In the present study, an analytical solution for nonlinear vibration problem of launch vehicle as a flexible beam
model carrying moving with constant velocity mass which is time function is proposed. The nonlinear third and
fifth order terms of partial differential equation with variable in time coefficients correspond to high amplitude
and mid-plane stretching beam and the right term of initial equation of the problem represents the concentrated
time-dependent moving mass effect. A hybrid asymptotic approach applied for an approximate analytical
solution problem at given boundary and initial conditions an given. The resulting (approximate) solution has a
form of a sum where each term consists of the product of two functions according to perturbation (on parameter
at nonlinear terms) and phase-integral-Galerkin technique (on singular parameter at higher derivative)
methods. The results of comparison of an approximate analytical solution and direct numerical integration of
initial equation have shown a good enough accuracy as for “small” as well for “large ” scalar parameters for
asymptotic expansion of the desired function

Gristchak, D. D.

Library of National Technical University "Kharkiv Polytechnic Institute"

Proceedings of the 5-th International Conference on Nonlinear Dynamics
ND -K hPI 2016 
September 2016, Kharkov, Ukraine
Nonlinear Vibration Problem of Launch Vehicle 
Carrying a Moving Time-Dependent Mass
D.D. Gristchak1*
Abstract
In the present study, an analytical solution for nonlinear vibration problem of launch vehicle as a flexible beam 
model carrying moving with constant velocity mass which is time function is proposed. The nonlinear third and 
fifth order terms of partial differential equation with variable in time coefficients correspond to high amplitude 
and mid-plane stretching beam and the right term of initial equation of the problem represents the concentrated 
time-dependent moving mass effect. A hybrid asymptotic approach applied for an approximate analytical 
solution problem at given boundary and initial conditions an given. The resulting (approximate) solution has a 
form of a sum where each term consists of the product of two functions according to perturbation (on parameter 
at nonlinear terms) and phase-integral-Galerkin technique (on singular parameter at higher derivative) 
methods. The results of comparison of an approximate analytical solution and direct numerical integration of 
initial equation have shown a good enough accuracy as for “small” as well for “large ” scalar parameters for 
asymptotic expansion of the desired function.
Keywords
Hybrid (P-WKB-G) asymptotic approach, nonlinear dynamic problem, Launch Vehicle, time dependent moving 
mass.
1
Zaporizhzhye National Univesity, Zaporizhzhye, Ukraine
* Corresponding author: grk@znu.edu.ua
Introduction
Nonlinear vibration problem of Launch Vehicle with concentrated moving time dependent 
mass as a continuously distributed system has broad applications in engineering [1], including 
airspace structures, space tethers, flexible manipulators, highway bridges and others. For an overview 
of the publications in this sphere we refer [2, 3]. In [4] it was shown that the moving load solution is 
not an upper bound for the moving mass solution for the given boundary conditions. The existence of 
nonlinear terms in equations governing the vibration of flexible body motion with concentrated 
moving time dependent mass is important from the point of view modeling of flight dynamics effects. 
Most of studies were devoted to investigation of nonlinear vibration. Nonlinear cubic function term 
was considered in order to obtain analytical solution [2-4]. Effects of high degree terms as a rule are 
ignored. Here we refer the publications [1-5] which are connected with the subject of discussion and 
special attention we have paid to paper [2], in which employing of combination the homotopy method 
and traditional perturbation procedure for nonlinear analysis of the beam carrying a moving mass.
In the present study hybrid asymptotic solution of the vibrational motion of Launch Vehicle 
beam model, in which a concentrated moving time dependent mass with employing of the 
perturbation and phase integral (WKB) methods combined with Galerkin othogonality principle (P- 
WKB-G method) has been obtained. An approximate analytical solution with direct numerical 
calculations of initial equation is compared. The results show that concentrated mass velocity and 
mass function from time are caused significant effects in nonlinear dynamic behavior of structure.
2. Formulation of the problem. Approximate analytical solution
The Launch Vehicle beam model with moving concentrated time dependent mass is shown in 
(Fig. 1).
294
D.D. Gristchak
Figure 1. Launch Vehicle carrying a concentrated mass
The nonlinear differential equation with variable in time coefficients governing of Launch 
Vehicle vibration in high amplitude and mid-plain stretching at simply supported boundary conditions 
in which time dependent mass with constant velocity is moving on is as follows [1]:
d2 w (x, t) ^ ^  d4 w (x, t) I EA a
"  I 2L 10
)  .
m -----+ EI
dt Ox
dw ( x, t)
x I 1
A 
' L
dw ( x, t)
dx 8L J0 dx
dx
d2 w ( x, t)
dx2 (1)
-M  (t) w ( x, t ) 5 [ x -  5] + Q0 SinCt
where w( x, t) -  transverse deformation of beam in time and coordinate.
If it is assumed [2] that deformation is large but performed slowly, the right part of initial 
equation (1) can be calculated as
, „ , N, d2w(x,t) , d 2w(x,t) 1 r 
w(x,t)= M ( t ) |-----^— + x ----- \ —- |5 [x - s]dt2 dx
In the new nondimensional variables initial equation (1) can be written as
(2)
d 2V + A - J l  p  I
d r2 dg4 | 2 Pl0
d^
~dg dg - ;  P i
d77 
~d%
d A ^ - -
\ d ?
(3)
- a  | u2g  + d 7  - Ao) + QoSinCir
where
w , x y s M  (t) M 0 - , .  iT,{\
l •g=~L •g =l •a (t )= i m r =—rM (t)=a M (t) •
P =— , u = J m vL  ^ r = f—J -L-J, C = c|— 1 L2I \E I I m J L 2 I EI I
(4)
Suppose that solution of the equation (2) satisfies to simply supported boundary conditions
n(£,r)  = q (T) Sin (ng) (5)
initial equation transforms to the equation in form
q ( t )  +  ©o2 ( u ,g o) q ( t )  +  f  ( u ,g o) q 3 ( t) -  r ( u ,g o) q 5 ( f )  = 0 (6)
qi (t) + ©o2 (t) qi (t) = f  (u,go) q3 (t) + Y (u,go) q5 (t)] + Qo (g)SinCr = n N [t} + Q (g,r)
where coefficients are
2
295
I(u,£0)
f  (to ) =
|K 4 - 2u2a (t)sin2 (n£,0) ]k 2 
1 + 2a (t) sin2 (k£0)
D.D. Gristchak
Pk
r{to ) =
4 1^ 1 + 2a(t ) sin2 (k£0)]
_________________ 3_ p K __________________
641^ 1 + 2a (t)Sin2 (k£0)]
(7)
(8) 
(9)
In order to obtain an approximate analytical solution equation (6) it will be employed the 
hybrid asymptotic approach [5] based on perturbation (with respect to parameter at nonlinear terms)
q (t) = % (t) + A  (t) (10)
and for the second step we employ phase integral (WKB) approximation for solutions of system 
singular non homogeneous linear equations with variable coefficients (with respect to the parameter 
near the highest derivative) [6-7]:
A : g2q,0 (t) + ®02(t)q0(t) = Q (£0 ) 
a 1: e 2q1 (t)+ ®02 (t ) ?1 (t ) = A j  (u,£0) q3 (t )+ f  (u,£0) q5 (t )] = a N  [ t}
(11)
(12)
where
®02 (t ) =
k 2 -  2u a(t)Sin2 (k£0 )
1 + 2a (t) Sin2 (k£0 ) j2 = K -, q (£,t)  = j Q (£o )
(13)
An asymptotic solution of initial equation (6) will be consider at initial conditions
q  (0) = 1 
q0 (0) = 0
(13)
The system of ordinary singular differential equations with variable in time coefficient ®02 (t) is 
solved by two terms WKB-approximation. Finally we have obtained the solution of nonlinear forced 
vibration problem on the basis of hybrid perturbation -  (two-term) WKB method in the form:
q (z)  =
1
K (z)]
+CosK (r)
SinK 0 ) , 1 rQ (Zo)CosK 0 ) , f N  (q0o )CosK (z) 
C1 + e J K  (t) + A  K  (t)
dz
1 rQ (£ ,t) SinK (z) fN (q0,z) SinK ( t ) j
: - 7J TT7\ A  dz
K  0 ) K  0)
where
K (r) = Je 1® 0 ( r )d r  
For the numerical calculations parameters of the system are:
a (t) = 10-3 (1 - r ) ,  p  = 1.2-105, u = 17.3, v = 10 m /sec 
Some results of numerical calculations for free vibration are presented at Fig. 2-4.
(14)
(15)
(16)
+
c
296
D.D. Gristchak
q q
a) b ) ..........................
Figure 2. Comparison of the approximate analytical solution and numerical simulation for different initial
values.
Figure 4. Influence of amplitude of concentrated mass Figure 5. Influence of non dimensional parameter of
function in time moving concentrated mass function in time
Conclusions
In the present study an approximate analytical solution for nonlinear forced vibration problem 
of Launch Vehicle beam model carrying a moving time-dependent mass has been obtained. A hybrid 
perturbation and phase integral methods was employing. Some results for influence of amplitude mass 
function in time and velocity parameter of concentrated mass for free vibration of model are 
presented.
References
[1] Quadrelli M.B., Cameron J., Balaram B., Baranwal Mayank, Bruno A.. Modeling and Simulation 
of Flight Dynamics of Variable Mass System. SPACE, Conference&amp; Exposition, San-Diego, 4-7 
August 2014.
[2] Poorjamshidian M., Sheikhi J., Mahjioub-Moghadas S., Nakhaie M. Nonlinear Vibration 
Analysis of the Beam Carrying a Moving Mass Using Modified Homotopy. Journal o f Solid 
Mechanics, 2014, Vol. 6 (4), p. 389-396.
[3] Pan L. Nonlinear Stability and Local Bifurcation in a simply supported beam carrying a moving 
mass. Acta Mechanica Solida Sinica, 2007, 20(2), p. 123-129.
[4] Sadiku S., Leipholz H. On the Dynamics of Elastic Systems with Moving concentration Masses. 
Arch. Appl. Mech., 1987, Vol. 57, p. 223-242.
[5] Geer J.F., Andersen C.V. Natural Frequency Calculations Using a Hybrid Perturbation Galerkin 
Technique. Pan American Congress on Appl. Mech., 1991, p. 571-574.
[6] Gristchak V.Z., Ganilova O.A. A Hybrid WKB-Galerkin Method Applied to a Piezoelectric 
Sandwich Plate Vibration Problem Considering Shear Force Effects. Journal o f Sound and Vibration, 
2008, Vol. 317 (1-2), p. 366-377.
[7] Azarskov V.F., Gristchak D.D. Oscillation Damping of Air-Space Structures with Joint-Up 
Dynamic Absorber. Proceedings the Sixth World Congress “Safety in Aviation and Space 
Technologies”, Kiev, Ukraine, September 23-25, 2013.
297


Nonlinear Vibration Problem of Launch Vehicle Carrying a Moving Time-Dependent Mass

eNTUKhPIIR Electronic National Technical University "Kharkiv Polytechnic Institute"; Institutional Repository

Proceedings of the 5-th International Conference on Nonlinear DynamicsND -K hPI 2016 September 2016, Kharkov, UkraineNonlinear Vibration Problem of Launch Vehicle Carrying a Moving Time-Dependent MassD.D. Gristchak1*AbstractIn the present study, an analytical solution for nonlinear vibration problem of launch vehicle as a flexible beam model carrying moving with constant velocity mass which is time function is proposed. The nonlinear third and fifth order terms of partial differential equation with variable in time coefficients correspond to high amplitude and mid-plane stretching beam and the right term of initial equation of the problem represents the concentrated time-dependent moving mass effect. A hybrid asymptotic approach applied for an approximate analytical solution problem at given boundary and initial conditions an given. The resulting (approximate) solution has a form of a sum where each term consists of the product of two functions according to perturbation (on parameter at nonlinear terms) and phase-integral-Galerkin technique (on singular parameter at higher derivative) methods. The results of comparison of an approximate analytical solution and direct numerical integration of initial equation have shown a good enough accuracy as for “small” as well for “large ” scalar parameters for asymptotic expansion of the desired function.KeywordsHybrid (P-WKB-G) asymptotic approach, nonlinear dynamic problem, Launch Vehicle, time dependent moving mass.1Zaporizhzhye National Univesity, Zaporizhzhye, Ukraine* Corresponding author: grk@znu.edu.uaIntroductionNonlinear vibration problem of Launch Vehicle with concentrated moving time dependent mass as a continuously distributed system has broad applications in engineering [1], including airspace structures, space tethers, flexible manipulators, highway bridges and others. For an overview of the publications in this sphere we refer [2, 3]. In [4] it was shown that the moving load solution is not an upper bound for the moving mass solution for the given boundary conditions. The existence of nonlinear terms in equations governing the vibration of flexible body motion with concentrated moving time dependent mass is important from the point of view modeling of flight dynamics effects. Most of studies were devoted to investigation of nonlinear vibration. Nonlinear cubic function term was considered in order to obtain analytical solution [2-4]. Effects of high degree terms as a rule are ignored. Here we refer the publications [1-5] which are connected with the subject of discussion and special attention we have paid to paper [2], in which employing of combination the homotopy method and traditional perturbation procedure for nonlinear analysis of the beam carrying a moving mass.In the present study hybrid asymptotic solution of the vibrational motion of Launch Vehicle beam model, in which a concentrated moving time dependent mass with employing of the perturbation and phase integral (WKB) methods combined with Galerkin othogonality principle (P- WKB-G method) has been obtained. An approximate analytical solution with direct numerical calculations of initial equation is compared. The results show that concentrated mass velocity and mass function from time are caused significant effects in nonlinear dynamic behavior of structure.2. Formulation of the problem. Approximate analytical solutionThe Launch Vehicle beam model with moving concentrated time dependent mass is shown in (Fig. 1).294D.D. GristchakFigure 1. Launch Vehicle carrying a concentrated massThe nonlinear differential equation with variable in time coefficients governing of Launch Vehicle vibration in high amplitude and mid-plain stretching at simply supported boundary conditions in which time dependent mass with constant velocity is moving on is as follows [1]:d2 w (x, t) ^ ^  d4 w (x, t) I EA a"  I 2L 10)  .m -----+ EIdt Oxdw ( x, t)x I 1A ' Ldw ( x, t)dx 8L J0 dxdxd2 w ( x, t)dx2 (1)-M  (t) w ( x, t ) 5 [ x -  5] + Q0 SinCtwhere w( x, t) -  transverse deformation of beam in time and coordinate.If it is assumed [2] that deformation is large but performed slowly, the right part of initial equation (1) can be calculated as, „ , N, d2w(x,t) , d 2w(x,t) 1 r w(x,t)= M ( t ) |-----^— + x ----- \ —- |5 [x - s]dt2 dxIn the new nondimensional variables initial equation (1) can be written as(2)d 2V + A - J l  p  Id r2 dg4 | 2 Pl0d^~dg dg - ;  P id77 ~d%d A ^ - -\ d ?(3)- a  | u2g  + d 7  - Ao) + QoSinCirwherew , x y s M  (t) M 0 - , .  iT,{\l •g=~L •g =l •a (t )= i m r =—rM (t)=a M (t) •P =— , u = J m vL  ^ r = f—J -L-J, C = c|— 1 L2I \E I I m J L 2 I EI I(4)Suppose that solution of the equation (2) satisfies to simply supported boundary conditionsn(£,r)  = q (T) Sin (ng) (5)initial equation transforms to the equation in formq ( t )  +  ©o2 ( u ,g o) q ( t )  +  f  ( u ,g o) q 3 ( t) -  r ( u ,g o) q 5 ( f )  = 0 (6)qi (t) + ©o2 (t) qi (t) = f  (u,go) q3 (t) + Y (u,go) q5 (t)] + Qo (g)SinCr = n N [t} + Q (g,r)where coefficients are2295I(u,£0)f  (to ) =|K 4 - 2u2a (t)sin2 (n£,0) ]k 2 1 + 2a (t) sin2 (k£0)D.D. GristchakPkr{to ) =4 1^ 1 + 2a(t ) sin2 (k£0)]_________________ 3_ p K __________________641^ 1 + 2a (t)Sin2 (k£0)](7)(8) (9)In order to obtain an approximate analytical solution equation (6) it will be employed the hybrid asymptotic approach [5] based on perturbation (with respect to parameter at nonlinear terms)q (t) = % (t) + A  (t) (10)and for the second step we employ phase integral (WKB) approximation for solutions of system singular non homogeneous linear equations with variable coefficients (with respect to the parameter near the highest derivative) [6-7]:A : g2q,0 (t) + ®02(t)q0(t) = Q (£0 ) a 1: e 2q1 (t)+ ®02 (t ) ?1 (t ) = A j  (u,£0) q3 (t )+ f  (u,£0) q5 (t )] = a N  [ t}(11)(12)where®02 (t ) =k 2 -  2u a(t)Sin2 (k£0 )1 + 2a (t) Sin2 (k£0 ) j2 = K -, q (£,t)  = j Q (£o )(13)An asymptotic solution of initial equation (6) will be consider at initial conditionsq  (0) = 1 q0 (0) = 0(13)The system of ordinary singular differential equations with variable in time coefficient ®02 (t) is solved by two terms WKB-approximation. Finally we have obtained the solution of nonlinear forced vibration problem on the basis of hybrid perturbation -  (two-term) WKB method in the form:q (z)  =1K (z)]+CosK (r)SinK 0 ) , 1 rQ (Zo)CosK 0 ) , f N  (q0o )CosK (z) C1 + e J K  (t) + A  K  (t)dz1 rQ (£ ,t) SinK (z) fN (q0,z) SinK ( t ) j: - 7J TT7\ A  dzK  0 ) K  0)whereK (r) = Je 1® 0 ( r )d r  For the numerical calculations parameters of the system are:a (t) = 10-3 (1 - r ) ,  p  = 1.2-105, u = 17.3, v = 10 m /sec Some results of numerical calculations for free vibration are presented at Fig. 2-4.(14)(15)(16)+c296D.D. Gristchakq qa) b ) ..........................Figure 2. Comparison of the approximate analytical solution and numerical simulation for different initialvalues.Figure 4. Influence of amplitude of concentrated mass Figure 5. Influence of non dimensional parameter offunction in time moving concentrated mass function in timeConclusionsIn the present study an approximate analytical solution for nonlinear forced vibration problem of Launch Vehicle beam model carrying a moving time-dependent mass has been obtained. A hybrid perturbation and phase integral methods was employing. Some results for influence of amplitude mass function in time and velocity parameter of concentrated mass for free vibration of model are presented.References[1] Quadrelli M.B., Cameron J., Balaram B., Baranwal Mayank, Bruno A.. Modeling and Simulation of Flight Dynamics of Variable Mass System. SPACE, Conference&amp; Exposition, San-Diego, 4-7 August 2014.[2] Poorjamshidian M., Sheikhi J., Mahjioub-Moghadas S., Nakhaie M. Nonlinear Vibration Analysis of the Beam Carrying a Moving Mass Using Modified Homotopy. Journal o f Solid Mechanics, 2014, Vol. 6 (4), p. 389-396.[3] Pan L. Nonlinear Stability and Local Bifurcation in a simply supported beam carrying a moving mass. Acta Mechanica Solida Sinica, 2007, 20(2), p. 123-129.[4] Sadiku S., Leipholz H. On the Dynamics of Elastic Systems with Moving concentration Masses. Arch. Appl. Mech., 1987, Vol. 57, p. 223-242.[5] Geer J.F., Andersen C.V. Natural Frequency Calculations Using a Hybrid Perturbation Galerkin Technique. Pan American Congress on Appl. Mech., 1991, p. 571-574.[6] Gristchak V.Z., Ganilova O.A. A Hybrid WKB-Galerkin Method Applied to a Piezoelectric Sandwich Plate Vibration Problem Considering Shear Force Effects. Journal o f Sound and Vibration, 2008, Vol. 317 (1-2), p. 366-377.[7] Azarskov V.F., Gristchak D.D. Oscillation Damping of Air-Space Structures with Joint-Up Dynamic Absorber. Proceedings the Sixth World Congress “Safety in Aviation and Space Technologies”, Kiev, Ukraine, September 23-25, 2013.297

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Nonlinear Vibration Problem of Launch Vehicle Carrying a Moving Time-Dependent Mass

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