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A note on optimal design of a beam with a mass at its end
Journal of Sound and Vibration (1982) 80(2), 203-208
A NOTE
ON OPTIMAL WITH
DESIGN OF A BEAM
A MASS AT ITS END? J. B~ACHUT
The Institute of Physics, Technical University of Cracow, ul. Podchoraiych 30-084 Krakdw, Poland
1,
(Received 2 February 1981, and in revised form 1 May 1981)
The problem of determining the optimal cross-sectional area of a column with a tip mass, which is such that volume is minimized under given external load, fixed frequency and geometrical constraints, is investigated by use of the Pontryagin maximum principle.
1. INTRODUCTION The determination of natural frequencies of a prismatic clamped-free beam, which carries a concentrated mass at a free end, was considered by Laura, Pombo and Susemihl [l]. The rotatory inertia effect on the natural frequencies was neglected. Both the shear and rotatory effects caused by a heavy tip mass attached at a free end of a prismatic cantilever were discussed by Swaminadham and Michael [2]. The problem of optimal design of a cantilever carrying an end mass, which for a given total mass yields the highest possible value of the first frequency under omission of the rotatory effect was studied by Dym [3] and Karihaloo and Niordson [4]. The main goal of this note is to determine the influence of a concentrated mass rotatory inertia moment on optimal solutions, under vibration and stability constraints. The cantilever discussed is loaded by a concentrated tip mass and an additional vertical force which does not change its direction of action (a so-called Eulerian force [5]). The results obtained can be considered as an additional illustration of a unified approach to optimal design of columns which was published by B?achut and Gajewski [5]. 2. FORMULATION OF THE PROBLEM Consider a vibrating, elastic cantilever of length 1 and mass m, carrying a concentrated mass m, at its end. The cantilever is subject to the Euler type force P and a weight m,g (see Figure l), where g is the gravitational constant. The equation describing small oscillations of the structure shown in Figure 1 may be written as (E1@“)“+(P+m,g)G”+pAri:
=O,
(1)
with the boundary conditions @(O,t) = 0, %7(0,f)=O,
-EW’(I,
t) =J d3#(l, r)/dx dr2,
[E1@“(1,t)]‘= m, d’$(/, t)/dt2-(P+m,g)
dc/dx,
(2)
where E is Young’s modulus, I is the second moment of area, A is the area of cross-section, p is the mass density and J is the rotatory inertia of the attached mass. ? This paper
was supported
by Grant
05.12.
203 0022-460X/82/020203+ 06 $02.00/O
@ 1982 Academic
Press Inc. (London)
Limited
204
J.
BLACHUT
Figure 1. Sketch of the cantilever with end mass and vertical load.
A steady motion of separable form, @(x, t) = w(x) exp (iot),
(3)
is assumed and the following dimensionless variables and parameters w=yll,
I = I&y,
@=A/Ao,
p = Pl*/E&,
are introduced:
0 = pAo1402/EIo,
c = gl*mf EIo.
(4)
Here @ is a dimensionless cross-section area, A0 is the cross-section area at point x0, usually arbitrarily chosen, but to simplify the calculations the point x0 can be chosen here to satisfy Aol = V,
(3
where V is the volume of the column only. The exponent v is such that Y = 1 for a plane-convergence bar of a constant height of cross-section, Y = 2 for a uniformly convergent bar and v = 3 for a plane-convergent bar of a constant width of cross-section. Introducing expressions (3) and (4) into equations (1) and (2) gives the differential equation (WY”)“+@ +(mJm)c)y”-R@y
=0
(6)
and the boundary conditions @“y”(l) = (J/m12).ny’(l).
y(O)=% Y‘(0) = 0,
[@“y”(l)]’ = -(m,lm)LW)
-(P + (m,lm)c)Y’(I).
(7)
The fourth order differential equation (6) for the deflection (6) can be broken down into a set of four first order differential equations (the state equations): Y’=Q,
Q’ = -kf/@v,
kf’ = Q + (P +
This is done by introducing the appropriate state variables y, (8) one has the boundary conditions y(O)=O,
M(l) =
-(J/W#)flQ,
Q’ = -R@y.
(&m)c)Q,
Q(0) = 0,
Q, M
(8)
and Q. With equations
Q(1) = (m,lm)Ry.
(9)
OPTIMAL
DESIGN
OF A BEAM
The problem of optimal design is formulated function G(x) that minimizes the functional
J
WITH
205
END-MASS
as follows: one wishes to determine
a
1
dx + min,
Q(x)
(10)
0
subject to the constraints (a) for the cross-section area @I =SCD(x)S @*,
(11)
(b) for the external load P+ (c)
mccO
=
const
<
Pcritical,
(12)
for the prescribed frequency, 0 = const,
(13)
and (d) fulfilment of the state equation (8) with the boundary conditions (9). 3. METHOD
OF SOLUTION: OPTIMALITY
CRITERION
One can make use of the Pontryagin maximum principle to investigate the problem specified by expressions (lo)-(13). Upon introducing a new variable y,,(x) satisfying the equation Yb = @(xl
(14)
Yom
(15)
with the boundary condition = 0,
the cost function (9) becomes of the form I= hjlom,
(16)
where A is the Lagrangian multiplier. The Hamiltonian for this problem is [5]
H=rjl,~+J/,(-~/~“)+~~[Q+(P+(~cl~)c)cpl+c/la(-.n~y)+ILo~.
(17)
From equation (17) one obtains the following set of adjoint equations: I& = -aH/ay I+& = -aH/aM
= *&@, = (I/@“)$,,
+-:, = -aHlap
= -+,y - (0 + (&m)cMM,
(/lb = -aH/aQ
= -*&f,
J/b = -aH/ax
= 0.
(18)
By adopting the notation CL,= kQ,
& =-k&i,
J/M = kR
40
= -kY,
(19)
the adjoint system (18) can be written in the same form as equations (8). According to the self-adjointness of equations (8) and (18) the Hamiltonian (17) has the form H=k{Q~+(M*/Q>“)+[Q+(P+(m,/m)c)rp]+~~y*+h~},
(20)
where I= i/k. The optimal configuration G(x) can be found from the optimality criterion aH/a@ = 0, and has the form (in the case Y = 1) 0(x) = J2M2/(h
+0y2).
(21)
206
J. BLACHUT
n
I.0 -
Figure 2. Maximization of natural frequencies: R maxas a function of Jf ml2 for optimal shapes. -, Optimal; - - -, PriSmatiC [l, 21. Values of parameter m,/m: (a) 0.0, 0.4; (b) O-2, 0.6; (c) 1.0. Selected optimal shapes shown in insets.
OPTIMAL
DESIGN
OF A BEAM
4. DUAL
WITH
207
END-MASS
FORMULATION
According to the desired method of solution [5], one can reformulate the problem specified by expressions (lo)-( 13) as follows: determine a cross-section area @(x), under the constraints
J
1
@*
2s 0(x)
c
cD2,
G(x)
dx = 1,
p + (WZ,/WI)C= const < @critical, (22-24)
0
which simultaneously satisfies the state equations (8) plus the boundary and maximizes a fundamental frequency of vibration, i.e., L! + max.
condition
(9)
(25)
In this case, there is also the possibility of another formulation:
namely,
/3 + (m,/m)c + max,
(26)
under the constraints (22) and (23), given 0 = const, with simultaneous satisfaction of the state equations (8) plus the boundary conditions (9). The appropriate results were obtained numerically on the basis of the iterative method described in reference [5]. 5. RESULTS
5.1. MAXIMIZATION OF NATURAL FREQUENCIES In Figures 2(a)-(c) R,,, is depicted as a function of the parameter J/ml* for optimal shapes. Every point of the heavy lines has its own optimal shape (in accordance with equation (21)). Some of them are shown in the insets on the figures and in Figures 2(a)-(c) the broken curves are for a prismatic cantilever. It is worth mentioning that Q,,,, for the optimal shapes strongly decreases with increasing values of J/m12.
.L
2k.,.fJ,
2
A
t
or+~maI
pz1.0,
v:,
shape for point A
Figure 3. Maximization of loading: p as a function of 0. Inset shows optimal shape for point A. J/ml*= 0.2, Values of m,/m for curves l-6: 1, 2.0;2,1.0;3,0+8; 4,0.6;5,0.4;6,0.2.
c = 0.005.
208 5.2.
J. BLACHUT MAXIMIZATION
OF LOADING
For different values of J/mZ2 and c = 0.005 appropriate 3. An optimal shape of cantilever is shown in the inset.
results
are shown
in Figure
REFERENCES 1. P. A. A. LAURA,J. L.POMBO and E. A. SUSEMIHL 1974 JournalofSound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end. 2. M. SWAMINADHAM and A. MICHAEL 1979 Journal of Sound and Vibration 66, 144-147. A note on frequencies of a beam with a heavy tip mass. 3. C. L. DYM 1974 Journal of Sound and Vibration 32, 49-70. On some recent approaches to structural optimization. and F. I.NIORDSON 1973 Journal of Optimization Theory and Applications 4. B. L. KARIHALOO 11,638-654. Optimum design of vibrating cantilevers. 5. J. BLACHUT and A. GAJEWSKI 1980 Solid Mechanics Archives 5,363-413. A unified approach to optimal design of columns.
A NOTE
ON OPTIMAL WITH
DESIGN OF A BEAM
A MASS AT ITS END? J. B~ACHUT
The Institute of Physics, Technical University of Cracow, ul. Podchoraiych 30-084 Krakdw, Poland
1,
(Received 2 February 1981, and in revised form 1 May 1981)
The problem of determining the optimal cross-sectional area of a column with a tip mass, which is such that volume is minimized under given external load, fixed frequency and geometrical constraints, is investigated by use of the Pontryagin maximum principle.
1. INTRODUCTION The determination of natural frequencies of a prismatic clamped-free beam, which carries a concentrated mass at a free end, was considered by Laura, Pombo and Susemihl [l]. The rotatory inertia effect on the natural frequencies was neglected. Both the shear and rotatory effects caused by a heavy tip mass attached at a free end of a prismatic cantilever were discussed by Swaminadham and Michael [2]. The problem of optimal design of a cantilever carrying an end mass, which for a given total mass yields the highest possible value of the first frequency under omission of the rotatory effect was studied by Dym [3] and Karihaloo and Niordson [4]. The main goal of this note is to determine the influence of a concentrated mass rotatory inertia moment on optimal solutions, under vibration and stability constraints. The cantilever discussed is loaded by a concentrated tip mass and an additional vertical force which does not change its direction of action (a so-called Eulerian force [5]). The results obtained can be considered as an additional illustration of a unified approach to optimal design of columns which was published by B?achut and Gajewski [5]. 2. FORMULATION OF THE PROBLEM Consider a vibrating, elastic cantilever of length 1 and mass m, carrying a concentrated mass m, at its end. The cantilever is subject to the Euler type force P and a weight m,g (see Figure l), where g is the gravitational constant. The equation describing small oscillations of the structure shown in Figure 1 may be written as (E1@“)“+(P+m,g)G”+pAri:
=O,
(1)
with the boundary conditions @(O,t) = 0, %7(0,f)=O,
-EW’(I,
t) =J d3#(l, r)/dx dr2,
[E1@“(1,t)]‘= m, d’$(/, t)/dt2-(P+m,g)
dc/dx,
(2)
where E is Young’s modulus, I is the second moment of area, A is the area of cross-section, p is the mass density and J is the rotatory inertia of the attached mass. ? This paper
was supported
by Grant
05.12.
203 0022-460X/82/020203+ 06 $02.00/O
@ 1982 Academic
Press Inc. (London)
Limited
204
J.
BLACHUT
Figure 1. Sketch of the cantilever with end mass and vertical load.
A steady motion of separable form, @(x, t) = w(x) exp (iot),
(3)
is assumed and the following dimensionless variables and parameters w=yll,
I = I&y,
@=A/Ao,
p = Pl*/E&,
are introduced:
0 = pAo1402/EIo,
c = gl*mf EIo.
(4)
Here @ is a dimensionless cross-section area, A0 is the cross-section area at point x0, usually arbitrarily chosen, but to simplify the calculations the point x0 can be chosen here to satisfy Aol = V,
(3
where V is the volume of the column only. The exponent v is such that Y = 1 for a plane-convergence bar of a constant height of cross-section, Y = 2 for a uniformly convergent bar and v = 3 for a plane-convergent bar of a constant width of cross-section. Introducing expressions (3) and (4) into equations (1) and (2) gives the differential equation (WY”)“+@ +(mJm)c)y”-R@y
=0
(6)
and the boundary conditions @“y”(l) = (J/m12).ny’(l).
y(O)=% Y‘(0) = 0,
[@“y”(l)]’ = -(m,lm)LW)
-(P + (m,lm)c)Y’(I).
(7)
The fourth order differential equation (6) for the deflection (6) can be broken down into a set of four first order differential equations (the state equations): Y’=Q,
Q’ = -kf/@v,
kf’ = Q + (P +
This is done by introducing the appropriate state variables y, (8) one has the boundary conditions y(O)=O,
M(l) =
-(J/W#)flQ,
Q’ = -R@y.
(&m)c)Q,
Q(0) = 0,
Q, M
(8)
and Q. With equations
Q(1) = (m,lm)Ry.
(9)
OPTIMAL
DESIGN
OF A BEAM
The problem of optimal design is formulated function G(x) that minimizes the functional
J
WITH
205
END-MASS
as follows: one wishes to determine
a
1
dx + min,
Q(x)
(10)
0
subject to the constraints (a) for the cross-section area @I =SCD(x)S @*,
(11)
(b) for the external load P+ (c)
mccO
=
const
<
Pcritical,
(12)
for the prescribed frequency, 0 = const,
(13)
and (d) fulfilment of the state equation (8) with the boundary conditions (9). 3. METHOD
OF SOLUTION: OPTIMALITY
CRITERION
One can make use of the Pontryagin maximum principle to investigate the problem specified by expressions (lo)-(13). Upon introducing a new variable y,,(x) satisfying the equation Yb = @(xl
(14)
Yom
(15)
with the boundary condition = 0,
the cost function (9) becomes of the form I= hjlom,
(16)
where A is the Lagrangian multiplier. The Hamiltonian for this problem is [5]
H=rjl,~+J/,(-~/~“)+~~[Q+(P+(~cl~)c)cpl+c/la(-.n~y)+ILo~.
(17)
From equation (17) one obtains the following set of adjoint equations: I& = -aH/ay I+& = -aH/aM
= *&@, = (I/@“)$,,
+-:, = -aHlap
= -+,y - (0 + (&m)cMM,
(/lb = -aH/aQ
= -*&f,
J/b = -aH/ax
= 0.
(18)
By adopting the notation CL,= kQ,
& =-k&i,
J/M = kR
40
= -kY,
(19)
the adjoint system (18) can be written in the same form as equations (8). According to the self-adjointness of equations (8) and (18) the Hamiltonian (17) has the form H=k{Q~+(M*/Q>“)+[Q+(P+(m,/m)c)rp]+~~y*+h~},
(20)
where I= i/k. The optimal configuration G(x) can be found from the optimality criterion aH/a@ = 0, and has the form (in the case Y = 1) 0(x) = J2M2/(h
+0y2).
(21)
206
J. BLACHUT
n
I.0 -
Figure 2. Maximization of natural frequencies: R maxas a function of Jf ml2 for optimal shapes. -, Optimal; - - -, PriSmatiC [l, 21. Values of parameter m,/m: (a) 0.0, 0.4; (b) O-2, 0.6; (c) 1.0. Selected optimal shapes shown in insets.
OPTIMAL
DESIGN
OF A BEAM
4. DUAL
WITH
207
END-MASS
FORMULATION
According to the desired method of solution [5], one can reformulate the problem specified by expressions (lo)-( 13) as follows: determine a cross-section area @(x), under the constraints
J
1
@*
2s 0(x)
c
cD2,
G(x)
dx = 1,
p + (WZ,/WI)C= const < @critical, (22-24)
0
which simultaneously satisfies the state equations (8) plus the boundary and maximizes a fundamental frequency of vibration, i.e., L! + max.
condition
(9)
(25)
In this case, there is also the possibility of another formulation:
namely,
/3 + (m,/m)c + max,
(26)
under the constraints (22) and (23), given 0 = const, with simultaneous satisfaction of the state equations (8) plus the boundary conditions (9). The appropriate results were obtained numerically on the basis of the iterative method described in reference [5]. 5. RESULTS
5.1. MAXIMIZATION OF NATURAL FREQUENCIES In Figures 2(a)-(c) R,,, is depicted as a function of the parameter J/ml* for optimal shapes. Every point of the heavy lines has its own optimal shape (in accordance with equation (21)). Some of them are shown in the insets on the figures and in Figures 2(a)-(c) the broken curves are for a prismatic cantilever. It is worth mentioning that Q,,,, for the optimal shapes strongly decreases with increasing values of J/m12.
.L
2k.,.fJ,
2
A
t
or+~maI
pz1.0,
v:,
shape for point A
Figure 3. Maximization of loading: p as a function of 0. Inset shows optimal shape for point A. J/ml*= 0.2, Values of m,/m for curves l-6: 1, 2.0;2,1.0;3,0+8; 4,0.6;5,0.4;6,0.2.
c = 0.005.
208 5.2.
J. BLACHUT MAXIMIZATION
OF LOADING
For different values of J/mZ2 and c = 0.005 appropriate 3. An optimal shape of cantilever is shown in the inset.
results
are shown
in Figure
REFERENCES 1. P. A. A. LAURA,J. L.POMBO and E. A. SUSEMIHL 1974 JournalofSound and Vibration 37, 161-168. A note on the vibrations of a clamped-free beam with a mass at the free end. 2. M. SWAMINADHAM and A. MICHAEL 1979 Journal of Sound and Vibration 66, 144-147. A note on frequencies of a beam with a heavy tip mass. 3. C. L. DYM 1974 Journal of Sound and Vibration 32, 49-70. On some recent approaches to structural optimization. and F. I.NIORDSON 1973 Journal of Optimization Theory and Applications 4. B. L. KARIHALOO 11,638-654. Optimum design of vibrating cantilevers. 5. J. BLACHUT and A. GAJEWSKI 1980 Solid Mechanics Archives 5,363-413. A unified approach to optimal design of columns.