- Email: [email protected]
Natural vibrations of a cantilever beam with support damping
Journal of Sound and Vibration (1988) 120(3), 511-515
NATURAL VIBRATIONS OF A CANTILEVER BEAM WITH SUPPORT DAMPING H. SAITO ANt) H. YAGUCHI
Department of Mechanical Engineering, Tohoku Gakuin Unirersi O" Tagajo, Aliyagi, Japan (Received 2 October 1986, and in revised form 2 April 1987)
The ellect of a support insert on the overall damping capacity of a cantilever beam is investigated under the condition of free vibrations. In the analysis, the support insert is considered to consist of massless line springs of the Winkler type having structural damping. The frequency equation is derived and the ettect of the length and the stiffness of a support insert on the natural frequency and logarithmic decrement of the system is discussed. I. INTRODUCTION Much of the recent research into the reduction of vibration in lightweight, high speed structural members has been directed toward the use of damping treatment applied to the surface of a vibrating member. Though these damping treatments are effective in vibration control, it is not always possible to use an external damping treatment. In such cases one o f the known methods of vibration control is to insert a damping material into the support part. MacBain and Genin [ 1] analyzed the steady state response of a materially damped beam with viscoelastic support inserts, which were represented by translational and rotational damped springs at a point of the support. Vibrations of beams on such damped flexible end supports have also been investigated by Miller [2], Piunkett [3], Genin and Sweet [4], Fu and Mentel [5], and Mead and Wilby [6]. in practical problems, however, when viscoelastic materials with damping properties are inserted into the support, one must attend to the fact that the beam is clamped in a viscoelastic support of finite length. Damping may be due to support damping attributable to viscoelastic inserts. It would be of interest to study the effects of viscoelastic properties and dimensions of the inserts. This p a p e r presents the results of studying the effect of a viscoelastic support insert on the natural vibrations of a cantilever beam. The insert is assumed to be the base consisting of closely spaced independent linear springs of the Winkler type having structural damping. Specifically, the effects of the insert stiffness, loss factor, and the insert length to the beam length ratio on the damping capacity of a flexibly supported cantilever beam under the condition of natural vibration are investigated in detail.
2. ANALYSIS Figure 1 shows a cantilever beam of thickness h and breadth b with support inserts of thickness H and breadth b having material damping. The support and the span lengths of the beam are it and 12. One assumes that in vibration the gap does not yield at the contact surfaces between inserts, beam and bases of the support. If one uses rubber or elastomeric materials which exhibit structural damping as the support insert, the longitudinal stress-strain relationship for the insert material can be written by using a Voigt 511 0022-460X/88/030511 + 05 $03.00/0 ~ 1988 Academic Press Limited
512
H. S A I T O
AND
I-I. Y A G U C I I I
L
o1" 0
I, I.
9 Yl
I
v/"/////~
Figure 1. C a n t i l e v e r b e a m with viscoelastic s u p p o r t inserts.
model in the form
cr = {E, +( Eo/~O~) a/Ot)e,
(!)
where E, and Eo are the Young's and loss modulus, oR is the actual radian frequency of the system, and t is time. The dynamic foundation modulus ka of the insert, then, becomes ka = k{I + (3,/toR) a/at},
(2)
where k and 3' are the Winkler's foundation modulus and loss factor of the insert, and k =2E, b/H,
(3)
3' = Eo/E,.
The damping of the beam material is neglected as small compared with the insert material damping. The various quantities for the support and the span will be distinguished from each other by the subscript i ( = 1,2). One may take the co-ordinate axes as shown in Figure I. The equations of motion governing the free vibrations of a uniform thin beam, which are based on the Bernoulli-Euler theory, are given by
E! a%v,/ax~+pA a2w,/at~+ T,k{l +(3,/~oR)a/at}w, = 0,
i = 1,2,
(4)
where to, is the deflection in the ),, direction, E is the Young's modulus of the material of the beam, I is the secondary moment o f inertia of the cross-section, 19 is the density, A is the cross-sectional area, and T, = 1,
T2 = 0.
(5)
Substitution of i = 1 and 2 into equation (4) gives the equations of motion of the support and the span parts of the beam. The boundary conditions of the beam can be written as follows: a2wt/Ox~=0,
a~w,i/Ox~=0
at
xt=0,
(6)
O:w2/Ox~ = O,
a3w~/ax~= 0
at
x2 = 12,
(7)
wt = w , ,
a2w,/ax~ = ah,.2/ax~,
aw,lOx, = a w , l a x , , a3w,/ax~=a3w2/ax~
at
x,=lt,x2=0.
(8)
Using the standard method of separation of variables, one assumes w, = W,(x,) ff~',
to = o~R+jcol,
(9)
where j = 4-ZT and ~ is the complex radian frequency of the system, of which real part ~aR represents the natural radian frequency and imaginary part w~ the decay constant. By substituting equations (9) into equations (4), W~(x;) is found as g~(x,) = C . sin Vx, + C.. cos r/x, + C,~ sinh r/x, + C~ cosh r/x,,
(10)
513
CANTILEVER BEAM WITH SUPPORT D A M P I N G
a~here
04=~'-T,k*/EI,
~=pAoJ:/El,
k*=k(l+jT~/oJR).
and
(11,12)
Substitution of equation (10) into equations (6), (7) and (8) yields eight homogeneous algebraic equations for C , - C,4(i = 1, 2). A non-trivial solution for them is possible only ,vhen the determinant of the coefficient matrix is equal to zero. Introducing nondimensional quantities
[i = 1,/12,
I(* = k*l~/El,
(~R,~b,)=(toR, w,)/(1/l~)~/-E-~-pa,
~5 = t o / ( l / i g ) ~ ,
( = (odR+jad,) '/z ,
fi = ( ~ - - R*) I/4,
(13)
gives the frequency equation [d,,,.,[ = 0,
m, n = 1 , 2 , . . . , 6 ,
(14)
where dta =sin # +sinh ~,
d|.2 = c o s # +cosh #,
dr,4 = - 1, d2.t = cos ~ +cosh ~,
dl.5 = 0,
dl.6 = - 1,
dLz = -sin ~ +sinh ~,
d2.,= -((/,~) r,,
d2, 4 = O,
d~.~ = - s i n fi + sinh "~,
ds.2 = 0,
d5. 6 = c o g h ~
d2.6= 0,
d,.~= -(r d5.3 = - s i n 6
d6. ! --- 0,
d~.3 = ( ( / ~ ) 3r3,
d4.6= 0, d d : = - c o s ~,
d6. 2 = 0,
d6.s = cosh ~
d3. 3 = 0,
d~.6=-r
d4.2 = sin ~ + s i n h fl,
d~:=o, d5.1 = 0,
d2.3 = - ( ~ / f / ) lt,
d3.2 = - c o s f/+ cosh "~,
d~.~=0, d4., = - c o s q + cosh n,
dl.3 = 0 ,
(/6. 3 = - c o s
(,
d6.6= sinh (.
d5.5 = sinh d6.4 = sin s/, (15)
Substitution of equation (12) into K* in equations (13) yields K* = K(1 +jTw/WR),
(16)
g = 2 4 ( E , / E ) (l~/Hh3).
(17)
where
If one keeps the ratios 12/h and h / H constant, the value of g depends only on the ratio E, to E. The roots a3a and 6, of equation (14) give the natural radian frequencies and the decay constants. The logarithmic decrement $ of the system is then determined from the relation 8 = 2,n'wfftoR.
(18)
3. NUMERICAL RESULTS Natural radian frequencies and logarithmic decrements versus support insert length ratios have been calculated. Results are presented in Figures 2-5 for the value y = 0 . 1 . Figures 2 and 3 show the values of the natural radian frequency t,dR versus the support
514
H. S A I T O A N t ) 35
.
.
.
.
I
'
3O
J
'
II. Y A G U C i l l
'
I
'
'
'
'
I
'
'
'
'
K = 106
25 2C 104
1-5 10
103
0.5 0
0 05
010
015
0 20
Figure 2. Natural radian frequencies d R vs. support length ratios ]t for the first mode.
2,5
'
'
'
I
i
,
J
,
I
'
'
'
*
I
'
'
'
'
9
,
i
i
2O
15
10
5
0O'5
010
015
0 20
7, Figure 3. Natural radian frequencies o~n t's. support length ratios ]t for the second mode.
04
'l
'
'
'
i
'
'
'
'
i
'
I
.
'
'
'
I
.
.
.
.
I
.
.
.
.
.10 2
g ,o" f,o"
.
0
.
.
.
I
0 05
.
.
.
.
010
.
.
.
015
0 20
Figure 4. Logarithmic decrements 8 t's. support length ratios [l for the first mode.
CANTILEVER . . . .
BEAM I
. . . .
~,VITII S U P P O R T I
. . . .
515
DAMPING I
. . . .
10 2
03
,~ O2
o,I/+ 0
1 0 05
010
015
0 20
r, Figure 5. Logarithmic decrements <5rs. support length ratios i~ for the second mode. length ratio [~( = 1~/12) for the first and the second modes, w i t h / ~ taken as the parameter. For small values o f loss factor, "y< 0.2, the values o f values o f ~R are not affected by the values o f loss factor and are the same as the ones in the case o f no damping. It is seen from the figures that the natural frequency increases with the increase o f ]'~ and /~ and the increment becomes large for large values of/~'. Figures 4 and 5 show the values o f the logarithmic decrement <5 versus the support length ratio [~ for the first and the second modes. The logarithmic decrement b e c o m e s m a x i m u m at a particular value o f it and decreases rapidly with the increase of/'~ as the values o f / ~ b e c o m e large.
4. CONCLUSIONS The d a m p i n g characteristics o f a cantilever beam with viscoelastic support inserts have been investigated. Numerical results show that the logarithmic decrement b e c o m e s m a x i m u m at a particular value o f the length o f the insert. In order to reduce the vibration levels, the stiffness and length o f a support insert can be selected according to the results obtained.
REFERENCES 1. J. C. MACBAIN and J. GENIN 1975 hlternational Journal of Mechanical Science 17, 255-265. Energy dissipation of a vibrating Timoshenko beam considering support and material damping. 2. D. F. MILLER 1953 Transactions of the American Society of Mechanical Engineers Journal of applied Mechanics 167-172. Forced lateral vibration of beams on damped flexible end supports. 3. R. PLUNKETT 1972 The Shock and Vibration Bulletin 42, 57-63. Optimum damping distribution for structural vibration. 4. J. GENIN and J. SWEET 1971 Zeitschriftfiir Angewandte Mathematik und Mechanik 51, 63-65. Response of a vibrating beam on viscoelastic supports. 5. C. C. Fu and T. J. MENTEL 1960 Wright Air Decelopment Division ( Wright-Patterson Air Force Base), WADD Technical Report 60-60. Steady-state response of beams with translational and rotational damping motions at the supports. 6. D. J. MEAD and J. F. WILBY 1964 Institute of Sound and Vibration Research, Unit'ersity of Southampton, Report No. 121. The damping of beam vibration by rotational damping at the supports.
NATURAL VIBRATIONS OF A CANTILEVER BEAM WITH SUPPORT DAMPING H. SAITO ANt) H. YAGUCHI
Department of Mechanical Engineering, Tohoku Gakuin Unirersi O" Tagajo, Aliyagi, Japan (Received 2 October 1986, and in revised form 2 April 1987)
The ellect of a support insert on the overall damping capacity of a cantilever beam is investigated under the condition of free vibrations. In the analysis, the support insert is considered to consist of massless line springs of the Winkler type having structural damping. The frequency equation is derived and the ettect of the length and the stiffness of a support insert on the natural frequency and logarithmic decrement of the system is discussed. I. INTRODUCTION Much of the recent research into the reduction of vibration in lightweight, high speed structural members has been directed toward the use of damping treatment applied to the surface of a vibrating member. Though these damping treatments are effective in vibration control, it is not always possible to use an external damping treatment. In such cases one o f the known methods of vibration control is to insert a damping material into the support part. MacBain and Genin [ 1] analyzed the steady state response of a materially damped beam with viscoelastic support inserts, which were represented by translational and rotational damped springs at a point of the support. Vibrations of beams on such damped flexible end supports have also been investigated by Miller [2], Piunkett [3], Genin and Sweet [4], Fu and Mentel [5], and Mead and Wilby [6]. in practical problems, however, when viscoelastic materials with damping properties are inserted into the support, one must attend to the fact that the beam is clamped in a viscoelastic support of finite length. Damping may be due to support damping attributable to viscoelastic inserts. It would be of interest to study the effects of viscoelastic properties and dimensions of the inserts. This p a p e r presents the results of studying the effect of a viscoelastic support insert on the natural vibrations of a cantilever beam. The insert is assumed to be the base consisting of closely spaced independent linear springs of the Winkler type having structural damping. Specifically, the effects of the insert stiffness, loss factor, and the insert length to the beam length ratio on the damping capacity of a flexibly supported cantilever beam under the condition of natural vibration are investigated in detail.
2. ANALYSIS Figure 1 shows a cantilever beam of thickness h and breadth b with support inserts of thickness H and breadth b having material damping. The support and the span lengths of the beam are it and 12. One assumes that in vibration the gap does not yield at the contact surfaces between inserts, beam and bases of the support. If one uses rubber or elastomeric materials which exhibit structural damping as the support insert, the longitudinal stress-strain relationship for the insert material can be written by using a Voigt 511 0022-460X/88/030511 + 05 $03.00/0 ~ 1988 Academic Press Limited
512
H. S A I T O
AND
I-I. Y A G U C I I I
L
o1" 0
I, I.
9 Yl
I
v/"/////~
Figure 1. C a n t i l e v e r b e a m with viscoelastic s u p p o r t inserts.
model in the form
cr = {E, +( Eo/~O~) a/Ot)e,
(!)
where E, and Eo are the Young's and loss modulus, oR is the actual radian frequency of the system, and t is time. The dynamic foundation modulus ka of the insert, then, becomes ka = k{I + (3,/toR) a/at},
(2)
where k and 3' are the Winkler's foundation modulus and loss factor of the insert, and k =2E, b/H,
(3)
3' = Eo/E,.
The damping of the beam material is neglected as small compared with the insert material damping. The various quantities for the support and the span will be distinguished from each other by the subscript i ( = 1,2). One may take the co-ordinate axes as shown in Figure I. The equations of motion governing the free vibrations of a uniform thin beam, which are based on the Bernoulli-Euler theory, are given by
E! a%v,/ax~+pA a2w,/at~+ T,k{l +(3,/~oR)a/at}w, = 0,
i = 1,2,
(4)
where to, is the deflection in the ),, direction, E is the Young's modulus of the material of the beam, I is the secondary moment o f inertia of the cross-section, 19 is the density, A is the cross-sectional area, and T, = 1,
T2 = 0.
(5)
Substitution of i = 1 and 2 into equation (4) gives the equations of motion of the support and the span parts of the beam. The boundary conditions of the beam can be written as follows: a2wt/Ox~=0,
a~w,i/Ox~=0
at
xt=0,
(6)
O:w2/Ox~ = O,
a3w~/ax~= 0
at
x2 = 12,
(7)
wt = w , ,
a2w,/ax~ = ah,.2/ax~,
aw,lOx, = a w , l a x , , a3w,/ax~=a3w2/ax~
at
x,=lt,x2=0.
(8)
Using the standard method of separation of variables, one assumes w, = W,(x,) ff~',
to = o~R+jcol,
(9)
where j = 4-ZT and ~ is the complex radian frequency of the system, of which real part ~aR represents the natural radian frequency and imaginary part w~ the decay constant. By substituting equations (9) into equations (4), W~(x;) is found as g~(x,) = C . sin Vx, + C.. cos r/x, + C,~ sinh r/x, + C~ cosh r/x,,
(10)
513
CANTILEVER BEAM WITH SUPPORT D A M P I N G
a~here
04=~'-T,k*/EI,
~=pAoJ:/El,
k*=k(l+jT~/oJR).
and
(11,12)
Substitution of equation (10) into equations (6), (7) and (8) yields eight homogeneous algebraic equations for C , - C,4(i = 1, 2). A non-trivial solution for them is possible only ,vhen the determinant of the coefficient matrix is equal to zero. Introducing nondimensional quantities
[i = 1,/12,
I(* = k*l~/El,
(~R,~b,)=(toR, w,)/(1/l~)~/-E-~-pa,
~5 = t o / ( l / i g ) ~ ,
( = (odR+jad,) '/z ,
fi = ( ~ - - R*) I/4,
(13)
gives the frequency equation [d,,,.,[ = 0,
m, n = 1 , 2 , . . . , 6 ,
(14)
where dta =sin # +sinh ~,
d|.2 = c o s # +cosh #,
dr,4 = - 1, d2.t = cos ~ +cosh ~,
dl.5 = 0,
dl.6 = - 1,
dLz = -sin ~ +sinh ~,
d2.,= -((/,~) r,,
d2, 4 = O,
d~.~ = - s i n fi + sinh "~,
ds.2 = 0,
d5. 6 = c o g h ~
d2.6= 0,
d,.~= -(r d5.3 = - s i n 6
d6. ! --- 0,
d~.3 = ( ( / ~ ) 3r3,
d4.6= 0, d d : = - c o s ~,
d6. 2 = 0,
d6.s = cosh ~
d3. 3 = 0,
d~.6=-r
d4.2 = sin ~ + s i n h fl,
d~:=o, d5.1 = 0,
d2.3 = - ( ~ / f / ) lt,
d3.2 = - c o s f/+ cosh "~,
d~.~=0, d4., = - c o s q + cosh n,
dl.3 = 0 ,
(/6. 3 = - c o s
(,
d6.6= sinh (.
d5.5 = sinh d6.4 = sin s/, (15)
Substitution of equation (12) into K* in equations (13) yields K* = K(1 +jTw/WR),
(16)
g = 2 4 ( E , / E ) (l~/Hh3).
(17)
where
If one keeps the ratios 12/h and h / H constant, the value of g depends only on the ratio E, to E. The roots a3a and 6, of equation (14) give the natural radian frequencies and the decay constants. The logarithmic decrement $ of the system is then determined from the relation 8 = 2,n'wfftoR.
(18)
3. NUMERICAL RESULTS Natural radian frequencies and logarithmic decrements versus support insert length ratios have been calculated. Results are presented in Figures 2-5 for the value y = 0 . 1 . Figures 2 and 3 show the values of the natural radian frequency t,dR versus the support
514
H. S A I T O A N t ) 35
.
.
.
.
I
'
3O
J
'
II. Y A G U C i l l
'
I
'
'
'
'
I
'
'
'
'
K = 106
25 2C 104
1-5 10
103
0.5 0
0 05
010
015
0 20
Figure 2. Natural radian frequencies d R vs. support length ratios ]t for the first mode.
2,5
'
'
'
I
i
,
J
,
I
'
'
'
*
I
'
'
'
'
9
,
i
i
2O
15
10
5
0O'5
010
015
0 20
7, Figure 3. Natural radian frequencies o~n t's. support length ratios ]t for the second mode.
04
'l
'
'
'
i
'
'
'
'
i
'
I
.
'
'
'
I
.
.
.
.
I
.
.
.
.
.10 2
g ,o" f,o"
.
0
.
.
.
I
0 05
.
.
.
.
010
.
.
.
015
0 20
Figure 4. Logarithmic decrements 8 t's. support length ratios [l for the first mode.
CANTILEVER . . . .
BEAM I
. . . .
~,VITII S U P P O R T I
. . . .
515
DAMPING I
. . . .
10 2
03
,~ O2
o,I/+ 0
1 0 05
010
015
0 20
r, Figure 5. Logarithmic decrements <5rs. support length ratios i~ for the second mode. length ratio [~( = 1~/12) for the first and the second modes, w i t h / ~ taken as the parameter. For small values o f loss factor, "y< 0.2, the values o f values o f ~R are not affected by the values o f loss factor and are the same as the ones in the case o f no damping. It is seen from the figures that the natural frequency increases with the increase o f ]'~ and /~ and the increment becomes large for large values of/~'. Figures 4 and 5 show the values o f the logarithmic decrement <5 versus the support length ratio [~ for the first and the second modes. The logarithmic decrement b e c o m e s m a x i m u m at a particular value o f it and decreases rapidly with the increase of/'~ as the values o f / ~ b e c o m e large.
4. CONCLUSIONS The d a m p i n g characteristics o f a cantilever beam with viscoelastic support inserts have been investigated. Numerical results show that the logarithmic decrement b e c o m e s m a x i m u m at a particular value o f the length o f the insert. In order to reduce the vibration levels, the stiffness and length o f a support insert can be selected according to the results obtained.
REFERENCES 1. J. C. MACBAIN and J. GENIN 1975 hlternational Journal of Mechanical Science 17, 255-265. Energy dissipation of a vibrating Timoshenko beam considering support and material damping. 2. D. F. MILLER 1953 Transactions of the American Society of Mechanical Engineers Journal of applied Mechanics 167-172. Forced lateral vibration of beams on damped flexible end supports. 3. R. PLUNKETT 1972 The Shock and Vibration Bulletin 42, 57-63. Optimum damping distribution for structural vibration. 4. J. GENIN and J. SWEET 1971 Zeitschriftfiir Angewandte Mathematik und Mechanik 51, 63-65. Response of a vibrating beam on viscoelastic supports. 5. C. C. Fu and T. J. MENTEL 1960 Wright Air Decelopment Division ( Wright-Patterson Air Force Base), WADD Technical Report 60-60. Steady-state response of beams with translational and rotational damping motions at the supports. 6. D. J. MEAD and J. F. WILBY 1964 Institute of Sound and Vibration Research, Unit'ersity of Southampton, Report No. 121. The damping of beam vibration by rotational damping at the supports.