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Vibration of a pinned-pinned beam forced by harmonic motion of lateral force
Journal of Sonnd and Vibration (1992) B(2),
VIBRATION
367-372
OF A PINNED-PINNED BEAM FORCED OF LATERAL FORCE
BY HARMONIC
MOTION
S. KUKLA
Institute of Mathematics,
Technical University of Czptochowa, Czptochowa, Poland (Received 26 September
1.
ui. Deglera 35, 42-200
1991)
INTRODUCTION
The problem of beam vibration forced by a load moving with a constant velocity has been analyzed by many authors [l-4]. References [l, 21 are concerned with the problem of vibrations of the beam under the action of a concentrated force, varying sinusoidally with time, the point of application of which moves along the beam. In reference [3] the problem of vibrations of an infinite beam, subjected to a harmonically alternating moving force, is investigated and the solution is given in the form of travelling waves. Reference [4] is concerned with the analysis of the dynamic deflection and acceleration of a concrete bridge which is subjected to a moving vehicle load (the bridge is modelled as a beam). In reference [ 51 the dynamic analysis of an elastic beam traversed by a concentrated mass is presented. In this work the vibration of a pinned-pinned beam under the action of a lateral force in harmonic motion around a selected point on this beam is considered (see Figure 1). The equation of motion of such a beam is EZ(~4~/~x4)(x, ~)+~A(~*w/c%*)(x, t)=PS{x-(x0--asin
of)},
(1)
where EI is the modulus of flexural ridigity of the beam, p is the density per unit volume, A is the cross-sectional area, P is a force in harmonic motion at frequency rp and of amplitude a around a point x0, and 6( . ) denotes the Dirac delta function, It is also assumed that the condition a
w(0, 1) = (a*w/ax*)(o, I) = 0,
t) = 0.
(2)
Zero initial conditions are assumed w(x, 0) = (aw/at)(X,
2.
SOLUTION
OF
THE
0) = 0.
(3)
PROBLEM
The solution technique consists of developing a Green function. The Green function in the case considered is of the form [5]
W, 5,~ r)=
sin y
n4 sin w,( t -
sin L
T),
(4)
where CO,=(m/L)*,/m denotes the frequency of natural vibration of the pinnedpinned beam. From the properties of the Green and Dirac delta functions one finds that 367 0022460)
+ 06 %03.00/O
Q
1992 Academic
Press Limited
368
LETTERS
Figure 1. Pinned-pinned
TO THE EDITOR
beam loaded by a moving lateral force.
the solution of the problem is expressed by the formula I w(x, t) = P
G(x, x0-a sin
dr.
~417, t, 7)
(5)
s0 The following functional relationships are then used [6] : cos (u sin
97)
=
Jo(u) +2 f
J&U) cos
2kq7,
Wa)
sin (UC- l)p7.
(6b)
k-l
sin (usin q7)=2
f
Jx-,(u)
k=l
Here J,( . ) denotes the Bessel function of the first kind and order v. By substituting the function G(x, 5, t, 7) given by equation (4) into integral (5) and using the relationships (6) one obtains w(x, t) = z
where A,,= ma/L
z, -+- T,(t) sin nxx/L, n n
and m
T,(t) =COS (nKXo/L)
c
hk- I@n)Skn(f)
k-l
+ sin (mxo/L)
$
Jo(&,)[ 1 - cos
(m,Ol+ f
Jzk(&)Ckfi(~)
[w, sin (2k- I)@--(UcIt cos (w& - (1 /w,) sin (w,t)
Ck,, =
-2wn [cos(2krpt) -cos 0;
(wJ)]
(8)
for (2k - l)q# w,
for (UC- l)~,=w,
4k2q2 -
t sin (wnr)
1)qsin (o,t)]
, I
k=I
J Pa) for 2ktp#w, (9b)
for 2kq= w, 3.
ANALYSIS
The relationship between the frequency of oscillations q of the force P around the point x0 and the first frequency of the beam vibration w, can be expressed as (P=aml,
(10)
369
LE’I-I-ERS TO THE EDITOR
where a is a positive real number. Moreover, w,=n2w,.
(11)
Hence, for any natural number n, the expression rnq- o,= (am-n2)wI (see equations (9)) can be equal to zero for some natural number m except when a is an irrational number. In fact, if a is an irrational number, the equality a = n2/m will not hold. However, if a is a positive rational number, i.e., a =p/q, where p and q are natural numbers, the equation mcp- CD,,= 0 will take the form pm - qn2 = 0. Therefore, natural numbers m and n satisfying this equation always have to exist (e.g., n=p, m=pq). Thus, if a is a rational number, then due to the form of the coefficients Q,, and Q,, determined by equations (9), the occurrence of resonance is possible. In the further part of this work it is assumed that a is a positive rational number. The solution (7) can be presented in the form (12)
w(x, t) = uJ/Ax>t) + W(X, 0,
where w,,(x, t) is a periodic function (with respect to the variable 1) and w,(x, t) denotes the “resonance part” of the solution. The “resonance part” is of the form 2P w,(x, t)=---t
m sin (nax/t) 1
PAL ,,=I
wn
J,n2,,,&,)&,(0,
(13)
where R,(t) = XA,(n) sin (n7rx0/L) sin (w,t) + Xc,(n) cos (nnxO/L) cos (0J).
(14)
has been assumed here that A,=(nEN:3kEN, 2ka-n’=O}, B,={nEN:3kEN, (2k-l)a-n2=0}. H ere N denotes a set of natural numbers and xA.( . ) and xB,( . ) are the characteristic functions of the sets A, and B, , respectively. For example,
It
if nrzA,\ if n$A,j
(15)
‘.
The function w,(x, t) disappears when the expression R,(t) vanishes for every t and every n. Such a situation can occur when the conditions xA,(n) sin (n7rx0/L) = 0,
X8,(n) cos (naxO/L) = 0
(16)
are simultaneously satisfied for every neN. In Table I the values of the functions XAO(n) and XB.(n) are presented for a a positive integer. It can be seen that conditions (I 6) are satisfied when a is an odd natural number and x0= L/2. Consider a pinned-pinned beam loaded by two forces PI and P2 which perform harmonic motions around the points xl and x2, respectively, with the same frequency rp. Furthermore, it is assumed that the amplitudes of the oscillatory motions of both forces TABLE I Values of the functions XAu(n) and xR,,(n) .for a EN x,+,(n)
a odd
a even
Xe,(~~)
n odd
II even
II odd
II even
0 0
0 or 1 0 or 1
0 or 1 0
0 0 or I
370
LETTERS
TO THE
EDITOR
are such that these forces load the beam at every instant of time. The displacement of the beam w(x, t) is the sum of the displacements produced by the forces Pi and Pz, due to the linearity of the problem. If the coefficient a of the relationship (10) is a rational number then, as in the former case, the oscillatory motion of the forces can produce a resonance. The conditions for nullifying the “resonance part” when PI= P2 become
I
4x, x,4.(n) sin
xe.(n)cos
+x2)
nn(xl-XZ)= cos
2L
na(x, +x2) 2L
2L
(17) COSwxl-x2)=
2L
for every neAC If a is an odd natural number, conditions xi +x2= L.
4.
NUMERICAL
0 0 I (17) will be satisfied when
EXAMPLES
Some examples of results of numerical calculation are to be presented here. The following geometrical and material data have been used :L = 6.0 m, EI= 1.3772 x lo4 Nm2, pA = 50.0 Ns2/m2. The pinned-pinned beam is loaded by the force P= 100.0 N, being in harmonic motion of amplitude a= 2.0 m. The first frequency of natural vibration wl = 4.55 rad/s. In Figure 2 the time histories of the first three components m(r) =zoJ,,(r), where T,(r) has been determined on the basis of formula (8) by assuming a = 1, are presented. The plots shown in Figures 3-5 present the displacement of the beam due to the oscillatory motion of force, determined at the cross-section x= I.0 m. Figure 3 is for the case of resonance a = 1 and x0 = 3.5 m, whereas Figure 4, which is for a resonance case as well, has been plotted for a =2 and x0= 3.5 m. In Figure 5 the plot of the periodic beam displacements motion is presented for a = 1 and x0= 3.0 m.
025 s b-T 0.00
-0.25
V
-
V
-oflo Figure 2. Components
of displacement
of vibrating beam.
Vi
371
LETTERS TO THE EDITOR 0.06
,,,,,,,,,(,,,,,,,,,,,,,,,,
I
I,,,,,,
!,,,I
,,,,I
III
(8,
,,,,,,,,,,,,,
0 06
0.04
Figure 3. Displacement of beam in the cross-section x = I.0 m in the case of resonance; a = 1, x0= 3.5 m
006
0 04
;
0.02
d L ;
0.00 -0
-
02
v
vv”vvv”~
“V
Figure 4. Displacement of beam in the cross-section x= I.0 m in the case of resonance; a = 2, x0 = 3.5 m.
0
I
2
3
4
5
6
t
Figure 5. Displacement x-g=3.0 m.
of beam in the cross-section
x= I.0 m in the case of periodic vibration;
a = I.
5. CONCLUSIONS
An analysis of a pinned-pinned beam vibration forced by a harmonic motion of the lateral force has been presented. For such forcing the occurrence of resonance is possible. However, for a selected range of parameters characterizing the motion of the force on the beam, the resonance will not occur. Similar conclusions may be drawn from analysis of
372
LETTERS TO THE EDITOR
the solution of the problem in the case when the beam is loaded by the harmonic motions of two lateral forces of the same frequency. REFERENCES 1. J. T. KENNEY 1954 Journal of Applied Mechanics 21, 359-364. Steady-state vibrations of beam on elastic foundation for moving load. 2. P. M. MATHEWS 1958 Zeitshrift fir Angewandte Mathematik und Mechanik 38, 105-I 15. Vibrations of a beam on elastic foundation, 3. R. BOGACZ, T. KRZY~ANSKI and K. POPP 1989 Zeitschrifi ftir Angewandte Mathematik und Mechunik 69, 243-252. On the generalization of Mathews’ problem of the vibrations of a beam on elastic foundation. 4. J. HINO, T. YOSHIMURA, K. KONISHIand N. ANANTHANARAYANA1984 Journal of Sound and Vibration %,45-53. A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. 5. S. SADIKU and H. H. E. LEIPHOLZ1987 Ingeneur Archiu 57,223-242. On the dynamics of elastic system with moving concentrated masses. 6. I. S. GRADSTEIN and I. M. RYZHIK 1980 Tables of Integrals, Series and Products. New York: Academic Press.
VIBRATION
367-372
OF A PINNED-PINNED BEAM FORCED OF LATERAL FORCE
BY HARMONIC
MOTION
S. KUKLA
Institute of Mathematics,
Technical University of Czptochowa, Czptochowa, Poland (Received 26 September
1.
ui. Deglera 35, 42-200
1991)
INTRODUCTION
The problem of beam vibration forced by a load moving with a constant velocity has been analyzed by many authors [l-4]. References [l, 21 are concerned with the problem of vibrations of the beam under the action of a concentrated force, varying sinusoidally with time, the point of application of which moves along the beam. In reference [3] the problem of vibrations of an infinite beam, subjected to a harmonically alternating moving force, is investigated and the solution is given in the form of travelling waves. Reference [4] is concerned with the analysis of the dynamic deflection and acceleration of a concrete bridge which is subjected to a moving vehicle load (the bridge is modelled as a beam). In reference [ 51 the dynamic analysis of an elastic beam traversed by a concentrated mass is presented. In this work the vibration of a pinned-pinned beam under the action of a lateral force in harmonic motion around a selected point on this beam is considered (see Figure 1). The equation of motion of such a beam is EZ(~4~/~x4)(x, ~)+~A(~*w/c%*)(x, t)=PS{x-(x0--asin
of)},
(1)
where EI is the modulus of flexural ridigity of the beam, p is the density per unit volume, A is the cross-sectional area, P is a force in harmonic motion at frequency rp and of amplitude a around a point x0, and 6( . ) denotes the Dirac delta function, It is also assumed that the condition a
w(0, 1) = (a*w/ax*)(o, I) = 0,
t) = 0.
(2)
Zero initial conditions are assumed w(x, 0) = (aw/at)(X,
2.
SOLUTION
OF
THE
0) = 0.
(3)
PROBLEM
The solution technique consists of developing a Green function. The Green function in the case considered is of the form [5]
W, 5,~ r)=
sin y
n4 sin w,( t -
sin L
T),
(4)
where CO,=(m/L)*,/m denotes the frequency of natural vibration of the pinnedpinned beam. From the properties of the Green and Dirac delta functions one finds that 367 0022460)
+ 06 %03.00/O
Q
1992 Academic
Press Limited
368
LETTERS
Figure 1. Pinned-pinned
TO THE EDITOR
beam loaded by a moving lateral force.
the solution of the problem is expressed by the formula I w(x, t) = P
G(x, x0-a sin
dr.
~417, t, 7)
(5)
s0 The following functional relationships are then used [6] : cos (u sin
97)
=
Jo(u) +2 f
J&U) cos
2kq7,
Wa)
sin (UC- l)p7.
(6b)
k-l
sin (usin q7)=2
f
Jx-,(u)
k=l
Here J,( . ) denotes the Bessel function of the first kind and order v. By substituting the function G(x, 5, t, 7) given by equation (4) into integral (5) and using the relationships (6) one obtains w(x, t) = z
where A,,= ma/L
z, -+- T,(t) sin nxx/L, n n
and m
T,(t) =COS (nKXo/L)
c
hk- I@n)Skn(f)
k-l
+ sin (mxo/L)
$
Jo(&,)[ 1 - cos
(m,Ol+ f
Jzk(&)Ckfi(~)
[w, sin (2k- I)@--(UcIt cos (w& - (1 /w,) sin (w,t)
Ck,, =
-2wn [cos(2krpt) -cos 0;
(wJ)]
(8)
for (2k - l)q# w,
for (UC- l)~,=w,
4k2q2 -
t sin (wnr)
1)qsin (o,t)]
, I
k=I
J Pa) for 2ktp#w, (9b)
for 2kq= w, 3.
ANALYSIS
The relationship between the frequency of oscillations q of the force P around the point x0 and the first frequency of the beam vibration w, can be expressed as (P=aml,
(10)
369
LE’I-I-ERS TO THE EDITOR
where a is a positive real number. Moreover, w,=n2w,.
(11)
Hence, for any natural number n, the expression rnq- o,= (am-n2)wI (see equations (9)) can be equal to zero for some natural number m except when a is an irrational number. In fact, if a is an irrational number, the equality a = n2/m will not hold. However, if a is a positive rational number, i.e., a =p/q, where p and q are natural numbers, the equation mcp- CD,,= 0 will take the form pm - qn2 = 0. Therefore, natural numbers m and n satisfying this equation always have to exist (e.g., n=p, m=pq). Thus, if a is a rational number, then due to the form of the coefficients Q,, and Q,, determined by equations (9), the occurrence of resonance is possible. In the further part of this work it is assumed that a is a positive rational number. The solution (7) can be presented in the form (12)
w(x, t) = uJ/Ax>t) + W(X, 0,
where w,,(x, t) is a periodic function (with respect to the variable 1) and w,(x, t) denotes the “resonance part” of the solution. The “resonance part” is of the form 2P w,(x, t)=---t
m sin (nax/t) 1
PAL ,,=I
wn
J,n2,,,&,)&,(0,
(13)
where R,(t) = XA,(n) sin (n7rx0/L) sin (w,t) + Xc,(n) cos (nnxO/L) cos (0J).
(14)
has been assumed here that A,=(nEN:3kEN, 2ka-n’=O}, B,={nEN:3kEN, (2k-l)a-n2=0}. H ere N denotes a set of natural numbers and xA.( . ) and xB,( . ) are the characteristic functions of the sets A, and B, , respectively. For example,
It
if nrzA,\ if n$A,j
(15)
‘.
The function w,(x, t) disappears when the expression R,(t) vanishes for every t and every n. Such a situation can occur when the conditions xA,(n) sin (n7rx0/L) = 0,
X8,(n) cos (naxO/L) = 0
(16)
are simultaneously satisfied for every neN. In Table I the values of the functions XAO(n) and XB.(n) are presented for a a positive integer. It can be seen that conditions (I 6) are satisfied when a is an odd natural number and x0= L/2. Consider a pinned-pinned beam loaded by two forces PI and P2 which perform harmonic motions around the points xl and x2, respectively, with the same frequency rp. Furthermore, it is assumed that the amplitudes of the oscillatory motions of both forces TABLE I Values of the functions XAu(n) and xR,,(n) .for a EN x,+,(n)
a odd
a even
Xe,(~~)
n odd
II even
II odd
II even
0 0
0 or 1 0 or 1
0 or 1 0
0 0 or I
370
LETTERS
TO THE
EDITOR
are such that these forces load the beam at every instant of time. The displacement of the beam w(x, t) is the sum of the displacements produced by the forces Pi and Pz, due to the linearity of the problem. If the coefficient a of the relationship (10) is a rational number then, as in the former case, the oscillatory motion of the forces can produce a resonance. The conditions for nullifying the “resonance part” when PI= P2 become
I
4x, x,4.(n) sin
xe.(n)cos
+x2)
nn(xl-XZ)= cos
2L
na(x, +x2) 2L
2L
(17) COSwxl-x2)=
2L
for every neAC If a is an odd natural number, conditions xi +x2= L.
4.
NUMERICAL
0 0 I (17) will be satisfied when
EXAMPLES
Some examples of results of numerical calculation are to be presented here. The following geometrical and material data have been used :L = 6.0 m, EI= 1.3772 x lo4 Nm2, pA = 50.0 Ns2/m2. The pinned-pinned beam is loaded by the force P= 100.0 N, being in harmonic motion of amplitude a= 2.0 m. The first frequency of natural vibration wl = 4.55 rad/s. In Figure 2 the time histories of the first three components m(r) =zoJ,,(r), where T,(r) has been determined on the basis of formula (8) by assuming a = 1, are presented. The plots shown in Figures 3-5 present the displacement of the beam due to the oscillatory motion of force, determined at the cross-section x= I.0 m. Figure 3 is for the case of resonance a = 1 and x0 = 3.5 m, whereas Figure 4, which is for a resonance case as well, has been plotted for a =2 and x0= 3.5 m. In Figure 5 the plot of the periodic beam displacements motion is presented for a = 1 and x0= 3.0 m.
025 s b-T 0.00
-0.25
V
-
V
-oflo Figure 2. Components
of displacement
of vibrating beam.
Vi
371
LETTERS TO THE EDITOR 0.06
,,,,,,,,,(,,,,,,,,,,,,,,,,
I
I,,,,,,
!,,,I
,,,,I
III
(8,
,,,,,,,,,,,,,
0 06
0.04
Figure 3. Displacement of beam in the cross-section x = I.0 m in the case of resonance; a = 1, x0= 3.5 m
006
0 04
;
0.02
d L ;
0.00 -0
-
02
v
vv”vvv”~
“V
Figure 4. Displacement of beam in the cross-section x= I.0 m in the case of resonance; a = 2, x0 = 3.5 m.
0
I
2
3
4
5
6
t
Figure 5. Displacement x-g=3.0 m.
of beam in the cross-section
x= I.0 m in the case of periodic vibration;
a = I.
5. CONCLUSIONS
An analysis of a pinned-pinned beam vibration forced by a harmonic motion of the lateral force has been presented. For such forcing the occurrence of resonance is possible. However, for a selected range of parameters characterizing the motion of the force on the beam, the resonance will not occur. Similar conclusions may be drawn from analysis of
372
LETTERS TO THE EDITOR
the solution of the problem in the case when the beam is loaded by the harmonic motions of two lateral forces of the same frequency. REFERENCES 1. J. T. KENNEY 1954 Journal of Applied Mechanics 21, 359-364. Steady-state vibrations of beam on elastic foundation for moving load. 2. P. M. MATHEWS 1958 Zeitshrift fir Angewandte Mathematik und Mechanik 38, 105-I 15. Vibrations of a beam on elastic foundation, 3. R. BOGACZ, T. KRZY~ANSKI and K. POPP 1989 Zeitschrifi ftir Angewandte Mathematik und Mechunik 69, 243-252. On the generalization of Mathews’ problem of the vibrations of a beam on elastic foundation. 4. J. HINO, T. YOSHIMURA, K. KONISHIand N. ANANTHANARAYANA1984 Journal of Sound and Vibration %,45-53. A finite element method prediction of the vibration of a bridge subjected to a moving vehicle load. 5. S. SADIKU and H. H. E. LEIPHOLZ1987 Ingeneur Archiu 57,223-242. On the dynamics of elastic system with moving concentrated masses. 6. I. S. GRADSTEIN and I. M. RYZHIK 1980 Tables of Integrals, Series and Products. New York: Academic Press.