SHEAR7 Version4.4 2005 TheoryManual A

7/23/2019 SHEAR7 Version4.4 2005 TheoryManual A

1/43

1

SHEAR7 V4.4 PROGRAM THEORETICAL MANUAL

March 25, 2005

J. Kim Vandiver and Li Li

Department of Ocean Engineering

Massachusetts Institute of Technology

Copyright 2005

Massachusetts Institute of Technology

All rights reserved.


2/43

2

INTRODUCTION

Many cylindrical structures in wind and marine applications are in areas of strong winds orcurrents. When a fluid flows about a cylinder, as shown in Fig. 1, there will be flow separation

due to the presence of the structure, resulting in shed vortices and periodic wakes. Because ofthe periodic shedding of the vortices, a force perpendicular to the flow direction is exerted on thecylinder, causing it to vibrate in the cross-flow direction. This is called vortex-induced vibration.

When a cylinder is subjected to a uniform flow, the vortex-induced vibration is well understoodand the structures response can be predicted rather accurately. However, this is not the casewhen the flow is not uniform but sheared, as is the case of most real flows. The vortex-inducedvibration of a cylinder in sheared flow is more complicated than that in a uniform flow, since, ingeneral, more than one structural mode participates in the vibration if the flow is sheared. Thestructures response to sheared flow may be multi-moded or single-mode dominated, as shown inFig. 2.

Since the mid 1970s, a series of field experiments have been conducted to investigate the flow-induced vibration of long, slender cylinders (Vandiver, 1993) and an early theoretical model topredict the vortex-induced vibration of a cylinder in sheared flow has been developed (Vandiverand Chung, 1988). Based on the theoretical and experimental work, a computer program, namedSHEAR, was developed at MIT for predicting the vortex-induced vibration response of beamsand cables (with linearly varying tension) in non-uniform flow.

In 1993 and 1994, an extensive revision of the program SHEAR was completed, resulting in twoprograms: SHEAR7 and SHEARINF. SHEAR7 Version 1.0 was entirely new and was based onmode superposition. It identifies which modes are likely to be excited and estimates the cross-flow VIV response in steady, uniform or sheared flows. It is capable of evaluating multi-modenon-lockin response as well as single mode lockin response. The program is capable ofmodeling natural frequencies, mode shapes, and response of cables and beams with linearlyvarying tension and a variety of boundary conditions. The user may also input the naturalfrequencies and mode shapes. The response prediction includes RMS displacement, RMS stress,and fatigue life as well as local lift and drag coefficients. It is also capable of modeling sectionsof risers with VIV suppression devices.

SHEARINF was based on a lift force cross-spectrum model and uses a Greens function solutiontechnique. This program is able to compute the random vibration response of a cylinder that hasthe dynamic characteristics of infinitely long structures. In addition to the RMS displacement,stress, and fatigue damage, SHEARINF also calculates the displacement, acceleration, and stressresponse spectra at user specified locations.

In general, if the dynamic behavior of the cylinder is of finite nature, i.e., the standing wavepatterns dominate, one should use the SHEAR7 program. If the cylinder is long in dynamicsense and wave propagation is the dominant dynamic characteristic, then a mode superpositionapproach is not appropriate and one should use a program such as SHEARINF.


3/43

3


4/43

4

The SHEARINF program still exists but has not been maintained and has not been calibratedagainst experimental data.

This manual presents the theoretical background for the SHEAR7 Version 4.4, referred to simplyas SHEAR7 in the rest of the text. The structure that is modeled by SHEAR7 is a one-dimensional cylinder that may have stop variations in diameter. It is assumed to be either a beamor a taut cable. The flow profile is assumed to be described as piece-wise linear.

Three calculation options are defined in the program. They are:

1. the program calculates natural frequency, mode shape, and curvature only2. the program calculates natural frequency, mode shape, curvature, and structural

response, such as RMS displacement, RMS stress, and fatigue damage rate, based onmode superposition and iteration

3. the program calculates structural response using user-input natural frequency, modeshape and curvature, based on mode superposition and iteration


5/43

5

THEORY AND FORMULATION

The following describes most calculations in the SHEAR7 program and the theoreticalbackground behind it. Many outputs are in terms of dimensionless parameters. This manualbegins with a discussion of several parameters referred to in the input and output.

1. A parameter characterizing the structural behavior,T

EIk2

The equation of motion for a tensioned beam (with no damping) is

m y EIy Tyt ' ' ' ' ' '+ = 0 (1)

where mt

is the linear density of the beam (including fluid added mass effect),EIis the bending

stiffness, andTis the tension. Let the displacement be y Aej kx t= +( ) withA the amplitude, kthe

wave number, the frequency, andx andtare spatial and temporal variables, respectively.

Substituting this expression in the beam governing equation leads to

2 4 2 0tm EIk Tk + + = (2)

A parameter which characterizes the behavior of the beam is defined as

PT

EIk=

2(3)

If P>> 1 the structure is essentially a taut string. Otherwise, the bending stiffness of the

structure should be taken into consideration. Eq. 2 can be solved fork, the wave number,yielding the dispersion relation:

kT T EI m

EI

t2

2 24

2=

+ (4)

Pcan be more conveniently expressed in terms of frequency as

PE

T

t

=

+ +

2

1 14

2

2Im

(5)

where the plus sign in Eq. 4 has been chosen. To evaluate a given structure based on excitationdata, one may use the maximum excitation frequency and the minimum tension so that theminimumPis obtained. This choice of parameters emphasizes beam importance when theminimumPis used to characterize the structural behavior. Numerical studies indicate that whenPis less than 30, the bending stiffness of the structure is important and therefore it isrecommended that the beam model should be chosen in this case.Pis reported in the programoutput.


6/43

6

2. The mass ratio

The mass ratio mr

is defined as

4times the ratio of the mass per unit length of the cylinder to

the mass per unit length of the displaced fluid (the added mass is not included in this

calculation).

mm

mr

s

f

=

4(6)

where ms

is the mass per unit length of the structure (including internal mass if there is any) and

mf

is the mass per unit length of the displaced fluid. Mass ratio is used in program

computations.

3. The Reynolds number

The Reynolds number is given by

Re( )( )

xV x D

v= (7)

where V x( ) is the flow vclocity at locationx, v is the kinematic viscosity of the fluid, and

D is cylinder diameter. This equation gives the local Reynolds number along the structure,which is used to adjust lift and drag coefficients in the program. Reynolds number is reported inprogram output.

4. The minimum and maximum excitation frequencies, min and max

From the Strouhal relationship, the minimum and maximum excitation frequencies are,respectively,

minmin

2 tS V

D

= , maxmax

2 tS V

D

=

(8)

where St

is the Strouhal number, which is a function of the Reynolds number and roughness in

the program, andVmin andVmax are the minimum and maximum flow velocities in the shearedflow profile, respectively. It should be pointed out that these two frequencies are derived fromthe Strouhal relationship and the velocity profile. The cylinder may or may not have significantresponse at these frequencies.


7/43

7

5. The lock-in double bandwith, dVR

RR

R

VdV

V

= (9)

which is the ratio of the lock-in bandwidth expressed as reduced velocity compared to the ideallock-in reduced velocity.[Vandiver, 2003].

1idealR

t

VS

= (10)

For example, if .4RdV = and 0.2tS = , then idealRV =5.0 and RV may vary from 4 to 6 or +20%.

6. The wave propagation parameter, nn

A parameter to characterize finite or infinite structural behavior is the wave propagation

parameter [Vandiver, 1993] which is given by nn where n is the nth modal damping ratio

(including both structural and hydrodynamic damping), and is given by

n

n

n n

R

M=

2, with nR

being the nth modal damping constant, n the nth natural frequency, and n the nth modal

mass. When nn is greater than 2.0, then infinite structural behavior is the dominant

characteristic. In such cases, it is recommended that one use a program such as SHEARINFinstead of the SHEAR7 program. When this value is less than 0.2, spatial attenuation is small

and standing wave behavior is common.

7. Types of structures and boundary conditions that SHEAR7 is able to model

Natural frequency and mode shape solutions are contained in SHEAR7 for the following types ofstructures. The nmodel variable is specified in the input data file to select the desired dynamicmodel.

nmodel definition

0 9: cylinder with linearly varying tension

nmodel=0, pinned-pinned cable, origin at minimum tension endnmodel=1, pinned-pinned beam, origin at minimum tension endnmodel=2, free-pinned beam, origin at free endnmodel=9, free-pinned (w/spring) beam, origin at free end

10-19: cylinder with constant tension


8/43

8

nmodel=10, pinned-pinned cable, origin at either endnmodel=11, pinned-pinned beam, origin at either endnmodel=19, free-pinned (w/spring) beam, origin at free end

20-29: cylinder with no tension

nmodel=22, free-pinned beam, origin at free endnmodel=23, clamped-free beam, origin at clamped endnmodel=24, clamped-pinned beam, origin at clamped endnmodel=25, clamped-clamped beam, origin at either endnmodel=26. sliding-pinned beam, origin at sliding end

(unused numbers have been reserved for future use)

The natural frequency, mode shape, and curvature expressions for each model are given in the

appendix. Modes not found in this table may be computed separately and then provided in anexternal data file for use in the program.

8. The potentially excited modesDue to the need of handling a variety of structural models in this program, a general method toidentify the resonantly excited modes is developed, as explained below. First, the programcalculates the natural frequency, mode shape, and curvature for modes from mode 1 to mode nf.nf is an estimated mode number which is based on the maximum excitation frequency. Theprogram next identifies the potentially excited modes (from n

min, the minimum excited mode

number, to nmax , the maximum excited mode number) by comparing the minimum and maximumexcitation frequencies with the natural frequencies. How the modes at the boundary of theexcitation frequency band are identified is illustrated in Fig. 3: if the maximum shedding

frequency is less than fj+1 (the natural frequency of modej + 1) but greater than

f fj j

+ +1

2(the

average of thejth andj + 1th natural frequencies), then it is assumed that thej + 1th mode is

excited. If the maximum shedding frequency is less thanf f

j j+ +1

2, then thej + 1th mode is not

excited. Similar calculations are made regarding the minimum excited mode.

A broader set of modes is used in the program to compute final riser response: from mode 1 to

the mode whose natural frequency is approximately 1.5 times the highest vortex sheddingfrequency. The purpose of this broader range of modes is to perform a model analysis which isable to correctly model spatial attenuation. This requires that non-resonant modes be included inthe response calculation.

Therefore, there are two sets of modes defined in this program. The first one is for thepotentially excited modes, from mode n

minto mode n

max, which lie within the range of vortex

shedding frequencies. The second set is the first set of modes plus additional ones above the first


9/43

9

set. In general, the second set has more modes, and contains the first set. For example, the firstset of modes may range from mode 1 to mode 6, and the second set may contain from mode 1 tomode 9. These additional non-resonant modes are essential in obtaining correct spatialattenuation in the response.

9. The average minimum wavelength corresponding to the maximum excited modeFrom the maximum shedding frequency one may estimate an average minimum wave length bythe following expression:


10/43

10

min

max

=2L

n(11)

whereL is the length of the cylinder, andnmax

is the maximum excited mode number. This

expression is exact when the cylinder is simply-supported and the tension is a constant, but is a

useful approximation for other cases. min is used to estimate spatial resolution needed incalculations.

10. The spatial resolutionThe spatial resolution is calculated by

xL

= (12)

whereNis the number of segments in the structure andL is the total length of the structure.

11. The number of segments in the spatial domainAssuming that 10 segments in each wavelength will be sufficient, then replacing

minby 10x in

Eq. 11 leads to

NL

xn= =

5

max(13)

whereNis the number of segments in the structure andnmax is the highest mode number in thevortex shedding frequency range.

12. The ratio of change of tension to average tension

The ratio of the change of the tension to the average of the tension in a cylinder,Tp

, is given by

( )( )minmax

minmax2

TT

TTT

p +

= (14)

where Tmin

andTmax

are, respectively, the minimum and maximum tensions. The bigger the ratio,

the larger the variation in tension. When tension varies along a structure, the wavelength in thelow tension region is shorter than that in the high tension region. If this variation is large, anaverage tension solution will be inaccurate in natural frequency and response predictions.


11/43

11

13. Mode superposition solutionThe major tasks of SHEAR7 are:

1. to estimate the dynamic properties of the given structure2.

to predict whether or not lock-in may occur3. to calculate iteratively the lift and damping coefficients

4. to estimate cylinder response and fatigue life.Figure 4 shows the flow chart of this calculation based on modal analysis.

The analysis proceeds as follows. First, from structural data, natural frequencies and modeshapes are calculated or read in from an external .mds file. Based on the minimum andmaximum excitation frequencies, the potentially excited modes (whose natural frequencies arewithin or close to the border of the excitation frequency band) are then identified. A roughestimate of the total power that could be generated by each vibration mode is obtained from the

expression: 2

r r, where Q modal force, R modal damping2

rr

r

Q

R = ,

The modal force Qris computed from the following expression. No iteration is conducted for thelift coefficient. It is for the purposes of this preliminary power computation only assumed to be afunction of the maximum lift coefficient possible given the local reduced velocity.

( )2( )1

( , ) ( ) ( )2r

r f L R x r

L

Q C x V D x V x Y x dx= (15)

The modal damping Rris estimated from the following equation:

( ) ( ) ( ) ( )2 2

h sWhere R and R are the modal hydrodynamic and structural damping.

r

L

r h r r s r r

oL L

R R x Y x dx R x Y x dx

= + (16)

No iterations are performed at this stage as only a measure of the relative strength of each modeis being sought. The damping and lift coefficients are computed at a fixed value of the modal

amplitude. Lr is the excitation region for mode r, and is determined by lower and upper limits,

which result from the user-selected, multi-mode, reduced velocity bandwidth. The resultingestimated power for each mode is divided by that of the mode with the largest power to create aratio that is 1.0 for the most powerful mode. In the preliminary power computation, there is noiteration to find the lift coefficient as a function of A/D. It is a local function of reduced velocityonly, with the center of the power-in region having a larger CL than the edges.

A comparison is then made as to the significance of the input power for each mode. Based on auser selected input cut-off value, modes with an input power ratio lower than the cutoff are


12/43

12

dropped. This leads to the identification of the principally excited modes (modes above the cut-off value), see Fig. 5.

If the number of modes above cutoff is one, then a single mode lock-in response is computed.Otherwise, multi-mode (multiple modes) response is anticipated.

Next, the length of the power-in region, in terms of nodes, for each excited mode is calculated.The basic equation used in this calculation is

VV x

f DR

r

=( )

(17)

where V(x) is the local flow velocity atx,D(x) is the local diameter of the cylinder, VR

(x) is the

reduced velocity atx for mode r, and fr

is the rth natural frequency of the structure (i.e., the

natural frequency of mode r). The natural modes under investigation at this stage are the modesabove the cutoff. Since V(x), in general, varies with the location, a reduced velocity power-in

region has to be defined in order to define the structural portion (of non-zero length) whichcontributes to the resonant response of one mode. It is assumed that whenever the reducedvelocity atx for mode ris within the reduced velocity bandwidth for moder, the fluid will excitethe structure and contribute to the structural response of this mode.

This process divides the entire structure into different power-in regions, which have the samerange of reduced velocity but vary in location because each region is defined by a differentnatural frequency, f

r(see Fig. 6). The number of these regions will equal the number of modes

above the cutoff. It is then clear that, for every excited mode, there are power-in (power input tothe structure) and power-out (power dissipated in the fluid) regions. For example, the power-in(excitation) region length for mode ris Lr and the length of the power-out (damping) region for

mode ris L Lr .


13/43

13


14/43

14


15/43

15

If it is determined that several modes participate in the vibration and there is overlap betweenadjacent power-in regions, the program will perform a mode overlap elimination. The criterionused in the elimination is that the power-in region length of each mode involved in the overlapshrinks equally until the overlap disappears. Within the power-in region for each mode,r, the liftforce is assumed to occur at the natural frequency of that mode. After the elimination of themode overlap, the program calculates the so-called excitation length ratio, which is defined asthe ratio of the length of the power-in region to the total length of the structure:

lL

Le

r

r

= (18)

where Lr is the power-in region length for moder.

If it is determined that only one mode is involved in vibration (this is the case when lock-inoccurs), the excitation length for this mode is then determined by the user-input single modereduced velocity bandwidth.


16/43

16

The following step is to calculate the input and output power for each mode. When the responseof the system (structure and fluid) reaches steady state, therth modal input power (by the liftforce in the power-in region) will balance the rth modal output power (through thehydrodynamic damping in the power-out region and the structural damping), as shown in Fig. 7.

In the following, using a taut string as an example, the procedure and the main formulas for thiscalculations are outlined. The governing equation for a taut string is given by

" ( , )tm y Ry Ty P x t + = (19)

where mt

is the mass per unit length (including the added mass), y is the acceleration of the

structure,R is the damping per unit length (including both structural and hydrodynamic), y is the

velocity of the structure,Tis the tension,y is the second derivative of the displacement of thestructure with respect to the spatial variable, andP(x, t) is the excitation force per unit length (liftforce distribution).

The system displacement response can be written as the superposition of modal responses:

y x t Y x q tr r

r

( , ) ( ) ( )= (20)

where Y xr( ) is the rth mode shape of the system. Substituting this relation into the governing

equation of the string and performing the standard modal analysis lead to


17/43

17

M q t R q t K q t P tr r r r r r r ( ) ( ) ( ) ( )+ + = (21)

where:

Mr

is the modal mass and is given by M Y x m dxr r

o

L

t=

z

2 ( )

Rr

is the modal damping and is given by R Y x R x dxr r

o

L

= z 2 ( ) ( )

Kr

is the modal stiffness and is given by K TY x Y x dxr r

o

L

r= z " ( ) ( )

Pr

is the modal force and is given by P t Y x P x t dxr r

o

L

( ) ( ) ( , )= z

In the power-in region for mode r, it is assumed that the local force andrth modal velocity arealways in phase (i.e., power into the structure). This is accomplished numerically in the programby using absolute values. Hence, the formula for the calculation of therth modal force in the rth

mode power-in region is P t Y x P x t dxr r

L

b g b g b g= z ,0

The lift force per unit length, with frequency r, can be written as [Blevins, 1990]

P x t DV x C x tf L r r

, ( ) ( ; ) sin( )b g = 12

2 (22)

where f

is the fluid volume density,D is the diameter of the cylinder, V(x) is the flow velocity,

andC xL r

( ; ) is the lift coefficient amplitude for mode r. Let the modal velocity for mode rbe

sinq t A t r r r r b g b g= (23)

where Ar

is the modal displacement amplitude of the structure for moder. Then the rth modal

input power is the rth modal excitation force times the rth modal velocity:

r

in

L

f L r r r r r

r

DV x C x A t Y x dx= z12

2 2 b g b g b g b g; sin (24)

where Lr represents the length of the power-in region for the rth mode. The time-average of the

modal input power over one period,P, is

< >= =z z rin rino

P

f

L

L r r r rP

dt DV x C x A Y x dxr

1 1

4

2 b g b g b g; (25)

The rth modal output power is the rth modal damping force times the rth modal velocity:


18/43

18

r

out

L

r r r r R x Y x A t dx= z b g b g b g2 2 2 2 sin (26)

The time-average of the modal output power over one period,P, is

< >= =z z r

out

r

out

o

P

L

r r rP dt R x Y x A dx

1 1

2

2 2 2

b g b g (27)

It is assumed that, for this mode, input and output power are in balance. Therefore, from< >=< >

r

in

r

out , the following expression is obtained:

( ) ( ) ( )

( ) ( ) ( ) ( )

2

2 2

1;

2 r

r

f L r r

r L

L

h r r s r r

oL L

V x C x Y x dxA

DR x Y x dx R x Y x dx

=

+

(28)

where the damping has been separated into the hydrodynamic part Rhb g and the structural part

Rsb g because they have different integration intervals, andL Lr denotes the length of the

power-out region.

In the program, an initial value is assigned to the lift coefficient. An iteration calculation is thenperformed, with the lift force and damping updating (as will be discussed in the following

sections), until convergence is reached (the difference in theA

Dvalue between two successive

calculations is within a pre-specified limit. This iterative process is completed for each modeabove the cutoff. After convergence, the modal responses are used to calculate the total RMS

response of the cylinder.

14. The lift coefficient C xL r

;b g

Shear7 4.3 and 4.4 have been updated with many improvements and new features that enhancethe computational accuracy and versatility of the program. The major improvements concern theway Shear7 assigns a lift coefficient to each point along the riser.

WARNING: Shear7 4.2 incorporates conservatism into its lift coefficient calculations, but withShear7 4.3 and 4.4 the user can choose to use NON-CONSERVATIVE lift coefficient curves. If

conservative results are needed, Shear7 4.3 or 4.4 should be run using the new lift coefficienttable number 1, which uses the new lift coefficient methodology by creating a parabola, which isa best fit to the SHEAR7 V4.2 lift coefficient table. As a test V4.3 or 4.4 may also be run in amode which actually uses the original V4.2 lift table.

New Lift Coefficient Methodology in V4.4: As with Shear7 4.2 the value of the lift coefficient


19/43

19

(designated CL ) in versions 4.3 and 4.4 is a function of the value of the non-dimensionalresponse amplitude (A/D). However, rather than using a piecewise linear table(as done in V4.2)

Versions 4.3 and 4.4 generate a function-based, smooth lift coefficient vs. A/D curve, as shownin Figure 1.

The modification introduced in Shear7 4.4 is adding value CLfloor to the lift calculations, whichcan be modified by the user, see Figure 1 and Shear 7 Version 4.4 User Manual.

Some key points regarding the new lift coefficient curves:

Lift Coefficient (CL) is still a function of non-dimensional response amplitude (A/D). Instead of having a look up table, CL is determined from a smooth curve constructed by

fitting two parabolas through three points, defined by the following four values:

1. The value of A/D when CL = 02. The value of A/D when CL is at its maximum3. The maximum value of CL4. The value of CL when A/D = 0

CLfloor 5


20/43

20

Using this method, CL curves are easily defined; thus, multiple curves can be used to makeCL a function of the non-dimensional frequency ratio, a quantity related to the inverse of thereduced velocity.

The four values are specified in an input file called common.cl that contains the followingsets of curves:

CL table1:A single curve, independent of frequency ratio, which approximates the Shear7 4.2lookup table.

CL table2:A non-conservative, experimental data fit, which is a three dimensional lift coefficientcurve that varies based on A/D and frequency ratio. This will give an expected valueprediction of response of the riser and is based on sub-critical Reynolds number data,including that of Gopalkrishna.

CL table3:A single, conservative curve, independent of frequency ratio, which can be used tomodel strakes.

CL table4:A space for a set of user-defined curves. A place holder set of values is provided for acase independent of frequency ratio.

The user can define multiple structural zones in a *.dat file, and assign a different CL tablefor each region.

Some key points regarding the common.cl file:

Figure 2 gives an example of the data contained in the common.cl file.

The common.cl file must be in the same directory as the Shear7 program. The first block of the .cl file gives the number of CL tables contained in the file. Each ensuing

block contains the data for one CL table.

Within a CL table block, the first line indicates the number of non-dimensional frequencyratio points for which the CL curves will be defined. The non-dimensional frequency,

n vof f , is the ratio of the vibration frequency of the riser, fn, to the vibration frequency of

the riser, fvo, which would promote lock-in in an optimum manner. For the purpose of thisprogram the optimum frequency would correspond to the center of a lock-in region, definedin terms of reduced velocity, based on the actual vibration frequency. For subcriticalReynolds numbers, this optimum reduced velocity is approximately 5.88, which has aninverse value of 0.17. In SHEAR7 the optimum lock-in frequency is computed from a user-provided Strouhal number. The Strouhal number is specified in the data for each zone. Itcan be in the form of a specific number or taken from a Reynolds number dependent table.In either case at every location on the riser there is an optimum vortex shedding frequency,fvo(z), which may be computed using the Strouhal number, the local flow velocity and the


21/43

21

diameter. The lift coefficient table is tabulated in terms of the ratio of the natural frequencyin each power-in zone to the optimum shedding frequency at every node inside the power-inregion according to the following equation.

( ) ( )

n n

vo t

f f

f z S U z D

= (29)

This ratio has a maximum value of 1.0 at the value the user wants to be the center of a lock-inrange. This is usually at or near the peak in the lift coefficient curve.

The CL table block contains six columns, giving the following data: Column 1:

The non-dimensional frequency ratio, n vof f , which Shear7 4.3 and 4.4 use to

automatically center the power-in region at or near the peak lift coefficient in the CL

table, which is defined at ( ) 1.0n vof f z = .

( )n vof f z is shown in the equation above, where fn is the frequency of the mode beingconsidered and fvo(z) is the local Strouhal frequency. n vof f does not provide any

information about the effect of mass ratio on natural frequency, but reflects only theability of the fluid to synchronize with the motion of the cylinder.

In a sheared flow the local Strouhal frequency, fvo, varies with the flow velocity, but eachmode has only one natural frequency fn. When the parameter 1.0n vof f = , the vibration

frequency matches the Strouhal frequency. Looking at a particular power-in region for agiven mode and assuming a constant St number, it is seen that for the faster flow regionthe local Strouhal frequency will be larger than the natural frequency, and

thus 1.0n vof f < . Similarly, for the slower flow region 1.0n vof f > . As long as the

peak lift coefficient in the CL table is centered on 1.0n vof f = , then the peak lift

coefficient will automatically be at the center of the power-in region, whatever theStrouhal frequency, or Strouhal number that has been specified in the input data.

Column 2 -6:Each column (2-6) is one of the five values defined in Figure 1 that give Shear7 thenecessary information to construct the lift coefficient curves. Column 2 corresponds to

the normalized amplitude corresponding to zero lift coefficient, column 3 to theamplitude at the maximum lift coefficient, column 4 to the maximum lift coefficient,column 5 to the lift coefficient at zero amplitude, and column 6 the minimum value of liftcoefficient used by the program. In Shear7 4.3 this minimum value was set constant atCLfloor= -1.0 inside version 4.3 fixed at -1.0 for all the lift coefficient tables. WithShear7v4.4, the user can modify this value by editing the common.CL file.


22/43

22

A single-line CL table (such as tables 1, 3, and 4 in the common.cl file which is provided)indicates a single lift coefficient vs. A/D curve that is applied at all frequency ratio values.This represents a lift coefficient model that is independent of the frequency ratio, and thuslooks like a constant ridge in 3 dimensions.

** * Cl i f t Dat a *** DATE VERSI ON: Febr uar y 7, 2005

4 Number of CLt ype = nCLt ype** * CLt ype 1 = mi mi c ol d SHEAR7 ( v4. 2f ) dat a ** *1 f r eq_r ati o(1): number of non- di mensi onal f r equenci es1. 0 1. 100 0. 300 0. 700 0. 300 - 1. 000 ndf r eq( ) , aCL0, aCLmax, CLmax, CLa0, CLf l oor*** CLt ype 2 = CML ext ended f r omGopal kr i shna data ***26 nndf r eq( 1) : number of non- di mensi onal f r equenci es

0. 70 0. 149 0. 100 0. 10 0. 000 - 1. 000 ndf r eq( ) , aCL0, aCLmax, CLmax, CLa0, CLf l oor0. 73 0. 266 0. 200 0. 10 0. 000 - 1. 0000. 74 0. 400 0. 214 0. 10 0. 016 - 1. 0000. 76 0. 451 0. 235 0. 10 0. 040 - 1. 0000. 78 0. 505 0. 270 0. 10 0. 080 - 1. 0000. 81 0. 530 0. 350 0. 14 0. 110 - 1. 0000. 87 0. 588 0. 450 0. 20 0. 180 - 1. 0000. 93 0. 658 0. 500 0. 35 0. 240 - 1. 0000. 96 0. 746 0. 500 0. 50 0. 300 - 1. 0000. 98 0. 890 0. 460 0. 78 0. 350 - 1. 0001. 00 0. 900 0. 430 0. 80 0. 400 - 1. 000

1. 02 0. 837 0. 400 0. 70 0. 200 - 1. 0001. 05 0. 761 0. 400 0. 40 0. 100 - 1. 0001. 08 0. 706 0. 400 0. 30 0. 000 - 1. 0001. 10 0. 666 0. 400 0. 20 0. 000 - 1. 0001. 16 0. 615 0. 380 0. 10 0. 000 - 1. 0001. 22 0. 592 0. 350 0. 10 0. 000 - 1. 0001. 28 0. 575 0. 313 0. 10 0. 000 - 1. 0001. 34 0. 539 0. 275 0. 10 0. 000 - 1. 0001. 40 0. 504 0. 238 0. 10 0. 000 - 1. 0001. 45 0. 420 0. 200 0. 10 0. 000 - 1. 0001. 57 0. 312 0. 160 0. 10 0. 000 - 1. 0001. 63 0. 247 0. 140 0. 10 0. 000 - 1. 0001. 69 0. 186 0. 120 0. 10 0. 000 - 1. 0001. 74 0. 160 0. 100 0. 10 0. 000 - 1. 0001. 80 0. 136 0. 090 0. 10 0. 000 - 1. 000

** * CLt ype 3 = Conser vat i ve St r ake model1 nndf r eq( 1) : number of non- di mensi onal f r equenci es1. 0 0. 3 0. 15 0. 20 0. 1 - 1. 000 ndf r eq( ) , aCL0, aCLmax, CLmax, CLa0, CLf l oor** * CLtype 4 = User i nput Dat a ( ** CHANGE NUMBERS I N BELOW TABLE**)1 nndf r eq( 1): number of non- di mensi onal f r equenci es1. 0 1. 100 0. 350 0. 750 0. 500 - 0. 5 ndf r eq( ) , aCL0, aCLmax, CLmax, CLa0, CLf l oor

The user is also allowed to put in a lift coefficient reduction factor in each specified structuralzone. The lift coefficient values in Figure 8 are multiplied by this factor prior to iteration.

15. The damping model: sectional damping

The total modal damping includes structural damping and hydrodynamic damping, which isusually the dominant form of damping in sheared flows. Hydrodynamic damping is an activesubject of research in the field of vortex-induced vibration. A new model for hydrodynamicdamping has been incorporated in SHEAR7. This model was developed in a doctoraldissertation by Madan Venugopal at MIT. It is briefly described below. The principalcharacteristics of the hydrodynamic damping model are that it accounts for local responseamplitude and local reduced velocity. A weighted average of the local values integrated over the


23/43

23

entire structure is used to produce an estimate of the total modal damping as described brieflybelow.

The modal damping ratio n

is defined as

n n s n h= +, , ' (30)

where n s,

is the structural modal damping andn h,

is the modal damping from hydrodynamic

sources.

The structural damping is typically a few tenths of a percent of for steel tubulars and is oftennegligible when compared to the hydrodynamic contribution.

The hydrodynamic contribution, n h, '

, is defined as

n h n h n n

R M, ,

/= 2b g (31)

where n

is the natural frequency in radians per second, Mn

is the modal mass, andRn h,

is the

modal hydrodynamic damping constant for moden.

If we define ( )nY z as the mode shape for mode n, then Mn andRn h, are each defined by integrals

over the length of the structure of the local mass per unit length or damping per unit length,weighted by the square of the mode shape as follows:

( ) ( )2

n nm z Y z dz = (32)

( ) ( )2

,n h h nR r z Y z dz= (33)

In his thesis, Venugopal synthesizes a broad variety of experimental evidence and recommendsthe following empirical model for the local hydrodynamic damping constant,r z

h b g . Thisdamping constant has units of force per unit velocity per unit length. The model depends on thelocal reduced velocity and is different for regions in which the reduced velocity is low (belowlock-in) and high (above lock-in).

The models for the two different regions are:

Low reduced velocity damping model: (34)

r z r C DV h sw rl b g = +

where rsw

is the still water contribution andCrl

is an empirical coefficient taken for the

present to be 0.18.


24/43

24

The still water contribution is given by:

22 2 20.

2 Resw sw

D AR C

D

= +

(35)

where Re /w

D v= 2 , a vibration Reynolds number. v is the kinematic viscosity of the

fluid and is the vibration frequency. The value of Csw in equation 33 above for the still

water damping has a default setting of 0.2 in SHEAR7. The coefficient may be varied toproduce more or less still water damping.

High reduced velocity damping model:

r C Vh rh

= 2 / (36)

where Crh

is another empirical coefficient which at present is taken to be 0.2. is the

fluid density.

The still water damping is dependent on both the vibration amplitude and frequency of vibration.The low reduced velocity model reduces to still water values when the local flow velocity goesto zero. The low reduced velocity damping is proportional to the local flow velocity and the highreduced velocity model of damping depends on the vibration frequency and the local velocitysquared. The hydrodynamic damping model used in SHEAR7 has been shown to be accurateand conservative in independent experiments conducted by Vikestad, 2000, and presented at the2000 Offshore Technology Conference.

Altering the default values of Csw, Crl, and Crh in SHEAR7

In SHEAR7 V4.2 there was a line in Block 5 of the input .dat file which normally has three zerosin it. i.e. 0.0 0.0 0.0. This resulted in the program assigning three default values to thesedamping coefficients. In V4.3 (and in V4.4) the location in the input data file has changed andthe user must specify the values for each zone. The new input data location is the last(5th) line ofthe input data for each zone. Three coefficients must be given which correspond to the threecoefficients in the damping models described above: Csw, Crl, and Crh respectively. The standardrecommended values are 0.2, 0.18 and 0.2 forCsw, Crl, and Crh, respectively. If the user insertsany other values the program will use the new values in computing the hydrodynamic damping.Hence to double the damping everywhere, one would use 0.4, 0.36 and 0.4 on this line of the .datfile. To double the still water contribution only one would double the first entry only.

It is useful to consider the circumstances in which each model is relevant. The mode excited byvortex shedding from the highest speed in the current profile will only experience damping ascomputed with the low reduced velocity model. This is because that mode will never encounterreduced velocities above its own lock-in band. Each lower mode will encounter high reducedvelocities due to currents above the lock-in speed for that mode. Each will also encounter lowreduced velocities on portions of the riser which have lower current speeds than that whichcreates the power-in region for the mode. In general the highest excited modes in a sheared flowwill tend to have the lowest damping.


25/43

25

The structural damping for tensioned marine structures susceptible to flow-induced

vibration is usually very small and for structures in water is rarely the deciding factor in thedetermination of whether or not lock-in occurs. However, when lock-in does not occur or occursover only a portion of the structure, then hydrodynamic sources of damping can be large and

become very important in determining dynamic response behavior. Whether or not a cylinderresponds dynamically as if it is of infinite length or is dominated by standing waves dependsheavily on hydrodynamic sources of damping.

16. A full modal analysis of the structural response.After convergence, the iteration calculation, based on power balance, provides the excitationforce (in terms of lift coefficients) and damping (in terms of damping ratio), for each excitedmode. These results are then used to calculate structural response. A full modal analysisformulation is adopted in the program to calculate RMS displacement, stress, and fatigue damage

rate. Both the resonant and non-resonant modes are included. This calculation is able to predictcorrectly the spatial attenuation when the wave propagation parameter, nn

, is not small.

When calculating the model force, Pnr , one must use a special formula to account for the effects

of lift force on the resonant and non-resonant modes. It is suggested that the following formulashould be used:

P Y x Y x P x dxnrr

L

n r= zsgn b g b g b g

0

(37)

where P xrb g is the distributed lift force in the rth power-in region:P x DV x C xr f L r

b g b g b g=

1

2

2 ; ,

Y xrb g is the rth mode shape, andsgn is a sign function which has the following values:sgn Y x

rb g = +1 ifY xrb g is greater than zero; sgn Y xrb g = 1 ifY xrb g is less than zero; sgn Y xrb g = 0 ifY x

rb g equals zero. It should be stressed that the integration is Eq. 32 is performed within theexcitation length L

ronly. C x

L r;b g is the converged lift coefficient obtained from the iteration

calculation.

From the mode superposition solution, the displacement response (in complex form) is obtainedas

y x y x Y x P Hr

r n

nr

nr nr

r

n

b g b g b g= = FHG IKJ ;

(38)

where rrepresents one of the excited modes, and the frequency response function is given by


26/43

26

HK

j

nr

r

n nr

n

n

r

n

FHG

IKJ

=

FHG

IKJ

+L

NMM

O

QPP

1 1

1 2

2(39)

where n is the damping ratio obtained from the iteration calculation. y xb g in Eq. 33 is thedisplacement response atx due to all modes, both resonant and non-resonant.

The commonly used RMS displacement is given by

( ) ( )

12 2

1

2r

nrrms n nr

r n n

y x Y x P H

=

(40)

The RMS acceleration is given by

( ) ( )

12 2

41

2r

nrrms r n nr

r n n

y x Y x P H

=

(41)

The RMS stress atx due to all modes is

( ) ( )

12 2

"1

8

rnrrms n s nr

r n n

S x Y x Ed P H

=

(42)

whereEis Youngs modulus, Y xn

" b g is the curvature for mode n, and ds is the strength diameter.

The damage rate at locationx due to all modes is given by the summation of the individualmodal damage rates.

D x D xr

r

b g b g= (43)

The damage rate D xrb g due to excitation frequency r is given by

DT

CS x

br

r

r rms

b

=+F

HGIKJ

22

2

2, b ge j (44)


27/43

27

with S x Y x Ed P H r rms n n s nr nr

r

n

,

"b g b g= FHGIKJ

LNMM

OQPP

1

2 2

andb andCare constants, and are defined by the S-N

curve expression: NS Cb = where Sis the stress amplitude. The program input data is in terms ofstress range (see user guide).

The above formulation is presently under review and will likely change in the very near future.

17. RMS response estimate in uniform flow cases.

FromA

Dfor the excited mode n, the RMS structural displacement response can be estimated

using the following formula:

yAY x

rms

n=b g2

(45)

where y xrms b g is the RMS displacement at locationx, n is the excited mode number, Y xn b g is the

nth mode shape, andA is the antinode displacement amplitude.

The maximum structural acceleration response can be estimated using the following formula:

y xAY x

rms

nb g b g= 22

(46)

where y xrms b g is the RMS acceleration at locationx, n is the excited mode number, n is the

natural frequency for the nth mode, andA is the antinode displacement amplitude.

The maximum structural stress can be estimated using the following formula:

S xEAY x d

rms

n sb g b g="

2 2(47)

where S xrms b g is the RMS stress at locationx, n is the excited mode number,Eis the Youngs

modulus, sd is the strength diameter of the cylinder, Y xn" b g is the curvature for mode n, andA is

the displacement amplitude.

The calculation of damage rate, Dn, is based on a period of one year. The inverse ofD

nwill be

the fatigue life in years. The damage rate caused by the excited moden can be estimated byusing the Miner rule, in its continuous form, as

Dn S

N SdS

n

o

=

z b gb g (48)


28/43

28

whereN(S) is the number of cycles to failure at stress amplitudeS. In one yearT x x= 365 24 3600 secondsb g , the number of cycles at stress amplitude betweenSandS+dSis

n STp S T

p Sn

nb g b g b g= =2 2

/(49)

where n

is the frequency for the nth mode (which can be viewed as an expected frequency of

the narrow-banded process) andp(S) is the probability density function for the stress amplitudeprocess. When a stochastic process is narrow-banded and Gaussian, its peak value is given bythe Rayleight distribution:

p SS

Se

rms

S

Srms( ) =

2

2

2

2(50)

By using the S-Ncurve expression, NS Cb = , with b andCbeing determined by material

properties, and the above expressions, the damage caused by moden in one year can be found as

DT

CS

bn

n

rms

b

=+F

HGIKJ

22

2

2e j (51)

However, Dn

can also be obtained by assuming the stress is a steady state sinusoidal function,

( )sin nS t + . In one year, the number of cycles at stress amplitude for moden is

n ST T

n

n( )/

= =2 2

(52)

The damage rate is then

Dn S

N S

T S

Cn

n rms

b

= =( )

( )

2

2(53)

Let DR be the ratio of the damage rate determined based on the Rayleigh distribution to the

damage rate based on a steady state sinusoidal function. RD

is a function: b +F

HGIKJ

2

2, whose

value depends on b, a material parameter. As b assumes 2, 3, 4, 5, and 6, RD

is equal to 1.0, 1.3,

2.0, 3.3, and 6.0, respectively. For a steel structure in sea water,b assumes a value of between 3

and 5, with 4 being a mean value at which RD equals 2.0. If the structure vibrates in a purelysinusoidal fashion with a constant amplitude, then the calculation based on the Rayleigh

distribution overestimates the damage rate by the factor DR . However, in reality, purely

harmonic, constant-amplitude motion seldom occurs. Therefore, the damage calculation basedon the Rayleigh assumption is more realistic, and is used in the program.


29/43

29

18. The drag amplification factor.

The drag coefficient during a VIV process is usually larger than that without VIV amplification.After the structural displacement is obtained, a drag amplification factor can be calculated by anempirical formula [Vandiver, 1983]:

C iy i

DD amp

rms

,

.

( ) . .( )

= +FHG

IKJ10 1043 2

0 65

(54)

where y irms

( ) is the RMS value of the structural displacement at node i. This amplification factor

is then multiplied by CD0

, the Reynolds number dependent stationary cylinder drag coefficient

given in Figure 10. This result is printed in the output file as a function of location. It should bepointed out that the above formula is for a smooth cylinder. It does not include marine growtheffects.

20. Global and local stress concentration factors.

Two factors are used in the program to modify the stress. They are the global and local stressconcentration factors. These factors are applied in the stress and the fatigue calculations. Therelationship between these two factors is explained below. The global stress concentration factoris applied to all nodes. The local stress concentration factor (SCF) is applied only at selectednodes. When using a local SCF, the user needs to input the stress concentration location, in

terms ofx

L, and the corresponding SCF. If a node is specified as having stress concentration, the

local stress concentration factor overrides the global stress concentration factor at this node.

21. VIV suppression simulation.

In practice, it is often desirable to have VIV suppression devices applied to the cylinderto reduce its vibration and to increase its fatigue life. When these devices are used, the regionwhere they are located has essentially very little excitation. It is therefore necessary to modifythe lift coefficient in this region. In the program, the VIV suppression region can be defined as aseparate structural zone and an appropriate lift coefficient table may be specified. For example,table 3 in the common.cl file distributed with the program is a conservative model for strakes.One may also use a lift coefficient reduction factor, which is specified in the zone input data toalter the lift in a region. An example would be to model fairings using a reduction factor of 0.01.

REFERENCES

1. Vandiver, J.K., Dimensionless parameters important to the prediction of vortex-inducedvibration of long, flexible cylinders in ocean currents,Journal of Fluids and Structures,July 1993.


30/43

30

2. Vandiver, J. K. and Chung, T. Y., Predicted and measured response of flexible cylindersin sheared flow, Proceedings of the ASME Winter Annual Meeting, Chicago, November1988.

3. Vandiver, J. K., Drag coefficients of long-flexible cylinders, OTC Paper 4490, OffshoreTechnology Conference, Houston, 1983.

4.

Blevins, R. D., Flow-induced vibration, Van Nostrand Reinhold, 2

nd

ed., New York,1990.5. Kim. Y. C., Nonlinear vibrations of long slender beams, Ph.D. Thesis, MIT, OE, 1983.6. Capozucca, P., Flow-induced vibration of a non-constant tension cable in a sheared flow,

MS Thesis, MIT, OT, 1988.7. Gopalkrishnan, R., Vortex-induced forces on oscillating bluff cylinders, Ph.D. thesis,

MIT, OE, 1993.8. Rudge, D. and Fei, C., Response of structural members to wind-induced vortex shedding,

M.S. Thesis, MIT, OE, 1991.9. Vandiver, J.K., Marcollo, H., High Mode Number VIV Experiments, IUTAM

Symposium On Integrated Modeling of Fully Coupled Fluid-Structure Interactions Using

Analysis, Computations, and Experiments, 1-6 June 2003, Kluwer Academic Publishers,Dordrecht.10. Vikestad, K., Larsen, C.M., &Vandiver, J.K., Norwegian Deepwater Program:

Damping of Vortex-Induced Vibration, OTC Paper 11998, Proceedings of the OffshoreTechnology Conference, May 2000, Houston

Nomenclature

An: nth modal displacement amplitude

,D ampC : drag coefficient amplification factor

CD0 : mean drag coefficient of a stationary cylinder as a function of Reynolds numberC i k

L( , ) : lift coefficient at node i for mode k

D : diameter of a cylinderEI: bending stiffness of a cylinder (Youngs modulus times area moment of inertia)f

min: minimum calculation frequency

fmax

: maximum calculation frequency

k: transverse wave numberL : total length of a cylinderL

n : excitation length for mode nM

n: nth modal mass

mt : mass per unit length of a structure (including added mass)m

s: mass per unit length of a cylinder

mf

: mass per unit length of displaced fluid

N: number of segments in a structureP: tension-bending stiffness ratio of a cylinderP x t,b g : distributed lift forceP

n: nth modal force


31/43

31

Rn: nth modal damping constant

Re( )x : Reynolds number at locationx

: bending stress amplitude

St: Strouhal number

Tav

: average tension in a cylinder

Tmax : maximum tension in a cylinder

Tmin

: minimum tension in a cylinder

Tp

: ratio of the change of tension to average tension

T xb g : tension at locationxV xb g : flow velocity at locationxV

max: maximum velocity in a flow profile

Vmin

: minimum velocity in a flow profile

VR

l : lower bound of the reduced velocity region

VR

u : upper bound of the reduced velocity region

xrms : root-mean-square (RMS) value of variablexY x

n b g : the nth mode shape of the systemY x

n

" b g : the nth curvature of the systemy x t,b g : cylinder displacement response ,y x tb g : cylinder velocity response ,y x tb g : cylinder acceleration response : lock-in bandwidth

x : spatial resolution =FHG

IKJ

L

N

L

: Reynolds number modification factor for lift coefficient

min

: minimum wave length

v : kinematic viscosity of a fluid

min: minimum excitation frequency due to vortex shedding (radians/sec)

max

: maximum excitation frequency due to vortex shedding (radians/sec)

n: nth natural frequency (radians/sec)

n: nth modal damping ratio

s: structural damping radio


32/43

32

APPENDIX: NATURAL FREQUENCY, MODE SHAPE, AND CURVATURE

The expressions of natural frequency, mode shape, and curvature for each structural model areprovided in this appendix. Caution, these formulas are very complex and may containtypographical errors. The formulas used in SHEAR7 have been very carefully verified for the

linearly varying and constant tension cable and beam models only. Other model boundaryconditions should be compared to known test cases the first time one uses them. This is to checkon numerical stability and accuracy of the equations implemented in the programs.

1. Pinned-pinned cable with varying tension (nmodel=0).The nth natural frequency is given by

n

t

L ds

T s m sn

b g b g/0z = (55)

where L is the total length of the structure, T(s) is the tension, m stb g is the total mass per unit

length, andn

is the nth natural frequency. The nth mode shape is

Y xds

T s m sn

n

t

x

b gb g b g

=L

NMM

O

QPPzsin /

0

(56)

wherex is the spatial location. The origin is at the minimum tension end.

The nth curvature is

Y xm

T x

ds

T s m sn

n t n

t

x

" sin/

b g b g b g b g=

L

NMM

O

QPPz

2

0

(57)

2. Pinned-pinned beam with varying tension (nmodel=1).

The nth natural frequency is given by:

+ LNMM OQPP+ =z 12 12 4

22

0

T sEI s

T sEI s

m sEI s

ds nt n

L

b gb g b gb g b gb g , n = 1 2 3, , ,... (58)

where T(s) is the tension, EI sb g is the bending stiffness, m stb g is the mass per unit length, andn is the nth natural frequency of the structure.

The nth mode shape is:


33/43

33

Y xT s

EI s

T s

EI s

m s

EI sds

n

t n

x

b g b gb gb gb g

b gb g

= +LNMM

OQPP

+

L

N

MMM

O

Q

PPP

zsin 12

1

24

22

0

(59)

wherex is the spatial location. The origin is at the minimum tension end.

The nth curvature is:

Y xT x

EI

T x

EI

m x

EI

T s

EI s

T s

EI s

m s

EI sdsn

n t t n

x

" sinb g b g b g b g b gb gb gb g

b gb g

= FHG

IKJ

+L

NMM

O

QPP +

LNM

OQP +

L

N

MMM

O

Q

PPP

z124 1

2

1

24

2 22

2

0

(60)

3. Free-pinned beam with varying tension (nmodel=2).

The restriction on the following formulation is that the minimum tension is zero. Thenth naturalfrequency is given by:

( ) ( ) ( )2 10 0 0

sin cos tanh 0L L L

sh s ds h s ds h s ds

= (61)

where h s1b g andh s2 b g are given by, respectively,

h sT s

EI

T s

EI

m

EI

t

1

22

2

1

2

4b g b g b g= +F

HG

I

KJ +

, h s

T s

EI

T s

EI

m

EI

t2

22

2

1

2

4b g b g b g= + F

HG

I

KJ+


Y x I I n b g = +1 2 (62)

where:

I T x c h s ds c h s dsx x

1 1 3 1

0

4 1

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b gcosh sinh


2 2 1 2

0

2 2

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b gsin cos

wherex is the spatial location. The origin is at the minimum tension (free) end. cs are given by


34/43

34

c4

1= , c

T L h s ds T L h s ds

T L h s ds T L h s ds

L L

L L3

1 1

0

2 2

0

1 1

0

2 2

0

=

LNM

OQP

+LNM

OQP

LNM

OQP

+LNM

OQP

z z

z z

b g b g b g b g

b g b g b g b g

cosh cos

sinh sin

, c c2 4

= , c c1 3

=

T L1b g andT L2 ( ) can be found from

( )( ) ( ) ( )

13 4

3 2 22 2

1

2 41 1

2 2t t

T x T x T xm mT x

EI EI EI EI EI

= + + +

T xT x

EI

T x

EI

m

EI

T x

EI

m

EI

t t

2

32

22

3

2

1

4

1

2

2 1

2

4b g b g b g b g= FHG

IKJ

+FHG

IKJ

+L

N

MM

O

Q

PP

RS|

T|

UV|

W|


Y x I I n

" b g = +3 4 (63)

where

I T x h x c h s ds c h s dsx x

3 1 1

2

3 1

0

4 1

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b g b gcosh sinh

I T x h x c h s ds c h s ds

x x

4 2 22

1 2

0

2 2

0

= LNM OQP + LNM OQPRST UVWz zb g b g b g b gsin cos

4. Pinned-pinned beam with varying tension and rotational springs a both ends (

nmodel=8).

The nth natural frequency is given by the roots of:

U U U U U U U U 1 2 3 4 5 6 7 8

0+ + + + + + + = (64)

where

U v b b v b b1 2 3

2

4

2

3 1

2

2

2= +c h c h , U v v v h L h L2 1 4 8 12 22= +b g b g

U v v v h h L h h L3 0 1 4 2 2 1 1

0 0= b g b g b g b g , U v v v h h L h h L4 0 1 7 2 1 1 20 0= +b g b g b g b g

U v v w h h L h h L5 1 5 1 1 1

2

1 2

20 0= +b g b g b g b g , U v v w h h L h h L6 1 6 1 2 12 2 220 0= +b g b g b g b g


35/43

35

U v v w h L h h L h7 1 5 2 1 1

2

1 2

20 0= +b g b g b g b g , U v v w h L h h L h8 1 6 2 2 12 2 220 0= +b g b g b g b g

where w1, w

2andv

0..8are given by

wK

EI

r

1

1= , wK

EI

r

2

2= , v w w0 1 2

=

v T T L T T L1 1 1 2 2

0 0= ( ) ( ) ( ) ( ) , v v h L h T L T o2 1 2 1

2

2

20 0= ( ) ( ) ( ) ( ) , v v h h L T T L3 0 1 2 1

2

2

20 0= ( ) ( ) ( ) ( )

v b b4 1 3

= , v b b5 1 4

= , v b b6 2 3

= , v b b7 2 4

=

v h h8 1

2

2

20 0= +( ) ( )

T T L1 1

0( ) ( ) andT T L2 2

0( ) ( ) can be found, respectively, from

T xT x

EI

T x

EI

m

EI

T x

EI

m

EI

t t

1

32

22

3

2

1

4

1

2

2 1

2

4b g b g b g b g=FHG

IKJ

+ +FHG

IKJ

+L

NMM

O

QPP

RS|

T|

UV|

W|

T xT x

EI

T x

EI

m

EI

T x

EI

m

EI

t t

2

32

22

3

2

1

4

1

2

2 1

2


IKJ

+FHG

IKJ

+L

NMM

O

QPP

RS|

T|

UV|

W|

h1

0

b g ,h L

1

( ),

h2

0

b g , andh L

2

( )can be found from

h sT s

EI

T s

EI

m

EI

t

1

22

2

1

2

4b g b g b g= +FHG

IKJ

+

, h sT s

EI

T s

EI

m

EI

t

2

22

2

1

2

4b g b g b g= +FHG

IKJ

+

b h s dsL

1 2

0

=LNM

OQPzsin b g , b h s ds

L

2 2

0

=LNM

OQPzcos b g , b h s ds

L

3 1

0

=LNM

OQPzsinh b g , b h s ds

L

4 1

0

=LNM

OQPzcosh b g , Kr1 and Kr2 are the

rotational spring constants forx = 0 andx L= , respectively.


Y x U U n b g = +9 10 (65)

where

U T x c h s ds c h s dsx x

9 1 3 1

0

4 1

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b gsinh cosh


36/43

36

U T x c h s ds c h s dsx x

10 2 1 2

0

2 2

0

=LNM

OQP

+LNM

OQP

RST


wherex is the spatial location. The origin is at the minimum tension end. cs are given by

c4

10= . , c

K

EI

T h

T L bT L b

T T L b

TT h h

K

EIT h T L b

K

EI

T h

T L b

r

r r

3

1 2 2

2 1

1 4

1 2 2

2

1 1

2

2

2

1

1 1 1 3

1 2 2

2 1

0 0 0

00 0 0

0 00 0

=

LNM

OQP + +

b g b gb g b g

b g b gb g b g b g b g

b g b g b g b g b gb g

cT

T2

1

2

0

0=

b gb g

, cc T L b c T L b c T L b

T L b1

2 2 2 3 1 3 4 1 4

2 1

= + +b g b g b g

b g


Y x U U n

" b g = +11 12 (66)

where

( ) ( ) ( ) ( )211 1 3 1 4 10 0

sinh coshx x

xU T x h x c h s ds c h s ds

= +

U T x h x c h s ds c h s ds

x x

12 2 2

2

1 20

2 20

= L

NM

O

QP+

L

NM

O

QP

R

ST

U

VWz zb g b g b g b gsin cos

5. Free-pinned beam with varying tension and rotational spring at x=L (nmodel=9).


I I I I1 2 3 4

0+ + + = (67)

where

I T L h LK

EIb h L b

K

EIb b h L b br r

1 1

2

1 4 1 3 3 5 1 4 5= + +

LNM

OQP

b g b g b g b g

I T L h LK

EIv h L b v

K

EIv b h L vr r

2 1 1 4 1 5 4 3 5 1 3= + +

LNM

OQP

b g b g b g b g

I T L e v e v b3 1 1 1 2 2 5

= b g


37/43

37

Iv v e v v e

b4

1 4 1 2 3 2

4

=

where T L1b g and T L2 b g can be found, respectively from

T xT x

EI

T x

EI

m

EI

T x

EI

m

EI

t t

1

32

22

3

2

1

4

1

2

2 1

2

4b g b g b g b g=FHG

IKJ

+ +FHG

IKJ

+L

NMM

O

QPP

RS|

T|

UV|

W|

T xT x

EI

T x

EI

m

EI

T x

EI

m

EI

t t

2

32

22

3

2

1

4

1

2

2 1

2


IKJ

+FHG

IKJ

+L

NMM

O

QPP

RS|

T|

UV|

W|

e T L h LK

EIb h L br

1 2 2 2 2 1= +

LNM

OQP

b g b g b g , e T L h L KEI

b h L br2 2 2 1 2 2

= +LNM

OQP

b g b g b g

vT h

T h1

1 1

3

2 2

3

0 0

0 0=

b g b gb g b g

, vT h

T h2

1 1

2

2 2

2

0 0

0 0=

b g b gb g b g

, vT L T h b

T h3

2 1 1

3

1

2 2

3

0 0

0 0=

b g b g b gb g b g

, vT L T h b

T h4

2 1 1

2

2

2 2

3

0 0

0 0=

b g b g b gb g b g

h1

0b g , h L1b g , h2 0b g , and h L2 b g can be found from

h sT s

EI

T s

EI

m

EI

t

1

2

2

2

1

2

4b g b g b g= + FHG

IKJ

+ . h s T sEI

T s

EI

m

EI

t

2

2

2

2

1

2

4b g b g b g= + FHG

IKJ

+

b h s dsL

1 2

0

=LNM

OQPzsin b g , b h s ds

L

2 2

0

=LNM

OQPzcos b g , b h s ds

L

3 1

0

=LNM

OQPzsinh b g , b h s ds

L

4 1

0

=LNM

OQPzcosh b g , b h s ds

L

5 1

0

=LNM

OQPztanh b g

Kr

is the rotational spring constant.


Y x I I n b g = +5 6 (68)where


5 1 3 1

0

4 1

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b gsinh cosh


38/43

38


6 2 1 2

0

2 2

0

=LNM

OQP

+LNM

OQP

RST


wherex is the spatial location. The origin is at the minimum tension (free) end. cs are given by

c4

10= . , cT h T L h L K

EIb h L T h h e b

T h T L h LK

EIh L b T h e b

r

r

3

2 2

3

1 1 5 1 1 1

2

2 2 4

2 2

3

1 1 1 5 1 1

3

1 4

0 0 0 0 0

0 0 0 0

=

LNM OQP+

LNM

OQP

+

b g b g b g b g b g b g b g b g

b g b g b g b g b g b g b g

/

/

cc T h

T h2

4 1 1

2

2 2

2

0 0

0 0=

b g b gb g b g

, cc T h

T h1

3 1 1

3

2 2

3

0 0

0 0=

b g b gb g b g


Y x I I n

" b g = +7 8 (69)

where


7 1 1

2

2 1

0

4 1

0

=LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b g b gsinh cosh


8 2 2

2

1 2

0

2 2

0

= LNM

OQP

+LNM

OQP

RST

UVWz zb g b g b g b gsin cos

6. Pinned-pinned cable with constant tension (nmodel=10).

The nth natural frequency is:

n

t

n

L

T

m= (70)

whereL is the total length, Tis tension, andmt

is mass per unit length.


Y xn x

Ln b g =

FHG

IKJsin

(71)

wherex is the spatial location. The origin can be at either end.



39/43

39

Y xn

L

n x

Ln

"` sinb g = FHGIKJ

FHG

IKJ

2

(72)

7. Pinned-pinned beam with constant tension (nmodel=11).


n

tL

EI

mn

n TL

EI= +

FHG

IKJ

2

2

4

2 2

2

1 2/

(73)


Y xn x

Ln b g =

FHG

IKJsin

(74)

wherex is the spatial location. The origin can be at either end.


Y xn

L

n x

Ln

" sinb g = FHGIKJ

FHG

IKJ

2

(75)

8. Free-pinned beam with constant tension and rotational spring at x=L (nmodel=19).


e e e e e e1 5

3

2

2

3 5 40

FHG

IKJ +

FHG

IKJ + + = (76)

where

= +FHG

IKJ +

n tm

EI

T

EI

T

EI

2 2

2 2,

= +

FHG

IKJ

n tm

EI

T

EI

T

EI

2 2

2 2

eK

EIL Lr

1

2= + cos sinb g b g , e KEI

L Lr2

2= + sin cosb g b g

eK

EIL Lr

3

2= cosh sinhb g b g , e KEI

L Lr4

2= sinh coshb g b g


40/43

40

( ) ( )

( ) ( )

2 3

5 3 3

cos cos

sin sinh

L Le

L L

+=

+

Kr

is the rotational spring constant.


Y x d x d x d x d xn b g b g b g b g b g= + + +1 2 3 4sin cos sinh cosh (77)

wherex is the spatial location. The origin is at the free end, andd4

10= . , d e d3 5 4

= , d d2

2

4=

FHG

IKJ

,

d d1

3

3=

FHG

IKJ

.


Y x d x d x d x d xn

" sin cos sinh coshb g b g b g b g b g= + + + 2 1 2 2 3 4 (78)

9. Free-pinned beam with no tension (nmodel=22).


n

n

tL

EI

m

=2

2(79)

whereL is the total length,EIis bending rigidity, andmt

is mass per unit length. s are given

by

1

392660231= . , 2

7 06858275= . , 3

10 21017612= . , 4

1335176878= . , 5

16 49336143= . , and

n

n=

+4 1

4

b g, n > 5 .


Y xx

L

x

L

x

L

x

Ln

n

n

n n

n

nb g = FHGIKJ

FHG

IKJ+

FHG

IKJ

FHG

IKJcosh sinh cos sin

(80)

wherex is the spatial location. The origin is at the free end. s are given by 1

1000777304= . ,

2

1000001445= . , 3

10= . , 4

10= . , 5

10= . , andn

= 10. , n > 5 .

The nth curvature is


41/43

41

Y xL

x

L

x

L

x

L

x

Ln

n n

n

n n

n

n" cosh sinh cos sinb g = FHGIKJ

FHG

IKJ

FHG

IKJ

FHG

IKJ+

FHG

IKJ

LNM

OQP

2

(81)

10. Clamped-free beam with no tension (nmodel=23).

The natural frequency equation is the same as that for the case when nmodel=22. s are given

by

1

187510407= . , 2

4 69409113= . , 3

7 85475744= . , 4

10 99554073= . , 5

1413716839= . , and

n

n=

2 1

2

b g, n > 5


Y xx

L

x

L

x

L

x

Ln

n

n

n n

n

nb g = FHGIKJ

FHG

IKJ

FHG

IKJ+

FHG


(82)

where x is the spatial location. The origin is at the clamped end. s are given by

1

0 734095514= . , 2

1018467319= . , 3

0 999224497= . , 4

1000033553= . , 5

0 999998550= . , and

n

= 10. , n > 5 .


Y xL

x

L

x

L

x

L

x

Ln

n n

n

n n

n


FHG

IKJ

FHG

IKJ+

FHG

IKJ

FHG

IKJ

LNM

OQP

2

(83)

11. Clamped-pinned beam with no tension (nmodel=24).


by

1

392660231= . , 2

7 06858275= . , 3

10 21017612= . , 4

1335176878= . , 5

16 49336143= . ,

and n

n= +4 14

b g , n > 5


Y xx

L

x

L

x

L

x

Ln

n

n

n n

n

nb g = FHGIKJ

FHG

IKJ

FHG

IKJ+

FHG


(84)


42/43

42

where x is the spatial location. The origin is at the clamped end. s are given by

1

1000777304= . , 2

1000001445= . , 3

10= . , 4

10= . , 5

10= . , andn

= 10. , n > 5 .


Y xL

xL

xL

xL

xL

n

n n

n

n n

n

n" cosh sinh cos sinb g = FHG IKJ FHG IKJ FHG IKJ+ FHG IKJ FHG IKJLNM OQP

2

(85)

12. Clamped-clamped beam with no tension (nmodel=25).


by

1

4 73004074= . , 2

7 85320462= . , 3

10 9956079= . , 4

141371655= . , and

5

17 2787597= . ,

n

n=

+2 1

2

b g, n > 5


Y xx

L

x

L

x

L

x

Ln

n

n

n n

n

nb g = FHGIKJ

FHG

IKJ

FHG

IKJ+

FHG


(86)

where x is the spatial location. The origin can be at either end. s are given by

1

0 982502215= . , 2

1000777312= . , 3

0 999966450= . , 4

1000001450= . , 5

0 999999937= . , and

n

= 10. , n > 5 .


Y xL

x

L

x

L

x

L

x

Ln

n n

n

n n

n


FHG

IKJ

FHG

IKJ+

FHG

IKJ

FHG

IKJ

LNM

OQP

2

(87)

13. Sliding-pinned beam with no tension (nmodel=26).


by

n

n= 2 12

b g (88)


Y xn x

Ln b g

b g=

LNM

OQPcos

2 1

2

(89)


43/43

wherex is the spatial location. The origin is at the sliding end.


Y x nL

n xL

n

" cosb g b g b g= LNM OQPFHG IKJ

2 12

2 12

2

(90)

Date post:	11-Feb-2018
Category:	Documents
Upload:	uktaindra
View:	250 times
Download:	0 times

SHEAR7 Version4.4 2005 TheoryManual A

Documents