炮身沿摇架大位移后坐系统的时空有限元建模方法研究
|
摘要
本文以某国防项目为研究背景,利用理论分析、数值计算和实验测试相结合的方法,按照循序渐进的思路对炮身沿摇架大位移后坐系统的时空有限元建模理论与方法进行探讨研究。
对时空有限元法的基本建模理论进行了讨论分析,结合经典的移动惯性载荷作用下弦的振动问题,编制了求解结构动力学问题的时空有限元算法程序,基于经典算例,将计算结果与精确解、Newmark法、Wilson-θ法、中心差分法等进行比较分析,验证该算法的精度和稳定性。
建立了移动力/移动质量作用下的简支梁和悬臂梁的时变力学模型,在传统有限元质量矩阵和刚度矩阵的基础上,利用迭代格式对位移和速度进行更新,推导出每个时间步上的时空质量矩阵和刚度矩阵,以简支梁和悬臂梁为例,计算移动力/移动质量以一定的速度匀速通过时的动力响应,并与精确解、Newmark法、Wilson-θ法等进行分析对比,进一步验证时空有限元法的有效性。
建立了炮身沿摇架大位移后坐的时变动力学模型,给出了相应的时空有限元数值算法,将摇架简化成悬臂梁,炮身分别用移动刚体、移动簧载质量、移动簧载刚体、移动梁等来模拟,利用时空有限元法进行时域和空间域的离散,获得了时变动力学方程组,通过数值计算得到了炮身沿摇架大位移后坐的动力响应。
对炮身沿摇架大位移后坐的振动特性进行了测试研究,将理论计算与实验结果进行对比分析,验证了理论建模的正确性。
本文取得的炮身沿摇架大位移后坐时变系统的时空有限元数值计算初步成果对深入开展火炮大位移时变动力学设计理论研究具有一定的借鉴作用。
Under the background of a National Defense Project, this thesis is concerned with space-time finite element modeling theory and method for a canon barrel with large displacement by combination of theoretical analysis, numeric calculation and experiment.
Basic modeling theory on space-time finite element method is analysised. Combining with the classical case of string vibration under moving inertial load, the algorithm program for solving structural dynamic problems is compiled. Based on classical examples, the results are compared with the exact solution, the Newmark method, Wilson-θmethod and Central difference method, which verifies the accuracy and stability of this algorithm.
A time-varying mechanical model of simply supported beam and cantilever beam with moving force or moving mass is established. Based on the traditional finite element mass matrix and stiffness matrix, space-time finite element mass matrix and stiffness matrix in each time step is derived by using the iterative scheme to update the displacement and velocity. The dynamic response is calculated with the examples of moving force or moving mass with a certain speed subjected to simply supported beam and cantilever beam. The results are compared with exact solution, Newmark method and Wilson-θmethod, which further demonstrates the effectiveness of space-time finite element method.
Time-varying dynamics model of the canon barrel with large displacement is established, the corresponding space-time finite element numerical algorithm is deduced. The cradle and barrel are simplified as cantilever beam, moving rigid body, moving spring mass, moving spring rigid body and moving beam. The time domain and space domain is discreted by space-time finite element method, time-varying dynamics equation group is derivated, the dynamic response of the canon barrel with large displacement is calculated through numerical calculation.
The vibration characteristics of the canon barrel with large displacement is tested and researched. The correctness of theoretical modeling is verified by the results comparing of theoretical calculation and experiment.
The preliminary research findings of numerical calculation using space-time finite element for the cannon barrel with large displacement achieved in this paper can be of certain reference to in-depth theoretical study on dynamic design of the large displacement time-varying.
对时空有限元法的基本建模理论进行了讨论分析,结合经典的移动惯性载荷作用下弦的振动问题,编制了求解结构动力学问题的时空有限元算法程序,基于经典算例,将计算结果与精确解、Newmark法、Wilson-θ法、中心差分法等进行比较分析,验证该算法的精度和稳定性。
建立了移动力/移动质量作用下的简支梁和悬臂梁的时变力学模型,在传统有限元质量矩阵和刚度矩阵的基础上,利用迭代格式对位移和速度进行更新,推导出每个时间步上的时空质量矩阵和刚度矩阵,以简支梁和悬臂梁为例,计算移动力/移动质量以一定的速度匀速通过时的动力响应,并与精确解、Newmark法、Wilson-θ法等进行分析对比,进一步验证时空有限元法的有效性。
建立了炮身沿摇架大位移后坐的时变动力学模型,给出了相应的时空有限元数值算法,将摇架简化成悬臂梁,炮身分别用移动刚体、移动簧载质量、移动簧载刚体、移动梁等来模拟,利用时空有限元法进行时域和空间域的离散,获得了时变动力学方程组,通过数值计算得到了炮身沿摇架大位移后坐的动力响应。
对炮身沿摇架大位移后坐的振动特性进行了测试研究,将理论计算与实验结果进行对比分析,验证了理论建模的正确性。
本文取得的炮身沿摇架大位移后坐时变系统的时空有限元数值计算初步成果对深入开展火炮大位移时变动力学设计理论研究具有一定的借鉴作用。
Under the background of a National Defense Project, this thesis is concerned with space-time finite element modeling theory and method for a canon barrel with large displacement by combination of theoretical analysis, numeric calculation and experiment.
Basic modeling theory on space-time finite element method is analysised. Combining with the classical case of string vibration under moving inertial load, the algorithm program for solving structural dynamic problems is compiled. Based on classical examples, the results are compared with the exact solution, the Newmark method, Wilson-θmethod and Central difference method, which verifies the accuracy and stability of this algorithm.
A time-varying mechanical model of simply supported beam and cantilever beam with moving force or moving mass is established. Based on the traditional finite element mass matrix and stiffness matrix, space-time finite element mass matrix and stiffness matrix in each time step is derived by using the iterative scheme to update the displacement and velocity. The dynamic response is calculated with the examples of moving force or moving mass with a certain speed subjected to simply supported beam and cantilever beam. The results are compared with exact solution, Newmark method and Wilson-θmethod, which further demonstrates the effectiveness of space-time finite element method.
Time-varying dynamics model of the canon barrel with large displacement is established, the corresponding space-time finite element numerical algorithm is deduced. The cradle and barrel are simplified as cantilever beam, moving rigid body, moving spring mass, moving spring rigid body and moving beam. The time domain and space domain is discreted by space-time finite element method, time-varying dynamics equation group is derivated, the dynamic response of the canon barrel with large displacement is calculated through numerical calculation.
The vibration characteristics of the canon barrel with large displacement is tested and researched. The correctness of theoretical modeling is verified by the results comparing of theoretical calculation and experiment.
The preliminary research findings of numerical calculation using space-time finite element for the cannon barrel with large displacement achieved in this paper can be of certain reference to in-depth theoretical study on dynamic design of the large displacement time-varying.
引文
[1]J.T.Oden. Finite Elements Of Nonlinear Continua[M]. London:McGRAW-HILL Book Company Limited,1971
[2]I.Fried. Finite Element Analysis of Time Dependent Phenomena[J]. AIAA J,1969, vol 7: 1170～1173
[3]J.H.Argyris, D.W.Scharpf. Finite Element in Time and Space[J]. Nucl Eng Des,1969, vol 10:456～459
[4]C.D.Bailey. A New Look at'Hamilton's Principle[J]. Foudations of Physics,1975, vol 5: 433～451
[5]C.I.Bajer. Triangular and tetrahedral space-time finite elements in vibration analysis[J]. Int J Numer Meth Eng,1986, vol 23:2031～2048
[6]C.I.Bajer. Notes on the stability of non-rectangular space-time elements[J]. Int J Numer Meth Eng,1987, vol 24:1721～1739
[7]黄川,魏泳涛,曹波,于建华.时空有限元法求解动力学问题[J].固体力学学报,2003,24：163-167
[8]毕继红,胡昌亮.由卷积型变分原理导出的时空有限元法[J].天津大学学报,1999,32(1)：15-18
[9]王骥,李宏,刘儒勋.时间依赖问题的时空有限元法[J].中国科学技术大学学报,1999,29(4)：440-446
[10]谢小平,胡兵,罗鲲.一类混合型问题的空间\时间有限元方法研究[J].四川大学学报(自然科学版),1999,36(4)：668-672
[11]李宏,刘儒勋.一类非线性问题的时空有限元方法的误差估计[J].中国科学技术大学学报,2000,30(1)：14-24
[12]李宏,刘儒勋.抛物方程的时空有限元方法[J].应用力学和数学,2001,22(6)：613-624
[13]毕继红,石朝花.基于卷积型变分原理的时空有限元法求解板的动力学初值问题[J].天津城市建设学院学报,1998,4(4)：63-68
[14]胡昌亮,毕继红.基于卷积型积分方程的时间有限元法[J].天津大学学报,1999,31(1)：77-80
[15]钟万勰,高强.时间—空间混和有限元[J].动力学与控制学报,2007,5(1)：1-7
[16]于开平,邹经湘.时间域有限元法[J].力学进展,1998,28(4)：461-468
[17]L.Fryba. Vibration of solids and structures under moving loads(3rd Edition) [M]. London:Thomas Telford,1999
[18]Jia-Jang Wu. Dynamic analysis of an inclined beam due to moving loads[J]. Sound and vibration,2005,288:107～131
[19]Bajer CI, Bartlomiej Dyniewicz. Numerical modelling of structure vibration under inertial moving load[J]. Arch Appl Mech,2009, vol 79:499～508
[20]Bajer CI. Space-time finite element formulation for the dynamical evolutionary process[J]. Applied Mathematics and Computer Science,1993,3(2):251～268
[21]郭保全,谢石林,张希农,潘玉田.高速移动载荷冲击作用下轴向运动梁的非线性振动[C].第九届全国振动理论及应用学术会议论文集,2007,2-127-2-133
[22]姜沐.移动质量载荷在梁中激起的振动[J].力学与实践,2002,24(6)：44-47
[23]林家浩,张守云,吕峰,张亚辉.移动简谐载荷作用下桥梁响应的高效计算[J].计算力学学报,2006,23(4)：385-390
[24]张志超,林家浩.移动质量作用下桥梁响应的精细积分[J].大连理工大学学报,2006,46：S22-S28
[25]沈火明,肖新标.求解车桥耦合振动问题的一种数值方法[J].西南交通大学学报,2003,38(6)：658-662
[26]蒲军平,王晓臣.梁桥受移动载荷作用的数值模拟[J].浙江工业大学学报,2007,35(2)：228-230
[27]彭献,刘子建,洪家旺.匀变速移动质量与简支梁耦合系统的振动分析[J].工程力学,23(6)：25-29
[28]姜沐,郭锡福.弹丸加速运动在身管中激发的振动[J].弹道学报,2002,14(3)：57-68
[29]何水清,张家泰,杨波,陈其廉.炮身的双时变横向振动分析[J].兵工学报,1995,21：3-18
[30]史跃东,王德石.考虑惯性效应的移动弹丸作用下身管振动特性[J].兵工学报,2011,32(4)：414-420
[31]史跃东,王德石.舰炮振动的刚柔耦合动力学分析[J].弹道学报,2010,22(1)：37-40
[32]康新中,吴三灵,马春茂.火炮系统动力学[M].北京：国防工业出版社,1999
[33]于会杰,王德石.身管在加速弹丸激励下的振动研究[J].海军工程大学学报,2009,21(2)：12-17
[34]王颖泽,张小兵.移动质量作用下膨胀波发射装置的振动响应分析[J].航空动力学报,2009,24(8)：1714-1719
[35]刘雷.弹丸-身管耦合系统动力学模型[J].振动与冲击,2007,26(6)：121-124
[36]何永,高树滋,毛保全.身管振动中摇架支承的分析[J].装甲兵工程学院学报,1996,10(1)：88-91
[37]蔡文勇,陈运生,杨国来.车载火炮炮口扰动影响因素分析[J].南京理工大学学报,2005,29(6)：658-661
[38]何永,高树滋,陈效华.弹性身管和弹丸耦合运动的研究[J].四川兵工学报,1995,16(3)：5-7
[39]刘宁,杨国来.弹管横向碰撞对身管动力响应的影响[J].弹道学报,2010,22(2)：67-70
[40]Bajer CI, Dyniewicz B. Space-time approach to numerical analysis of a string with a moving mass[J]. International Journal for Numerical Methods in Engineering,2008, 76(10):1528～43
[41]刘晶波,杜修力.结构动力学[M].北京：机械工业出版社,2007
[42]刘祥庆,刘晶波,丁桦.时域逐步积分算法稳定性与精度的对比分析[J].岩石力学与工程学报,2007,26：3000-3008
[43]黄川.时域协调的动力时空有限元算法及其应用[D].四川大学硕士学位论文,2003
[44]王勖成,邵敏.有限单元法基本原理和数值方法[M].清华大学出版社,2003
[45]姚昌仁,唐国梁,宋廷伦.火箭导弹发射动力学[M].北京：北京理工大学出版社,1996：169
[46]陈姣.移动载荷作用下桥梁的车桥耦合振动分析[D].合肥工业大学硕士学位论文,2009
[47]Bajer CI, Dyniewicz B. Virtual functions of the space-time finite element method in moving mass[J]. Computers and Structures,2009,76:444～455
[48]Bajer CI, Dyniewicz B. Numerical modelling of structure vibrations under inertial moving load[J]. Arch Appl Mech,2009,79:499～508
[49]李海斌,毕世华,方远乔羽.运动刚体激励下弹性梁的振动响应分析[J].工程力学,1998,15(4)：74-80
[50]彭献,游福贺.车桥耦合系统固有频率的研究[J].动力学与控制学报,2010,8(3)：258-262
[51]郭锡福.远程火炮武器系统射击精度分析[M].北京：国防工业出版社,2004
[52]葛建立.车载炮动态非线性有限元仿真研究[D].南京理工大学博士学位论文,2007
[53]郭锡福.弹丸发射动力学[M].南京：华东工学院,1988
[54]Bajer CI, Bohatier C. The soft way method and the velocity formulation[J]. Computers and Structures,1995,55(6):1015～1025
[55]Dyniewicz B, Bajer CI. New feature of the solution of a Timoshenko beam carrying the moving mass particle[J]. Arch Appl Mech,2010,62(5):327～341
[56]Dyniewicz B, Bajer CI. Paradox of the particle's trajectory moving on a string[J]. Arch Appl Mech,2008, Vol.79:213～223
[57]Dyniewicz B, Bajer CI. Numerical methods for vibration analysis of Timoshenko beamsubjected to inertial moving load[J]. Symposium Vibrations in Physical Systems, 2010, Vol.24
[58]Jia-Jang Wu, A.R.Whittaker, M.P.Cartmell. The use of finite element techniques for calculating the dynamic response of structures to moving loads[J]. Computers and Structures,2000, Vol.78:789～799
[59]J.J.Wu, Wu, A.R.Whittaker, M.P.Cartmell. Dynamic responses of structures to moving bodies using combined finite element and analytical methods[J]. International Journal of Mechanical Sciences,2001, Vol.43:2555～2579
[60]A.Bonell, O.S.Bursi. Explicit predictor-multicorrector time discontinuous Galerkin methods for linear dynamics[J]. Journal of Sound and Vibration,2001, Vol.246, No.4: 625～652
[61]X.D.Li, N-E.Wiberg. Structural dynamic analysis by a time-discontinuous Galerkin finite element method [J]. International Journal For Numerical Methods in Engineering, 1996, Vol.39:2131～2152
[62]陈彦辉,郭旻,何宗颖,王宝元.炮口振动测试方法及实践[J].火炮发射与控制学报,2010,(1)：80-83
[63]王宝元,邵小军.炮口振动响应实验测试方法综述[J].火炮发射与控制学报,2010,(3)：112-116
[64]武瑞文,王兆胜.自行火炮武器系统射击精度研究[J].兵工学报,2004,25(4)：407-409
[65]彭献,张普,刘子建.基座运动情况下弹炮系统的振动分析[J].振动与冲击,2009,28(3)：23-26
[66]谈乐斌,张相炎,管红根等.火炮概论[M].北京：北京理工大学出版社,2005：1-6
[67]张相炎.火炮设计理论[M].北京：北京理工大学出版社,2005：10-22
[68]马福球,陈运生,朵英贤火炮与自动武器[M].北京：北京理工大学出版社,2003：12-13
[69]吴三灵,温波,于永强火炮动力学试验[M].北京：国防工业出版社,2004：71
[70]王靖君,郝信鹏.火炮概论[M].北京：兵器工业出版社,1992：123
[71]王宝元,陈顺德,周发明,衡刚,郭旻.火炮振动与射击密集度实验方法研究[J].火炮发射与控制学报,2009,(4)：89-9
[72]李海斌,毕世华,方远乔羽,袁曾风.移动刚体激励的梁的振动分析和实验[J].北京理工大学学报,1999,19(1)：18-22
[73]毕世华,李海斌,方远乔,袁曾凤.弹性发射梁振动的主动控制及实验研究[J].兵工学报,1998,19(4)：310-314
[74]P.I.Kattan,韩来彬MATLAB有限元分析与应用[M].北京：清华大学出版社,2004：92-94
[75]徐斌,高跃飞,余龙. MATLAB有限元结构动力学分析与工程应用[M].北京：清华大学出版社,2009：107
[2]I.Fried. Finite Element Analysis of Time Dependent Phenomena[J]. AIAA J,1969, vol 7: 1170～1173
[3]J.H.Argyris, D.W.Scharpf. Finite Element in Time and Space[J]. Nucl Eng Des,1969, vol 10:456～459
[4]C.D.Bailey. A New Look at'Hamilton's Principle[J]. Foudations of Physics,1975, vol 5: 433～451
[5]C.I.Bajer. Triangular and tetrahedral space-time finite elements in vibration analysis[J]. Int J Numer Meth Eng,1986, vol 23:2031～2048
[6]C.I.Bajer. Notes on the stability of non-rectangular space-time elements[J]. Int J Numer Meth Eng,1987, vol 24:1721～1739
[7]黄川,魏泳涛,曹波,于建华.时空有限元法求解动力学问题[J].固体力学学报,2003,24：163-167
[8]毕继红,胡昌亮.由卷积型变分原理导出的时空有限元法[J].天津大学学报,1999,32(1)：15-18
[9]王骥,李宏,刘儒勋.时间依赖问题的时空有限元法[J].中国科学技术大学学报,1999,29(4)：440-446
[10]谢小平,胡兵,罗鲲.一类混合型问题的空间\时间有限元方法研究[J].四川大学学报(自然科学版),1999,36(4)：668-672
[11]李宏,刘儒勋.一类非线性问题的时空有限元方法的误差估计[J].中国科学技术大学学报,2000,30(1)：14-24
[12]李宏,刘儒勋.抛物方程的时空有限元方法[J].应用力学和数学,2001,22(6)：613-624
[13]毕继红,石朝花.基于卷积型变分原理的时空有限元法求解板的动力学初值问题[J].天津城市建设学院学报,1998,4(4)：63-68
[14]胡昌亮,毕继红.基于卷积型积分方程的时间有限元法[J].天津大学学报,1999,31(1)：77-80
[15]钟万勰,高强.时间—空间混和有限元[J].动力学与控制学报,2007,5(1)：1-7
[16]于开平,邹经湘.时间域有限元法[J].力学进展,1998,28(4)：461-468
[17]L.Fryba. Vibration of solids and structures under moving loads(3rd Edition) [M]. London:Thomas Telford,1999
[18]Jia-Jang Wu. Dynamic analysis of an inclined beam due to moving loads[J]. Sound and vibration,2005,288:107～131
[19]Bajer CI, Bartlomiej Dyniewicz. Numerical modelling of structure vibration under inertial moving load[J]. Arch Appl Mech,2009, vol 79:499～508
[20]Bajer CI. Space-time finite element formulation for the dynamical evolutionary process[J]. Applied Mathematics and Computer Science,1993,3(2):251～268
[21]郭保全,谢石林,张希农,潘玉田.高速移动载荷冲击作用下轴向运动梁的非线性振动[C].第九届全国振动理论及应用学术会议论文集,2007,2-127-2-133
[22]姜沐.移动质量载荷在梁中激起的振动[J].力学与实践,2002,24(6)：44-47
[23]林家浩,张守云,吕峰,张亚辉.移动简谐载荷作用下桥梁响应的高效计算[J].计算力学学报,2006,23(4)：385-390
[24]张志超,林家浩.移动质量作用下桥梁响应的精细积分[J].大连理工大学学报,2006,46：S22-S28
[25]沈火明,肖新标.求解车桥耦合振动问题的一种数值方法[J].西南交通大学学报,2003,38(6)：658-662
[26]蒲军平,王晓臣.梁桥受移动载荷作用的数值模拟[J].浙江工业大学学报,2007,35(2)：228-230
[27]彭献,刘子建,洪家旺.匀变速移动质量与简支梁耦合系统的振动分析[J].工程力学,23(6)：25-29
[28]姜沐,郭锡福.弹丸加速运动在身管中激发的振动[J].弹道学报,2002,14(3)：57-68
[29]何水清,张家泰,杨波,陈其廉.炮身的双时变横向振动分析[J].兵工学报,1995,21：3-18
[30]史跃东,王德石.考虑惯性效应的移动弹丸作用下身管振动特性[J].兵工学报,2011,32(4)：414-420
[31]史跃东,王德石.舰炮振动的刚柔耦合动力学分析[J].弹道学报,2010,22(1)：37-40
[32]康新中,吴三灵,马春茂.火炮系统动力学[M].北京：国防工业出版社,1999
[33]于会杰,王德石.身管在加速弹丸激励下的振动研究[J].海军工程大学学报,2009,21(2)：12-17
[34]王颖泽,张小兵.移动质量作用下膨胀波发射装置的振动响应分析[J].航空动力学报,2009,24(8)：1714-1719
[35]刘雷.弹丸-身管耦合系统动力学模型[J].振动与冲击,2007,26(6)：121-124
[36]何永,高树滋,毛保全.身管振动中摇架支承的分析[J].装甲兵工程学院学报,1996,10(1)：88-91
[37]蔡文勇,陈运生,杨国来.车载火炮炮口扰动影响因素分析[J].南京理工大学学报,2005,29(6)：658-661
[38]何永,高树滋,陈效华.弹性身管和弹丸耦合运动的研究[J].四川兵工学报,1995,16(3)：5-7
[39]刘宁,杨国来.弹管横向碰撞对身管动力响应的影响[J].弹道学报,2010,22(2)：67-70
[40]Bajer CI, Dyniewicz B. Space-time approach to numerical analysis of a string with a moving mass[J]. International Journal for Numerical Methods in Engineering,2008, 76(10):1528～43
[41]刘晶波,杜修力.结构动力学[M].北京：机械工业出版社,2007
[42]刘祥庆,刘晶波,丁桦.时域逐步积分算法稳定性与精度的对比分析[J].岩石力学与工程学报,2007,26：3000-3008
[43]黄川.时域协调的动力时空有限元算法及其应用[D].四川大学硕士学位论文,2003
[44]王勖成,邵敏.有限单元法基本原理和数值方法[M].清华大学出版社,2003
[45]姚昌仁,唐国梁,宋廷伦.火箭导弹发射动力学[M].北京：北京理工大学出版社,1996：169
[46]陈姣.移动载荷作用下桥梁的车桥耦合振动分析[D].合肥工业大学硕士学位论文,2009
[47]Bajer CI, Dyniewicz B. Virtual functions of the space-time finite element method in moving mass[J]. Computers and Structures,2009,76:444～455
[48]Bajer CI, Dyniewicz B. Numerical modelling of structure vibrations under inertial moving load[J]. Arch Appl Mech,2009,79:499～508
[49]李海斌,毕世华,方远乔羽.运动刚体激励下弹性梁的振动响应分析[J].工程力学,1998,15(4)：74-80
[50]彭献,游福贺.车桥耦合系统固有频率的研究[J].动力学与控制学报,2010,8(3)：258-262
[51]郭锡福.远程火炮武器系统射击精度分析[M].北京：国防工业出版社,2004
[52]葛建立.车载炮动态非线性有限元仿真研究[D].南京理工大学博士学位论文,2007
[53]郭锡福.弹丸发射动力学[M].南京：华东工学院,1988
[54]Bajer CI, Bohatier C. The soft way method and the velocity formulation[J]. Computers and Structures,1995,55(6):1015～1025
[55]Dyniewicz B, Bajer CI. New feature of the solution of a Timoshenko beam carrying the moving mass particle[J]. Arch Appl Mech,2010,62(5):327～341
[56]Dyniewicz B, Bajer CI. Paradox of the particle's trajectory moving on a string[J]. Arch Appl Mech,2008, Vol.79:213～223
[57]Dyniewicz B, Bajer CI. Numerical methods for vibration analysis of Timoshenko beamsubjected to inertial moving load[J]. Symposium Vibrations in Physical Systems, 2010, Vol.24
[58]Jia-Jang Wu, A.R.Whittaker, M.P.Cartmell. The use of finite element techniques for calculating the dynamic response of structures to moving loads[J]. Computers and Structures,2000, Vol.78:789～799
[59]J.J.Wu, Wu, A.R.Whittaker, M.P.Cartmell. Dynamic responses of structures to moving bodies using combined finite element and analytical methods[J]. International Journal of Mechanical Sciences,2001, Vol.43:2555～2579
[60]A.Bonell, O.S.Bursi. Explicit predictor-multicorrector time discontinuous Galerkin methods for linear dynamics[J]. Journal of Sound and Vibration,2001, Vol.246, No.4: 625～652
[61]X.D.Li, N-E.Wiberg. Structural dynamic analysis by a time-discontinuous Galerkin finite element method [J]. International Journal For Numerical Methods in Engineering, 1996, Vol.39:2131～2152
[62]陈彦辉,郭旻,何宗颖,王宝元.炮口振动测试方法及实践[J].火炮发射与控制学报,2010,(1)：80-83
[63]王宝元,邵小军.炮口振动响应实验测试方法综述[J].火炮发射与控制学报,2010,(3)：112-116
[64]武瑞文,王兆胜.自行火炮武器系统射击精度研究[J].兵工学报,2004,25(4)：407-409
[65]彭献,张普,刘子建.基座运动情况下弹炮系统的振动分析[J].振动与冲击,2009,28(3)：23-26
[66]谈乐斌,张相炎,管红根等.火炮概论[M].北京：北京理工大学出版社,2005：1-6
[67]张相炎.火炮设计理论[M].北京：北京理工大学出版社,2005：10-22
[68]马福球,陈运生,朵英贤火炮与自动武器[M].北京：北京理工大学出版社,2003：12-13
[69]吴三灵,温波,于永强火炮动力学试验[M].北京：国防工业出版社,2004：71
[70]王靖君,郝信鹏.火炮概论[M].北京：兵器工业出版社,1992：123
[71]王宝元,陈顺德,周发明,衡刚,郭旻.火炮振动与射击密集度实验方法研究[J].火炮发射与控制学报,2009,(4)：89-9
[72]李海斌,毕世华,方远乔羽,袁曾风.移动刚体激励的梁的振动分析和实验[J].北京理工大学学报,1999,19(1)：18-22
[73]毕世华,李海斌,方远乔,袁曾凤.弹性发射梁振动的主动控制及实验研究[J].兵工学报,1998,19(4)：310-314
[74]P.I.Kattan,韩来彬MATLAB有限元分析与应用[M].北京：清华大学出版社,2004：92-94
[75]徐斌,高跃飞,余龙. MATLAB有限元结构动力学分析与工程应用[M].北京：清华大学出版社,2009：107