制动器试验台的控制方法分析.doc

制动器试验台的控制方法分析摘 要本文对模拟路试时的试验台中的控制方法进行了详细分析，通过运用能量守恒定律、定轴转动的动能定理和连续问题的离散化模型等物理、数学知识，针对不同问题进行了相应的分析并求解。针对第一问，利用物理知识，可求得单个前轮的滚动半径为0.286 m，制动时承受的载荷为6230 N的等效转动惯量为51.9988857。针对第二问，可以看成先分别求空心圆柱体的的机械惯量，再进行组合可得到8种组合，以及进一步要求的相应组合：1、基础惯量为10，厚度为0.0392m的1号飞轮，需要用电动机补偿的惯量为11.99058； 2、基础惯量为10，厚度为0.0784m的2号飞轮，需要用电动机补偿的惯量为-18.0177。针对第三问，可建立驱动电流依赖于瞬时转速的非线性数学模型，采用控制电流的方法对问题进行求解。运用已建立模型与问题二求解得出两种组合的机械惯量（分别是40和70），可计算得出对应的电流分别为-174.78A和262.17A。针对第四问，以误差分析的方法，先根据刚体定轴转动时的动能定理，可求出路试时消耗的总能量为52150.1983J，再求出实验时所消耗的总能量为49270.3971J,求出它们的相对误差为5.52%，从而判断此方法是科学合理的；针对第五问，按照第三问导出的数学模型，得到模型，能根据给出的前一个时间段的观测值，求出本时间段电流值。针对第六问，在第五问中，不能保证角加速度一直是不变的，为了克服此不足，可利用动力学知识，只是重新设计方法，得到模型：，此方法有科学理论支持，并且很好地解决了第五问所不能解决的问题。关键词： 刚体定轴转动动能定理 能量守恒定律 转动方式牛顿定律 能量误差1.问题重述：汽车的行车制动器（以下简称制动器）使行驶时使车辆减速或者停止，是车辆设计中的重要环节，直接影响着人身和车辆的安全。为了检验设计的优劣，必须进行相应路试，以及在试验台上进行对所设计的路试进行大量细微的模拟试验。在试验台上，飞轮组可提供相应的机械惯量，而剩余不可精确提供的惯量由电动机在一定规律的电流控制下参与工作，从而补偿由于机械惯量不足而缺少的能量，来满足模拟试验。在工程实际中，常用的计算机控制方法是把整个制动时间离散化为许多小的时间段，比如10 ms为一段，然后根据前面时间段观测到的瞬时转速与/或瞬时扭矩，设计出本时段驱动电流的值，这个过程逐次进行，直至完成制动，最后根据能量误差的大小来评价控制方法优劣。现在要求解答以下问题：1. 设车辆单个前轮的滚动半径为0.286 m，制动时承受的载荷为6230 N，求等效的转动惯量。2. 飞轮组由3个外直径1 m、内直径0.2 m的环形钢制飞轮组成，厚度分别为0.0392 m、0.0784 m、0.1568 m，钢材密度为7810 kg/m3，基础惯量为10 kgm2，问可以组成哪些机械惯量？设电动机能补偿的能量相应的惯量的范围为 -30, 30 kgm2，对于问题1中得到的等效的转动惯量，需要用电动机补偿多大的惯量？3. 建立电动机驱动电流依赖于可观测量的数学模型。在问题1和问题2的条件下，假设制动减速度为常数，初始速度为50 km/h，制动5.0秒后车速为零，计算驱动电流。4. 对于与所设计的路试等效的转动惯量为48 kgm2，机械惯量为35 kgm2，主轴初转速为514转/分钟，末转速为257转/分钟，时间步长为10 ms的情况，用某种控制方法试验得到的数据见附表。请对该方法执行的结果进行评价。5. 按照第3问导出的数学模型，给出根据前一个时间段观测到的瞬时转速与/或瞬时扭矩，设计本时间段电流值的计算机控制方法，并对该方法进行评价。6. 第5问给出的控制方法是否有不足之处？如果有，请重新设计一个尽量完善的计算机控制方法，并作评价。2问题的条件和假设：1.在本题中，假设所有的观测数据均有效，即不考虑人为因素造成的无效数据或误差。2.假设路试时轮胎与地面的摩擦力为无穷大，因此轮胎与地面无滑动。3. 模拟实验中，可认为主轴的角速度与车轮的角速度始终一致。4. 忽略车轮自身转动具有的能量。5. 假设试验台采用的电动机的驱动电流与其产生的扭矩成正比；6.车辆的车轮及实验台上的飞轮均可看作刚体。3符号说明：第个时间段消耗的能量：第个时间段的瞬时扭矩：第个时间段的瞬时角速度：k=1.5 A/Nm:电动机的驱动电流与其产生的扭矩成正比的比例系数；4模型分析与求解:4.1问题一的分析与求解:在此，车辆制动时承受的载荷可理解为车辆的重力G将车轮看作质点，其平均平动动能为 （1）再将其看作刚体旋转，其旋转能量为 （2）利用能量相等，有 （3）由 代入得 车辆单个前轮的滚动半径为0.286 m，制动时承受的载荷为6230 N，则等效的转动惯量为51.99888574.2问题二的分析与求解:首先我们将空心圆柱体可看成一个空心圆柱体，而对于空心圆柱体，我们可以先求出其质量：（4）而空心圆柱体的飞轮转动惯量： （5）将所提供的数据代入上式可求得各个环形钢环飞轮的转动惯量（见表1）环形钢环编号厚度（m）外直径（m）内直径（m）密度（）飞轮转动惯量（）10.039210.278103020.078410.278106030.156810.27810120（表1）即可组成以下8种飞轮组所能提供的机械惯量，由第一问知，所等效的转动惯量为51.9988857，所以需要用电动机补偿惯量相应为（见表2）飞轮组编号组合飞轮组所能提供机械惯量（）所要求等效的转动惯量（）需要用电动机补偿多大的惯（）01052420+14052120+27052-180+313052-780+1+210052-480+1+316052-1080+2+319052-1380+1+2+322052-1680代表基础惯量（表2）又因为电动机能补偿的能量相应的惯量的范围为 -30, 30 kgm2 即可得到符合要求的相应组合（见表3）飞轮组编号0+10+2飞轮组所能提供机械惯量（）4070需要用电动机补偿多大的惯量（）12-18（表3）即相应组合为；1、基础惯量为10，厚度为0.0392m的1号飞轮，需要用电动机补偿的惯量为12；2、基础惯量为10，厚度为0.0784m的2号飞轮，需要用电动机补偿的惯量为-18。4.3问题三的分析与求解:扭矩相当于是物理学中的力矩，主轴的扭矩是由制动扭矩与由驱动电流产生的扭矩两部分组成的。即有 （6）由转动方式牛顿定律知制动扭矩（7）主轴的扭矩 （8） （9）根据假设试验台采用的电动机的驱动电流与其产生的扭矩成正比（本题中比例系数取为1.5 A/Nm，即） 所以 （10）由第一二问可知：，为12或-18假设汽车东西方向行驶（向东为正方向），又因制动减速度为常数，初始速度为50 km/h，制动5.0秒后车速为零，且车辆单个前轮的滚动半径为0.286 m，则又因k=1.5 A/Nm，将各个数据代入（10）式可求得：电动机的驱动电流为-174.78A（电机补偿惯量为12）或262.17A（电机补偿惯量为-18），正负号表示电流方向。4.4问题四的分析与求解:所设计的路试时的制动器在制动过程中消耗的能量，为电动机产生的动能，即（11）等效的转动惯量为48 kgm2，机械惯量为35 kgm2，主轴初转速为514转/分钟，末转速为257转/分钟，时间步长为10 ms根据 可求得主轴初角速度为53.826m/s ，末角速度为26.913m/s代入数据求得E=52150.1983J实验台上制动器在制动过程中第个时间段消耗的能量为在整个实验过程中消耗的总能量为 （各个时间段消耗能量的数据表见附录） 可求得W=49270.3971J所设计的路试时的制动器与相对应的实验台上制动器在制动过程中消耗的能量之差为 相对误差为 故可认为这种控制方法在不考虑观测误差、随机误差和连续问题离散化所产生的误差的情况下是可行的。4.5问题五的分析与求解:按照第3问导出的数学模型，给出根据前一个时间段观测到的瞬时转速与/或瞬时扭矩，先求得角速度（12）再求出第时刻的角加速度（13）第时刻驱动电流产生的扭矩=将（7）式代入得： =-（14）即： （是已观测的值）（图4）评价：此方法能根据给出根据前一个时间段的观测量，利用动力学公式求出由电流产生的扭矩，进而根据，求出本时间段电流值，有科学理论依据，方法可行。4.6问题六的分析与求解:一、问题五的不足之处：做出角速度关于时间的图像发现不能保证一直是匀速的（图像见附录）；二、新的设计方法：先根据前一个时间段观测到的瞬时转速与/或瞬时扭矩，求得角速度，根据第三问， （7） （8）由可得 故有 （15）由，又因k=1.5 A/Nm，将第三问中各个数据代入（15）式可求得：电动机的驱动电流为-174.78A（电机补偿惯量为12）或262.17A（电机补偿惯量为-18），即这种方法和第三问情况的求出的结果是相同的，说明这种设计控制方法是可行的。评价：1、此方法也能根据给出根据前一个时间段观测到的观测量，求出本时间段电流值，也是利用动力学公式和 得到的，有科学理论依据，方法可行。 2、此方法能解决问题五的不足之处。5模型的评价及改进建议：一、优点：1、 数据处理时，是在比较合理的假设基础上进行的。2、 较逼真地建立了由电流来补偿由于机械惯量不足而缺少的能量的模型，且模型的建立是有一定的科学基础的，所以有可信度。3、 第六问更真实地模拟了现实中的情况，在简化计算的同时也成功模拟了制动过程。二、缺点：1、 没有考虑系统本身也会损失能量，以及空气阻力对飞轮的影响。2、 机械制动过程中也会散热等因素也会导致模拟制动过程中的能量误差。参考文献（1）赵近芳 大学物理学（上册） 北京邮电大学出版社 2002年08月（2）姜启源 谢金星 叶俊 数学模型（第三版） 高等教育出版社 2003年8月第3版附录1、角速度与时间图像2、用某种控制方法试验得到的数据见下表：扭矩(N.m)转速(rpm)角速度(rad/s)时间(s)消耗能量（J）40514.3353.8605107040513.7953.80396210.0121.521584840513.2453.74636620.0221.498546541.25513.7953.80396210.0322.194134443.75513.7953.80396210.0423.539233445513.7953.80396210.0524.211782947.5513.2453.74636620.0625.529523950513.2453.74636620.0726.873183153.75512.6953.68877030.0828.857714155512.6953.68877030.0929.528823757.5512.1553.63222170.130.838527558.75512.1553.63222170.1131.508930262.5512.1553.63222170.1233.520138562.5512.6953.68877030.1333.555481567.5512.1553.63222170.1436.201749667.5512.1553.63222170.1536.201749672.5511.653.57462580.1638.841603775511.653.57462580.1740.180969481.25511.0653.51807710.1843.483437786.25511.653.57462580.1946.208114891.25511.653.57462580.248.88684696.25510.5153.46048130.2151.4557132101.25510.5153.46048130.2254.1287373105510.5153.46048130.2356.1335053110511.0653.51807710.2458.8698849115510.5153.46048130.2561.4795535120509.4253.34633670.2664.0156041127.5509.4253.34633670.2768.0165793133.75509.4253.34633670.2871.3507254143.75509.4253.34633670.2976.6853591150509.4253.34633670.380.0195051157.5508.8753.28874090.3183.9297669161.25508.3353.23219220.3285.8369099168.75507.7853.17459630.3389.7321313172.5507.7853.17459630.3491.7261787181.25507.2353.11700050.3596.2745634186.25507.2353.11700050.3698.9304134193.75507.2353.11700050.37102.914188198.75507.2353.11700050.38105.570038203.75506.6953.06045180.39108.110671208.75505.652.94630730.4110.525416211.25505.0552.88871140.41111.727403216.25504.552.83111560.42114.247287218.75503.9652.77456690.43115.444365222.5503.4152.7169710.44117.295261226.25502.8752.66042240.45119.144206230502.8752.66042240.46121.118971233.75502.3252.60282650.47122.959107237.5502.3252.60282650.48124.931713238.75501.2352.4886820.49125.316728242.5500.1452.37453740.5127.008253242.5499.5952.31694160.51126.868583247.5499.0452.25934570.52129.341881246.25499.0452.25934570.53128.688639245498.552.2027970.54127.896853241.25498.552.2027970.55125.939248245497.9552.14520120.56127.755743248.75497.9552.14520120.57129.711188256.25497.4152.08865250.58133.477172257.5497.4152.08865250.59134.12828262.5496.8652.03105660.6136.581524262.5496.3151.97346080.61136.430335266.25495.7751.91691210.62138.228778266.25495.2251.85931620.63138.07543266.25494.6851.80276760.64137.924869266.25493.5851.68757590.65137.618171266.25492.4951.57343130.66137.314261266.25491.451.45928680.67137.010351265490.8551.40169090.68136.214481266.25490.3151.34514230.69136.706441268.75489.7651.28754640.7137.835281272.5489.2251.23099770.71139.604469273.75488.6751.17340190.72140.087188276.25488.1251.1158060.73141.207414277.5488.1251.1158060.74141.846362277.5487.0351.00166150.75141.529611272.5486.4950.94511280.76138.825432272.5485.3950.82992110.77138.511535268.75484.350.71577650.78136.298649272.5483.2150.6016320.79137.889447267.5482.1250.48748750.8135.054029272.5481.5750.42989160.81137.421455270481.5750.42989160.82136.160707277.5481.5750.42989160.83139.942949278.75481.0350.37334290.84140.415693282.5479.9350.25815120.85141.979277282.5479.3950.20160260.86141.819527282.5479.3950.20160260.87141.819527282.5478.8450.14400670.88141.656819280478.350.0874580.89140.244882277.5477.7550.02986220.9138.832867276.25477.249.97226630.91138.048386273.75477.249.97226630.92136.799079273.75476.6649.91571760.93136.644277275476.1149.85812180.94137.109835276.25475.0249.74397720.95137.417737280474.4749.68638140.96139.121868280473.9349.62983270.97138.963532282.5471.249.34394780.98139.396652281.25468.4749.05806280.99137.975802283.75465.1948.71458211138.227627282.5463.5548.54284171.01137.133528278.75461.9248.37214851.02134.837364276.25461.3748.31455261.03133.468952275460.2848.20040811.04132.551122277.5459.7348.14281221.05133.596304280458.6448.02866771.06134.480269281.25458.0947.97107181.07134.918639285457.5547.91452311.08136.556391283.75457.5547.91452311.09135.957459283.75456.4647.80037861.1135.633574282.5455.9147.74278271.11134.873361281.25455.3647.68518691.12134.114588278.75455.3647.68518691.13132.922458277.5455.3647.68518691.14132.326394273.75454.8247.62863821.15130.383397277.5454.2747.57104231.16132.009643277.5453.1847.45689781.17131.692891281.25452.6347.3993021.18133.310537281.25451.5447.28515741.19132.989505283.7545147.22860881.2134.011177285450.4547.17101291.21134.437387285449.947.1134171.22134.273239283.75449.3647.05686841.23133.523864282.5448.8146.99927251.24132.772945282.5448.2746.94272381.25132.613195281.25448.2746.94272381.26132.026411278.75447.7246.8851281.27130.692294275447.1746.82753211.28128.775713276.25446.6346.77098341.29129.204842276.25446.0846.71338761.3129.045733280445.5446.65683891.31130.639149281.25444.9946.5992431.32131.060371285443.946.48509851.33132.482531283.75443.3546.42750261.34131.738039285442.8146.3709541.35132.157219282.5442.2646.31335811.36130.835237285441.7146.25576221.37131.828922280441.7146.25576221.38129.516134281.25441.7146.25576221.39130.094331276.25441.7146.25576221.4127.781543275441.1746.19921361.41127.047837273.75441.1746.19921361.42126.4703472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