· ~-. - -' '-- -. 

/[l!/JJI--cft!- / S s-t; 7 

CSDL-R-1675 

NASA-CR-175567 

19850013384 

TRANSVERSE VIBRATION AND BUCKLING OF A 

CANTILEVERED BEAM WITH TIP BODY UNDER 

CONSTANT AXIAL BASE ACCELERATION 

by 

Joel Storch and Stephen Gates 

October 1983 

/ 

'J'.NGLEY RESEr,,", I cp 

tI8R,c8" '< ... s· 

H .... '~ °TOIl. VIP·~lr!'A 

The Charles Stark Draper Laboratory, Inc. 

Cambridge, Massachusetts 02139 

Approved for public release; 

distributIOn unlimited. 

NASA(PA), 12/9/83 

11\\111\1 \IU IIII UIII UIII IUI\ IIUI 1\11 lUI 

NF00728 

" 
TECHNICAL REPORT STANDARD TITLE PAGE 

1. Report No. 12. Government Accession No. 3. ReCIpient's Catalog No. 

4. Title and Subtitle !I. f(eport Date 

TRANSVERSE VIBRATION AND BUCKLING OF A October 1983 

CANTILEVERED BEAM WITH TIP BODY UNDER 6. Performing Organu:atlon Code 

CONSTANT AXIAL BASE ACCELERATION 

7. Author(s) 8. Performing Organization Report No. 

Joel Storch and Stephen Gates CSDL-R-1675 

9. Performing Organization Name and Address 10. Work Unit No. 

The Charles Stark Draper Laboratl ~y, Inc. NAS9-l6023 

555 Technology Square 11. Contract or Grant No. 

Carnbr1dge, Massachusetts 02139 

13. Type of Report and Period Covered 

12. Sponsoring Agency Name and Address Techn1cal Report Nat10nal Aeronaut1cs and Space Adm1n1strat10n 

Johnson Space Center 

Houston, Texas 77058 14. Sponsoring Agency Code 

15. Supplementary Notes 

16. Abstract , 

Analyses of the planar transverse bending behav10r of a un1form cant1levered 

. beam w1th rig1d t1P body subJect \0 constant aX1al base accelerat10n are presented. The beam 1S 1nextens1ble and capable of small elastic transverse bend1ng deformations only. Two classes of t1P bod1es are recogn1zed, those w1th 

mass centers located along the beGm t1P tangent l1ne (restr1cted mass center 

offsets) and those w1th mass centers hav1ng an arb1trary offset w1th respect to 

the beam attachment P01nt. The s~eady-state response 1S stud1ed for the beam 

end-cond1t10n cases: free, t1P me. ss, t1P body w1th restricted mass center 

offset, and t1P body w1th arb1trary mass center offset. The f1rst three cases 

constitute class1cal Euler buckl1ng problems, and for each, the character1st1c 

equat10n for the crit1cal loads/accelerat10ns are determ1ned. A nonhomogeneous 

boundary value problem ar1ses for the last case and 1t 1S shown that a un1que 

steady-state solut10n eX1sts, except at certa1n d1screte values of base accelerat10n for wh1ch no such solutior exists. The free vibrat10n response 1S exam1ned for the two classes of tip )ody. The character1st1c equation, e1genfunct10ns and the1r orthogona11ty propert1es are obta1ned for the case of restr1cted - mass center offset. The vibrat10n problem 1S nonhomogeneous for the case of 

arb1trary mass center offset. The exact solut10n 1S obta1ned as a sum of the 

steady-state solut10n and a superpos1t10n of s1mple harmon1c mot10ns. Approximate solutions to the free v~rat1on and buckl1ng characteristics for the 

problem of restr1cted mass center offset are obta1ned uS1ng an assumed modes 

17 Key Words Suggested by Author 18. Distribution Statement {cont1nu ed) 

V1brat10ns Mechan1cal Stab1l1ty 

Buck11ng Dynam1cs 

Beams Parametr1c Excitat10n 

19. Security Classif. (of thiS report) 20. Security Classlf. (of this page) 21. No of Pages 22. Price 

UNCLASSIFIED UNCLASSIFIED 61 
16. ABSTRACT (Cont.) 

formulation. A method for recovering the stra~n energy from the govern~ng 

part~al d~fferent~al equat~on and boundary condit~ns ~s set forth. Selected 

FORTRAN programs and numer~cal results are prov~ded. 
CSDL-R-1675 

TRANSVERSE VIBRATION AND BUCKLING OF A 

CANTILEVERED BEAM WITH TIP BODY UNDER 

CONSTANT AXIAL BASE ACCELERATION 

by 

Joel Storch and Stephen Gates 

October 1983 

The Charles Stark Draper Laboratory, Inc. 

Cambridge, Massachusetts 02139 
ACKNOWLEDGMENT 

The authors wish to acknowledge the helpful discussions w1th 

Gary Rosen of The Charles Stark Draper Laboratory, Inc. which were 

invaluable in obta1ning the correct strain energy express10n. 

1i 
TABLE OF CONTENTS 

sectl.on 

INTR.OnTJCTION •••••••••••••••••••••••••••••••••••••••••••••••••• 

2 MOTION EOUATIONS •••••••••••••••••••••••••••••••••••••••••••••• 3 

2.1 nerl.vatl.on of the partl.al Differential 

Equatl.on and Boundary Condl.tions........................ 3 

3 STEADY-STATE SOLUTIONS........................................ 11 

3.1 Buckll.ng of a Cantl.levered-Free Beam 

under AX1al 'I'hrust...................................... '1 

3.2 Buckll.nq of cantl.levered Beam wl.th 

Tl.P Mass under Axl.al ~rust ••••••••••••••••••••••••••••• 13 

3.3 Buckll.nq of Cantl.levered Beam with 

Tl.p Body Under Axial 'l'hrust (y = 0) ••••••••••••••••••••• 16 

3.4 steady-State solutl.ons of a cantl.levered 

Beam wl.th Tip Body Under Axl.al 'l'hrust (y = 0) ••••••••••• 17 

4 VIBRATION SOLUTIONS........................................... 22 

4.1 Na tural Frequencl.es and Mode Shapes 

(y = O)--power Serl.es Solutl.on •••••••••••••••••••••••••• 22 

4.2 Free vl.bratl.on of a Cantl.levered Beam 

Wl.th Tl.p Body under Constant Base 

Accelerat10n (y * 0) •••••••••••••••••••••••••••••••••••• 28 

l.l.l. 
TABLE OF CONTENTS (Cant.) 

Section 

5 NATURAL FREQUENCIES AND MODE SHAPES (y = 0) 

ASSUMED mDES SOLUTION........................................ 31 

5.1 Kinetic and 1tral.n Energl.es............................. 31 

5.2 Eigenvalue ~ oblem •......•..•..•....••.•••••••.•.•.•...• 35 

6 FORTRAN PR()GRAMS................................................. 36 

Appendix 

A WORK -ENERGY BALANCl':........................................... 3 8 

B INDEFINITE INTEGRA.S OF EIGENFUNCTIONS •••••••••••••••••••••••• 43 

C NATURAL FREQUENCIES AND MODE SHAPES PROGRAM LISTING ••••••••••• 46 

D BUCKLING LOAD/ACCEI.ERATION PROGRAM LISTING.................... 53 

E COMPARISON OF SELECTED NUMERICAL RESULTS •••••••••••••••••••••• 56 

REFERENCES. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 58 

LV 
CHAPTER 1 

INTRODUCTION 

Beam-like structures cantilevered to the Shuttle cargo bay will, 

in general, experience axial loads as a result of attitude maneuvers. 

In our prev10us \lork (Reference 1), we developed dynamics models for 

such a class of orbiter-payload systems with the effects of axial loads 

ignored for reasons of simp11city and tractab11ity. In this report, we 

exam1ne that approximation in a more simple sett1ng which clearly exposes the phenomena of interest. Presented are analyses of the planar 

transverse bendirg behaV10r of a uniform cantilevered beam w1th rig1d 

tip body under the action of constant axial base acceleration. Exact 

steady-state and free v1bration solutions are obtained for var10US 

forms of th1s problem with add1tional approximate solut10ns for selected cases. While the results of this study are of limlted dlrect applicab1lity to the problem of Reference 1, they are of general interest, provide much useful ins1ght, and serve as a basis from which to 

address more complicated situations. 

The beam il taken to be long, slender, and 1nextensible with 

uniform mass, stiffness, and cross section. Only small elast1c transverse bend1nq deformat10ns in a plane of symmetry of the cross section 

are considered. The tip body is rigidly attached to the beam, with its 

mass center hav1ng an arbitrary offset w1th respect to the attachment 

p01nt. In all cases, the base acceleration is constant in magnitude 

and d1rected along the undeformed longitudlnal axis of the beam. The 

governing partial d1fferential equation is shown to be l1near with 
variable coeff1cients, accompan1ed by nonhomogeneous boundary conditions for the general problem of arbitrary mass center offset. The 

steady-state response is examined for a number of beam end conditions. 

For the cases of free-end, tip mass, and tip body with mass center 

along the beam t1P tangent line (henceforth referred to as restr1cted 

mass center offset), the boundary value problems are homogeneous. 

These class1cal Euler buckl1ng problems are solved e(actly for the 

cr1t1cal buckling loads/accelerat10ns. For the problem of arb1trary 

t1P body mass center offset, it is shown that a unique steady-state 

solut10n eX1sts, except for certa1n cr1t1cal values of the base accelerat10n for which no such solut10n eX1sts. The treatment of transverse 

v1brat10n begins with the case of restricted mass ce.ter offset. The 

boundary value problem 1S homoqeneous and admits the usual separat10n 

of var1ables solut10n with harmon1c time dependence. The ensu1ng var1-

able coeffic1ent ordinary d1fferential equation for the spat1al funct10ns is solved exactly in terms of a power ser1es. Appl1cat10n of the 

boundary conditions yields the character1st1c equat1c~ and eigenfunctions. For the case of arbitrary tip body mass center offset, the 

v1bration problem 1S nonhomogeneous. It is shown tha~ the exact solution may be written as the sum of the steady-state solution obtained 

previously, and a superposition of simple harmonic mccions which correspond to the exact solut10n of the assoc1ated homogeneous problem. 

Th1S approach, wh1ch takes advantage of the steady-state solut10n, 

yields a greatly s1mpl1fied form of the f1nal solut10n. An assumed 

modes formulat10n 1S deta1led for the restricted mass center offset 

case. The approx1mate solutions to the free vibr at lor. and buckl1ng 

character1st1cs serve to check the exact analyses. A particularly 

useful and general method for recover1nq the stra1n energy from the 

governinq partial d1fferent1al equat10n and boundary cond1t10ns 1S set 

forth. Selected FORTRAN programs and numer1cal results are prov1ded. 

2 
CHAPTER 2 

MOTION EQUATION~ 

2.1 Derivation of the Partial Differential Equation 

and Boundary Condltions 

In this chapter, we der i ve the partie;l dlfferential equat lon and 

boundary conditl0ns governlng the planar transverse vibratl0n of a beam 

wlth tip body under the action of an axial force. Flgure 2-1 deplcts 

the situatl0n ln the undeformed configuration. At one end of the beam, 

(x = 0), an aXlal force is applied lmparting a veloclty, vO(t), along 

the x·axis. The beam is assumed inextensibl~ and of length t. At the 

other end of the beam, (x = t), a tip body is rlgldly attached at point 

P. The rlgld tlP body has mass m and moment of inertia about ltS mass 

center It. The dlstance from P to the mass center of the 'tip body 1S 

c, and this d1rected 11ne segment makes an a,gle, y, w1th the beam t1P 

tangent at P. Figure 2-2 shows the system in the deformed state. The 

base 0 of the beam moves along the x d1rect1on while the deformat1on 

u(x, t), occurs along the y d1rect1on. Note that the angle y is maintained constant. 

y 

INERTIAL 

FRAME 

x 

y 

Flgure 2-1. 

3 
y y 

~--------~x INERTIAL O~==;=====~~=-------~-----L------~ FRAME Q x 

F1.qure 2-2. 

F1.qure 2-3 loS a free-body d1.aqram of a beam element between x 

and x + ~x. ~(x, t) loS the l.nternal tens1.on, Sex, t) loS the shear 

force, and M(X, t) is the bend1.nq moment. S1.nce the beam loS assumed to 

have no deformat1.on alonq the x axis, the accelerat1.on component alonq 

x loS ao(t) for any po1.nt l.n the beam, where aO(t) loS the prescr1.bed 

base accelerat1.on at O. 

M(X,t) 

TI··"rB---

u(x, tl SIx, tl 

x 

SIx + Ax, t) 

H---f-'T--i~ T(x + .:l.x, t I 

u(x + .:l.x, tl 

I x +.:l.x 

F1.qure 2-3. 

4 
Equil1brium of the beam element along the x direction gives 

3T 

ax = pa (t) 

o 

where p 1S the mass per un1t length of the beam. 

Similarly, equilibrium along the y-direction gives 

(2-1 ) 

(2-2) 

Applying rotational equilibr1um to the beam element and neglecting 

rotary 1nertia we obtain 

3M 3u ax + S (x, t) - T (x, t) ax = 0 (2-3 ) 

D1fferentiat1ng Eg. (2-3) and usinq Eq. (2-2), we obtain 

Assum1ng an Euler-Bernoulli beam 

M (x, t) 

we arrive at the part1al d1fferent1al equat10n for the beam deflect10n 

(2-4) 

5 
We also have the d~fferent~al relat~ons for the natural boundary cond~­

tions 

M (x, t) 

s (x, t) = 

Kinematics of the Tip Body 

-+- -+-

2 

T(x,t) au a [EI(X)~] ax - ax ax2 

(2-5) 

(2-6) 

Let v and w denote the ~nert1al veloc~ty of P and the angular p p 

velocity of the tip body, respectively 

-+- -+- au -+- v = v (t) ~ +- (t,t)] P 0 at (2-7) 

-+- a

2

u -+- w = atax (t, t) k 

P (2-8 ) 

-+- -+- -+- where i, j, k denote unit vectors along the directions of x, y, z, respectively. 

-+- If we let c be the vector from P to the mass center of the t1P body and 

-+-

ve the velocity of the t~p body mass center then 

= 

= 

(2-9) 

6 
a a2

u where we have dropped the nonlinear term a~ (t,t) atax (t,t). 

Differentiating Egs. (2-8) and (2-9), we have the acceleration expressl.ons 

+ 

a

EB 

+ 

. 

w = p (2-10) 

a

3

u ]+ a

2

u a

3

u]+ [a (t) - c sin y ---2--(t,t) l. + [---2 (t,t) + c cos y ---2--(t,t) J 

o at ax at at ax 

(2-11) 

Natural Boundary Conditions at x = t 

Figure 2-4 is a free body diagram of the tip body l.ndicating the 

force and moment exerted by the beam at p. 

-T(Q.t)---H~ 

r .. x 

Figure 2-4. 

7 
The equation of motion of the tip body when resolved along the 

directions of x and y yields the two scalar equat10ns 

-T(R.,t) a3

u 

= rna (t) - mc sin y ----- (R.,t) 

o axat2 

(2-12) 

-S(R.,t) (2-13) 

Using Eqs. (2-6) and (2-12) allows us to write the first boundary 

condition at x = t as 

a

3

u a

3

u au 

mc cos y ----- (R.,t) - EI --- (R.,t) - rna (t) (R.,t) 

axat2 ax3 0 ax 

(2-14) 

where we have ass\~ed a un1form EI and have dropped the non11near term 

au a

3

u 

mc S1n y ax (R.,t) axat 2 (R.,t). 

It can be shown that the z component of the time rate of chanqe 

of the angular mo~entum of the t1P body about its mass center is g1ven 

by 

+ 

3 

I ~ (R.,t) 

t axat2 

If Me is the net moment about the mass center of the t1P body 

+ + + + + 

= -M(R.,t)k + c x S(R.,t)] + c x T(R.,t)l 

8 
US1ng Eqs. (2-5) and (2-6) and dropp1ng non11near terms we obta1n 

2 a3u 

-[EI ~ (i,t) + EI c cos Y --- (i,t) + c sin y T(i,t) ]k 

ax

2 

ax

3 

(2-15) 

US1ng~. (2-12) for T(i,t) in Eq. (2-15), the second boundary cond1t1on 

at x = i is 

- mc sin y a (t) = 0 

o (2-16) 

The part1al different1al equation govern1ng u(x,t), Eq. (2-4), 

requires a spec1f1cation of the internal tens10n T(x,t) for 0 < x < i. 

Integrating Eq. (2-1) and uS1ng Eq. (2-12) for a boundary condit1on w~ 

obta1n 

T (x, t) a3

u = -[p(i - x) + m]a (t) + mc sin y (t,t) 

o axat2 

(2-17) 

Assume we have a compressive load, poet) > 0 applied at x = o. Then 

Hence 

p (t) 

o 

T (0, t) = - p (t) 

o 

a3u = (pi + m)a (t) - mc S1n y (i,t) 

o axat2 

which relates the app11ed axial force to the base acceleration. 

9 

(2-18) 
The governing equat10n of mot10n, Eq. (2-4), can be rewritten 

with the a1d of Eqs. (2-17) and (2-18) as 

EI o (2-19) 

We assume the beam to be cantilevered at x o so that 

u(O,t) au 

= ax (0, t) = 0 for all t (2-20) 

The part1al different1al Eq. (2-19) 's to be solved subject to 

the geometr1c boundary cond1t10ns of Eq. (2 ,20) and the natural boundary 

cond1t10ns of Eqs. (2-14) and (2-16). 

In Reference 1, we stud1ed the transverse vibration of a clamped 

beam w1th t1P body w1th no base acceleration. On page 10 of that report 

1t was stated that y must be restr1cted to ?ero in order that no aX1al 

loads be 1ntroduced 1nto the problem. W1th the present analys1s, one 

can accommodate any ax1al loads that may be 1ntroduced by y * 0, and 

w111 discover, surprisingly, that the result.s given in Reference 1 can 

be directly extended to the case of y * O. 

10 
CHAPTER 3 

STEADY-S ,'ATE SOLUTIONS 

3.1 Buckl~nq of a Cant~levered-Free Beam Under Axial Thrust 

In this section, the possibility of a cantilevered-free beam (no 

t~p body) bucklinq under axial thlust is investigated. The governing 

equat~on is Eq. (2-1q) with no time dependence and m = o. 

d

4 d d 

EI .-Y.. + P r ( 1 - xl!!.) dxY 1 = 0 dx4 cr dx (3-1 ) 

where Pcr denotes those values of thrust which lead to buckling. 

The two qeometric boundary conditions at x = 0 are 

yeo) = yl(O) = 0 

The boundary conditions at x = t are obta~ned from F~s. (2-14) and 

(2-16) with m = c = 0 

y"(9') = y"'(9.) = 0 

Inteqratinq F~. (3-1) and sett~ng the constant of integration to zero we 

obtain 

3 P d v cr dy - + - (1 - xl t) - = 0 

dx3 EI dx ( 3-2) 

11 
subJect to the three boundary conditions 

y (0) = y I (0) = y" (R.) = 0 

making the substitution 

dy = 

dx w, 

transforms Eq. (3-2) 1nto 

with boundary conditions 

Z = (pC~:2r3 (1 _ x/tJ 

w = 0 at z 

dw 

dz = o at z = o 

The general solution of the different1al equat10n can be written 

w = r- (2 3/2) r- (2 3/2) C1YZ J 1/ 3 3 Z + C2YZ J_ 1/ 3 3 Z 

where c1 and c2 are arb1trary constants and J1/3 denotes the Bessel funct10n of the f1rst kind of order 1/3. 

12 
The boundary cond~tion ~: = 0 at z = 0 implies that c 1 = o. 

Let In denote the roots of the Bessel funct~on of the f~rst kind of 

order (-1/3), i.e., 

Applying the rema~ning boundary condition we f~nd that the cr~tlcal 

buckllng loads are given by 

P 

cr,n 

9 .2 EI 

= "4 I n 2 

1. 

The first three buckllng loads are 

P crl = 7.8664 E; 

1. 

n = 1,2,3, ••• 

55.977 E; 

1. 

P cr3 

3.2 Bucklinq of Cantilevered Beam with Tip Mass 

Under Axial Thrust 

= 

(3-3) 

148.51 EI 

1. 2 

The govern~ng equation is Eq. (2-19) with no time dep~ndenee and 

the boundary eondit~ons are Eqs. (2-20), (2-14), and (2-16), with It = 0 

and e = o. 

(3-4) 

mP 

y (0) = y' (0) = 0 y' '(1.) = 0 EI y"'(1.) + p~ ~rm y' (1.) = 0 

where we have used Eq. (2-18) to express ao in terms of Per. 

13 
Integrat1ng Eq. (3-4) and sett1ng the constant of 1ntegration to 

zero, we will satisfy the last boundary cond1tion. 

3 P 

d Y + ~ (1 _ px ) dy = 0 

dx3 EI pt + m dx 

h . dy d If we make t e subst1tut1ons dx = w an 

z 

th1S equ,lt1on transforms to 

w1th boul,dary cond1t1ons 

and 

where 

and 

P 

[

--.£E. (pt + m) 2] 1/3( 1 _ px ) EI P pt + m 

w = 0 at z = zl 

dw 

dz = o at z = 

= 

P 

= m [ cr ]1/3 

2/3 

p EI(pt + m) 

14 

(3-5) 
]6 has already been shown 

w 

dw 

dz 

= 

= 

,- r (2 3/2) J (1. 3/2)1 z c l J l / 3 3 Z + c 2 -1/3 3 Z 

APplyinq the boundary cond1t1ons, we obta1n a homogeneous system 1n c l 

and c 2 • sett1ng the determinant to zero (we want Vex) * 0), we obta1n 

the follow1ng transcendental equation for the cr1t1cal buck11ng loads. 

* 

J l / 3 ((1 + m*)l) • J 2/ 3 ( 

m 1) 

f1 + l/m* 

* 

+ J -1/3 (( 1 + m*>l) ( m 1) o (3-6) • J -2/3 = 

f1 + l/m* 

where 

m* m/p9. 

and 

= # 9.2 

2 cr 

3 F.I 

Note that the roots of F'..q. (3-6) depend only upon the ratio of the t1P 

n, 'iSS to the beam mass. 

In the l1m1t as m* + 0, the cr1tical huck11nq loads 1n Eq. (3-6) 

should approach those found prev10usly for a cant1levered-free beam 

(F,q. 3-3). Th~s ~s read~lv established if we recall the facts 

and 

lim 

x+o 

lim 

x+O 

J 2

/ 3 (x) 

15 

= o 

= co 
3.3 Buck11nq of Cant1levered Beam w1th T1P Body 

Under AX1al Thrust (y = 0) 

The governinq equat10n 1S Eq. (2-19) with no time dependence and 

the boundary conditions are Eels. (2-14), (2-16), and (2-20) w1th y = o. 

= o 

y (0) = Y I (0) = 0 EI C y'" (1) + EI y"(I) = 0 (3-6) 

Pcr EI Y I I I (1) + IT. y I (1) = 0 pi + m (3-7) 

Inteqrating the differential equat10n and settinq the constant of 

1ntegrat10n to zero, we will satisfy the boundary cond1t10n Eq. (3-7). 

pX ) dy = 

pi + m dx o 

If we make the same Subst1tut10ns a9 1n the previous sect10n we obta1n 

w = 0 at z = 

subst1tut1nq Eq. (3-7) 1nto Eq. (3-~) allows us to wnte the second 

boundary cond1t10n as 

(pi + m) EI y" (1) - mc P yl (1) = 0 

cr 

or 1n terms of w(z) 

16 

(3-8) 

(3-9) 
US1ng the results from the prev10us sect1on, Eq. (3-9) can be wr1tten as 

J_ 2/3 ( m* ).) + % rrri* c* ). J1/3 ( m* ).~Cl 11 + l/m* 11 + l/m* J 

+ ~ 1M*" c* , ( m* ).) _ 11 

J -1/3 .~':;:1 ==:;:;::;: 

+ l/m* 

+ m* J 21 3 ( ',..-1----,--). m* ~ c 2 

+ l/m* 

= 

(3-10) 

where c* = cit. 

The first boundary cond1tlon at z = zl can be written as 

= o (3-1 1) 

The homogeneous system, Eqs. (3-10) and (3-11), w1ll have a nontrlv1al 

Solutlon 1f 

o (3-12) 

I 1 where a = 1 + m* and a = m*/v 1 + m*' Note that the critlcal bucklinq 

loads are lndependent of the tip body inertla. 

3.4 Steady-State Solutions of a Cantllevered Beam 

w1th T1P Body under Axlal Thrust (r * 0) 

o 

In the previous section, we investiqated the buckllnq of a un1-

form cantllevered beam wlth tlP body. It wlll be recalled that in that 

section, the mass center of the tip body was restricted to lle along the 

beam tlP tangent line. Presently, we wish to investlgate the POSS1-

blllty of bucklinq for the case of nonzero r. 

17 
As before, we wish to determ1ne solut1ons of Eq. (2-19) wh1ch are 

indeoendent of t1me, u = y(x). The govern1ng equat10n is 

o (3-13) 

w1th geometric boundary cond1t1ons y(O) = ;; (0) = O. The natural 

boundary condit10ns at x = ~ may be wr1tten 

p 

cr EI c cos y y'" (~) + EI y" (~) - mc S1n y P~ + m 

p 

cr EI v'" (~) + my' (~) = 0 P~ + m 

= o (3-14) 

(3-15) 

Inteqrat1ng Eq. (3-13) and sett1ng the constant of 1ntegrat10n ~o zero, 

we w1ll sat1sfy the boundary cond1t1on Eq. (3-15). 

= o 

If we make the change of var1ables 

and 

z = 

dy 

dx = w 

p 

~ (Pi. + m)2]1/3(1 _ px ) 

EI P P~ + m 

18 

(3-16) 
the different1al equat10n (3-16) transforms 1nto 

d2w 

--+ zw = 0 (3-17) 

dz 2 

p 

(pt + m ,2/3 x = o goes into z = z1 (~)1/3 - EI P 

P 

t goes 1nto z 

m [ cr 11/3 x = = z2 - 2!3 p EI(pt+m) 

substituting Eq. (3-15) into Eq. (3-14) and uS1ng the above transformat10n allows us to write the remaining natural boundar: cond1t10n as 

[CPt + m)EI1 2/ 3 (p p)1/3 dw (z ) + mc cos y P w(z) 

cr dz 2 cr 2 

we can wr1te the general solution to Eq. (3-17) as 

w = 

,.,here c1 and c2 are arb1 trary constants. 

= -mc S1n y P 

cr 

(3-18) 

ApplY1ng the boundary cond1t10ns w(z1) = 0 and Eq. (3-18) we 

obta1n the follow1ng nonhomogeneous system on c1 and c2 

= o 

(3-19) 

where the parameters m*, c*, A, a, B are as def1ned prev10usly. 

19 
In general these equat10ns will have a un1que solut10n. If A 

(axial load) assumes a value such that the coefficient matrix in 

Eq. (3-19) 1S singular, it can be shown that the equat10ns are incons1stent. Note that the transcendental equat10n obtained by sett1ng the 

determinant of the system (3-19) to zero is exactly the same as 

Eq. (3-12) except for the replacement c* + c* cos y. Hence the system 

(3-19) can become singular for an infinite number of real values of A. 

Fbr those cases where a unique solution eX1sts, we can 1ntegrate 

once more and obtain (recall y = 0 at x = 0) 

s y(x) = (3-20) 

where 

s 

2 

(

.2. ~) 1/3 

1 4 a 

and 

on the interval 0 < x < 1. 

F1gure 3-1 illustrates the steady-state solutions given by 

Eq. (3-20) at values of A lying between the first four consecutive cr1-

tical values. Numerical exper1ments revealed that the overall shapes of 

these curves are insens1tive to m*, c*, 1, and y w1thin the respect1ve 

ranges of A. 

We have shown that for the case y * 0, a un1que steady-state 

solution to the vibrat10n equation eX1sts, except when the aX1al thrust 

assumes certain cr1tical values (such that the l1near system (3-19) 

becomes singular) for which no solut10n exists. 

20 
A1 

"2 

A1 +A2 

2 

A2 +A3 

2 

., 

0 

~ 

.... 

"-0 

-0 

;: ., 

0 

"b 00 13 '0 25 eo '0 20 S3 50 67 00 eo '0 93 80 107 20 120 60 13' 00 

X {FT I 

.. 

, 

.... 

"-

-~ 

>< 

;: 

.. 

":"0 00 13 .0 25 eo '0 20 S3 50 57 00 80 '0 93 80 107 20 120 50 134.00 

X (FT) 

: 

.... 

"-

0 

N 

>< 

... 

"b 00 13 '0 28 80 '0.20 S3 eo 67 00 80 '0 93 80 107 20 120 50 134 00 

X {FT I 

N 

;~00~~-1+3-'-0~--2+5-e-0~--'+0-2-0~--S+3-5-0~--5+7-0-0~--8+0-'-0~--9+3-8-0~--1+0-7-2~0---1+2-0-6~0--~134.00 

X (FT I 

PARAMETERS m" = 2 778. c" = 0 5, l! = 1340 ft, 'Y = 45 O· 

FIRST FOUR CRITICAL VALUES OF A 

A1 = 0 89 A2 = 2 74 A3 = 4 81 A4 = 6 97 

Figure 3-1. Representat1ve steady-state solut10ns for y * o. 

21 
CHAPTER 4 

VIBRATION SOLUTIONS 

4.1 Natural Frequenc1es and Mode Shapes (y = O)--Power 

Ser1es Solution 

In th1S chapter, we w1ll formulate the eigenvalue problem for the 

t:ansverse v1brat1on of a cantilevered beam with tip body under the act10n ofaX1al thrust. Since an e1genvalue problem requires the Solut1on 

of a homogeneous different1al equation with homogeneous boundary cond1-

t1ons, we need to restrict y to zero (see boundary cond1tion (2-16). The 

a.'ial thrust Po 1S assumed constant. 

Using Eqs. (2-18) through (2-20), (2-14) and (2-16), we have 

4 

a 

m) ~~] a

2

u EI ~+ [ ( 1 px P 0 ax - + p -- = 

ax4 0 pR, + at2 (4-1 ) 

u(O,t) au 

= (0, t) 0: t > 0 

ax = (4-2) 

(4-3) 

22 
Assuming a solution eLwt~(x), we arrive at the eigenvalue 

problem 

d4p d [(1 - px ) ~l 2 EI + P 

o dx - pw ~(x) 0 

dx4 pR. + m dx (4-5) 

~(O) = "* (0) = 0 (4-6) 

2 P 2 EI ~"'(R.) + m(cw 0 

+ )~' (R.) pR. + 

+ mw ~(R.) 0 

m (4-7) 

(4-8) 

We wLsh to fLnd those values of w for whLch the dLfferentLal equatLon (4-5), subJect to the boundary condLtLons (4-6) throuqh (4-8) has 

nontrivLal solutLons. 

scaling of Eigenvalue problem 

Let ~ = x/R. and defLne y(~) _ ~(t~). Also define the dimens10nless parameters 

2 l' R. 2 

b 0 4 ~ 2 

a = w (4-9) EI EI 

It + 

2 

mc 

m* m/pt, c* c/9., J* 

pR.3 

23 
The e~genvalue problem will then transform ~nto the convenient form 

d

4

y + a2 d [( ~) dy ] 4 ~ d~4 d~ 1 - 1 + m* d~ - b yes) = 0 (4-10) 

y 0) = y' (0) = 0 (4-11) 

4 2 4 

y' I 1(1) + m*(c*b + 1 : m*)y' (1) + m*b y(1) = 0 (4 -12) 

4 (4 m*c* a 2)y' (1) _ yl I (1) m*c*b y(1) + J*b + 1 + m* = 0 (4-13) 

Here (I) denotes differentiation with respect to ~. 

Let Yi(~) and YJ(~) be two eigenfunctions correspond~ng to 

distinct eigenvalues. It can be shown that the following orthogonality 

condition holds 

1 f y~ (~)YJ (~)d~ + Jofryi (1)yl (1) + m*y (1)y (1) 

o J ~ J 

+ m *c * [y (1) Y I (1) + y! (1) y. (1)] = 0 

1 J ~ J 

cf. Eq. (2-16) of Reference • 

The qeneral solution of Eq. (4-10) can be wr~tten as a l~near comb~nat~on of four linearly lndependent solutions. 

y( ~) = 

For conven~ence, we spec~fy the ~n~t~al values of these four funct~ons 

and the~r first three der~vatives as 

24 
y(j-l) (0) = C .. 1 1] 

i = 1,2,3,41 j = 1,2,3,4 

(c .. - Kronecker delta) 1] 

Applying the geometric boundary conditions (4-11) we find 

= o and = o 

App11cat10n of the natural boundary conditions (4-12) and (4-13) Y1elds 

the homogeneous system of equations on c3 and c4 

where 

+ m* (C*b 4 + 1 

2 4 

all = y"'(1) : m*) yj (1) + m*b y 3 (1) 3 

(c*b4 + 

2 4 

a12 = y I I I (1) + m* : m*) Y4 (1 ) + m*b Y4 (1) 4 

(4-14) 

4 

+ (J*b 4 + 

m*c* 

a

2

a )yj(1) - y I 1(1) 21 = m*c*b y (1) 3 1 + m* 3 

4 + (J*b4 + m*c* a

2

)y4 (1) a - y I I (1) 22 = m*c*l y (1) 4 1 + m* 4 

In order for th1s system to have a nontr1v1al solution we require 

= 0 (4-1 5) 

Equation (4-15) determines the natural frequenc1es wi25 
Power Ser1es Expansion for Eigenfunctions 

It still rema1ns to f1nd the funct10ns Y3 and Y4. A purely 

numerical techn1que wh1ch 1S easy to apply cons1sts of integrat1ng two 

initial value problems and 1terat1ng on w unt1l values are found which 

sat1sfy Eq. (4-15). The well-known natural frequenc1es of a cantilevered-free beam serve to bracket the frequencies 1n the present case of 

a cant1levered beam w1th a tip body on an accelerating base. We choose 

to address the problem analytically by solv1ng the d1fferent1al 

Eq. (4-10) 1n the form of a power series, thus obta1ning the e1qenfunctions in functional form. 

Note that for Po = 0, the differential equat10n has constant 

coeff1c1ents and can therefore be solved in closed form 1n terms of elementary functions (see Reference 1). Fbr the more general case, Po * 

0, the equation is linear w1th analytic coefficients and has a solution 

which is regular at ~ = o. The series represent1nq the solution w1l1 

converge for all ~ (see Ince., E.L., nOr d1nary Differential Equations"). 

If we assume a solution of Eq. (4-10) 1n the form 

co 

y = L 

k=O 

insert this expans10n and 1ts corresponding der1vc+-1ves 1nto Eq. (4-10), 

then we obtain the following recursion formulas for the coefficients 

= 

= (4-16) 

b

4

Ak + a

2

/(1 + m*) • (k + 1)2Ak+1 - a

2 (k + 1) (k + 2)Ak+2 

(k + 1) (k + 2) (k + 3) (k + 4) 

(4-17) 

26 
These conditions determine A4 , AS' A6 , ••• once we presr~be values for 

AO' A1, A2 , and A3 • 

To generate Y3(~)' we must have AO 

Tb generate Y4(~)' we must have AO = 0, A1 = 0, A2 = 0, A3 = 1/6 

The eigenfunct~on Yi(~) correspond~ng to the e~genvalue wi ~s 

Y. (~) ~ = 

4 4 2 2 a + b. 

a ~4 + a ~5 + ~ ~6 + 

24 60 (1 + m*) 720 ... ] 

2 2 a4 + b4

. a 5 a 6 7 

120 ~ + 240(1 + m*) ~ + 5040~ ~ + ••• ] 

(4-18) 

00 

Given the ~nfinite series ~ Ak~k, then by the ratio test the series 

k=O 

converges (absolutely) for those values of ~ for wh1ch 

A 

~~: I ~:11 I~I < 1 

We noted above that the d~fferent~al equation (4-10) has solutions wh~ch are regular at ~ = 0 and that these series converge for 

all~. Using the recurrence formula ~q. (4-17) we can verify this 

fact. This more elementary procedure was used by Lamb ("Hydrodynamics", p. 335). 

Wrlte the recurrence formula in the form 

a 2 Ak+2 ~--~~~--~---- - (k + 3) (k + 4) ~ 

[a2/(1 + m*)] (k + 1) 

(k + 2) (k + 3) (k + 4) 

(k + 1) (k + 2) (k + 3) (k + 4) = 0 

27 
If lim lk+1 = L then 11m ~+2 = L 2 and 11m ~+4 = L 4 • 

k+oo ~ k+oo ~ k+oo , 

rrak1nq the 

11m1t 1n the above formula as k+oo we obta1n the cond1t10n L4 = 0 or 

lim ~+1 = o. qence the ser1es converges for all E in aqreement with 

k+oo ~ 

the general theory. 

4.2 Free vibrat10n of a Cant1levered Beam W1th T1P Body 

under Constant Base Accelerat10n (y * 0) 

In th1s sect10n we 1nvest1gate the v1brat10nal response of the 

acceleratinq beam w1th y * O. It is 1mportant to note that 1n all 

cases considered thus far, y = 0 or steady-state response, a constant 

axial force 1mp11ed a constant base accelerat10n. In the present 

case this 1S no longer true (see Eq. (2-18). We will assume here that 

a base force po(t) is applied such that the base accelerat10n ao 

is constant. 

Neglect1nq non11near terms in u(x,t), the equat10n of moton 1S 

(4-19) 

~he boundary cono1t10ns are q1ven by Eqs. (2-14), (2-16), and 

(2-20). The boundary cond1t10n (2-16) 1S 1nhomoqeneous. In order to 

solve the d1fferent1al equat10n (4-19) we write 

u(x,t) = v(x,t) + f(x) (4-20) 

and choose f(x) so that the boundary cond1t10ns on v are rendered 

homoqeneous. We then have the followinq requirements on f(x) 

f( 0) = 0 

f'(O) = 0 (4-21) 

28 
EI fl I I (R.) + rna fl (R.) = 0 

o (4-21 ) 

c EI cos Y fill (R.) + EI fll(R.) = mc Sl.n y a

O 

The partl.al differential equation on v(x,t) can be written as 

4 

{EI d ~ + a

o ;x [(PR. + m - px) ;;]} 

dx (4-22) 

If, l.n addl.tion to satl.sfying the conditions (4-21), we requ1re that 

f(x) satl.sfies 

4 

EI ~ + d [(PR. + m - px) ddxf] = 0 dx4 ao dx (4-23) 

then the d1fferential equat10n on v(x,t), as well as the boundary 

cond1t10ns, are rendered homogeneous. 

The condit10ns on f(x), i.e., Eq. (4-23) w1th boundary conditions (4-21), are prec1sely those governl.ng the steady-state 

solution of Eq. (4-19) w1th the boundary cond1t10ns (2-14), (2-16), 

and (2-20). It was shown in Section 3.4 that f(x) exists for all 

values of aO' except when the system (3-19) becomes sl.ngular. f(x) 

1S g1ven by (3-20) w1th 

1 JR.

2 

= 3 EI (pt + m)aO 

29 
It ~s seen that v(x,t) ~s governed by the system of Equat~ons 

(4-1) through (4-4) w~th the replacements 

thus 

c -+- c cos y, 

<XI 

2 s~n y 

v( x, t) = E (l\k cos W

k t + Bk sin w

k t)Yk ( F) 

k=1 

where the frequenc~es wk are solut~ons of Bq. (4-15) w~th 

c* -+- c* cos y, ~o -+- ao(pt + m), and the funct~ons Yk(~) are 

q~ven by Eq. (4-18). 

All frequenc~es ar~ pos~t~ve for suff~c~ently small values of 

aO. when aO assumes a cr~tical value such that Eq. (3-12) ~s satisf~ed (w~th c* -+- c* cos ~), a frequency wk qoes to zero. Th~s cprresponds to the s~tuat~on where f(x) fa~ls to ex~st. 

30 
CHAPTER 5 

NATURAL FREQUENCIES AND MODE SHAPES ('Y = 0) 

ASSUMED MODES SOLUTION 

In this chapter, approx1mate solut10ns to the natural 

frequenc1es and mode shapes of a cantilevered beam with t1P body 

subJect to constant axial base acceleration are determ1ned via an 

assumed modes fcrmulation. As ind1cated 1n Section 4.1, 'Y must be 

restricted to zero. The mass per un1t length, p, and bend1ng 

stiffness, EI, are assumed to be constant. 

5.1 Kinetic and Stra1n Enerq1es 

Let Tb , Tt , and Ts denote the kinet1c energy of the beam, tip 

body and total system respect1vely. Using Eqs. (2-7), (2-8), and 

(2-9) we have in general 

T 

s 

= 

= 

1 a2u 2 1 a 2u 2 

- I [- (t,t)] + - m {[v - c sin 'Y "x"t (t,t)] 2 t axat 2 0 a a 

au a2u 2 

+ [at (t,t) + c cos 'Y axat (t,t)] } 

V 2(t) + 1. 2 [a

2

u ]2 1 [aU ]2 2 (pt + m) 0 2 (It + mc ) ax3t (t,t) + 2" m at (t,t) 

1 

+-2 

t 

J a ~tU (x,t) ]2 p dx 

o 

(5-1 ) 

31 
For the case of y = 0, this reduces to 

T 

s = 1 2 1 JJl. (aU)2 1 [a

2

U ]2 2 (pJl. + m) Vo (t) + 2" 0 at p dx + 2" It atax (Jl.,t) 

1 [aU a

2

u 2 

+ 2" m at (Jl.,t) + C axat (Jl.,t)] (5-2) 

In Appendix A it is shown that the stra1n energy expression for 

the problem at hand is Eg. (A-11). Using Eq. (2-17) we obtain 

- 1 a mc[~ (Jl.,t)]2 2 0 ax 

Discretization 

Expressing the beam deformation as 

n 

u (x, t) = L 4>i (X)q1 (t) 

i=l 

the energy exprr~s10ns (5-2) and (5-3) become 

n n 

(5-3) 

. . v

2 (t) 1 L L T = (pJl. + m) +- MiJ qi qj s 2 0 2 1=1 J=1 

where 

Jl. 

+ mc

2 M = p ) 4>' (Jl.) <j>~ (Jl.) J 4> (x) 4>. (x) dx + (It + m$ (Jl.) 4> ( Jl.) 1J 0 1 J 1 J 1 J 

+ mc[4>. (Jl.) 4>' (Jl.) + 4>.' (Jl.)4> (Jl.)] 

1 J 1 J (5-4) 

32 
where 

K. 1J 

where 

= 

v = 

1 n n 

2 L L 

1=1 j=1 

K .. q.q. 1J 1 J 

t t 

EI J cP~' (x) CP!' (x) dx + aopt Jo[f - (1 + m*)] cP~ (x) cpJ' (x) dx 

o 1 J 

- a mc CP! (R.) CP! (R.) o 1 J (5-5) 

(') and (-) denote d1fferentiation w1th respect to x and t, 

respect1vely 

According to the assumed modes method, the spatial functions 

cp. (x) may be any arbitrary functions which satisfy the qeometric 1 

boundary conditions of the problem and are two times different1able on 

the interval 0 < x < t. Two sets of functions immediately suggest 

themselves as cand1dates: the fixed base eigenfunct10ns of a clampedfree beam and those of a clamped b('am with t1P body. Whereas the 

clamped-free e1genfunctions have the advantage that they can serve for 

beams with any tip body, superior ~onvergence will be real1zed w1th 

use of the clamped-tip body e1genfunct1ons. Furthermore, the latter, 

Wh1Ch were derived in Reference 1, diagonalize the mass matrix, (5-4), 

as a result of the1r orthogonality property. 

E1ther set of e1genfunction~ satisfy d1fferent1al equat10ns of 

the form 

4 

= a.cp (x) 1 1 (5-6) 

For distinct eigenvalues a: and a;, the followlng indef1nlte integral 

holds 

f cp! (x) CP! (x) dx 

1 J = (a~ 1 

[a 4 cP CP' - CP!" CP!' + CP" CP!" - e 4 <l>' cP ] iij 1 J 11 jij 

(5-7) 

33 
Th~s permits the analyt~cal evaluation of the second ~ntegral ~n Eq. (5-5) 

via integrat~on by parts 

R. R. J[T - (1 + m*)] <I>! (x)<I>~ (x) dx = 

o ~ J J[T - (1 + m*)] ~ [f<l>! (x) <1>' (x) dx1 dx 

o x ~ J 

= [~- (1 + m*)][a·<I>.<I>~ _ <I>~" <I>!' + tj>~' <1>'" _ a~<I>'<I> ]~IR. (a~ - a~) ~ l ~ J ~ J ~ J J ~ J ) 0 

~ J 

for i :I: J. 

Appendix B demonstrates a useful meahs by which to spec~alize the 

~ndefinite integral Eq. (5-7) to the case of ~ = j 

f[<I>!(x)fdX = .! [(<1>,)2 _ 2<1>.~~' +-' (<1>",)2] __ '_<I>~'<I>'." +l<l>.<I>' ~ 4 ~ 1 ~ a~ ~ 4a~ ~ ~ 4 1 ~ 

1 1 

(5-8) 

This permits the evaluat~on 

R. 

f[T- (' +m*)][<I>i.1 2dX = 

o 

\[f - (1 + m')1(~[(4>p2 - 24>i4>[' + :~(4)[ .. )21 

1 

_ -'- <1>" <I>~" + l <I> <I>!)IIR. 4a 4 ~ ~ 4 ~ ~ { 0 

~ 

_ 1. ~ x

2 

[( <1>') 2 _ 2<1>. <1>" + _1 (<I>! " ) 2] R. l 8 ~ ~ ~ a~ ~ 1 

34 
5.2 Eigenvalue Problem 

The application of Lagrange's equations yields the free vibration motion equations 

[Ml{ci} + [Kl{q} {OJ 

Seeking harmonlC solutions, {q} = {U}eint leads to the elqenvalue 

problem 

= {oJ (5-9) 

The demand of nontrivlal solutions of the assumed form Ylelds the set 

of natural frequencies nr and eigenfunctions 

y(r) (x) = 

n 

L 

i=1 

~. (x) u~r) 1 1 (r = 1,2, ••• , n) 

The natural frequencies, Or and eigenfunctions, y(r) are the 

sought approximatlons to those of Sectlon 4.1. Note that the values 

of base acceleration for WhlCh 

det [K (ao) 1 = 0 

are approximations to t'le critical accelerations of Section 3.3. 

35 
CHAPTER 6 

FORTRAN PROGRAMS 

Two FORTRAN programs have been created to evaluate the free 

v1bration and buckling characteristics of a uniform cantilevered beam 

with tip body under constant axial base acceleration. For each, the 

tip body mass center is restr1cted to l1e along the beam tip tangent 

line. Complete listings of each program, accompan1ed by annotated 

sample input/output data, are provided in Appendices C and D. 

The program of Appendix C computes the natural frequencies and 

mode shapes by implementing the assumed modes formulation of Chapter 

5. The eigenfunct10ns of a clamped-free beam on a fixed base are used 

as the adm1ss1ble spat1al funct10ns. The e1genvalues of the clamped4 free beam, (S.), are obta1ned from the roots of the correspond1ng char- l. 

ter1st1c equat10n 

The first fifty of these roots (valid for any beam), appear following 

the sample NAMELIST input data 1n the appendix. The program functions 

in one of two poss1ble ways, depending upon the values of the parameters AI, AF, and AINC (wh1ch denote the 1n1t1al, f1nal, and 1ncremental values of the base acceleration, respect1vely). If AI = AF 

(AINC arbitrary), the natural frequenc1es and mode shapes are computed 

for the s1ngle value of base accelerat10n. If AI < AF, then (only) the 

natural frequenc1es are computed for the values of base acceleration 

start1ng w1th AI and terminating w1th AF in 1ncrements of AINC. The 

general algebraic e1genvalue problem Eq. (5-9) is solved via IMSL 

subrout1ne EIGZS. 

36 
The program of Appendix 0 computes the first crltlcal buckllng 

load and acceleration by finding the first root of the buckling characteristic equatlon. Slnce available IMSL subroutines evaluate only 

Bessel functlons of posltlve order, the actual characterlstic equatlon 

lmplemented is not Eq. (3-12) but, rather, an equivalent form expressed 

in terms of posltive order Bessel functions 

The IMSL subroutines, MMBSJR and ZBRENT, are used to evaluate the Bessel functlons and the first root of the above characteristic equation, 

respectlvely. 

37 
APPENDIX A 

WORK-ENERGY BALANCE 

Starting w1th the general part1al differential eqUat10n governing transverse beam vibration under aX1al loading, we derive an 

erergy-balance relat10nship. We then apply the boundary cond1t1ons 

an] 1dent1fy the potential energy express10n. Th1S der1vat10n 

obv1ates the more common geometr1c arguments which lack a certa1n degree of rigor, and reveal the influence of the boundary cond1tions on 

the strain energy. 

The starting point of our der1vation 1S Eq. (2-4) 

f(x,t) (A-1 ) 

where we have inserted the term f(x,t) on the r1ght-hand s1de to accommodate external loading/length on the beam perpendicular to the x 

axis. The reader can verify that Eg. (A-l) can be written in the alte.nate form 

a [ au aU] 1 aT (~)2 - ax T(x,t) at ax - 2" at ax 

38 

= 

au f(x,t) at 

(A-2) 
Th1S equat10n holds for 0 < x < 1 and t > O. We can then 1nteh . . h I JO < x < 1 b .. grate over t e reg10n 1n text pane to < t < t 0 ta1n1ng 

t 2 +\J [~~ (EI~) at ax " 2 o aX 

= 

au f(x,t) at dx dt 

(A-3) 

The right-hand side of Eq. (A-3) represents the work performed by the 

external load1ng f(x,t) on the beam between t = 0 and t = t. 

The work performed by the axial force Po(t) at the base of 

the beam between t = 0 and t = t is given by 

= 

2 t t a3u 2 (p1 + m) vo(t) It=o - mc S1n y f 2 (1,t)v dt 

o axat 0 

(A-4) 

where we have used Eq. (2-18) (p assumed constant) • 

Now the total work done on the system by the external forces f and 

Po is equal to the change 1n kinetic plus potent1al energies. Using 

Eq. (5-1) for the kinet1c energy and Eq. (A-3) and (A-4), th1s work-energy 

balance can be wr1tten as 

39 
+ Q(t) 

t R. "IT 

1 f f a (~)2 dx dt 1 2 a2u 2 

2 at ax = 2 (It + mc )[axat (R.,t)] I~=o o 0 

1 [aU ] 21t ~ 2u 

+ 2 m at (R.,t) t=O + mc axat (R.,t) 

[ au ] t 

• cos Y at (R.,t) - vo(t) S1n Y It=o 

t 

+ Vet) I t=O (A-S) 

Vet) is the potential energy and Q(t) is given by 

Q(t) = 

(A-G) 

Assurn1ng a un1form EI and usu.g Eq. (2-12) aloog with the clamped 

boundary cond1tions at x = 0 we have 

Q (t) = 

a

3

u au au - mc S1n Y 2 (R.,t) ax (R.,t)] at (R.,t) 

axat 

(A-7) 

40 
I 

The natural boundary conditions (2-14) and (2-16) are eauivalent 

to 

a

3

u a

3 

au 

me cos y --- (t,t) - EI ~ (t,t) - rna (t) (t,t) 

axat 2 ax 3 0 ax 

(A-a) 

2 a

3

u au a

2

u (It + me ) --2 (t,t) - me cos y a (t) ax (t,t) + me cos y -- (t,t) 

axat 0 at 2 

a2u 

+ EI (t,t) = me sin y • a (t) ax2 0 

(A-9) 

with the aid of (A-a), (A-9), Eq. (A-7) further slmpllfies to 

a~ a

2

u t 1 [aU ]21t Q(t) = me cos y at (t,t) axat (t,t) It=o + 2 m at (t,t) t=O 

- me cos y 

t 

a

2

u - me Sln y f ao (t) axat (t,t) dt 

0 

t a3u au au - me sin y 

[ axat 2 (t,t) ax (t,t) at (t, t) dt 

41 
Now 

US1ng the above results, Eg. (A-5) can be put in the form 

V(t) - V(O) 1 = 2 

- mc cos y 

t a3 au au - mc Sln y f ~(R.,t) -ax(R.,t)a-t(R.,t) dt 

o axat 

(A-l0) 

Equatl0n (A-l0) gives the potentlal energy of a clamped beam w1th t1P 

body under axial acceleration ao(t)~ El, and p are assumed constant. Recall that potent1al energy is defined only up to an arb1-

trary add1tive conE~~nt~ hence we can set V(O) = o. 

For the spec1al case y = 0 and constant ao 

2 

R. 2 £ [El (:x~)2 + T(x) (~~)2] dx - ~ mc a V(t) = o [~~(R.,t) f 

(A-l1) 

Adm1ttedly, this derivat10n of the stra1n energy does not appear to be well motivated and 1ndeed was developed with h1ndsiqht. A 

more direct but mathematically sophisticated derivat10n can be found 

1n Reference 2. 

42 
APPENDIX B 

INDEFINITE INTEGRALS OF EIGENFUNCTIONS 

In this appendlx we shall present a method for evaluatlng the 

indefinite integral 

where ~i(x) is the Solutlon of the differential equatlon 

d4,f. 4 'f' a ~ (x) . dx 4 = (B-1 ) 

corresponding to a = ai. This technlque is extremely useful in the 

normallzation of eigenfunctlons and the evaluatlon of certain lntegrals arlsing in vibratlon problems in general. One of the earliest 

references to this method is S. Timoshenko ("Vibration Problems in 

Englneering," 2nd edltion, p. 335). 

The solution of the dlfferential equation (B-1) depends updn 

the value of a as well as x. We indicate thlS b~ the notation 

~ = 4>(x~ a). Formally expandlng ~ in a Taylor serles we have 

4>(x~ a + 013) = a~ ~ (x~ a) + ae 013 + ••• 

Denote by ~i(x)·the value of ~ correspondlng to a = ai and by 

~j(x) the value correspondlng to ai + 013. Thus 

43 

(B-2) 
From Eq. (B-1) one can readily show 

f 4>. 4>. dx ~ J = 4>.4>!" - 4>.4>'" + 4>'4>" - 4>!4>" J ~ ~ J ~ J J ~ 

whe~e a~ and a J are dist~nct (not necessarily eigenvalues). 

Substituting Eq. (B-2) ~nto Eq. (B-3) 

- 4a ioa f4>~ (x) [4>i (x) + (;:) a. 013 + ••• ] dx = 

~ 

- 4>. [4>!" + L (;~)Q 013 + ... ] + 4>' [4>" + d

2

2 (;~)Q 013 + ... ] ~ ~ dx 3 I-' I-' ~ ~ ~ dx I-' I-' ~ 

where we have used the binomial expansion 

(a. + 013)4 ~ = a? + 4a~oa ~ ~ 

+ ••• 

(B-3) 

(B-4) 

We observe from Eq. (B-1) that the funct~ons 4> depend upon x and a only 

through the argument ax. Therefore 

= a 4> a (ax) 

a (ax) -aaa4> = x -:---'-- a(ax) 

S~m~larly 

a4> 

a (ax) = .!. ~ a ax 

44 
Hence 

x 

= - <P! (x) S. 1 1 (B-5) 

Neglect1ng h1gher order terms in aS and using Eq. (B-5), allows us to wr1' e 

Eq. (B-4) as 

d 2 1 d 

+ <P' - -- (x<P.') - cjl." - - (Xcjl1!) 1 Si dx2 1 1 Si dx 

Expanding derivat1ves we obta1n the final desired result 

= 

x 

+ _3_ cjl. <P ~ " 4B~ 1 1 

1 

<P! <P" 1 1 

Indeed, one can verify this result by direct differentiation. 

Related integrals, e.g. 

can be evaluated in the same fash1on. 

45 

(B-G) 
ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

APPENDIX C 

NATURAL FREQUENCIES AND MODE SHAPES 

PROGRAM LISTING 

c···.··· •• ·············.················································00000100 

C THIS PROGRAM COMPUTES THE NATURAL FREOUENCIES AND MODE SHAPES OF A 00000200 

C UNIFORM CANTILEVERED BEAM WITH TIP BODY SUBuECT TO CONSTANT AXIAL 00000300 

C BASE ACCELERATION. AN ASSUMED MODES FORMULATION IS IMPLEMENTED USING 00000400 

C THE EIGENFUNCTIONS OF A CLAMPED-FREE BEAM ON A FIXED BASE AS THE 00000500 

C BASIS THE TIP BODY MASS CENTER IS RESTRICTED TO LIE ALONG THE BEAM 00000600 

C TIP TANGENT LINE. ( WRITTEN BY uOEL STORCH & STEPHEN GATES BASED 00000700 

C UPON CSDL R-1675 OCTOBER 1983 ). 00000800 

C····.······· ••• ~··.······.· •• ·········.·····.····.··.·.····.···········00000900 

C 00001000 

C··· •• •••••••••••••••••• INPUT/OUTPUT FILES ••••••••••••••••••••••• •• ···00001100 

C 00001200 

C FILEN5. NAMELIST INPUT & "BETA • L" VALUES 00001300 

C FILEN6 OUTPUT ACCELERATIONS & NATURAL FREOUENCIES 00001400 

C FILEN7: OUTPUT EIGENFUNCTIONS EVALUATED AT DISCRETE LOCATIONS 00001500 

C 00001600 

C······ •• •••••••• DESCRIPTION OF NAMELIST INPUT ITEMS ··················00001700 ~ 00001800 

C "EI" BEAM BENDING STIFFNESS 00001900 

C "RHO" MASS PER UNIT LENGTH OF BEAM 00002000 

C "L" BEAM LENGTH 00002100 

C "MT" TIP BODY MASS 00002200 

C "IT" TIP BODY INERTIA ABOUT IT'S MASS CENTER 00002300 

C OCT" TIP BODY MASS CENTER OFFSET FROM BEAM ATTACHMENT POINT 00002400 

C "AI" INITIAL VALUE OF BASE ACCELERATION 00002500 

C "AF" FINAL VALUE OF BASE ACCELERATION 00002600 

C "AINC" INCREMENT OF BASE ACCELERATION FOR SWEEP 00002700 

C "N" NUMBER OF CLAMPED-FREE EIGENFUNCTIONS USED 00002800 

C 00002900 

C············.·.····.··~·······.··· •• • •••• • •••• •••••••• ••• ···.······.···00003000 

C 00003100 

C NOTE ARRAYS DIMENSIONED FOR A MAXIMUM OF 50 DEGREES OF FREEDOM 00003200 

C 00003300 

C·······································.·······························00003400 

C 00003500 

IMPLICIT REAL·8 (A-H,O-Z) 00003600 

REAL.8 L,M,K,MT,IT,MSTAR 00003700 

DIMENSION K(50,50),M(50,50),BETAL(50),B(50),B4(50),FOL(50), 00003800 

+F1L(50),F20(50),F30(50),GAMMA(50,50),AA(1275),BB(1275),0(50), 00003900 

+Z(50,50),WK(26oo),PHI(51,50),Y(51,5) 00004000 

NAMELIST /INPUT/ EI,L,RHO,N,MT,IT,CT,AI,AF,AINC 00004100 

READ(5,INPUT) 00004200 

RHOL=RHO·L 00004400 

MSTARsMT/RHOL 00004500 

SRHOL=DSORT(RHOL) 00004600 

C 

C READ BETA·L VALUES 

C 

00004700 

00004800 

00004900 

00005000 

00005100 

00005200 

00005300 

00005400 

1 

2 

DO 1 l-l,N 

READ(5,2) BETAL(I) 

B(I)=BETAL(I)/L 

B4(1)=B(I)··4 

FORMAT(E17 8) 

46 
.. 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 0020 

ISN 0021 

ISN 0022 

ISN 0023 

ISN 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

ISN 0032 

ISN 0033 

ISN 0034 

ISN 0035 

ISN 0036 

ISN 0037 

ISN 0038 

ISN 0039 

ISN 0040 

ISN 0041 

ISN 0042 

ISN 0043 

ISN 0044 

ISN 0045 

ISN 0046 

ISN 0047 

ISN 0048 

ISN 0049 

ISN 0050 

ISN 0051 

ISN 0052 

ISN 0053 

ISN 0054 

ISN 0055 

C 

C ECHO PRINT INPUT DATA 

C 

3 

4 

5 

6 

7 

C 

C 

C 

WRITE(6,3) 

FORMAT(1H1,24X,'OATA FROM NAMELIST INPUT') 

WRITE(6,4) EI,RHO,L 

FORMAT(1HO,' EI - ',1PE13,6,2X,' RHO" ',1PE13 6.2X,' L-

+1PE13 6) 

WRITE(6,5) MT,IT,CT 

FORMAT(1HO,' MT - ',1PE13 6,2X.' IT - ',1PE13 6,2X,' CT - , +1PE13.6) 

WRITE(6,6) AI,AF,AINC 

FORMAT(1HO.' AI - ',1PE13,6,2X.' AF" '.1PE13.6.2X,'AINC .. 

+1PE13.6) 

WRITE(6,7) N 

FORMAT(1HO.'NO. OF CLAMPED-FREE MODES USED - ',12) 

EVALUATE CLAMPED-FREE EIGENFN & DERIVATIVES. 0 AND L 

DO 10 I-1.N BL~BETAL(I ) 

S-DSIN(BL) 

C=DCOS(BL) 

SH=DSINH(BL) 

CH=DCOSH(BL) 

FOL(I)- -2.0DO/SRHOL F1L(I)~FOL(I)*B(I)*(S*CH + C*SH)/(S*SH) 

F20(I)=FOL(I)*(B(I)**2)*(C+CH)/(S*SH) 

10 F30(1)=FOL(I)*(B(I)**3)*(S-SH)/(S*SH) 

C 

C ASSEMBLE MASS MATRIX 

C 

20 

21 

C 

DO 20 1-1,N 

DO 20 J-I,N 

M(I,J)- (IT+(MT*CT*CT»*F1L(I)*F1L(J) + MT*FOL(I)*FOL(J) 

++ MT*CT*(FOL(I)*F1L(J) + F1L(I)*FOL(J» 

M(J,I)-M(I,J) 

DO 21 I -1,N M(I.I)~M(I.I)+1 000 

C ASSEMBLE GAMMA MATRIX 

C 

DO 30 I-1,N 

IP1"I+1 

DO 30 J-IP1,N 

BMB"B4(I )-B4(J) 

T1-(1 ODO+MSTAR)*(F20(1)*F30(J) - F30(1)*F20(J» T2~MSTAR*(B4(I)*FOL(I)*F1L(J) - B4(J)*F1L(I)*FOL(J» 

T3-2 ODO*(B4(1)+B4(J»*F20(I)*F20(J)/(L*BMB) 

T4-4.0DO*B4(1)*B4(J)*FOL(I)*FOL(J)/(L*BMB) 

GAMMA(I.J)=(T1-T2-T3+T4)/BMB 

30 GAMMA(J.I)-GAMMA(I,J) 

DO 40 I-1.N 

T1-MSTAR*(L*(F1L(I)**2) + 3 ODO*FOL(I)*F1L(I»/4 000 

T2"(1 ODO+MSTAR)*F20(1)*F30(I)/(4 ODO*B4(1» 

T3-(L*F1L(I»**2 + 3 ODO*(FOL(I).*2) + (F20(I)**2)/B4(I) 

40 GAMMA(I,I)--T1-T2-T3/(8 ODO*L) 

C 

47 

00005500 

00005600 

00005700 

00005800 

00005900 

00006000 

00006100 

00006200 

00006209 

0000621B 

00006227 

00006236 

00006245 

00006254 

00006263 

00006272 

00006400 

00006500 

00006600 

00006700 

00006800 

00006900 

00007000 

00007100 

00067200 

00007300 

00007400 

00007500 

00007600 

00007610 

00007700 

00007710 

00007800 

00007900 

00008000 

00008100 

00008200 

00008300 

00008400 

00008410 

00008500 

00008510 

00008600 

00008700 

00008800 

00008900 

00009000 

00009100 

00009200 

00009300 

00009400 

00009500 

00009600 

00009700 

00009800 

00009900 

00010000 

00010010 
ISN 0056 

ISN 0051 

ISN 0058 

ISN 0059 

ISN 0060 

ISN 0061 

ISN 0062 

ISN 0063 

ISN 0064 

ISN 0065 

ISN 0066 

ISN 0061 

ISN 0068 

ISN 0069 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0011 

ISN 0018 

ISN 0019 

lSN 0080 

ISN 0081 

ISN 0082 

ISN 0083 

ISN 0085 

ISN 0081 

ISN 0088 

ISN 0089 

ISN 0090 

ISN 0091 

ISN 0092 

ISN 0093 

ISN 0094 

ISN 0095 

ISN 0096 

ISN 0091 

ISN 0098 

C ASSEMBLE STIFFNESS MATRIX 

C 

50 

60 

AX"AI 

00 60 1"1,N 

IP1-I+1 

00 60 ..I"IP1,N 

K(I,..I)"AX RHOL*GAMMA(I,..I)-MT*CT*AX*F1L(I)*F1L(..I) 

K(..I,O=K( • ..1) 

00 10 1=1 N 

10 

C 

K(I,I)=EI'S4(I)/RHO + AX*RHOL*GAMMA(I,I)-NT*CT*AX*(F1L(I)**2) 

C LOAD K INTO AA & N INTO BB 

C 

BOTH SYMNETRIC STORAGE 

II" 1 

00 80 I-1,tl 

00 80 v-1,I 

AA(II )"K(I ,v) 

BB( II )"N(I ,v) 

II=II+1 

80 CONTINUE 

C 

C COMPUTE EIGENVALUES & EIGENVECTORS 

C 

I..IOB=2 

IZ"50 

CALL EIGZS(AA,BB,~,I..IOB,O,Z,IZ,WK,IER) 

C 

C OUTPUT FIRST 5 EIGENVALUES & ACCELERATION 

C 

C 

WRITE(6,90 AX 

90 FORMAT(1HO,'BASE ACCELERATION - ',1PE13.6) 

WRITE(6,91 ) 

91 FORMAT(1HO 4X,'NATURAL FREOUENCIES') 

WRITE(6,92. 

92 FORNAT(1H ,1X,'N',5X,'ONEGA(N)**2 (SEC**-2)') 

DO 93 1-1,5 

93 WRITE(6,94) 1,0(1) 

94 FORMAT(1H I2,5X.1PE136) 

C TEST FOR EIGENVECTOR COMPUTATION OR ACCEL SWEEP 

C 

C 

IF( AI .EO AF) GO TO 100 

IF( AX .GE. AF ) GO TO 1000 

AX"AX+AINC 

GO TO 50 

C EVALUATE CLANPLJ-FREE EIGENFUNCTIONS AT 51 STATIONS 

C 

100 oX"L/50 000 

00 110 I"1.N 

BL=BETAL( I) 

SS=OSIN(BL)*OSINH(BL) 

SMS"OSIN(BL) - OSINH(BL) 

CPC"OCOS(BL) + OCOSH(BL) 

X"O.OOO 

DO 105 v"1,51 

BX=B(I )*X 

PHI(v,I)-SMS*(OSIN(BX)-OSINH(BX» + CPC*(OCOS(BX)-OCOSH(BX» 

48 

00010100 

00010110 

00010200 

00010300 

00010400 

00010500 

00010600 

00010100 

00010800 

00010900 

00010910 

00011000 

00011010 

00011100 

00011200 

00011300 

00011400 

00011500 

00011600 

00011100 

00011110 

00011800 

00011810 

00011900 

00012000 

00012100 

00012101 

00012200 

00012210 

00012300 

00012400 

00012500 

00012600 

00012610 

00012620 

00012630 

00012640 

00012650 

00012100 

00012800 

00012810 

00012900 

00013000 

00013100 

00013200 

00013210 

00013300 

00013310 

00013400 

00013500 

00013600 

00013100 

00013800 

00013900 

00014000 

00014100 

00014200 

00014300 
.. 

ISN 0099 PHI(~,I)·PHI(~,I)/(SRHOL·SS) 00014400 

ISN 0100 105 X=X+OX 00014500 

ISN 0101 110 CONTINUE 00014600 

C 00014610 

C COMPUTE FIRST 5 EIGENVECTORS 00014700 

C 00014710 

ISN 0102 DO 125 IP",5' 00014800 

ISN 0103 DO 120 IM",5 00014900 

ISN 0104 SUM=O 000 00015000 

ISN 0105 DO 115 ~=I,N 00015100 

ISN 0106 PROO=PHI(IP,~)·Z(~,IM) 00015200 

ISN 0107 115 SUM"SUM+PROO 00015300 

ISN 0108 120 Y(IP,IM)"SUM 00015400 

ISN 0109 125 CONTINUE 00015500 

C 00015510 

C OUTPUT EIGENVECTORS FOR PLOTTING 00015600 

C 00015610 

ISN 0110 X=O 000 00015700 

ISN 0111 DO 130 1·',51 00015800 

ISN 0112 WRITE(7,131) X,(Y(I,~),~·,,5) 00015900 

ISN 0113 130 X"X+OX 00016000 

ISN 0114 131 FORMAT(6(IX,EI2 5» 00016100 

ISN 0115 1000 STOP 00016200 

ISN 0116 END 00016300 

49 
EXAMPLE PROBLEM PARAMETERS 

BEAM PARAMETERS TIP BODY PARAMETERS 

7 2 EI = 625 • 10 Ib-ft m = 155 28 slugs 

It = 0 0 sl Ug_ft2 

L=1340ft 

p = 04172slug/ft • =OOft 

NAMELIST INPUT AND CLAMPED-FREE BEAM "EIGENVALUES" 

&INPUT EI=6 25E7,L=134 O,RHO=.4172,N=20 

MT=155.28,IT zO O,CT=O.O 

AI=O O,AF=60 O,AINC=20 O,&END 

o 18751041E+Ol 

o 46940911E+Ol 

o 78547574E+Ol 

o 10995541E+02 

o 14137168E+02 

0.17278760E+02 

o 20420352E+02 

0.23561945E+02 

0.26703538E+02 

o 29845130E+02 

o 32986723E+02 

o 36128316E+02 

0.39269908E+02 

0.42411501 E+02 

o 45553093E+02 

o 48694686E+02 

o 51836279E+02 

0.54977871E+02 

o 58119464E+02 

o 61261057E+02 

o 64402649E+02 

o 67544242E+02 

o 70685835E+02 

o 73827427E+02 

o 76969020E+02 

o 80110613E+02 

o 83252205E+02 

o 86393798E+02 

o 89535391E+02 

o 92676983E+02 

o 95818576E+02 

o 98960169E"02 

o 10210176E+03 

o 10524335E+03 

o 10838495E+03 

o 11152654E+03 

o 11466813E+03 

o 11780972E+03 

o 12095132E+03 

o 12409291E+03 

o 12723450E+03 

0.13037610E+03 

0.13351769E+03 

o 13665928E+03 

o 13980087E+03 

o 14294247E+03 

o 14608406E+03 

o 14922565E+03 

o 15236724E+03 

o 15550884E+03 

so 

FIRST FIFTY ROOTS (/3/11 OF 

cos /3,1l cosh {J,ll = -1 
... 

PROGRAM OUTPUT 

ACCELERATION SWEEP RUN 

DATA FROM NAMELIST INPUT 

EI. 6. 250000E+07 RHO· 4 172000E-01 L· 1 340000E+02 

MT. 1.552800E+02 IT· OOOOOOOE+OO CT· OOOOOOOE+OO 

AI 0 OOOOOOE+OO AF· 6 OOOOOOE+01 AINC· 2 OOOOOOE+01 

NO OF CLAMPED-FREE MODES USED • 20 

BASE ACCELERATION· 0 OOOOOOE+OO 

NATURAL FREQUENCIES 

N OMEGA(N)**2 (SEC**-2) 

1 4.625102E-01 

2 1 151594E+02 

3 1 176526E+03 

4 5 085561E+03 

5 1 483397E+04 

BASE ACCELERATION· 2.000000E+01 

NATURAL FREOUENCIES 

N QMEGA(N)**2 (SEC**-2) 

1 2 785320E-01 

2 1 094116E+02 

3 1 155686E+03 

4 5 039933E+03 

5 1 475390E+04 

BASE ACCELERATION· 4 OOOOOOE+01 

NATURAL FREQUENCIES 

N DMEGA(N)**2 (SEC**-2) 

1 9.289338E-02 

2 1 036581E+02 

3 1 134843E+03 

4 4 994303E+03 

5 1 467382E+04 

BASE ACCELERATION· 6 OOOOOOE+01 

NATURAL FREQUENCIES 

N OMEGA(N)**2 (SEC**-2) 

1 -9 466231E-02 

2 9.789877E+01 

3 1 113999E+03 

4 4 948672E+03 

5 1 459374E+04 

51 
PROGRAM OUTPUT 

MODE SHAPES RUN 

DATA FROM NAMELIST INPUT 

EI· 6 25OOO0E401 RHO· 4 112000E-Ol L· 1 340000E402 

NO OF CLAMPED-FREE MODES USED • 20 

BASE ACCELERATION. 3 220000E401 

NATURAL FREOUENCIES 

N OMEGA(N)"2 (SEC"-2) 

I I 655051E-01 

2 I 059021E402 

3 I 142912E403 

4 5 012099E403 

5 1 410505E404 

EIGENFUNCTIONS (NON-NORMALIZED) 

yO) V(2) V(3) (4) V(S) x (x) (x) (x) V(x) (x) 

o 00000£400 o 00000E4OO o OOOOOE 4OO o 00000E4OO o 00000£400 000000£400 

o 26800E401 0.41121£-04 -0 80237£ -03 o 25418E-02 -0 52092E-02 o 86654£-02 

o 53600E401 0.16641E-03 -0 31241E-02 o 96715£-02 -0 19296E-01 o 31340£-01 

o 80400E+Ol o 31331E-03 -0 68381E-02 o 20664£-01 -0 39984E-01 o 63036E-01 

o 10120E402 o 66154£-03 -0 11819E-01 o 34148£-01 -0 65064E-Ol o 98844E-Ol 

o 13400E402 o 10302E-02 '0 11939£ -0 1 o 51180E-01 -0 92400E-Ol o 13415E'00 

o 16080E402 o 14182E-02 -0.25012E-01 o 69230E-Ol -0. 11993E 400 o 16493E400 

o 19160E 402 o 20041E-02 -0 33096E-Ol o 98199E-Ol -0 14574£>00 o 18797£>00 

o 21440E 402 o 26095E-02 -0 41853E-01 o 10741E+00 -0.16814£400 o 20046£400 

o 24120£402 o 32885E-02 -0 51248E-01 o 12620E'00 -0.18570E·00 o 20108E'00 o 26900E402 o 4043!JE-02 o 61145E-Ol o 14395E400 -0 19733E'00 o 19908E'00 o 29480E402 o 48722E-02 o 71419E-01 o 16012E'00 -0 20219E'00 o 16494E'00 

o 32160E'02 o 57133E-02 o 81947E-01 o 17422£'00 -0 19980E4OO o 13012E4OO 

o 34840E402 o 67456E-02 o 92607£-01 o 18581E4OO -0 19001£400 o 86888E-01 

o 37520P02 o 71875E-02 0.10328E·00 o 19452E'00 -0 17307E'00 o 38085£-01 0 40200[+02 0 88976[-02 0 11386[+00 o 20006[-00 -0 14959[-00 -0 13019[-01 

0 42880E'02 0 10074E-01 0 12424£'00 o 20222E+00 -0 12041E+00 -0 62886E-01 

0 45560£+02 0 11317[-01 0 13430E+00 o 2oo88E'00 -0 86620E-Ol -0 10798E'00 

0 48240E-02 0 12622£ -0 1 0 14394£'00 o 19600£<00 -0 49540£-01 -0 14514£400 

0 50920E'02 0 13990E-01 0 15308E'00 o 18760E*00 -0 10676E-01 -0 11180E*00 

0 53600E*02 0 15418E-01 0 16161E*00 o 11579£*00 o 28371£-01 -0 18620[*00 

0 56280£-02 0 16905£-01 0 16947£'00 o 16076E-00 o 66022E-Ol -0 18130£>00 

0 58960£402 0 18449E-Ol 0 11656E*00 o 14279E+00 o 10078£*00 -0 17499E-00 

0 61640E*02 o 20048E-Ol 0 18283E-00 o '2219E*00 o 13123E*00 -0 15012E-00 

0 64320£402 0 21700£-01 0 18819E*00 o 9\<",;OE-Ol o 15610E*00 -0 11453E*00 

0 67000E*02 o 23404E-Ol 0 19261£400 o 74706E-Ol o 17434£*00 -0 70829£-01 

0 69680£*02 o 25158£-01 0 19602POO o 48708£-01 o 18521£*00 -0 22121E-Ol 

0 72360£>02 o 26960£-01 0 19840£*00 o 21882£-01 o 18832£*00 o 28218E-01 

0 75040£+02 o 28808E-01 0 19969E+00 -0 52418E-02 o 18356£'00 o 76647[-01 

0 77720£402 o 30700£-01 0 19989£*00 -0 32143£-01 o 17111£*00 o 11965E*00 

0 80400£-02 o 32634E-01 0 19897E*00 -0 58310£-01 o 15144£*00 o 15405£-00 

0 83080E+02 o 34608E-01 -0 19693E*00 '0 8322IE-Ol o 12541£>00 o 17745E+00 

0 85760£>02 o 36620£-01 -0 19375E+00 -0 10634E+00 o 94224£-01 o 18846£*00 

0 88440£*02 o 38669£-01 -0 18945£+00 '0 12125E*00 o 59078E-01 o 18605E-00 

0 91120£*02 o 40152£-01 -0 18405£+00 -0 14555£+00 o 21375E-01 o 17023£*00 

0 93800E*02 o 42868E-01 -0 11756E*00 -0 16086E*00 -0 1720IE-01 o 14236E*00 

0 96480£*02 o 45014£-01 -0 17001E*00 -0 11282£*00 -0 54938£-01 o 10464E+00 

0 99160E-02 o 41188£'01 -0 16144£>00 -0 19125£*00 -0 90385E-Ol o 59478E-01 

0 10184£403 o 49388E-01 -0 15190E4OO -0 19604E400 -0 12225E 4OO o 96639E-02 

0 10452E -03 o 51612E-Ol -0 14143E+00 -0 19701£400 -0 14896E 4OO -0 4077IE-01 

0 10120E-03 o 53859E-Ol -0 13009E-00 -0 18410E+00 -0 16925E*00 -0 81855£-01 

0 10988E-03 o 56126£-01 -0 11792£*00 -0 17739£ +00 -0 I 8237E +00 -0 12853E+00 

0 11256E-03 o 58411£-01 -0 10502£+00 -0 16705E+00 -0 18787E+00 -0 16008E+00 

0 11524E-03 o 60712£-01 -0 91462E-Ol -0 15338E+00 -0 18579E+00 -0 18089[<00 

0 11192E+03 o 63026E-01 -0 77315£-01 -0 13660£>00 -0 17603£-00 -0 18911£-00 

0 12060£+03 o 65353£-01 -0 62631£-01 -0 11690£+00 -0 15865E-00 -0 18335E+00 

0 12328E-03 o 67690E-Ol -0 47501E-01 -0 94618E-01 -0 13414E+00 -0 16359E-00 

0 12596E+03 o 70036E-01 -0 32000E -0 I -0 70237£-01 -0 10368£+00 -0 13153£*00 

0 12964£-03 o 72397E-01 -0 16262E-01 -0 44410E-01 o 69197E-01 -0 90955£-01 

0 13132E*03 o 7414IE-OI -0 37928£-03 -0 17859£-01 -032217E-Ol -0 44939E-01 

0 13400£-03 o 77096E-Ol o 15543E-Ol o 89914E-02 o 54750£-02 o 26399£-02 

S2 
ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0014 

ISN 0015 

ISN 0016 

ISN 0017 

ISN 0018 

ISN 0019 

ISN 0020 

ISN 0021 

ISN 0022 

ISN 0023 

APPENDIX D 

BUCKLING LOAD/ACCELERATION 

PROGRAM LISTING 

c································································.···.··00000100 

C THIS PROGRAM COMPUTES THE FIRST CRITICAL BUCKLING LOAD/ACCELERATION 00000200 

C OF A UNIFORM CANTILEVERED BEAM WITH TIP BODY THE "BUCKLING LOAD" 00000300 

C CORRESPONDS TO THE BEAM ROOT AXIAL FORCE AND IS DIRECTLY PROPOR- 00000400 

C TIONAL TO THE "BUCKLING ACCELERATION". THE TIP BODY MASS CENTER IS 00000409 

C RESTRICTED TO LIE ALONG THE BEAM TIP TANGENT LINE ( WRITTEN BY 00000418 

C uOEL STORCH & STEPHEN GATES BASED UPON CSDL R-1675 OCTOBER 1983) 00000427 

C······················································· •••• ••••••••• ••• 00000436 

C 00000445 

C···············.······ INPUT/OUTPUT FILES ••••••••••••••••••••••••••• •• 00000454 

C 00000463 

C FILEN5: INPUT NAMELIST INPUT DATA 00000472 

C FILEN6: OUTPUT CRITICAL BUCKLING LOAD/ACCELERATION 00000481 

C 00000490 

C·····.···.·· •• • DESCRIPTION OF NAMELIST 

C· 

INPUT ITEMS ···················00000500 

00000509 

00000518 

00000527 

00000536 

C "EI" 

C "RHO" 

C "L" 

C "MT" 

C OCT" 

C 

BEAM BENDING STIFFNESS 

MASS PER UNIT LENGTH OF BEAM 

BEAM LENGTH 

MASS OF TIP BODY 

TIP BODY MASS CENTER OFFSET FROM BEAM ATTACHMENT POINT 

00000545 

00000554 

00000563 

C··················.········ •• • •• • ••• • •• •• •• • •• • •••• •• •••• •••• •• • ••••••• fOO00572 

C (0001000 

C 

C 

C 

C 

IMPLICIT REAL·8(A-H,O-Z) 00001100 

REAL·8 L,LAMS,MSTAR,MT C0001200 

DIMENSION LAMS(2) 00001300 

DATA LAMS/1 00-8,1 8700/ 00001400 

COMMON ALPHA,SETA,C1,C2,C3,D1,D2 00001500 

EXTERNAL F 00001600 

NAMELIST /INPUT/ EI,L,RHO,MT,CT (0001610 

READ(5,INPUT) 00001620 

MSTAR=MT/(RHO·L) 00001630 

CSTAR=CT/L 00001640 

ECHO PRINT INPUT DATA 

00001650 

00001653 

00001656 

00001660 

2 

WRITE(6,1) EI,RHO,L 

FORMAT(1H1,24X,'DATA FROM NAMELIST INPUT' ,II,' 

+,' RHO - ',1PE13.6,2X,' L - ',1PE13.6) 

WRITE(6,2) MT,CT 

EI • ',1PE13.6,2X 00001670 

FORMAT(1HO,' MT· ',1PE13 6,2X,' CT· ',1PE13.6,) 

C COMPUTE CONSTANTS 

C 

C. 1001680 

0(001690 

00001700 

00001710 

00001720 

00001730 

00002000 

00002100 

00002200 

00002300 

00002400 

00002500 

00002600 

00002900 

ALPHA-1 ODO+MSTAR 

BETA-MSTAR/DSQRT(1 000+1 ODO/MSTAR) 

C1-4 000/(3 ODO·ALPHA) 

C2-DSQRT(ALPHA) 

C3=2.0DO.C2/(3 ODO·BETA) 

C4-1 5DO.DSORT(MSTAR) 

C5-2 ODO.DSQRT(MSTAR)/BETA 

D1=C4·CSTAR 

53 
ISN 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0031 

ISN 0032 

ISN 0033 

ISN 0034 

ISN 0035 

ISN 0036 

ISN 0037 

ISN 0038 

ISN 0039 

ISN 0002 

ISN 0003 

ISN 0004 

ISN 0005 

ISN 0006 

ISN 0007 

ISN 0008 

ISN 0009 

ISN 0010 

ISN 0011 

ISN 0012 

ISN 0013 

ISN 0015 

ISN 0017 

ISN 0018 

ISN 0020 

ISN 0021 

ISN 0022 

ISN 0023 

ISN 0024 

ISN 0025 

ISN 0026 

ISN 0027 

ISN 0028 

ISN 0029 

ISN 0030 

ISN 0031 

C 

D2=C2-C5·CSTAR 

A=LAMS(1) 

B-LAMS(2) 

MAXFN=80 

C COMPUTE FIRST ROOT OF BUCKLING CHARACTERISTIC EOUATION 

C 

3 

4 

5 

CALL ZBRENT(F.1 00-8.5.A.B.MAXFN.IER) 

IF(IER EO. 0) GO TO 4 

WRITE(6.3) IER 

FORMAT(1HO.'IQROR IN ZBRENT ALGORITHM IER - '.13) 

STOP 

PCR=(2.25DO·( /L··2).B.·2 

ACR=PCR/(RHO· + MT) 

WRITE(6.5)PCR.ACR 

FORMAT(lHO.'FIRST CRITICAL BUCKLING LOAD - '.1PE13.6./. 

+'FIRST CRITICAL BUCKLING ACCELERATION - '.lPE13 6) 

STOP 

END 

00003000 

00003200 

00003300 

00003400 

00003410 

00003420 

00003430 

00003500 

00003600 

00003700 

00003800 

00004000 

00004100 

00004200 

00004300 

00004400 

00004500 

00005400 

00005500 

C·················· • •• ••••• ••• ··.·································.····00005510 

C THIS FUNCTION SUBPROGRAM EVALUATES THE BUCKLING CHARACTERISTIC 00005520 

C EOUATION. EO (3-12) EXPRESSED IN TERMS OF BESSEL FUNCTIONS OF 00005530 

C POSITIVE ORDER. 00005540 

C························· •• •••••••••••••••••••••••••••• •• ··············00005550 

C -

FUNCTION F( LAM) 

IMPLICIT REAL·8(A-H.0-Z) 

REAL·8 LAM 

COMMON ALPHA.BETA.Cl.C2.C3.Dl.D2 

DIMENSION T(8).R~(2).WK(4) 

Al"'ALPHA·LAM 

A2-BETA·LAM 

C COMPUTE BESSEL F.INCTIONS OF ORDERS 1/3.2/3.4/3.5/3 

C AT ARGUEMENTS A1 AND A2 

c 

10 

20 

30 

40 

C 

C 

C 

IS=O 

DO 40 1=1.2 

OR=DFLOAT(I)/3.0DO 

DO 30 NA=1.2 

IF(NA EO 1) ARG-Al 

IF(NA EO 2) ARG=A2 

CALL MMBSuR(ARG.OR.2.Ru.WK.IER) 

IF( IER EO 0) GO TO 20 

WRITE(6.10) IER.ARG.OR 

FORMAT(1HO.5X,'ERROR IN EVALUATING BESSEL FUNCTION'.3X,'IER-'. + 13.2X.'ARGUEM~~T-'.E13.5.2X.'ORDER·'.F7 5) 

STOP 

IS-IS+l 

T(IS)·Ru( 1) 

IS-IS+l 

T(IS)"R~(2) 

CONTINUE 

CONTINUE 

COMPUTE FUNCTION "F" 

F=(Cl/LAM·T(5)-T(6»·(C3/LAM·T(3)-C2·T(4)+Dl·LAM·T(3»+ 

1 T(1).(D2·T(7)+Dl·LAM.T(8» 

RETURN 

END 

54 

00005600 

00005100 

00005800 

00005900 

00006000 

00006100 

00006200 

00006300 

00006400 

00006500 

00006600 

00006100 

00006800 

00006900 

00007000 

00007100 

00007200 

00007300 

00007400 

00007500 

00007600 

00007700 

00007800 

00007900 

00008000 

00008100 

00008200 

00008300 

00008400 

00008500 

00008600 

00008700 

00008800 

00008900 

00009000 

00009100 
.,. 

SAMPLE INPUT/OUTPUT FOR BUCKLING PROGRAM 

BEAM PARAMETERS 

7 2 

EI '" 625 • 10 Ib-ft 

L • 134.0 ft 

p '" 04172 slug/ft 

NAME LIST INPUT 

TIP BODY PARAMETERS 

m '" 15528 slugs 

c - OOft 

&INPUT EI=6.25E1,RHO- 4112,La 134 O,MT=155.28,CT-OO.O,&END 

PROGRAM OUTPUT 

DATA FROM NAMELIST INPUT 

EI - 6 250000E+07 RHO - 4 172000E-01 L. 1 340000E+02 

MT - 552800E+02 CT. O.OOOOOOE+OO 

FIRST CRITICAL BUCKLING LOAD· 1 054506E+04 Ib 

FIRST CRITICAL BUCKLING ACCELERATION - 4 993284E+01 fth2 

BEAM PARAMETERS 

SAME AS ABOVE 

PROGRAM OUTPUT 

TIP BODY PARAMETERS 

m = 15528slugs 

C ,. 670ft 

DATA FROM NAMELIST INPUT 

EI - 6 250000E+07 RHO· 4 172000E-01 L - 1.340000E+02 

NT - 552800E+02 CT· 6.100COOE+01 

FIRST CRITICAL BUCKLING LOAD· 5 261190E+03 Ib 

FIRST C"ITICAL BUCKLING ACCELERATION. 2. 491213E+01 ft/s2 

55 
APPENDIX E 

COMPARISON OF SELECTED NUMERICAL R, SULTS 

In this appendix we compare the numer1cal values for the fundamental natural frequencies obtained from four d1stinct numerical 

procedures. The results are for the case of a t1{ body with mass 

center along the beam t1P tangent line and the particular parameters 

llsted below. 

Beam Parameters T1P Body Parameters 

EI = 6.25 . 107 slug-ft2 m = 155.2J slug 

t = 134.0 ft It = 5.0 10 5 slug-ft2 

p = 0.4172 slug/ft c = 67.0 tt 

The first two columns of natural frequencies are based upon the 

assumed modes formulation of Chapter 5, usinq the f1rst 10 f1xed base 

eigenfunctiors of a clamped-free beam and clamped beam w1th tip body, 

respect1vely. The spline based.Galerkin method used to qenerate the 

results of the th1rd column, 1S described in Refer~,ce 2. The results 

of the fourth column are based upon the power series Solut10n of Rect10n 4.1, uS1ng SlX terms 1n each series. 

56 
VI 

...., 

Constant 

Ax1al Base 

Acceleration 

(ft/s2) 

0.0 

2.4913 

4.9825 

7.4738 

9.9651 

12.4564 

14.9476 

17.4389 

19.9302 

22.4215 

24.9127 -- - - - --

Assumed-Modes Method 

10 Clamped--Free 

E1genfunct1ons 

0.1329 

0.1198 

0.1067 

0.0936 

0.0805 

0.0674 

0.0543 

0.0411 

0.0280 

0.0148 

0.0017 - - --- _._-----

Fundamental Natural Frequencies (s-2) 

Assumed-Modes Method Spline-Based Power-Series 

10 Clamped--Tip Body Galerkin Method Solution 

Eigenfunctions 10-cubic Splines 6-Terms in Each Series 

• 0.1312 0.1312 0.1312 

0.1181 0.1181 0.1181 

0.1050 0.1050 0.1050 

0.0919 0.0919 0.0919 

0.0788 0.0788 0.0788 

0.0657 0.0657 0.0657 

0.0526 0.0526 0.0526 

0.0395 0.0395 0.0395 

0.0263 0.0263 0.0263 

0.0132 0.0132 0.0132 I 

I 

0 0 0 I 

I ---- -_._- ----- ---_._- -------- I 
REFERENCES 

1. Storch, J., and S. Gates, Planar Dynamics of a Uniform Beam 

with Rigid Bodies Affixed to the Ends, Report CSDL-R-1629, The 

Charles Stark Draper Laboratory, Inc., cambridge, MA, May 1983. 

2. Rosen, G., "Spline Based Galerk1n Method for Comput1ng the 

Natural Frequencies for the Transverse Vibration of a Cantilevered Beam with Tbp Body Under Constant Axial Base Accelerat10n,n Intralab Memo DYN-83-2, The Charles Stark Draper Laboratory, Inc., Cambr~dge, MA, November 1983. 

58 
End of Document