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THE EFFECT OF A MOVING MASS AND OTHER PARAMETERS ON THE DYNAMIC RESPONSE OF A SIMPLY SUPPORTED BEAM
Journal of Sound and Vibration (1996) 191(3), 357–362
THE EFFECT OF A MOVING MASS AND OTHER PARAMETERS ON THE DYNAMIC RESPONSE OF A SIMPLY SUPPORTED BEAM G. M, D. S A. N. K National Technical University of Athens, Athens, Greece (Received 30 November 1994, and in final form 9 May 1995) This paper deals with the linear dynamic response of a simply supported uniform beam under a moving load of constant magnitude and velocity by including the effect of its mass. Using a series solution for the dynamic deflection in terms of normal modes the individual and coupling effect of the mass of the moving load, of its velocity and of other parameters is fully assessed. A variety of numerical results allows us to draw important conclustions for structural design purposes. 7 1996 Academic Press Limited
1. INTRODUCTION
A lot of work has been reported during the past 100 years dealing with the dynamic response of railway bridges, and later on highway bridges, under the influence of moving loads. Extensive references to the literature on the subject can be found in the book by Fry´ba [1]. Two early interesting contributions in this area are due to Stokes [2] and Zimmermann [3]. In 1950 Krylov [4, 5], gave a complete solution of the problem of the dynamic behaviour of a prismatic bar acted upon by a load of constant magnitude, moving with a constant velocity. In 1922 Timoshenko solved the same problem but for a moving harmonic pulsating force. A pioneering work on the subject was presented in 1934 by Inglis [7], in which numerous parameters were taken into account. In 1951 Hillerborg [8] gave an analytical solution of the previous problem by means of Fourier’s method. Despite the availability of high speed computers, most of the methods used today for analyzing bridge vibration problems are essentially based on the early techniques of Inglis or Hillerborg. Relevant studies are those of Saller [9], Jeffcott [10], Steuding [11] and Sophianopoulos and Kounadis [12]. The problem of the forced motion of a beam, subjected to a moving load, is associated with serious difficulties when the effect of the mass of the load is accounted for. In the present work, by using as a first approximation the solution of the corresponding problem without the effect of the mass, a closed form solution is successfully derived.
2. MATHEMATICAL FORMULATION
We consider the simply supported beam shown in Figure 1, of length l, mass per unit length m and flexural rigidity EI, made from a uniform homogeneous and isotropic 357 0022–460X/96/130357 + 06 $18.00/0
7 1996 Academic Press Limited
. .
358
Figure 1. A moving load of mass M, with constant velocity v.
material. The beam is subjected to a load P of mass M, moving with a constant velocity v. The force acting on the beam is equal to P = Mg − Mw¨ (a)
(1)
with a = vt, and where g is the gravitational acceleration. Thus, the equation of motion can be written as EIw00(x, t) + mw¨ (x, t) = P d(x − a) = Mg d(x − a) − Mw¨ (a, t) d(x − a),
(2)
with 0 E t E l/v, and where the prime denotes differentiation with respect to x, dot differentiation with respect to t and d(· · ·), is the Dirac delta function. A series solution of equation (2) is of the form w(x, t) = s Xn (x)Tn (t),
(3)
n
where Xn = sin (npx/l) is the shape function of a freely vibrating beam and Tn (t) is the modal amplitude to be found. Due to expression (3) equation (2) can be written as
$
%
n (t) = Mg d(x − a) − M s Xn (a)Tn (t) d(x − a). EI s X2 n (x)Tn (t) + m s Xn (x)T n
n
n
(4)
Since for a free vibration one can assume that 2 EI s X2 n (x)Tn (t) − m s vn Xn (x)Tn (t) = 0, n
n
Figure 2. The ratio of dynamic to static middle-span deflection, as the load moves along the length of the beam (when the effect of the mass load is neglected).
359
Figure 3. The ratio of dynamic to static middle-span deflection versus load position: (a) when the moving load mass is included (———); (b) when it is neglected (............).
(with vn2 = (n 4p 4EI/ml 4 )) equation (4) becomes
$
%
m s vn2 Xn (x)Tn (t) + m s Xn (x)Tn (t) = Mg d(x − a) − M s Xn (a)Tn (t) d(x − a). n
n
n
Multiplying this expression by Xk and integrating from 0 to l gives a
Tn (t) + vn2 Tn (t) = (2Mg/ml)Xn (a) + (2M/ml)Xn (a) s Xk (a)Tk (t), k=1
or a
Tn (t)+vn2 Tn (t)=(2Mg/ml) sin (npvt/l)−(2M/ml) sin (npvt/l) s sin (kpvt/l)Tk (t).
(5)
k=1
Neglecting the second term on the right side of the last equation one can obtain the solution [13] Tn (t) = (2Mg/ml){1/(vn2 − Vn2 )}{sin Vn t − (Vn /vn ) sin vn t},
(6)
where Vn = npv/l. A first approximation to the solution of equation (5) can be obtained as follows. Introducing the value of Tn (t) given in equation (6) into the right side of equation (5) gives a
1 2 v − Vn2 k k=1
Tn (t) + vn2 Tn (t) = (2Mg/ml) sin Vn t − (2M 2g/m 2l 2 ) sin Vn t s × (−Vk2 sin Vk t + Vk vk sin vk t) sin Vk t.
(7)
Then one can obtain Tn (t) =
2Mg mlvn
g$ t
0
a M 1 sin Vn *t − sin Vn *t s 2 (−Vk2 sin Vk *t ml vk − Vk2 k=1
%
+Vk vk sin vk *t ) sin vk *t sin vn (t − *t ) dt*.
(8)
Using the dimensionless parameters x¯ = x/l,
v¯ = (2/p(zml 2/EI)v,
t = t/T1,
T1 = (2/p)zml 4/EI ,
(9)
. .
360 one can write Vn t = npv¯t,
vn t = 2n 2pt,
Vn /vn = v¯ /2n,
Vk /vn = kv¯ /2n 2.
Integrating equation (8) and using equations (3) and (9) yields w¯n (x¯ , t) = wn (x¯ , t)/(2Mgl 3/p 4EI) =
6
0
1
M v¯ 2ml sin (npx¯ ) An [sin (npv¯t) − sin (2n 2pt)] − s Bk 2ml 2n M k
−s k
+s k
−s k
+s k
+s k
−s k
1 Bk [sin (2k + n)pv¯t − a1k sin 2n 2pt] 2 2 1 − a1k 1 Bk [sin (2k − n)pv¯t − a2k sin 2n 2pt] 2 1 − a2k 2 1 Bk [sin b2k pt − g2k sin 2n 2pt] 2 2 1 − g2k 1 Bk [sin b4k pt − g4k sin 2n 2pt] 2 1 − g4k 2 1 Bk [sin b1k pt − g1k sin 2n 2pt] 2 1 − g1k 2
7
1 Bk [sin b3k pt − g3k sin 2n 2pt] , 2 1 − g1k 2
with An =
1 , n (1 − v¯ 2/4n 2 ) 4
Bk =
(v¯ /2k)2 , 1 + (v¯ /2k)2
a1k =
kv¯ v¯ + , n 2 2n
b 1k = kv¯ + 2k 2 + nv¯ ,
b2k = kv¯ − 2k 2 + nv¯ ,
b3k = kv¯ + 2k 2 − nv¯ ,
b4k = kv¯ − 2k 2 − nv¯ ,
a2k =
kv¯ v¯ − , n 2 2n
Figure 4. The influence of the ratio M/ml (for various velocities) on the ratio w¯M /w¯p (%) of the dynamic middle-span deflection, where w¯M includes the load mass M while w¯p does not.
361
T 1 w¯M /w¯p (%) for M/ml = 0·25 and 0·50 and various velocities M/ml
v¯ = 0·10
v¯ = 0·16
v¯ = 0·22
v¯ = 0·25
0·25 0·50
3·1 8·3
4 12·5
5 15
6·2 22·5
g1k =
kv¯ k2 v¯ , 2+ 2+ 2n 2n n
g2k =
kv¯ k2 v¯ , 2− 2+ 2n 2n n
g3k =
kv¯ k2 v¯ , 2+ 2− n 2n 2n
g4k =
kv¯ k2 v¯ . 2− 2− n 2n 2n
Note that 2Mgl 3/p 4EI can be taken to be approximately equal to the static deflection of the beam at x = l/2 due to the middle-span load Mg.
3. NUMERICAL RESULTS AND DISCUSSION
In this section, numerical results in graphical and tabular form are presented. The individual and coupling effects, on the dynamic response of various parameters such as load mass and velocity of the moving load, are discussed in detail. From Figure 2 one can see the classical case of various ratios of dynamic to static middlespan deflection as the load moves with various velocities along the length of the beam. In this case the effect of the load mass on the dynamic response of the beam is neglected. From this plot one sees that the maximum dimensionless deflection w¯ = 1·54 occurs for the dimensionless velocity v¯ = 0·75 when the load is located at x/l = 0·55, while for v¯ = 0·50 the deflection becomes w¯ = 1·30 (occurring when the load is located at x/l = 0·40). On can now consider how the above results are affected when the effect of the load mass is included. For the case in which M/ml = 0·30 the corresponding variations of the ratio dynamic to static middlespan deflection (as the load moves with the above velocities) are shown in Figure 3. Clearly, the maximum deflection w¯ = 1·72 occurs for v¯ = 0·75 when the load is located at x/l = 0·55. From Figure 4, one can see the influence of the ratio M/ml on the various values of the ratios of the dynamic middle-span deflection (when the effect of the load mass is included) to the dynamic middle-span deflection w¯M /w¯p (when the effect of the load mass is ignored) for various velocities. As the ratio M/ml increases, depending on the value of the velocity, the ratio w¯M /w¯p increases appreciably. This effect is more pronounced as the velocity increases. From Table 1, one can see the variation of the ratio w¯M /w¯p for two characteristic values of M/ml ( = 0·25, and 0·50) and four different values of velocities v¯ (0·10, 0·16, 0·22, 0·25).
REFERENCES 1. L. F´ 1972 Vibrations of Solids and Structures under Moving Loads. Groningen; Nordhoff International Publication Co. 2. G. G. S 1849 Transactions of the Cambridge Philosophical Society, Part 5, 707–735. Discussion of a differential equation relating to the breaking of railway bridges. 3. H. Z 1896 Centralblatt der Bauverwaltung 16(23), 249–251; (23A), 257–260; (24), 264–266; (26), 288. Die Schwingungen eines Tra¨gers mit bewegter Last. 4. A. N. K´ 1905 Mathematical Collection of Papers of the Academy of Sciences 61, St. Petersburg.
362
. .
5. A. N. K´ 1905 Mathematishce Annalen 61, 211. Uber die erzwungenen Schwingungen von gleichfo¨rmigen elastischen Sta¨ben. 6. S. P. T 1922 Philosophical Magazine Series 6, 43 1018. On the forced vibration of bridges. 7. C. E. I 1934 A Mathematical Treatise on Vibration in Railway Bridges. Cambridge: Cambridge University Press. 8. A. H 1951 Dynamic influences of smoothly running loads of simply supported girders. Stockholm: Kungl. Tekhn. Ho¨gskolan. 9. H. S 1921 Einfluss bewegter Last auf Eisenbahnoberbau und Bru¨cken. Berlin; Kreidels Verlag. 10. H. H. J 1929 Philosophical Magazine, Series 7, 8(48), 66–67. On the vibration of beams under the action of moving loads. 11. H. S 1934 Ingenieur Archiv, 5(4), 275–305; 1935 6(4), 265–270. Die Schwingungen von Tra¨gern bei bewegten Lasten I, II. 12. D. S. S and A. N. K. 1989 Acta Mechanica 79, 277–294. The axial motion effect on the dynamic response of the laterally vibrating frame subject to a moving load. 13. A. K 1983 Dynamics of Continuous Elastic Systems. Athens: Edition of NTVA.
THE EFFECT OF A MOVING MASS AND OTHER PARAMETERS ON THE DYNAMIC RESPONSE OF A SIMPLY SUPPORTED BEAM G. M, D. S A. N. K National Technical University of Athens, Athens, Greece (Received 30 November 1994, and in final form 9 May 1995) This paper deals with the linear dynamic response of a simply supported uniform beam under a moving load of constant magnitude and velocity by including the effect of its mass. Using a series solution for the dynamic deflection in terms of normal modes the individual and coupling effect of the mass of the moving load, of its velocity and of other parameters is fully assessed. A variety of numerical results allows us to draw important conclustions for structural design purposes. 7 1996 Academic Press Limited
1. INTRODUCTION
A lot of work has been reported during the past 100 years dealing with the dynamic response of railway bridges, and later on highway bridges, under the influence of moving loads. Extensive references to the literature on the subject can be found in the book by Fry´ba [1]. Two early interesting contributions in this area are due to Stokes [2] and Zimmermann [3]. In 1950 Krylov [4, 5], gave a complete solution of the problem of the dynamic behaviour of a prismatic bar acted upon by a load of constant magnitude, moving with a constant velocity. In 1922 Timoshenko solved the same problem but for a moving harmonic pulsating force. A pioneering work on the subject was presented in 1934 by Inglis [7], in which numerous parameters were taken into account. In 1951 Hillerborg [8] gave an analytical solution of the previous problem by means of Fourier’s method. Despite the availability of high speed computers, most of the methods used today for analyzing bridge vibration problems are essentially based on the early techniques of Inglis or Hillerborg. Relevant studies are those of Saller [9], Jeffcott [10], Steuding [11] and Sophianopoulos and Kounadis [12]. The problem of the forced motion of a beam, subjected to a moving load, is associated with serious difficulties when the effect of the mass of the load is accounted for. In the present work, by using as a first approximation the solution of the corresponding problem without the effect of the mass, a closed form solution is successfully derived.
2. MATHEMATICAL FORMULATION
We consider the simply supported beam shown in Figure 1, of length l, mass per unit length m and flexural rigidity EI, made from a uniform homogeneous and isotropic 357 0022–460X/96/130357 + 06 $18.00/0
7 1996 Academic Press Limited
. .
358
Figure 1. A moving load of mass M, with constant velocity v.
material. The beam is subjected to a load P of mass M, moving with a constant velocity v. The force acting on the beam is equal to P = Mg − Mw¨ (a)
(1)
with a = vt, and where g is the gravitational acceleration. Thus, the equation of motion can be written as EIw00(x, t) + mw¨ (x, t) = P d(x − a) = Mg d(x − a) − Mw¨ (a, t) d(x − a),
(2)
with 0 E t E l/v, and where the prime denotes differentiation with respect to x, dot differentiation with respect to t and d(· · ·), is the Dirac delta function. A series solution of equation (2) is of the form w(x, t) = s Xn (x)Tn (t),
(3)
n
where Xn = sin (npx/l) is the shape function of a freely vibrating beam and Tn (t) is the modal amplitude to be found. Due to expression (3) equation (2) can be written as
$
%
n (t) = Mg d(x − a) − M s Xn (a)Tn (t) d(x − a). EI s X2 n (x)Tn (t) + m s Xn (x)T n
n
n
(4)
Since for a free vibration one can assume that 2 EI s X2 n (x)Tn (t) − m s vn Xn (x)Tn (t) = 0, n
n
Figure 2. The ratio of dynamic to static middle-span deflection, as the load moves along the length of the beam (when the effect of the mass load is neglected).
359
Figure 3. The ratio of dynamic to static middle-span deflection versus load position: (a) when the moving load mass is included (———); (b) when it is neglected (............).
(with vn2 = (n 4p 4EI/ml 4 )) equation (4) becomes
$
%
m s vn2 Xn (x)Tn (t) + m s Xn (x)Tn (t) = Mg d(x − a) − M s Xn (a)Tn (t) d(x − a). n
n
n
Multiplying this expression by Xk and integrating from 0 to l gives a
Tn (t) + vn2 Tn (t) = (2Mg/ml)Xn (a) + (2M/ml)Xn (a) s Xk (a)Tk (t), k=1
or a
Tn (t)+vn2 Tn (t)=(2Mg/ml) sin (npvt/l)−(2M/ml) sin (npvt/l) s sin (kpvt/l)Tk (t).
(5)
k=1
Neglecting the second term on the right side of the last equation one can obtain the solution [13] Tn (t) = (2Mg/ml){1/(vn2 − Vn2 )}{sin Vn t − (Vn /vn ) sin vn t},
(6)
where Vn = npv/l. A first approximation to the solution of equation (5) can be obtained as follows. Introducing the value of Tn (t) given in equation (6) into the right side of equation (5) gives a
1 2 v − Vn2 k k=1
Tn (t) + vn2 Tn (t) = (2Mg/ml) sin Vn t − (2M 2g/m 2l 2 ) sin Vn t s × (−Vk2 sin Vk t + Vk vk sin vk t) sin Vk t.
(7)
Then one can obtain Tn (t) =
2Mg mlvn
g$ t
0
a M 1 sin Vn *t − sin Vn *t s 2 (−Vk2 sin Vk *t ml vk − Vk2 k=1
%
+Vk vk sin vk *t ) sin vk *t sin vn (t − *t ) dt*.
(8)
Using the dimensionless parameters x¯ = x/l,
v¯ = (2/p(zml 2/EI)v,
t = t/T1,
T1 = (2/p)zml 4/EI ,
(9)
. .
360 one can write Vn t = npv¯t,
vn t = 2n 2pt,
Vn /vn = v¯ /2n,
Vk /vn = kv¯ /2n 2.
Integrating equation (8) and using equations (3) and (9) yields w¯n (x¯ , t) = wn (x¯ , t)/(2Mgl 3/p 4EI) =
6
0
1
M v¯ 2ml sin (npx¯ ) An [sin (npv¯t) − sin (2n 2pt)] − s Bk 2ml 2n M k
−s k
+s k
−s k
+s k
+s k
−s k
1 Bk [sin (2k + n)pv¯t − a1k sin 2n 2pt] 2 2 1 − a1k 1 Bk [sin (2k − n)pv¯t − a2k sin 2n 2pt] 2 1 − a2k 2 1 Bk [sin b2k pt − g2k sin 2n 2pt] 2 2 1 − g2k 1 Bk [sin b4k pt − g4k sin 2n 2pt] 2 1 − g4k 2 1 Bk [sin b1k pt − g1k sin 2n 2pt] 2 1 − g1k 2
7
1 Bk [sin b3k pt − g3k sin 2n 2pt] , 2 1 − g1k 2
with An =
1 , n (1 − v¯ 2/4n 2 ) 4
Bk =
(v¯ /2k)2 , 1 + (v¯ /2k)2
a1k =
kv¯ v¯ + , n 2 2n
b 1k = kv¯ + 2k 2 + nv¯ ,
b2k = kv¯ − 2k 2 + nv¯ ,
b3k = kv¯ + 2k 2 − nv¯ ,
b4k = kv¯ − 2k 2 − nv¯ ,
a2k =
kv¯ v¯ − , n 2 2n
Figure 4. The influence of the ratio M/ml (for various velocities) on the ratio w¯M /w¯p (%) of the dynamic middle-span deflection, where w¯M includes the load mass M while w¯p does not.
361
T 1 w¯M /w¯p (%) for M/ml = 0·25 and 0·50 and various velocities M/ml
v¯ = 0·10
v¯ = 0·16
v¯ = 0·22
v¯ = 0·25
0·25 0·50
3·1 8·3
4 12·5
5 15
6·2 22·5
g1k =
kv¯ k2 v¯ , 2+ 2+ 2n 2n n
g2k =
kv¯ k2 v¯ , 2− 2+ 2n 2n n
g3k =
kv¯ k2 v¯ , 2+ 2− n 2n 2n
g4k =
kv¯ k2 v¯ . 2− 2− n 2n 2n
Note that 2Mgl 3/p 4EI can be taken to be approximately equal to the static deflection of the beam at x = l/2 due to the middle-span load Mg.
3. NUMERICAL RESULTS AND DISCUSSION
In this section, numerical results in graphical and tabular form are presented. The individual and coupling effects, on the dynamic response of various parameters such as load mass and velocity of the moving load, are discussed in detail. From Figure 2 one can see the classical case of various ratios of dynamic to static middlespan deflection as the load moves with various velocities along the length of the beam. In this case the effect of the load mass on the dynamic response of the beam is neglected. From this plot one sees that the maximum dimensionless deflection w¯ = 1·54 occurs for the dimensionless velocity v¯ = 0·75 when the load is located at x/l = 0·55, while for v¯ = 0·50 the deflection becomes w¯ = 1·30 (occurring when the load is located at x/l = 0·40). On can now consider how the above results are affected when the effect of the load mass is included. For the case in which M/ml = 0·30 the corresponding variations of the ratio dynamic to static middlespan deflection (as the load moves with the above velocities) are shown in Figure 3. Clearly, the maximum deflection w¯ = 1·72 occurs for v¯ = 0·75 when the load is located at x/l = 0·55. From Figure 4, one can see the influence of the ratio M/ml on the various values of the ratios of the dynamic middle-span deflection (when the effect of the load mass is included) to the dynamic middle-span deflection w¯M /w¯p (when the effect of the load mass is ignored) for various velocities. As the ratio M/ml increases, depending on the value of the velocity, the ratio w¯M /w¯p increases appreciably. This effect is more pronounced as the velocity increases. From Table 1, one can see the variation of the ratio w¯M /w¯p for two characteristic values of M/ml ( = 0·25, and 0·50) and four different values of velocities v¯ (0·10, 0·16, 0·22, 0·25).
REFERENCES 1. L. F´ 1972 Vibrations of Solids and Structures under Moving Loads. Groningen; Nordhoff International Publication Co. 2. G. G. S 1849 Transactions of the Cambridge Philosophical Society, Part 5, 707–735. Discussion of a differential equation relating to the breaking of railway bridges. 3. H. Z 1896 Centralblatt der Bauverwaltung 16(23), 249–251; (23A), 257–260; (24), 264–266; (26), 288. Die Schwingungen eines Tra¨gers mit bewegter Last. 4. A. N. K´ 1905 Mathematical Collection of Papers of the Academy of Sciences 61, St. Petersburg.
362
. .
5. A. N. K´ 1905 Mathematishce Annalen 61, 211. Uber die erzwungenen Schwingungen von gleichfo¨rmigen elastischen Sta¨ben. 6. S. P. T 1922 Philosophical Magazine Series 6, 43 1018. On the forced vibration of bridges. 7. C. E. I 1934 A Mathematical Treatise on Vibration in Railway Bridges. Cambridge: Cambridge University Press. 8. A. H 1951 Dynamic influences of smoothly running loads of simply supported girders. Stockholm: Kungl. Tekhn. Ho¨gskolan. 9. H. S 1921 Einfluss bewegter Last auf Eisenbahnoberbau und Bru¨cken. Berlin; Kreidels Verlag. 10. H. H. J 1929 Philosophical Magazine, Series 7, 8(48), 66–67. On the vibration of beams under the action of moving loads. 11. H. S 1934 Ingenieur Archiv, 5(4), 275–305; 1935 6(4), 265–270. Die Schwingungen von Tra¨gern bei bewegten Lasten I, II. 12. D. S. S and A. N. K. 1989 Acta Mechanica 79, 277–294. The axial motion effect on the dynamic response of the laterally vibrating frame subject to a moving load. 13. A. K 1983 Dynamics of Continuous Elastic Systems. Athens: Edition of NTVA.