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Vibrations of a beam with non-linear elastic constraints
Journal
qf
Soundand Vibration (1979) 66(l), 1-8
VIBRATIONS OF
A
BEAM WITH NON-LINEAR
ELASTIC
CONSTRAINTS H. SAITO AND K. MORI Department
qf Mechanical Engineering. Tohoku Uniwrsity. Sendai, Japan (Receiced 4 December 1978)
The transverse vibrations of a beam, with both ends supported on non-linear elastic constraints, carrying a concentrated mass and subjected to a transverse periodic force are analyzed. The non-linear constraints are represented by rotational and translational springs having both linear and cubic non-linear behavior. The harmonic responses of a beam involving the third order superharmonic and the one-third order subharmonic are considered. The beam responses to the effects of non-linear elastic constraints are discussed by taking two cases.
1. INTRODUCTION In the analysis of the dynamic behavior of a beam with elastically supported ends, it is usually assumed that the beam has linear boundary conditions. Recently, however, in many practical cases it has become necessary to analyze the dynamic responses of a beam with non-linear support constraints. Paslay and Gurtin [1] and Srinivasan [2] have treated the problem of a vibrating elastic system resting on a non-linear spring by using the Fourier technique. Investigations of situations in which a beam is restrained with non-linear springs at the ends have been made by Porter and Billett [3], who have studied the longitudinal vibrations of a bar with a non-linear spring at the end, and by Dokainish and I&mar [4, 51, who have dealt with the transverse vibrations of a beam having one end clamped and the other supported on a non-linear spring and carrying a concentrated mass. If the boundary conditions are non-linear, the system shows a non-linear response although the beam behavior can still be described by the usual linear equation of motion. The analysis of this problem is unavoidably approximate as Porter and Billet have already shown in their paper, since it involves the solution of so-called non-linear boundary value problems. This paper describes a study of the steady state transverse vibrations of a beam with both ends supported on non-linear elastic constraints, carrying a concentrated mass and subjected to a transverse periodic force. Non-linear constraints are represented by rotational and translational springs which have both linear and cubic non-linear behavior. By substituting the solution for a beam vibration which involves the one-third order subharmonic and the third order superharmonic as well as the fundamental into the boundary conditions and applying the harmonic balance method, cubic algebraic equations are obtained for the unknowns and these are solved by using the Newton-Raphson technique. In numerical calculations, the following two examples are treated; (a) a beam
2
H. SAITO AND K. MORI
with a central mass and symmetrical translational springs at both ends and (b) a beam with one end spring hinged and carrying a concentrated mass at the other end.
2. GOVERNING EQUATION Figure 1 shows a beam of length 1, supported with rotational and translational springs at each end, and carrying a concentrated mass m at x = a. The beam is subjected to a concentric transverse periodic force p cos l2t at x = b. For a Bernoulli-Euler beam with a concentrated mass and a concentric transverse periodic force, which are expressed by using Dirac delta functions, 6, the equation of motion is EI(84y/dx4) + {PA + m6(x - a)} J2y/dt2 = p6(x -b) cos Qt,
Figure
1. Beam with non-linear
(1)
constraints.
where y(x, t) is the transverse displacement of the beam, EZ is the flexural rigidity, p is the mass density, A is the cross-sectional area and t is the time. As the rotational and translational springs have both linear and cubic non-linear behavior, any solution of equation (1) must satisfy the following boundary conditions at all times:
&Lk ax~
ol{g
El!!?ax~ = -k tz{g
+ %i(g>‘},
+ sties},
EI$
= -k,i(y+s,,y3)
E12
= ktz(y+ct2y3)
at
x = 0,
(2)
at
x = 1,
(3)
where k,, and k,, (i = 1,2) are the rotational and translational linear spring constants, and .sgiand sti are the non-linear spring constants.
3. ANALYSIS Equation (1) and boundary conditions (2) and (3) are written in dimensionless forms as (a4j+W) + 54 {1 + ~6 (x - ii)) a2jjarz
= ps (2 - 6) cos 7,
(4)
A BEAM WITH
NON-LINEAR
3
CONSTRAINTS
where
z = i-h,
< = (pAQ2/E1)1’41,
x = x/l,
j = y/l,
ji = p12/EI,
y = m/pAl, kti = kti13jEI,
Loi = k,,l/EI,
~7= a/l,
b = b/l,
Eti = &J2,
i = 1,2. (7)
In order to investigate the one-third order subharmonic and the third order superharmanic as well as the fundamental harmonic, a solution of the form _ y(x, r) = X,,,(X) cos fr + X,(X) cos z + X,(X) cos 30 is assumed, of equation
where X,(x)(n = $1,3) are functions of X. If equation (4) it follows that X,(X) must satisfy the equations (d4X,/dX4) - n2C4{1 +$(x-a)}
x,(X) =
(8) is to be the solution
X, = j%(%6)6T,,,
where the ST,, are Kronecker deltas. Equations Laplace transforms [6], the results being
(8)
n = 3,1,3,
(9) can be solved simultaneously
S(F,,(x)X,(O)+Cl/jn41 G,,(Z)X;(0)-[li(fi{)2]
(9) by using
F,,(x)X;(O)
-C11(~5)31Gn2(X)X~(0) -J~SYL~~(X)X~(~)U(X-~) -ClIsY311s~2(x)u(x-h)s:,,},
n = +,1,3,
(10)
where
L,,,(X) = sin &
<(x -a)
- sinh 4
5(X -ii),
N2(%) = sint(x-b)
- sinh<(x--b),
(11)
and U(X-ti) is the unit step function. The primes denote differentiation with respect to .?. Equations (10) involve the unknowns X,(O) N X:(O) and X,(Z). Substituting X = ti in equations (10) yields X,(a) = ~(F,,(~)X,(O)+C1/~51G,~(~)X:,(O)-C1l(~5)21Fn2(ii)X~(Oj -
CM&
5)“l Gn2(4 X(O)1
(12a)
in the case where a < 6, and X,(a) = ~~F,,(~)X,(O)+C1l~51G,~(~)~:,(O)-C1l(~5)2lFn2(~)X:(O) - C11t,/%)31 G2(4X’(WC1/‘531
P~2C4~T.J
(12b)
in the case where ii > 6. The remaining unknowns X,,(O) N X,“(O) can be determined from the boundary conditions (5) and (6). Substituting equation (8) into equations (5) and (6) and applying the harmonic balance method gives the following equations:
n = f,1,3,
(13b:l
4
H. SAITO AND K. MORI
where f&(X)
= t(3X;:,(X)+~X;~~(X)X;(X)+X;2(X)X;,3(X)+X;~(X)X;,3(X)},
f;“(X) = ~{~x;~,(x)+~x;3(x)+x;~3(x)x;(x)+~x;~(x)x;(x)+xj2(x)x;(x)}, f;“(X) = ~{~x;3(x)+~x~~(x)+x;2(x)x;(x)+x;~3(x)x;(x)}, J-1,3(4 = ~~~~:,3(~)+~~:,3(~)~l(~)+~:(~)~1,3(x)+~:.(x)x~,3(x)}, fi(X)= ~{~x:,3(x)+~x:(x)+x:,3(x)x1(x)+~x:(x)x3(x)+x~(x)x1(x)}, f3(X)= ~{~x:(x)+~x~(x)+x:(x)x3(x)+x:,3(x)x3(x)}.
(14)
If one differentiates equations (10) with respect to X and puts X = 1, X,(l) expressed in terms of X,(O) - X;(O) as follows: X,(l)
- X;(l)
can be
= 3{~,,(1)X,(O)+C1/~51G,~(l)x:,(O)-C1/(~~)21~n2(1)X::(O) -C1M~~)3lG,2(1)X::(O)-~~y~,2(1)X,(~)-C1/531~N2(1)~T,.},
X,(l) = ~{-~5Gn2(1)Xn(O)+Fn1(1)X:,(O)+C1/~51G,,(1)X:(O) -C1/(~5)21~"2(1)X,"(O)-(~~)2~~,2(~)X,(~)-C~/521~~2(~)~T,.~, X(l) = 3{-(~5)2~,2(1)X.(O)--\/tf5Gn2(1)X~(O)+~n1(1)X~(O) +C1/~51G,~(1)x::(o)+(~5)3~~,,(1)X,(a)+C1/~l~N,(1)~T,.},
/
X;(l) = 5{(~r)3G,1(1)Xn(O)-(~~)2Fn2(1)X:,(O)--~G,2(l)X~(O) +F,,(l)X~(0)+(~5)4yH",(1)X,(a)+~M,(1)6T,,},
(15)
where cos$@-ii)
f cosh&&-ii),
M, (3
= cost(x-6)+cosht(x-6),
M2(4 L,,(X)
= sin,,/‘%r(%-ii)
+ sinh&l(Z-a),
N,(Z)
= sin r(X-6)
+ sinhc(x-6). (16)
Hence, equations (13) form twelve coupled algebraic equations for the twelve unknowns X,(O), X”(O), X:(O) and X:(O) (n = i, 1,3), which can be solved by using the NewtonRaphson technique iteratively with the aid of an electronic computer. 4. NUMERICAL
For beam beam forced mass which
RESULTS
AND DISCUSSION
the purpose of numerical calculations, we have considered two specific cases; (a) a with a central mass and symmetrical translational springs at both ends, and (b) a with one end spring hinged and carrying a concentrated mass at the other end. The situation considered is that in which a periodic force acts at the point where the is attached. The dimensionless amplitude root mean square (ARMS) at the mass, is defined as follows, is taken as the response of the beam: ARMS
= J t{X:,,(a)+X:(ii)+X:(a)}
.
(17)
4.1. CASE (a) This case corresponds to Et1 = kt2 = k,, Et1 = Et2 = EC, i&r = k,, = 0 and a = b = 0.5. For the free vibration, one may set the dimensionless amplitude of the exciting force p
A BEAM WITH
NON-LINEAR
CONSTRAINTS
E Figure 2. Backbone curves of a beam with a central mass and symmetrical translational springs at both ends. ARMS at the mass (symmetrical mode); k, = 1, y = 1, li = 0.5: $: a, -50; b, 0; c, 100; d, 500; e, 1000. p, ---, ARMS at the end (antisymmetrical mode).
equal to zero. The relations between the natural frequency parameter 5 and ARMS, known as the backbone curve, are shown in Figure 2, for the dimensionless values & = 1 and y = 1 and with the dimensionless non-linear constant E, as a variable parameter. The second mode of vibration is antisymmetrical to the center of the beam. Therefore, in the figure, ARMS at the end of the beam is shown by the chained line for the second mode. The dynamic response curves are shown in Figure 3, for the parameter values & = 1, Et = 100, y = 1, p = 0.05 and 0.1. For comparison, the linear response curve for & = 1, E, = 0, y = 1 and p = 0.1 is also shown, by the chained line. Numbers “r-s” on the response curves indicate “the rth mode of vibration-the sth order harmonic of a resonance”; e.g., “l- 3” means the third order superharmonic resonance of the first mode. In the figure, the solid lines indicate stable branches and the dashed lines unstable ones. The determination of the stability has been carried out by adopting the argument that the region where the amplitude decreases as the periodic force increases, which is contrary to physical facts, must be unstable and that the point having the vertical tangent to the response curve provides a limiting point; no analytical investigation of the stability has been made. No resonances corresponding to the antisymmetrical even order modes can be seen in Figure 3, because all the forced vibration modes to be expected are symmetrical with respect to the middle point of the beam.
Figure 3. Forced vibrations of a beam with a central mass and symmetrical translational springs at both ends. E, = 1, E, = 100, y = 1, ci = 0.5, 6 = 0.5. -, ARMS at the mass (stable), ---~ (unstable): ---, linear response curve, I, = 0.1.
6
H. SAITO AND K. MORI
x
FigFre 4. Forced vibration modes-of a beam with a central ends. k, = 1, E, = 100, y = 1, Ci= 0.5, b = 0.5, p = 0.05.
mass and symmetrical
translational
springs
at both
Figures 4 and 5 show the forced vibration modes and the wave forms at the points a w g on the response curve for @= 0.05 shown in Figure 3. In Figure 4 the forced vibration modes are those at the values z = 6mn (m = 0, 1,2, . . .). The beam bends slightly in the first mode because the translational springs are relatively soft. The wave forms in Figure 5 are those at the concentrated mass which correspond to the temporal behavior of the displacement. Driving force vs. time curves are also indicated by the chained line in order to show the phase relation between the displacement and the driving force.
Figure 5. Wave forms of center deflection of a beam with a central mass and symmetrical translational springs at both ends. k* = 1, E, = 100, y = 1, L?= 0.5, 6 = 0.5, p = 0.05, a, 5 = 0.50; b, 5 = 0.62; C, 5 = 1.00; d, 5 = 1.25: e, 5 = 1.8O;j’, 5 = 2.41; g, r = 4.17.
4.2.
CASE
(b)
This case corresponds to k,, = CC and k,, = kt2 - 0. The backbone curves for a tip mass are shown in Figure 6 for the values I&~= 1 and y = 1, with the non-linear constant se1 as a variable parameter. It can be seen that the non-linearity of the relatively soft spring has a great influence upon the second mode. Figure 7 shows the dynamic response
A BEAM WITH
NON-LINEAR
7
CONSTRAINTS
curves for the values I&,,= 1, soI = 100, y = 1, p = 0.02 and 0.04. As can be seen in the figure, the significant effect of the non-linearity of the spring can be observed at small amplitudes for the resonances of higher modes. Figure 8 displays the forced vibration modes at the values z = 6mn (m = 0, 1,2, . . .) which are calculated at the points a w g on the response curve for p = 0.02 shown in Figure 7.
Figure 6. Backbone curves of a beam with one end spring k,, = 1, y = 1, 15= 1: soI: a, -SO; b, 0: c, 100; d, 500; e, 1000.
hinged
and carrying
a mass at the other
Figure 7. Forced vibrations of a beam with one end spring hinged and carrying LoI = 1, soI = 100, 7 = 1, a = 1, 6= 1. --, ARMS at the mass (stable), ~-~~ response curve, ji = 0434.
end.
a mass at the other end. (unstable): ---, linear
-0.04
Figure 8. Forced vibration modes of a beam with one end spring hinged and carrying I;B1= 1, so1 = 100, y = 1, (i = 1, 6 = 1, p = 0.02.
a mass at the other end.
H. SAITO AND K. MORI
8
5. CONCLUSIONS
A study has been made of the natural and forced transverse vibrations of a beam carrying a concentrated mass, the ends of which are flexibly supported on non-linear elastic constraints which have cubic non-linearities. The analysis has been carried out with the non-linear constraints modeled by rotational and translational springs. In numerical calculations, the effects of the non-linearities of the springs upon the transverse vibrations of a beam have been investigated by using two examples. The results presented are only those for the region of lower modes, but one can see that the inclusion of non-linear elastic constraints at boundaries yields fairly different beam responses as compared with those obtained in linear cases.
REFERENCES 1. P. R. PASLAY and M. E. GURTIN 1960 Journal of Applied Mechanics
27, 212-274. The vibration
response of a linear undamped system resting on a nonlinear spring. 2. A. V. SRINIVASAN1966 American
Institute
of Aeronautics
and Astronautics
Steady state response of beams supported on nonlinear springs. 3. B. PORTER and R. A. BILLETT 1965 International Journal of Mechanical
Journal 4, 1863-1864.
Science 7, 431-439. Harmonic and sub-harmonic vibration of a continuous system having non-linear constraint. 4. M. A. DOKAINISHand R. KUMAR 1971 Experimental Mechanics 11, 263-270.Experimental and theoretical analysis of the transverse vibrations of a beam having bilinear support. 5. M. A. DOKAINISHand R. KUMAR 1972 Journal of Engineering Mechanics Division, American Societ?, of Civil Engineers 98, 1381-1395. Transverse beam vibrations-nonlinear constraint. 6. R. P. GOEL 1976 Journal of Sound and Vibration 47, 9-14. Free vibrations of a beam-mass system with elastically restrained ends.
qf
Soundand Vibration (1979) 66(l), 1-8
VIBRATIONS OF
A
BEAM WITH NON-LINEAR
ELASTIC
CONSTRAINTS H. SAITO AND K. MORI Department
qf Mechanical Engineering. Tohoku Uniwrsity. Sendai, Japan (Receiced 4 December 1978)
The transverse vibrations of a beam, with both ends supported on non-linear elastic constraints, carrying a concentrated mass and subjected to a transverse periodic force are analyzed. The non-linear constraints are represented by rotational and translational springs having both linear and cubic non-linear behavior. The harmonic responses of a beam involving the third order superharmonic and the one-third order subharmonic are considered. The beam responses to the effects of non-linear elastic constraints are discussed by taking two cases.
1. INTRODUCTION In the analysis of the dynamic behavior of a beam with elastically supported ends, it is usually assumed that the beam has linear boundary conditions. Recently, however, in many practical cases it has become necessary to analyze the dynamic responses of a beam with non-linear support constraints. Paslay and Gurtin [1] and Srinivasan [2] have treated the problem of a vibrating elastic system resting on a non-linear spring by using the Fourier technique. Investigations of situations in which a beam is restrained with non-linear springs at the ends have been made by Porter and Billett [3], who have studied the longitudinal vibrations of a bar with a non-linear spring at the end, and by Dokainish and I&mar [4, 51, who have dealt with the transverse vibrations of a beam having one end clamped and the other supported on a non-linear spring and carrying a concentrated mass. If the boundary conditions are non-linear, the system shows a non-linear response although the beam behavior can still be described by the usual linear equation of motion. The analysis of this problem is unavoidably approximate as Porter and Billet have already shown in their paper, since it involves the solution of so-called non-linear boundary value problems. This paper describes a study of the steady state transverse vibrations of a beam with both ends supported on non-linear elastic constraints, carrying a concentrated mass and subjected to a transverse periodic force. Non-linear constraints are represented by rotational and translational springs which have both linear and cubic non-linear behavior. By substituting the solution for a beam vibration which involves the one-third order subharmonic and the third order superharmonic as well as the fundamental into the boundary conditions and applying the harmonic balance method, cubic algebraic equations are obtained for the unknowns and these are solved by using the Newton-Raphson technique. In numerical calculations, the following two examples are treated; (a) a beam
2
H. SAITO AND K. MORI
with a central mass and symmetrical translational springs at both ends and (b) a beam with one end spring hinged and carrying a concentrated mass at the other end.
2. GOVERNING EQUATION Figure 1 shows a beam of length 1, supported with rotational and translational springs at each end, and carrying a concentrated mass m at x = a. The beam is subjected to a concentric transverse periodic force p cos l2t at x = b. For a Bernoulli-Euler beam with a concentrated mass and a concentric transverse periodic force, which are expressed by using Dirac delta functions, 6, the equation of motion is EI(84y/dx4) + {PA + m6(x - a)} J2y/dt2 = p6(x -b) cos Qt,
Figure
1. Beam with non-linear
(1)
constraints.
where y(x, t) is the transverse displacement of the beam, EZ is the flexural rigidity, p is the mass density, A is the cross-sectional area and t is the time. As the rotational and translational springs have both linear and cubic non-linear behavior, any solution of equation (1) must satisfy the following boundary conditions at all times:
&Lk ax~
ol{g
El!!?ax~ = -k tz{g
+ %i(g>‘},
+ sties},
EI$
= -k,i(y+s,,y3)
E12
= ktz(y+ct2y3)
at
x = 0,
(2)
at
x = 1,
(3)
where k,, and k,, (i = 1,2) are the rotational and translational linear spring constants, and .sgiand sti are the non-linear spring constants.
3. ANALYSIS Equation (1) and boundary conditions (2) and (3) are written in dimensionless forms as (a4j+W) + 54 {1 + ~6 (x - ii)) a2jjarz
= ps (2 - 6) cos 7,
(4)
A BEAM WITH
NON-LINEAR
3
CONSTRAINTS
where
z = i-h,
< = (pAQ2/E1)1’41,
x = x/l,
j = y/l,
ji = p12/EI,
y = m/pAl, kti = kti13jEI,
Loi = k,,l/EI,
~7= a/l,
b = b/l,
Eti = &J2,
i = 1,2. (7)
In order to investigate the one-third order subharmonic and the third order superharmanic as well as the fundamental harmonic, a solution of the form _ y(x, r) = X,,,(X) cos fr + X,(X) cos z + X,(X) cos 30 is assumed, of equation
where X,(x)(n = $1,3) are functions of X. If equation (4) it follows that X,(X) must satisfy the equations (d4X,/dX4) - n2C4{1 +$(x-a)}
x,(X) =
(8) is to be the solution
X, = j%(%6)6T,,,
where the ST,, are Kronecker deltas. Equations Laplace transforms [6], the results being
(8)
n = 3,1,3,
(9) can be solved simultaneously
S(F,,(x)X,(O)+Cl/jn41 G,,(Z)X;(0)-[li(fi{)2]
(9) by using
F,,(x)X;(O)
-C11(~5)31Gn2(X)X~(0) -J~SYL~~(X)X~(~)U(X-~) -ClIsY311s~2(x)u(x-h)s:,,},
n = +,1,3,
(10)
where
L,,,(X) = sin &
<(x -a)
- sinh 4
5(X -ii),
N2(%) = sint(x-b)
- sinh<(x--b),
(11)
and U(X-ti) is the unit step function. The primes denote differentiation with respect to .?. Equations (10) involve the unknowns X,(O) N X:(O) and X,(Z). Substituting X = ti in equations (10) yields X,(a) = ~(F,,(~)X,(O)+C1/~51G,~(~)X:,(O)-C1l(~5)21Fn2(ii)X~(Oj -
CM&
5)“l Gn2(4 X(O)1
(12a)
in the case where a < 6, and X,(a) = ~~F,,(~)X,(O)+C1l~51G,~(~)~:,(O)-C1l(~5)2lFn2(~)X:(O) - C11t,/%)31 G2(4X’(WC1/‘531
P~2C4~T.J
(12b)
in the case where ii > 6. The remaining unknowns X,,(O) N X,“(O) can be determined from the boundary conditions (5) and (6). Substituting equation (8) into equations (5) and (6) and applying the harmonic balance method gives the following equations:
n = f,1,3,
(13b:l
4
H. SAITO AND K. MORI
where f&(X)
= t(3X;:,(X)+~X;~~(X)X;(X)+X;2(X)X;,3(X)+X;~(X)X;,3(X)},
f;“(X) = ~{~x;~,(x)+~x;3(x)+x;~3(x)x;(x)+~x;~(x)x;(x)+xj2(x)x;(x)}, f;“(X) = ~{~x;3(x)+~x~~(x)+x;2(x)x;(x)+x;~3(x)x;(x)}, J-1,3(4 = ~~~~:,3(~)+~~:,3(~)~l(~)+~:(~)~1,3(x)+~:.(x)x~,3(x)}, fi(X)= ~{~x:,3(x)+~x:(x)+x:,3(x)x1(x)+~x:(x)x3(x)+x~(x)x1(x)}, f3(X)= ~{~x:(x)+~x~(x)+x:(x)x3(x)+x:,3(x)x3(x)}.
(14)
If one differentiates equations (10) with respect to X and puts X = 1, X,(l) expressed in terms of X,(O) - X;(O) as follows: X,(l)
- X;(l)
can be
= 3{~,,(1)X,(O)+C1/~51G,~(l)x:,(O)-C1/(~~)21~n2(1)X::(O) -C1M~~)3lG,2(1)X::(O)-~~y~,2(1)X,(~)-C1/531~N2(1)~T,.},
X,(l) = ~{-~5Gn2(1)Xn(O)+Fn1(1)X:,(O)+C1/~51G,,(1)X:(O) -C1/(~5)21~"2(1)X,"(O)-(~~)2~~,2(~)X,(~)-C~/521~~2(~)~T,.~, X(l) = 3{-(~5)2~,2(1)X.(O)--\/tf5Gn2(1)X~(O)+~n1(1)X~(O) +C1/~51G,~(1)x::(o)+(~5)3~~,,(1)X,(a)+C1/~l~N,(1)~T,.},
/
X;(l) = 5{(~r)3G,1(1)Xn(O)-(~~)2Fn2(1)X:,(O)--~G,2(l)X~(O) +F,,(l)X~(0)+(~5)4yH",(1)X,(a)+~M,(1)6T,,},
(15)
where cos$@-ii)
f cosh&&-ii),
M, (3
= cost(x-6)+cosht(x-6),
M2(4 L,,(X)
= sin,,/‘%r(%-ii)
+ sinh&l(Z-a),
N,(Z)
= sin r(X-6)
+ sinhc(x-6). (16)
Hence, equations (13) form twelve coupled algebraic equations for the twelve unknowns X,(O), X”(O), X:(O) and X:(O) (n = i, 1,3), which can be solved by using the NewtonRaphson technique iteratively with the aid of an electronic computer. 4. NUMERICAL
For beam beam forced mass which
RESULTS
AND DISCUSSION
the purpose of numerical calculations, we have considered two specific cases; (a) a with a central mass and symmetrical translational springs at both ends, and (b) a with one end spring hinged and carrying a concentrated mass at the other end. The situation considered is that in which a periodic force acts at the point where the is attached. The dimensionless amplitude root mean square (ARMS) at the mass, is defined as follows, is taken as the response of the beam: ARMS
= J t{X:,,(a)+X:(ii)+X:(a)}
.
(17)
4.1. CASE (a) This case corresponds to Et1 = kt2 = k,, Et1 = Et2 = EC, i&r = k,, = 0 and a = b = 0.5. For the free vibration, one may set the dimensionless amplitude of the exciting force p
A BEAM WITH
NON-LINEAR
CONSTRAINTS
E Figure 2. Backbone curves of a beam with a central mass and symmetrical translational springs at both ends. ARMS at the mass (symmetrical mode); k, = 1, y = 1, li = 0.5: $: a, -50; b, 0; c, 100; d, 500; e, 1000. p, ---, ARMS at the end (antisymmetrical mode).
equal to zero. The relations between the natural frequency parameter 5 and ARMS, known as the backbone curve, are shown in Figure 2, for the dimensionless values & = 1 and y = 1 and with the dimensionless non-linear constant E, as a variable parameter. The second mode of vibration is antisymmetrical to the center of the beam. Therefore, in the figure, ARMS at the end of the beam is shown by the chained line for the second mode. The dynamic response curves are shown in Figure 3, for the parameter values & = 1, Et = 100, y = 1, p = 0.05 and 0.1. For comparison, the linear response curve for & = 1, E, = 0, y = 1 and p = 0.1 is also shown, by the chained line. Numbers “r-s” on the response curves indicate “the rth mode of vibration-the sth order harmonic of a resonance”; e.g., “l- 3” means the third order superharmonic resonance of the first mode. In the figure, the solid lines indicate stable branches and the dashed lines unstable ones. The determination of the stability has been carried out by adopting the argument that the region where the amplitude decreases as the periodic force increases, which is contrary to physical facts, must be unstable and that the point having the vertical tangent to the response curve provides a limiting point; no analytical investigation of the stability has been made. No resonances corresponding to the antisymmetrical even order modes can be seen in Figure 3, because all the forced vibration modes to be expected are symmetrical with respect to the middle point of the beam.
Figure 3. Forced vibrations of a beam with a central mass and symmetrical translational springs at both ends. E, = 1, E, = 100, y = 1, ci = 0.5, 6 = 0.5. -, ARMS at the mass (stable), ---~ (unstable): ---, linear response curve, I, = 0.1.
6
H. SAITO AND K. MORI
x
FigFre 4. Forced vibration modes-of a beam with a central ends. k, = 1, E, = 100, y = 1, Ci= 0.5, b = 0.5, p = 0.05.
mass and symmetrical
translational
springs
at both
Figures 4 and 5 show the forced vibration modes and the wave forms at the points a w g on the response curve for @= 0.05 shown in Figure 3. In Figure 4 the forced vibration modes are those at the values z = 6mn (m = 0, 1,2, . . .). The beam bends slightly in the first mode because the translational springs are relatively soft. The wave forms in Figure 5 are those at the concentrated mass which correspond to the temporal behavior of the displacement. Driving force vs. time curves are also indicated by the chained line in order to show the phase relation between the displacement and the driving force.
Figure 5. Wave forms of center deflection of a beam with a central mass and symmetrical translational springs at both ends. k* = 1, E, = 100, y = 1, L?= 0.5, 6 = 0.5, p = 0.05, a, 5 = 0.50; b, 5 = 0.62; C, 5 = 1.00; d, 5 = 1.25: e, 5 = 1.8O;j’, 5 = 2.41; g, r = 4.17.
4.2.
CASE
(b)
This case corresponds to k,, = CC and k,, = kt2 - 0. The backbone curves for a tip mass are shown in Figure 6 for the values I&~= 1 and y = 1, with the non-linear constant se1 as a variable parameter. It can be seen that the non-linearity of the relatively soft spring has a great influence upon the second mode. Figure 7 shows the dynamic response
A BEAM WITH
NON-LINEAR
7
CONSTRAINTS
curves for the values I&,,= 1, soI = 100, y = 1, p = 0.02 and 0.04. As can be seen in the figure, the significant effect of the non-linearity of the spring can be observed at small amplitudes for the resonances of higher modes. Figure 8 displays the forced vibration modes at the values z = 6mn (m = 0, 1,2, . . .) which are calculated at the points a w g on the response curve for p = 0.02 shown in Figure 7.
Figure 6. Backbone curves of a beam with one end spring k,, = 1, y = 1, 15= 1: soI: a, -SO; b, 0: c, 100; d, 500; e, 1000.
hinged
and carrying
a mass at the other
Figure 7. Forced vibrations of a beam with one end spring hinged and carrying LoI = 1, soI = 100, 7 = 1, a = 1, 6= 1. --, ARMS at the mass (stable), ~-~~ response curve, ji = 0434.
end.
a mass at the other end. (unstable): ---, linear
-0.04
Figure 8. Forced vibration modes of a beam with one end spring hinged and carrying I;B1= 1, so1 = 100, y = 1, (i = 1, 6 = 1, p = 0.02.
a mass at the other end.
H. SAITO AND K. MORI
8
5. CONCLUSIONS
A study has been made of the natural and forced transverse vibrations of a beam carrying a concentrated mass, the ends of which are flexibly supported on non-linear elastic constraints which have cubic non-linearities. The analysis has been carried out with the non-linear constraints modeled by rotational and translational springs. In numerical calculations, the effects of the non-linearities of the springs upon the transverse vibrations of a beam have been investigated by using two examples. The results presented are only those for the region of lower modes, but one can see that the inclusion of non-linear elastic constraints at boundaries yields fairly different beam responses as compared with those obtained in linear cases.
REFERENCES 1. P. R. PASLAY and M. E. GURTIN 1960 Journal of Applied Mechanics
27, 212-274. The vibration
response of a linear undamped system resting on a nonlinear spring. 2. A. V. SRINIVASAN1966 American
Institute
of Aeronautics
and Astronautics
Steady state response of beams supported on nonlinear springs. 3. B. PORTER and R. A. BILLETT 1965 International Journal of Mechanical
Journal 4, 1863-1864.
Science 7, 431-439. Harmonic and sub-harmonic vibration of a continuous system having non-linear constraint. 4. M. A. DOKAINISHand R. KUMAR 1971 Experimental Mechanics 11, 263-270.Experimental and theoretical analysis of the transverse vibrations of a beam having bilinear support. 5. M. A. DOKAINISHand R. KUMAR 1972 Journal of Engineering Mechanics Division, American Societ?, of Civil Engineers 98, 1381-1395. Transverse beam vibrations-nonlinear constraint. 6. R. P. GOEL 1976 Journal of Sound and Vibration 47, 9-14. Free vibrations of a beam-mass system with elastically restrained ends.