Project Report of DOE

Project Report of DOE Research on the working principle of the spinning top 3/6/8 Group 3 Leader: Li Zheran 李哲然 Members: Xu Zan 许赞 She Manrong 佘曼榕 Lei Chen 雷晨

目录. Motivation... 3. Literature Review... 3. The description of spinning top... 3. The process of turning spinning top... 4 3. Factor Decomposition... 4 3. Assumption... 4 3. Response Variable... 3.3 Control Variable... 3.4 Nuisance Variable... 4. Design of preliminary experiment... 6 4. Preliminary experiment implementation... 6 4. Data analysis... 7. Design of formal experiment.... First experiment implementation.... Data analysis....3 Second experiment implementation... 6.4 Data analysis... 7 6. Conclusion and Discussion...

. Motivation The Spinning top is one of the most popular sports in China. The action of a top relies on the gyroscopic effect for its operation. Typically the top will at first wobble until the shape of the tip and its interaction with the surface force it upright. After spinning upright for an extended period, the angular momentum, and therefore the gyroscopic effect will gradually lessen, leading to ever increasing precession, finally causing the top to topple in a frequently violent last thrash. To understand the working principle of the spinning top, we designed an experiment for spinning tops. The purpose of the experiment is as following: ) Find out the effect on spinning top -Spinning time, direction, weight, height of fall, ground-friction and etc ) Design a model between the effect and the response -Define the response and control variable 3) Help understand the work principle of spinning top -Analyze the conclusion and observe significant factors -Improve experiment design. Literature Review 明代刘侗 于奕正 帝经物略 二春场 载 : 陀螺者, 木制, 如小空钟, 中实而无柄 绕 以鞭之绳, 卓于地, 争制其鞭 8 年, 博科 (Foucault) 首先将高速旋转的刚体称为陀螺. The description of spinning top A spinning top is a toy designed to be spun rapidly on the ground, the motion of which causes it to remain precisely balanced on its tip because of inertia. Such toys have existed since antiquity. Traditionally tops were constructed of wood, sometimes with an iron tip, and would be set in motion by aid of a string or rope coiled around its axis which, when pulled quickly, caused a rapid unwinding that would set the top in motion. Today they are often built of plastic and modern materials and manufacturing processes allows tops to be constructed with such precise balance that they can be set in motion by a simple twirl of the wrist without need for string or rope.

. The process of turning spinning top Two kinds of spinning tops were used in experiment: ) Torsion spring control style Characteristic: Heavy, high center of gravity, but easy to control (spring-controlled) ) Rack control style Characteristic: Light, low center of gravity, but difficult to control (manually) 3. Factor Decomposition 3. Assumption The action of the spinning top is affected by various variable, including force level, ground surface material, weight and height when it was relieving, besides, there are also some hidden variable such as wind speed, surface homogeneous and force direction. We choose ground surface material, weight and height when it was relieving to be the control variables. And the other effect variable, like force level, wind speed and surface homogeneous is hard to control, so we let them to be noise variable which we must serious considered. To let the experiment more possible to implement and find out our purpose, we can make the following assumptions. ) There exists a certain optimum condition that makes the spinning top turning longer

and stable with some important variable effects. ) The force keep constant every time when we turn the spinning top. 3) The weight we changes and the height we changes every time should be fixed. 3. Response Variable Our dependent variable is the action of the spinning top, so we choose the slewing time of the spinning top to be the response variable. The response variable can be measured by a stopwatch, and we assume to ignore the measuring error. 3.3 Control Variable ) Ground surface: a) Cement We choose the ground behind the 4 th Zi Jing building. b) Plastic We choose the ground of the Zi Jing playground. c) Ceramic We choose the ground of the C building. ) Height: a) High b) Low 3) Weight : a) Light Using the original spinning top. b) Heavy Plus the weight of the spinning top by add some plasticine on it. 3.4 Nuisance Variable ) Force: Assuming to be constant and try to keep stable ) Surface homogeneous: Try to keep using one piece of ground as continuous as possible 3) Wind speed: Waiting to do the experiment when there is no wind.

4. Design of preliminary experiment The force to drive the gyro is a held constant variable in the experiment, guaranteeing the control variable s effect be shown on the response. But the force is hard to measure in time. So the constant force is dependent on the individual. So the preliminary experiment is designed to train the operators to hold the driving force constant and measure the time of spinning top to test whether the force is held in the acceptance range. 4. Preliminary experiment implementation The preliminary experiments run for 7 to 3 times, under the condition of concrete surface, low height and without weight added, and the spinning time is recorded as the response. It is an experiment under basic conditions somehow. Table variable codes Control variable Code Levels Level code Ground surface A Cement ground Height B Low(H = cm) Gyro weight C Light The target is to see whether the spinning time can be controlled.

Ground Surface cement Height H=cm Weight 4. Data analysis Figure time series plot X axis is set under the order of experiments running. Y axis shows spinning time of every experiment, with the unit of second. As the figure above shows, the spinning time after exp comes to a relative

constant range. So we choose the spinning time after order to make analysis. Summary statistics are as below Table statistics summary 变量 N N* 均值 标准误 标准差 最小值 水泥 cm ( 选择 ) 3 7.8.73.6 4. 下四分位数 中位数 上四分位数 最大值.6 7...7 To test whether the force is under operators control, try an I-MR chart. Mean in the chart is sample mean of the last 3 records. To make the constraint stricter, the StDev is chosen using formula below. StDev = R / ( * z.97) It means that we assume that the sample is chosen just under the 9% confidence interval. The formula is chosen to give a good control of the spinning time.

Figure I-MR control chart As the figure shows, the process after th experiment is almost controlled, though the process before not. It means that after training, we can say that, the operator can control the force within a measurable range. At next, we apply this result to the whole experiment process. For we are to control the variance of the force, the variance of the spinning time is a critical parameter. After observing the whole experiment data, it is found that the variance of the spinning time is increased with the average spinning time in the different condition. At the hypothesis of the experiment is under control, suppose the coefficient of variance CV = sigma / mu is constant is more realistic. So we make the I-MR chart for the whole formal experiment, regarding CV of the preliminary experiment as standard, the mean to be the sample mean, the StDev to be the sample mean timing the standard CV. One of the I-MR chart shows as below

Figure 3 I-MR control chart Covering all the charts, there are totally 3 out of 3 experiment points out of control, which can be seen as something accident. It means that the whole experiments run at a controlled level, guaranteeing the result meaningful for the further study.. Design of formal experiment After implementing the primary experiment, we can conclude that the force which could be a nuisance factor is been controlled in an accepted level. And then we design the formal experiment which is used to find out the main effect of the action of the spinning top. To realize the randomness and make our experiment more reasonable, we choose to use a 3 full fractional design. And we have designed two kinds of experiments, one is using Rotating handle gyro, and the other is using Clockwork gyro. Here are our experiments.. First experiment implementation We first designed an experiment for Rotating handle gyro, and here is the codes for variables and their levels.

Table 3 variable codes Control Variable Code Levels Level Code Cement Ground Ground Surface A Plastic Ground(playground) Height Gyro Weight B C Ceramic Ground 3 Low(H=cm) High(H=cm) Light Heavy(with plasticine) And each combination of levels we have replications runs. Here is our design showed by Minitab. Table 4 run design. Data analysis We first draw the main effect plots to get a preliminary view of the three factors.

. Figure 4 main effect plot Slewing Time 主效应图数据均值 A B 3 均值 C 3 3 From the main effect plot of A, we can find that with different type of ground surface, the slewing time is different. To be detailed, on ceramic ground, the gyro slews for the longest time, next is on the cement ground, and the 3 rd is on the plastic playground. The main effect plot of B shows that height has no evident effect on the slewing time. The main effect plot of C shows that with the increase of the gyro weight, the slewing time decreases.

Figure interaction plot Slewing Time 交互作用图数据均值 4 3 A A 3 B 4 3 B 4 3 C C 3 From the interaction plot in figure, we find that the lines of A and B are nearly parallel, which indicates A and B have no evident interaction effect. B and C have no interaction effect, either. But the lines between A and C are not parallel, indicating A and C do have interaction effect. More specifically, when the gyro is on the ceramic ground, the difference in slewing time between a light gyro and a heavy one is bigger than on the other two types of ground.

The model obtained from Minitab confirms our finding above, i.e. factor A, C and the interaction between them have significant effect on the slewing time. That is, the ground surface and gyro weight do influence the slewing time at a significant level. The height, however, almost have no effect on the slewing time.

Figure 6 residual plot Slewing Time 残差图 正态概率图 与拟合值 99.9 99 9 百分比 残差 -. - - 残差 - 3 拟合值 4 直方图 与顺序 4 8 频率 残差 6 - -8-4 残差 4 8-3 3 4 观测值顺序 4 6 From the residual plot in figure 6, we find that in the normal probability plot and the histogram, the residuals are approximately normally distributed, which confirms our assumption of normality. For the residual versus fitted value plot, we find that with the increase of fitted value, the residuals becomes larger like a trumpet, meaning that the residuals may not have equal variance. To solve the problem of unequal variance, we try to attach a weight to each value of slewing time the reciprocal of it. Then we refine the model and get the residual plot as shown in figure 4. Unfortunately, the residual versus fitted value plot doesn t show any improvement. What s worse, the residual versus observation order plot shows a larger fluctuation than the original one. However, it doesn t mean that the method of attaching weight is infeasible. We shall explain it further in the analysis of another experiment with plastic gyro.

Figure 7 residual plot by attaching weights Slewing Time 残差图 正态概率图 与拟合值 99.9 99 9 百分比 残差. - - 残差 - - 3 拟合值 4 频率 直方图 残差 与顺序 - -8-4 残差 4 8-3 3 4 观测值顺序 4 6.3 Second experiment implementation We then designed an experiment for Clockwork gyro, and here is the codes for variables and their levels. Table variable codes Control Variable Code Levels Level Code Cement Ground Ground Surface A Plastic Ground(playground) Height Gyro Weight B C Ceramic Ground 3 Low(H=3cm) High(H=cm) Light Heavy(with plasticine)

.4 Data analysis

Figure 8 main effect plot Slewing Time 主效应图数据均值 A B 3 均值 C 3 3 Figure 9 interaction plot Slewing Time 交互作用图数据均值 4 3 A A 3 B 4 3 B 4 3 C C 3 From the model, main effect plot and interaction plot of the second experiment, we can draw the absolutely same conclusion as we draw from the first experiment. At the same time, we find the same problem of unequal variance in residual in figure.

We also use the method of attaching weights to each value of slewing time, and the results are shown in figure. We find that the trumpet -shape has been corrected a lot. Figure residual plot Slewing Time 残差图 正态概率图 与拟合值 百分比 99.9 99 9. -4 - 标准化残差 4 标准化残差 4 - -4 3 拟合值 4 直方图 与顺序 3 4 频率 标准化残差 - -3. -.6. 标准化残差.6 3. -4 3 3 4 观测值顺序 4 6 Figure residual plot by attaching weight Slewing Time 残差图 正态概率图 与拟合值 百分比 99.9 99 9 标准化残差 3... -.. -4 - 标准化残差 4-3. 3 拟合值 4 直方图 3. 与顺序 频率 标准化残差.. -. -.4 -.. 标准化残差..4-3. 3 3 4 观测值顺序 4 6

6. Conclusion and Discussion After implementing these two kinds of experiments and analyzing data we achieved, we can easily draw a conclusion that the ground surface and gyro weight do influence the slewing time at a significant level. When we play spinning top on the ceramic ground, it lasts longest. And when the weight increases, the spinning top becomes turning shorter. The height, however, almost have no effect on the slewing time. This may be unbelievable, since we will instinctively feel that when the height increases, the slewing time should be shorter with the impact of force. However, we explain our model that the reason why the result is like this may be that our scope of the height distance is not wider enough that the difference may be not obvious. There are also many restrictions on our experiment which is hard to get over. ) Force control: Though we have practice to keep force constant, there is still personal error. ) Weight is difficult to control: To control the weight, we choose to cover plasticine to plus weight. 3) Data acquisition: Each run we use stopwatch to record data, which may have measuring error. 4) Wearing restriction: With the increasing of run times, the gyro wearing increases.