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Comments on “dynamic response of a beam with intermediate point constraints subject to a moving load”
Journal of Sound and Vibration (1995) 180(5), 809–812
LETTERS TO THE EDITOR COMMENTS ON ‘‘DYNAMIC RESPONSE OF A BEAM WITH INTERMEDIATE POINT CONSTRAINTS SUBJECT TO A MOVING LOAD’’ Y.-H. L Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung City, Taiwan 20224, Republic of China (Received 28 June 1994)
1. This letter addresses some of the subtle points in dealing with the analysis of moving load problems, as reported in a recent article by Lee [1], and clarifies under what circumstances the conclusions drawn in reference [1] are correct or are otherwise invalid. In reference [1], the assumed mode approach combined with the energy method was used to formulate the equation of motion for a multi-span Bernoulli–Euler beam traversed by a moving concentrated force. The intermediate pinned supports were simulated by using springs of large stiffness. First of all, in reference [1] it is inappropriate to state that the dynamic behavior of the beam can be governed by the Euler beam theory from the assumption of transverse deflections being small. Even with the assumption of small vibrations, the Bernoulli–Euler beam theory may still be in serious error if the slenderness ratio, defined as r/L, where r is the radius of gyration and L the beam length, is large or vibration of higher modes is considered. The use of springs of large stiffness to simulate the intermediate pinned support can be tricky. For the numerical example discussed in reference [1], a spring stiffness of 1010 N/m was found to function well in the dynamic response analysis for the beam under a moving load. However, the magnitudes of the springs are problem dependent and in practice one has to scan the magnitude of the elements in the original structural stiffness matrix to select springs of appropriate higher order of magnitudes. Springs of low stiffness result in poor representation of the intermediate pinned supports, whereas springs of too large stiffness may yield erroneous results and may even fail to yield a solution due to numerical difficulties, which will be illustrated shortly. Therefore, the use of springs to represent the intermediate pinned supports not only is cumbersome and prone to needless error but also has an undesirable feature of affecting the execution time for the numerical integrations, as reported in reference [1]. All of these deficiencies can be avoided if the general finite element procedure dealing with various types of moving load problems is applied [2]. In reference [2], a general finite element approach was formulated for moving loads of various forms, from the most simple case of a moving concentrated force to the very complex situation of a two-foot moving sprung dynamic system travelling with a general motion profile. When the approach in reference [2] is applied, the intermediate pinned supports can be conveniently accommodated by removing the corresponding rows and columns in the structural matrices, and the total number of degrees of freedom of the system analyzed is reduced. In the case of flexible intermediate supports as represented by springs, the stiffnesses of the springs can be directly added to corresponding diagonal positions in the structural stiffness matrix if the springs are located at the element nodes. If the springs are not located at element nodes, equivalent stiffness can be computed and combined with the 809 0022–460X/95/100809 + 04 $08.00/0
7 1995 Academic Press Limited
810
original structural stiffness matrix. The effect of the spring stiffness on the dynamic response under the moving load is illustrated in Figure 1 for a four-span simply supported beam. The same set of data as in reference [1] was used. As can be seen, the use of spring stiffness of 1026 N/m results in serious error. Further increase of spring stiffness yields inaccurate results and eventually fails to obtain a solution due to numerical difficulties in the MATLAB environment [3], which uses double precision for computation. Although springs of the magnitude may not be selected for the present numerical example, one has to realize what the spring stiffness can do to the solution accuracy and takes proper precautions, as discussed above. Since the system considered in reference [1] is linear and time invariant, mode superposition can be applied efficiently to solve the beam dynamic response. The execution time was found to be within 10 s in the double precision computing environment in MATLAB installed on a low end 386-33 MHz personal computer with the use of 16 beam elements. The first seven modes with 100 solution steps were used. Each modal response was computed by using a point-wise Duhamel integral expression. The solution time of 1 minute magnitude, obtained by using a higher end 486-33 MHz personal computer, which runs much faster than the 386-33 MHz computer, is rather long for a problem of this size. In reference [1], it was commented that for a multi-span beam, the dimensionless beam deflection under the moving load was found to be insensitive to the magnitude of the moving load used in the numerical solution. As discussed earlier, the system considered is linear and thus the principle of superposition applies. Since the beam deflection is normalized by the maximum static deflection, which includes the magnitude of the load, the dimensionless deflection under the moving load is independent of the value of the load. Sensitivity is of no concern for the linear system analyzed here. Finally, it was stated in conclusion that, for a relatively fast moving load, the reduction of deflection under the moving load caused by the addition of intermediate supports is less significant. The conclusion does not capture the whole picture and may be misleading. In Figure 2 is depicted the maximum dimensionless deflection under the moving load for oneand two-span beams respectively versus the moving speed parameter, a, defined as pv/Lv1 , where v1 = (p/L)2zEI/m, which was misprinted as (p/L)4(EI/m) in reference [1]. As shown in Figure 2, the maximum normalized deflection for analysis of the one-span beam
Figure 1. The normalized deflection under the moving load for a four-span beam at a = 1·5 obtained by using 32 beam elements and the first nine modes; ----, k = 1010 N/m; – – –, k = 1025 N/m; · · · , k = 1026 N/m.
811
Figure 2. The maximum normalized deflection under the moving load. ----, One-span beam; – – –, two-span beam.
reaches a maximal value of 1·6108 at a = 0·45 and decreases monotonically with increasing moving load speed. Since the two-span beam has higher rigidity, with the fundamental natural frequency being four times that of the one-span beam, the moving load speed required to make an appreciable impact on the support beam is expected to be higher than that for the one-span beam, and this is illustrated in the lower curve in Figure 2 with a maximum value 0·2577 at a = 1·4875. As to the percentage of reduction in the deflection under the moving load, Figure 3 clearly illustrates its variation with respect to the moving load speed parameter, a, in which w1 and w2 denote the maximum deflections under the moving load for the one- and two-span beams, respectively. The author’s conclusions in reference [1] is correct for a moving load speed up to a = 1·715, except in the very low speed range as shown in Figure 3. For a moving load speed above a = 1·715 and up to a = 2·3525, the conclusion in reference [1] is invalid, and it is observed that the addition
Figure 3. The percentage reduction in the deflection under the moving load due to the addition of an intermediate point support at the beam center.
812
of the intermediate support at the beam center for the reduction of deflection under the moving load becomes significant in this speed range. A similar analysis can be conducted for beams with more spans. Therefore, conclusions drawn from a few simulation results may be misleading and an analysis considering a broad speed range such as done in this work should be conducted to establish sound statements. What one can conclude from this analysis is that for very high moving load speeds the deflection under the moving load is considerably less than the maximum static deflection, and for practical applications it may not be warranted to engineer additional intermediate supports to reduce further the deflection under the moving load. 2. The linear Bernoulli–Euler beam theory is valid for lower modes analysis of small vibration of slender beams. The use of springs to simulate the intermediate pinned supports is rather inconvenient and an alternative scheme of removing the corresponding rows and columns in the structural matrices eliminates many problems associated with the use of springs. The dimensionless deflection under the moving load is independent of the magnitude of the moving load regardless of the number of beam spans since a linear system is considered. The percentage of reduction in the deflection under the moving load due to the addition of intermediate supports may decrease or increase, depending on the moving load speed range. 1. H. P. L 1994 Journal of Sound and Vibration 171, 361–368. Dynamic response of a beam with intermediate point constraints subject to a moving load. 2. Y.-H. L and M. W. T 1990 Journal of Sound and Vibration 136, 323–342. Finite element analysis of elastic beams subjected to moving dynamic loads. 3. T MW, I. 1991 MATLAB, User’s Guide, edition 3.51, Massachusetts.
LETTERS TO THE EDITOR COMMENTS ON ‘‘DYNAMIC RESPONSE OF A BEAM WITH INTERMEDIATE POINT CONSTRAINTS SUBJECT TO A MOVING LOAD’’ Y.-H. L Department of Mechanical and Marine Engineering, National Taiwan Ocean University, Keelung City, Taiwan 20224, Republic of China (Received 28 June 1994)
1. This letter addresses some of the subtle points in dealing with the analysis of moving load problems, as reported in a recent article by Lee [1], and clarifies under what circumstances the conclusions drawn in reference [1] are correct or are otherwise invalid. In reference [1], the assumed mode approach combined with the energy method was used to formulate the equation of motion for a multi-span Bernoulli–Euler beam traversed by a moving concentrated force. The intermediate pinned supports were simulated by using springs of large stiffness. First of all, in reference [1] it is inappropriate to state that the dynamic behavior of the beam can be governed by the Euler beam theory from the assumption of transverse deflections being small. Even with the assumption of small vibrations, the Bernoulli–Euler beam theory may still be in serious error if the slenderness ratio, defined as r/L, where r is the radius of gyration and L the beam length, is large or vibration of higher modes is considered. The use of springs of large stiffness to simulate the intermediate pinned support can be tricky. For the numerical example discussed in reference [1], a spring stiffness of 1010 N/m was found to function well in the dynamic response analysis for the beam under a moving load. However, the magnitudes of the springs are problem dependent and in practice one has to scan the magnitude of the elements in the original structural stiffness matrix to select springs of appropriate higher order of magnitudes. Springs of low stiffness result in poor representation of the intermediate pinned supports, whereas springs of too large stiffness may yield erroneous results and may even fail to yield a solution due to numerical difficulties, which will be illustrated shortly. Therefore, the use of springs to represent the intermediate pinned supports not only is cumbersome and prone to needless error but also has an undesirable feature of affecting the execution time for the numerical integrations, as reported in reference [1]. All of these deficiencies can be avoided if the general finite element procedure dealing with various types of moving load problems is applied [2]. In reference [2], a general finite element approach was formulated for moving loads of various forms, from the most simple case of a moving concentrated force to the very complex situation of a two-foot moving sprung dynamic system travelling with a general motion profile. When the approach in reference [2] is applied, the intermediate pinned supports can be conveniently accommodated by removing the corresponding rows and columns in the structural matrices, and the total number of degrees of freedom of the system analyzed is reduced. In the case of flexible intermediate supports as represented by springs, the stiffnesses of the springs can be directly added to corresponding diagonal positions in the structural stiffness matrix if the springs are located at the element nodes. If the springs are not located at element nodes, equivalent stiffness can be computed and combined with the 809 0022–460X/95/100809 + 04 $08.00/0
7 1995 Academic Press Limited
810
original structural stiffness matrix. The effect of the spring stiffness on the dynamic response under the moving load is illustrated in Figure 1 for a four-span simply supported beam. The same set of data as in reference [1] was used. As can be seen, the use of spring stiffness of 1026 N/m results in serious error. Further increase of spring stiffness yields inaccurate results and eventually fails to obtain a solution due to numerical difficulties in the MATLAB environment [3], which uses double precision for computation. Although springs of the magnitude may not be selected for the present numerical example, one has to realize what the spring stiffness can do to the solution accuracy and takes proper precautions, as discussed above. Since the system considered in reference [1] is linear and time invariant, mode superposition can be applied efficiently to solve the beam dynamic response. The execution time was found to be within 10 s in the double precision computing environment in MATLAB installed on a low end 386-33 MHz personal computer with the use of 16 beam elements. The first seven modes with 100 solution steps were used. Each modal response was computed by using a point-wise Duhamel integral expression. The solution time of 1 minute magnitude, obtained by using a higher end 486-33 MHz personal computer, which runs much faster than the 386-33 MHz computer, is rather long for a problem of this size. In reference [1], it was commented that for a multi-span beam, the dimensionless beam deflection under the moving load was found to be insensitive to the magnitude of the moving load used in the numerical solution. As discussed earlier, the system considered is linear and thus the principle of superposition applies. Since the beam deflection is normalized by the maximum static deflection, which includes the magnitude of the load, the dimensionless deflection under the moving load is independent of the value of the load. Sensitivity is of no concern for the linear system analyzed here. Finally, it was stated in conclusion that, for a relatively fast moving load, the reduction of deflection under the moving load caused by the addition of intermediate supports is less significant. The conclusion does not capture the whole picture and may be misleading. In Figure 2 is depicted the maximum dimensionless deflection under the moving load for oneand two-span beams respectively versus the moving speed parameter, a, defined as pv/Lv1 , where v1 = (p/L)2zEI/m, which was misprinted as (p/L)4(EI/m) in reference [1]. As shown in Figure 2, the maximum normalized deflection for analysis of the one-span beam
Figure 1. The normalized deflection under the moving load for a four-span beam at a = 1·5 obtained by using 32 beam elements and the first nine modes; ----, k = 1010 N/m; – – –, k = 1025 N/m; · · · , k = 1026 N/m.
811
Figure 2. The maximum normalized deflection under the moving load. ----, One-span beam; – – –, two-span beam.
reaches a maximal value of 1·6108 at a = 0·45 and decreases monotonically with increasing moving load speed. Since the two-span beam has higher rigidity, with the fundamental natural frequency being four times that of the one-span beam, the moving load speed required to make an appreciable impact on the support beam is expected to be higher than that for the one-span beam, and this is illustrated in the lower curve in Figure 2 with a maximum value 0·2577 at a = 1·4875. As to the percentage of reduction in the deflection under the moving load, Figure 3 clearly illustrates its variation with respect to the moving load speed parameter, a, in which w1 and w2 denote the maximum deflections under the moving load for the one- and two-span beams, respectively. The author’s conclusions in reference [1] is correct for a moving load speed up to a = 1·715, except in the very low speed range as shown in Figure 3. For a moving load speed above a = 1·715 and up to a = 2·3525, the conclusion in reference [1] is invalid, and it is observed that the addition
Figure 3. The percentage reduction in the deflection under the moving load due to the addition of an intermediate point support at the beam center.
812
of the intermediate support at the beam center for the reduction of deflection under the moving load becomes significant in this speed range. A similar analysis can be conducted for beams with more spans. Therefore, conclusions drawn from a few simulation results may be misleading and an analysis considering a broad speed range such as done in this work should be conducted to establish sound statements. What one can conclude from this analysis is that for very high moving load speeds the deflection under the moving load is considerably less than the maximum static deflection, and for practical applications it may not be warranted to engineer additional intermediate supports to reduce further the deflection under the moving load. 2. The linear Bernoulli–Euler beam theory is valid for lower modes analysis of small vibration of slender beams. The use of springs to simulate the intermediate pinned supports is rather inconvenient and an alternative scheme of removing the corresponding rows and columns in the structural matrices eliminates many problems associated with the use of springs. The dimensionless deflection under the moving load is independent of the magnitude of the moving load regardless of the number of beam spans since a linear system is considered. The percentage of reduction in the deflection under the moving load due to the addition of intermediate supports may decrease or increase, depending on the moving load speed range. 1. H. P. L 1994 Journal of Sound and Vibration 171, 361–368. Dynamic response of a beam with intermediate point constraints subject to a moving load. 2. Y.-H. L and M. W. T 1990 Journal of Sound and Vibration 136, 323–342. Finite element analysis of elastic beams subjected to moving dynamic loads. 3. T MW, I. 1991 MATLAB, User’s Guide, edition 3.51, Massachusetts.