the plate area).
A static prestress analysis is run using a DC bias voltage on the input and output fixed comb drives. A
harmonic analysis sweep is then performed using a small AC voltage. The displacement (magnitude
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31
Slide Film Damping
and phase angle) of the central mass is shown in Figure 5.4: Displacement of Central Mass (p. 32). Figure 5.5: Real and Imaginary Current (p. 32) shows the current.
Figure 5.4: Displacement of Central Mass
Figure 5.5: Real and Imaginary Current
/prep7
/title Comb drive resonator with FLUID139 Couette damping
!
!
!
!
!
!
!
DRIVE COMB
|
|
CRAB LEG SPRING --- CENTRAL MASS --- CRAB LEG SPRING
|
|
PICK-UP COMB
! Reference:
! "Microelectromechanical Filters for Signal Processing",
! Lin, L., Howe, R.T., J. Microelectromechanical Systems,
! VOL 7, No. 3, Sept 1998
! beam parameters (uMKS units)
E = 190e3
d = 2.33e-15
hb = 2
wb = 2
Lb = 100
k = 5.699
!Young modulus
! density
! beam thickness
! beam width
! beam length
! mechanical stiffness
! comb parameter
n= 9
np=n*2
! number of finger
! number of finger pairs
32
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Slide Film Damping Example
x0= 10
L0= 26
L=20
gi= L0-x0
w= 4
h= hb
gp=3
g= 3
Mr = 1.96563e-11
!
!
!
!
!
!
!
!
!
initial finger overlap
finger length (stator comb)
finger length (moving comb)
initial gap
finger width
finger thickness
finger gap (lateral)
comb gap (vertical)
mass
! Lateral gap area
areal=np*h*x0
! Vertical gap area
areav=n*w*x0
! Stroke vs. Capacitance data
x1
cx1
x2
cx2
x3
cx3
x4
cx4
x5
cx5
=
=
=
=
=
=
=
=
=
=
14
0.97857e-2
15
0.96669e-2
16
0.95445e-2
17
0.94185e-2
18
0.93384e-2
! Voltage conditions
Vi = 40
Vo = 60
Vac = 10
! input dc bias
! output dc bias
! input ac bias
! Damping parmeters
nu=18.3e-12
! dynamic viscosity
po=0.1
! reference pressure
dens=1.17e-18
! density
fc=nu/(2*3.14159*dens*(g**2))
! cutoff frequency
daduv=w*n
! change in plate area - vertical
dadul=h*np
! change in plate area - lateral
! Model
n,1,0
n,2,0
n,3,0
n,4,0
et,1,126,,0,1
!Trans126, UX-Volt DOF
r,1,,,gi,,
rmore,x1,cx1,x2,cx2,x3,cx3
rmore,x4,cx4,x5,cx5
e,1,2
e,2,3
et,2,21,,,4
r,2,Mr
type,2
real,2
e,2
et,3,14,,1
r,3,k
type,3
real,3
e,2,4
! Mass element
! linear spring, UX DOF
! Lateral dampers
et,4,139
! Slide-film damper, Couette flow, Continuum theory
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33
Slide Film Damping
mp,visc,4,nu
mp,dens,4,dens
r,4,g,areal,-dadul,po
type,4
mat,4
real,4
e,1,2
e,2,3
! Vertical dampers
r,5,g,areav,-daduv,po
real,5
e,1,2
e,2,3
d,1,ux,0
d,3,ux,0
d,4,ux,0
d,2,uy,0
d,2,volt,0
! Prestress static analysis
/solu
d,1,volt,Vi
d,3,volt,Vo
pstres,on
solve
fini
! Harmonic Frequency Sweep
fr0=60000
fr1=120000
/solu
antyp,harm
d,1,volt,Vac
d,3,volt,0
pstres,on
harfrq,fr0,fr1
nsubs,60
outres,all,all
kbc,1
solve
! AC voltage component on capacitor
! prestress
/post26
nsol,2,2,u,x,ux2
esol,3,2,,nmisc,24,ir
esol,4,2,,nmisc,25,ii
/axlab,x,Frequency (Hz)
! label of x axis
plcplx,0
/axlab,y, Displacement (um)
plvar,2
! plot displacement versus time (frequency)
plcplx,1
/axlab,y, Phase angle
plvar,2
plcplx,2
/axlab,y, Current (pA)
plvar,3,4
fini
34
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Index
C
Couette flow, 29
cross-talk, 23
E
example problems
modal projection method, 24
slide film, 31
squeeze film, 10
F
flow regime
thin film, 7
fluid analyses
slide film, 29
squeeze film, 5
thin film, 1
K
Knudsen number, 7
L
large signal cases, 28
M
modal projection method, 21
S
slide film, 1, 29
squeeze film, 1
squeeze film analysis
harmonic analysis, 6
static analysis, 6
transient analysis, 7
squeeze stiffness effect, 5
Stokes flow, 29
T
thin film analysis, 1
Couette vs. Stokes flow, 29
cross-talk, 23
flow regime, 7
large signal cases, 28
modal projection method, 21
slide film, 29
squeeze film, 5
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35
36
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of ANSYS, Inc. and its subsidiaries and affiliates.